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# K-12 Blended and Online Learning
## Name of Unit/Course: Relationships between Quantities and Expressions/ GSE Algebra 1
Overall Unit Information (Required for MOOC participants)
Self-Check
Unit or Course
Goal(s)
See A1
Lokey-Vega (2014)
Relationships Between Quantities and Expressions- In this unit, students will interpret
expressions and solve problems that are related to unit analysis. Students will learn
properties of rational and irrational numbers. All material will provide a basis for
subsequent units. This unit is for a class entirely online.
Standards
## Name of standards: Georgia Standards of Excellence
See A2
Subject: Math- GSE Algebra I
Standard (as written):
MGSE9-12.N.RN.2 Rewrite expressions involving radicals (i.e., simplify and/or use the
limited to square roots).
MGSE9-12.N.RN.3 Explain why the sum or product of rational numbers is rational; why the
sum of a rational number and an irrational number is irrational; and why the product of a
nonzero rational number and an irrational number is irrational.
Reason quantitatively and use units to solve problems. MGSE9-12.N.Q.1 Use units of
measure (linear, area, capacity, rates, and time) as a way to understand problems:
1. Identify, use, and record appropriate units of measure
within context, within data displays, and on graphs;
2. Convert units and rates using dimensional analysis
(English-to-English and Metric-to-Metric without conversion factor provided and between
English and Metric with conversion factor);
3. Use units within multi-step problems and formulas; interpret units of input and
resulting units of output.
MGSE9-12.N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.
Given a situation, context, or problem, students will determine, identify, and use
appropriate quantities for representing the situation. MGSE9-12.N.Q.3 Choose a level of
accuracy appropriate to limitations on measurement when reporting quantities. For
example, money situations are generally reported to the nearest cent (hundredth). Also, an
answers precision is limited to the precision of the data given.
Interpret the structure of expressions.
MGSE9-12.A.SSE.1 Interpret expressions that represent a quantity in terms of its context.
MGSE9-12.A.SSE.1a Interpret parts of an expression, such as terms, factors, and
coefficients, in context. MGSE9-12.A.SSE.1b Given situations which utilize formulas or
expressions with multiple terms and/or factors, interpret the meaning (in context) of
Lokey-Vega (2014) individual terms or factors.
Perform arithmetic operations on polynomials. MGSE9-12.A.APR.1 Add, subtract, and
multiply polynomials; understand that polynomials form a system analogous to the integers
in that they are closed under these operations.
Learner
Characteristics
## These students are 9th graders in a suburban Georgia high school.
English learners: 1%
Free or reduced lunch: 11%
Testing: 80% met expectations in math
See B1
Technology
requirements
## Computer with internet
Webcam
Telephone
TI-84 Graphing Calculator
See D5
Prerequisite
Skills
Basic computer skills
Safe internet research
Telephone communication skills
See A4 & D6
Introductory
Communication
Plans
Each module will have the same format, so students understand the expectations and
there is clear alignment. Students are expected to answer a discussion question
associated with each module and comment on peers posts. Also the teacher will check in
with the student once every other week via phone call or webcam.
Universal Design
Principles
Considered
## Multiple forms or representation: videos, textbook, websites
Multiple forms of expressions: quizzes, discussions, assignments
Multiple forms of engagement: phone calls, synchronous sessions with webcams,
discussion posts
See B4
Number of
Modules or
Weeks
According to the stat standards, this unit will take 4 to 5 weeks online. The next module will
be released every Friday afternoon. Students will have one week to complete all module
assignments.
See A3
Lokey-Vega (2014)
## K-12 Blended and Online Learning
Module 1 Plan (Note: module and lesson used interchangeably) (Required for MOOC participants)
Self-Check
Module
Objective(s)
MGSE9-12.N.RN.2 Students will be able to do basic operations with radicals and be able
to simplify them earning at least an 80% on the quiz.
MGSE9-12.N.RN.3 Students will learn properties of rational numbers, be able to identify
real world examples through discussion posts, and earn at least on 80% on quiz.
See A1 & A2
Module
Assessment(s)
1. There will be a quiz on simplifying radicals and another one on properties of rational
functions.
2. Discussion post: Give an example where you would find and use radicals and irrational
numbers in real life. Comment on two other peers posts.
3. Complete the assignment posted on my website.
See A2 A3 C1 C2 &
C5
Description of
Learning
Activities
1. Students will read chapter from textbook that corresponds to current material.
2. Students will reinforce material by watching selected videos from Math Tube.
3. Student will complete assignment posted on my website.
4. Student will answer discussion question and comment on two peers.
5. Students will attend synchronous session.
See A2 A3 B3 B4 &
B10
Formative
Evaluation &
Feedback
As previously mentioned, students will communicate with the teacher every other week via
phone call or webcam.
Students mandatory discussion posts will be commented on.
Quizzes will contain corrections with guided comments and feedback.
There will be one synchronous class session via webcam for each unit.
See A3 C1 C3 & C5
Lokey-Vega (2014)
Physical
Learning
Materials
Holt McDougal.
## See A3, A9, B1, B4,
& B6
Digital Learning
Objects
In the textbook above, there is a CD in the back cover so students also have a digital copy
of the material.
Students will also need to check my website daily for updates, notes, and posts.
Students will frequent the website Math Tube, as well.
## See A3, A9, B1, B4,
& B6
Plans for
Differentiation
calls and web cam session.
Students who need accommodations as stated in their IEP can receive a copy of teacher
notes.
During synchronous sessions, students will be grouped in such a way that their skill sets
are comparable and the strengths/weaknesses are matched.
See B1 B4 & B6
Lokey-Vega (2014)
## K-12 Blended and Online Learning
Module 2 Plan (Optional for MOOC participants)
Self-Check
Module
Objective(s)
See A1 & A2
Module
Assessment(s)
See A2 A3 C1 C2 &
C5
Description of
Learning
Activities
See A2 A3 B3 B4 &
B10
Formative
Evaluation &
Feedback
See A3 C1 C3 & C5
Physical
Learning
Materials
& B6
Digital Learning
Objects
## See A3, A9, B1, B4,
& B6
Plans for
Differentiation
See B1 B4 & B6
Lokey-Vega (2014)
## K-12 Blended and Online Learning
Module 3 Plan (Optional for MOOC participants)
Self-Check
Module
Objective(s)
See A1 & A2
Module
Assessment(s)
See A2 A3 C1 C2 &
C5
Description of
Learning
Activities
See A2 A3 B3 B4 &
B10
Formative
Evaluation &
Feedback
See A3 C1 C3 & C5
Physical
Learning
Materials
& B6
Digital Learning
Objects
## See A3, A9, B1, B4,
& B6
Plans for
Differentiation
See B1 B4 & B6
Lokey-Vega (2014)
## K-12 Blended and Online Learning
Module 4 Plan (Optional for MOOC participants)
Self-Check
Module
Objective(s)
See A1 & A2
Module
Assessment(s)
See A2 A3 C1 C2 &
C5
Description of
Learning
Activities
See A2 A3 B3 B4 &
B10
Formative
Evaluation &
Feedback
See A3 C1 C3 & C5
Physical
Learning
Materials
& B6
Digital Learning
Objects
## See A3, A9, B1, B4,
& B6
Plans for
Differentiation
See B1 B4 & B6
Lokey-Vega (2014)
## K-12 Blended and Online Learning
Module 5 Plan (Optional for MOOC participants)
Self-Check
Module
Objective(s)
See A1 & A2
Module
Assessment(s)
See A2 A3 C1 C2 &
C5
Description of
Learning
Activities
See A2 A3 B3 B4 &
B10
Formative
Evaluation &
Feedback
See A3 C1 C3 & C5
Physical
Learning
Materials
& B6
Digital Learning
Objects
## See A3, A9, B1, B4,
& B6
Plans for
Differentiation
See B1 B4 & B6
Lokey-Vega (2014)
## K-12 Blended and Online Learning
Module 6 Plan (Optional for MOOC participants)
Self-Check
Module
Objective(s)
See A1 & A2
Module
Assessment(s)
See A2 A3 C1 C2 &
C5
Description of
Learning
Activities
See A2 A3 B3 B4 &
B10
Formative
Evaluation &
Feedback
See A3 C1 C3 & C5
Physical
Learning
Materials
& B6
Digital Learning
Objects
## See A3, A9, B1, B4,
& B6
Plans for
Differentiation
See B1 B4 & B6
Lokey-Vega (2014)
10
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https://www.extramarks.com/ncert-solutions/cbse-class-11/mathematics-permutations-and-combinations
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# Permutations and Combinations
## Why do we find students nowadays perplexed and unsure where to go for academic problems? Perhaps, books aren’t sufficient and there are too many online tutorials available on the net. The best way to resolve the issue is to rely on a trustworthy source that can give students the best direction to score more. What better than Extramarks! The NCERT solutions for Class 11 Maths consists of a chapter called Permutations and Combinations that has been explained by Extramarks with Animations, Mindmaps and other learning modules. Students who are weak at Maths don’t have to feel anxious about the subject anymore. Extramarks gives all necessary study help to make even the most difficult concepts easy and fun to learn. Permutations and Combinations in Class 11 is a chapter that can really help the students stretch few good scores if understood properly. All thanks to Extramarks! A permutation is an arrangement in a definite order of a number of objects, taken some or all at a time. Fundamental principle of counting states that m×n is the total number of ways in which two events occur where m and n are the number of ways in which first and second event occur respectively. If there are two mutually exclusive events, such that they can be occurred independently in m and n ways respectively, then either of the two events can be occurred in (m + n) ways. Product of the first ‘n’ natural numbers is n!. It is read as ‘n factorial’ or ‘factorial n’. The number of all permutations of n different objects, taken all at a time is n!. The number of permutations of n different objects taken r at a time, where repetition is not allowed, is given by nPr = n! / (n – r)! , 0 ≤ r ≤ n The number of permutations of n different objects taken r at a time, where repetition is allowed, is nr. The number of permutations of n objects, where p objects are of the same kind and rest are all different is (n! / p!). The number of permutations of n objects taken all at a time, where p1 objects are of first kind, p2 objects are of the second kind, …, pk objects are of kth kind and rest, if any, are all different is (n! / p1!p2!...pk!). A combination is a selection made by taking some or all of a number of objects, irrespective of their arrangements. The number of combinations of n different things taken r at a time is given by nCr = n! / r!(n – r)! , 0 ≤ r ≤ n.
To Access the full content, Please Purchase
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http://mathhelpforum.com/pre-calculus/119077-fixed-real-numbers-part-2-a.html
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# Thread: Fixed Real Numbers...Part 2
1. ## Fixed Real Numbers...Part 2
Suppose a, b, and c are fixed real numbers such that
b^2 - 4ac ≥ 0. Let r and s be the solutions of
ax^2 + bx + x = 0.
(a) Use the quadratic formula to show that r + s = -b/a and
rs = c/a.
(b) Use part (a) to verify that ax^2 + bx + c =
a(x - r)(x - s).
2. ax^2 + bx + x = 0. Solutions: r,s
I think you mean
ax^2 + bx + c = 0.
$x=\frac{-b\pm{\sqrt{b^2-4ac}}}{2a}$
Let $r=\frac{-b+{\sqrt{b^2-4ac}}}{2a}$
And $s=\frac{-b-{\sqrt{b^2-4ac}}}{2a}
$
Hence
$(x-r)(x-s)=x^2+\frac{b}{a}x+\frac{c}{a}=0$
$x^2-(r+s)x+rs=x^2+\frac{b}{a}x+\frac{c}{a}=0$
Hence $r+s=\frac{-b}{a}$ and $rs=\frac{c}{a}$
2nd part should fall into place now
3. ## well...
Originally Posted by I-Think
ax^2 + bx + x = 0. Solutions: r,s
I think you mean
ax^2 + bx + c = 0.
$x=\frac{-b\pm{\sqrt{b^2-4ac}}}{2a}$
Let $r=\frac{-b+{\sqrt{b^2-4ac}}}{2a}$
And $s=\frac{-b-{\sqrt{b^2-4ac}}}{2a}
$
Hence
$(x-r)(x-s)=x^2+\frac{b}{a}x+\frac{c}{a}=0$
$x^2-(r+s)x+rs=x^2+\frac{b}{a}x+\frac{c}{a}=0$
Hence $r+s=\frac{-b}{a}$ and $rs=\frac{c}{a}$
2nd part should fall into place now
Part 2 does not fall into place but I will play with it later today.
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https://nhsrcommunity.com/external-posts/create-your-machine-learning-library-from-scratch-with-r-2-5-pca/
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Promoting the use of R in the NHS
#### Blog Article
This post was originally published on this site
(This article was first published on Enhance Data Science, and kindly contributed to R-bloggers)
This is this second post of the “Create your Machine Learning library from scratch with R !” series. Today, we will see how you can implement Principal components analysis (PCA) using only the linear algebra available in R. Previously, we managed to implement linear regression and logistic regression from scratch and next time we will deal with K nearest neighbors (KNN).
## Principal components analysis
The PCA is a dimensionality reduction method which seeks the vectors which explains most of the variance in the dataset. From a mathematical standpoint, the PCA is just a coordinates change to represent the points in a more appropriate basis. Picking few of these coordinates is enough to explain an important part of the variance in the dataset.
## The mathematics of PCA
Let be the observations of our datasets, the points are in . We assume that they are centered and of unit variance. We denote the matrix of observations.
Then, can be diagonalized and has real and positive eigenvalues (it is a symmetric positive definite matrix).
We denote its ordered eigenvalues and the associated eigenvectors. It can be shown that is the cumulative variance explained by .
It can also be shown that is the orthonormal basis of size which explains the most variances.
This is exactly what we wanted ! We have a smaller basis which explains as much variance as possible !
### PCA in R
The implementation in R has three-steps:
1. We center the data and divide them by their deviations. Our data now comply with PCA hypothesis.
2. We diagonalise and store the eigenvectors and eigenvalues
3. The cumulative variance is computed and the required numbers of eigenvectors to reach the variance threshold is stored. We only keep the first eigenvectors
```###PCA
my_pca<-function(x,variance_explained=0.9,center=T,scale=T)
{
my_pca=list()
##Compute the mean of each variable
if (center)
{
my_pca[['center']]=colMeans(x)
}
## Otherwise, we set the mean to 0
else
my_pca[['center']]=rep(0,dim(x)[2])
####Compute the standard dev of each variable
if (scale)
{
my_pca[['std']]=apply(x,2,sd)
}
## Otherwise, we set the sd to 0
else
my_pca[['std']]=rep(1,dim(x)[2])
##Normalization
##Centering
x_std=sweep(x,2,my_pca[['center']])
##Standardization
x_std=x_std%*%diag(1/my_pca[['std']])
##Cov matrix
eigen_cov=eigen(crossprod(x_std,x_std))
##Computing the cumulative variance
my_pca[['cumulative_variance']] =cumsum(eigen_cov[['values']])
##Number of required components
my_pca[['n_components']] =sum((my_pca[['cumulative_variance']]/sum(eigen_cov[['values']]))<variance_explained)+1
##Selection of the principal components
my_pca[['transform']] =eigen_cov[['vectors']][,1:my_pca[['n_components']]]
attr(my_pca, "class") <- "my_pca"
return(my_pca)
}
```
Now that we have the transformation matrix, we can perform the projection on the new basis.
```predict.my_pca<-function(pca,x,..)
{
##Centering
x_std=sweep(x,2,pca[['center']])
##Standardization
x_std=x_std%*%diag(1/pca[['std']])
return(x_std%*%pca[['transform']])
}
```
The function applies the change of basis formula and a projection on the principals components.
## Plot the PCA projection
Using the predict function, we can now plot the projection of the observations on the two main components. As in the part 1, we used the Iris dataset.
```library(ggplot2)
pca1=my_pca(as.matrix(iris[,1:4]),1,scale=TRUE,center = TRUE)
projected=predict(pca1,as.matrix(iris[,1:4]))
ggplot()+geom_point(aes(x=projected[,1],y=projected[,2],color=iris[,5]))
```
## Comparison with the FactoMineR implementation
We can now compare our implementation with the standard FactoMineR implementation of Principal Component Analysis.
```library(FactoMineR)
pca_stats= PCA(as.matrix(iris[,1:4]))
projected_stats=predict(pca_stats,as.matrix(iris[,1:4]))\$coord[,1:2]
ggplot(data=iris)+geom_point(aes(x=projected_stats[,1],y=-projected_stats[,2],color=Species))+xlab('PC1')+ylab('PC2')+ggtitle('Iris dataset projected on the two mains PC (FactomineR)')
```
When running this, you should get a plot very similar to the previous one. This ensures the sanity of our implementation.
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2
Q:
# What was the day on 31st Dec,2011 ?
A) Monday B) Saturday C) Sunday D) Friday
Explanation:
31st Dec, 2011 = (2010 years + Period from 1.1.2011 to 31.12.2011)
Odd days in 2000 years = 0
10 years has 2 leap years + 8 ordinary years.
Number of odd days in 10 years ( 2 x 2 + 8) = 5 odd days.
31st,Dec => complete year of 2011 (non-leap year) = 1 odd day.
Total number of odd days = (0 + 5 + 1) = 6.
Given day is saturday.
Q:
What is two weeks from today?
A) same day B) previous day C) next day D) None
Explanation:
We know that the day repeats every 7 days, 14 days, 21 days,...
So if today is Monday, after 7 days it is again Monday, after 14 days again it is Monday.
Hence, after 2 weeks i.e, 14 days the day repeats and is the same day.
1 272
Q:
How old are you if you are born in 1995?
A) 22 B) 23 C) 24 D) 25
Explanation:
Calculating Age has 2 conditions. Let your Birthday is on January 1st.
1. If the month in which you are born is completed in the present year i.e, your birthday, then
Your Age = Present year - Year you are born
As of now, present year = 2018
i.e, Age = 2018 - 1995 = 23 years.
2. If the month in which you are born is not completed in the present year i.e, your birthday, then
Your Age = Last year - Year you are born
As of now, present year = 2018
i.e, Age = 2017 - 1995 = 22 years.
2 244
Q:
The day before yesterday, I was 25 years old, and next year I will turn 28. How is it possible?
On carefully inspecting this question, one can understand that there are two days which are important and these are:
A. My Birthday.
B. The day when I am making this statement.
If you think for a while, you will understand that such statements can be made only around the year’s end. So, if my birthday is on 31 December, then I will be making this statement on 1 January.
I will further explain using the following example:
1. Consider that today is 01 January 2017.
2. Then, the day before yesterday was 30 December 2016 and according to the question I was 25 then.
3. Yesterday was 31 December 2016, which happens to be my birthday too (Woohoo!), and my age increases by one to become 26.
4. I will turn 27 on my birthday this year (31 December 2017).
5. I will turn 28 on my birthday next year (31 December 2018).
Now, if you read the question again, it will make more sense:
The day before yesterday(30 December 2016), I was 25 years old and next year(31 December 2018) I will be 28.
257
Q:
The calendar for the year 2018 will be the same for the year
A) 2023 B) 2027 C) 2029 D) 2022
Explanation:
How to find the years which have the same Calendars :
Leap year calendar repeats every 28 years.
Here 28 is distributed as 6 + 11 + 11.
Rules:
a) If given year is at 1st position after Leap year then next repeated calendar year is Givenyear+6.
b) If given year is at 2nd position after Leap year then next repeated calendar year is Givenyear+11.
c) If given year is at 3rd position after Leap year then next repeated calendar year is Givenyear+11.
Now, the given year is 2018
We know that 2016 is a Leap year.
2016 2017 2018 2019 2020
Lp Y 1st 2nd 3rd Lp Y
Here 2018 is at 2 nd position after the Leap year.
According to rule b) the calendar of 2018 is repeated for the year is 2018 + 11 = 2029.
4 588
Q:
How many days in 4 years?
A) 1460 B) 1461 C) 1462 D) 1459
Explanation:
Days in 4 years =>
Let the first year is Normal year i.e, its not Leap year. A Leap Years occurs once for every 4 years.
4 years => 365 + 365 + 365 + 366(Leap year)
4 years => 730 + 731 = 1461
Therefore, Number of Days in 4 Years = 1461 Days.
6 583
Q:
How many weekends in a year?
A) 52 B) 53 C) 103 D) 104
Explanation:
In normal we have 104 Weekend Days.
We know that a Each normal year has 365 days or 52 weeks plus one day, and each week has two weekend days, which means there are approximately 104 weekend days each year.
Whereas in a leap year we have 366 days it adds one more day to the year. And what makes the change is the starting day of the year.
9 555
Q:
What is 90 days from today?
Hints : Today is 20th January 2017, Sunday
A) 18th April, Friday B) 20th April, Saturday C) 21th April, Sunday D) 19th April, Saturday
Explanation:
Given Today is 20th January 2017, Sunday
In january, we have 31 days
February - 28 days (Non leap year)
March - 31 days
April - 30 days
=> Remaining days => 31 - 20 = 11 in Jan
+ 28 in Feb + 31 in Mar = 11 + 28 + 31 = 70 days
More 20 days to complete 90 days => upto 20th April
Therefore, after 90 days from today i.e, 20th Jan 2017 is 20th Apr 2017.
Now, the day of the week will be
90/7 => Remainder '6'
As the day starts with '0' on sunday
6 => Saturday.
Required day is 20th April, Saturday.
6 992
Q:
The calendar for the year 1988 is same as which upcoming year ?
A) 2012 B) 2014 C) 2016 D) 2010
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# 3678 hours to minutes
## Result
3678 hours equals 220680 minutes
## Conversion formula
Multiply the amount of hours by the conversion factor to get the result in minutes:
3678 hr × 60 = 220680 min
## How to convert 3678 hours to minutes?
The conversion factor from hours to minutes is 60, which means that 1 hours is equal to 60 minutes:
1 hr = 60 min
To convert 3678 hours into minutes we have to multiply 3678 by the conversion factor in order to get the amount from hours to minutes. We can also form a proportion to calculate the result:
1 hr → 60 min
3678 hr → T(min)
Solve the above proportion to obtain the time T in minutes:
T(min) = 3678 hr × 60 min
T(min) = 220680 min
The final result is:
3678 hr → 220680 min
We conclude that 3678 hours is equivalent to 220680 minutes:
3678 hours = 220680 minutes
## Result approximation
For practical purposes we can round our final result to an approximate numerical value. In this case three thousand six hundred seventy-eight hours is approximately two hundred twenty thousand six hundred eighty minutes:
3678 hours ≅ 220680 minutes
## Conversion table
For quick reference purposes, below is the hours to minutes conversion table:
hours (hr) minutes (min)
3679 hours 220740 minutes
3680 hours 220800 minutes
3681 hours 220860 minutes
3682 hours 220920 minutes
3683 hours 220980 minutes
3684 hours 221040 minutes
3685 hours 221100 minutes
3686 hours 221160 minutes
3687 hours 221220 minutes
3688 hours 221280 minutes
## Units definitions
The units involved in this conversion are hours and minutes. This is how they are defined:
### Hours
An hour (symbol: h; also abbreviated hr.) is a unit of time conventionally reckoned as 1⁄24 of a day and scientifically reckoned as 3,599–3,601 seconds, depending on conditions. The seasonal, temporal, or unequal hour was established in the ancient Near East as 1⁄12 of the night or daytime. Such hours varied by season, latitude, and weather. It was subsequently divided into 60 minutes, each of 60 seconds. Its East Asian equivalent was the shi, which was 1⁄12 of the apparent solar day; a similar system was eventually developed in Europe which measured its equal or equinoctial hour as 1⁄24 of such days measured from noon to noon. The minor variations of this unit were eventually smoothed by making it 1⁄24 of the mean solar day, based on the measure of the sun's transit along the celestial equator rather than along the ecliptic. This was finally abandoned due to the minor slowing caused by the Earth's tidal deceleration by the Moon. In the modern metric system, hours are an accepted unit of time equal to 3,600 seconds but an hour of Coordinated Universal Time (UTC) may incorporate a positive or negative leap second, making it last 3,599 or 3,601 seconds, in order to keep it within 0.9 seconds of universal time, which is based on measurements of the mean solar day at 0° longitude.
### Minutes
The minute is a unit of time or of angle. As a unit of time, the minute (symbol: min) is equal to 1⁄60 (the first sexagesimal fraction) of an hour, or 60 seconds. In the UTC time standard, a minute on rare occasions has 61 seconds, a consequence of leap seconds (there is a provision to insert a negative leap second, which would result in a 59-second minute, but this has never happened in more than 40 years under this system). As a unit of angle, the minute of arc is equal to 1⁄60 of a degree, or 60 seconds (of arc). Although not an SI unit for either time or angle, the minute is accepted for use with SI units for both. The SI symbols for minute or minutes are min for time measurement, and the prime symbol after a number, e.g. 5′, for angle measurement. The prime is also sometimes used informally to denote minutes of time. In contrast to the hour, the minute (and the second) does not have a clear historical background. What is traceable only is that it started being recorded in the Middle Ages due to the ability of construction of "precision" timepieces (mechanical and water clocks). However, no consistent records of the origin for the division as 1⁄60 part of the hour (and the second 1⁄60 of the minute) have ever been found, despite many speculations.
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Web Results
www.freemathhelp.com/q6-right-triangle.html
Answer. Finding the missing side of a right triangle is a pretty simple matter if two sides are known. One of the more famous mathematical formulas is \(a^2+b^2=c^2\), which is known as the Pythagorean Theorem.The theorem states that the hypotenuse of a right triangle can be easily calculated from the lengths of the sides.
handymath.com/cgi-bin/angle4.cgi?submit=Entry
This calculator calculates for the length of one side of a right triangle given the length of the other two sides. A right triangle has two sides perpendicular to each other. Sides "a" and "b" are the perpendicular sides and side "c" is the hypothenuse. Enter the length of any two sides and leave the side to be calculated blank.
www.mathsisfun.com/algebra/trig-finding-side-right-triangle.html
Finding a Side in a Right-Angled Triangle Find a Side when we know another Side and Angle. We can find an unknown side in a right-angled triangle when we know:. one length, and; one angle (apart from the right angle, that is).
www.mathwarehouse.com/geometry/triangles/right-triangles/find-the-side-length...
There are many ways to find the side length of a right triangle. We are going to focus on two specific cases. Case I. When we know two sides of the right triangle, in which case, we will use the Pythagorean theorem. Case II.
sciencing.com/side-lengths-triangles-5868750.html
High school or college geometry students may be asked to find the lengths of a triangle's sides. Engineers or landscapers may also need to determine the lengths of a triangle's sides. If you know some of the sides or angles of the triangle, you can figure out the unknown measurements.
www.calculator.net/triangle-calculator.html
Law of sines: the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. Using the law of sines makes it possible to find unknown angles and sides of a triangle given enough information. Where sides a, b, c, and angles A, B, C are as depicted in the above calculator, the law of sines can be written as shown ...
There are many methods available when it comes to discovering the sides and angles of a triangle. To find the length or angle of a triangle, one can use formulas, mathematical rules, or the knowledge that the angles of all triangles add up to 180 degrees. Tools to Discover the Sides and Angles of a Triangle. Pythagoras' theorem; Sine rule ...
www.varsitytutors.com/.../how-to-find-the-length-of-the-side-of-a-right-triangle
Explanation: . In order to find the missing side of a right triangle you must use one of two things: 1. Pythagorean Theorem. 2. Trigonometry. Since we only know what the side lengths are we must use the Pythagorean Theorem.
sciencing.com/rules-length-triangle-sides-8606207.html
According to the first triangle inequality theorem, the lengths of any two sides of a triangle must add up to more than the length of the third side. This means that you cannot draw a triangle that has side lengths 2, 7 and 12, for instance, since 2 + 7 is less than 12.
www.csgnetwork.com/righttricalc.html
This calculator is designed to give the two unknown factors in a right triangle, assuming two factors are known. This calculator is for a right triangle only! The factors are the lengths of the sides and one of the two angles, other than the right angle. All values should be in positive values but decimals are allowed and valid.
Related Search
Related Search
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# Background
I found this interesting question Formula for dropping dice (non-brute force) and excellent answer https://stats.stackexchange.com/a/242857/221422, but couldn't figure out how to generalize a generating function for when more than one die is dropped. Similarly, I'm having difficulty figuring out a related mechanic for when the highest roll is dropped.
# Description of the Problem
Suppose you have $$N$$ fair dice each with $$S$$ sides. Roll all the dice and then remove the lowest [or highest, alternatively] $$M$$ (where $$M > 0$$ and $$M < N$$) dice and then sum the remainder. What is the probability distribution of the sum? Specifically, how does one go about finding the generating polynomial?
I found whuber's answer to be incredibly thorough. I thought it might be nice to see how to actually implement it in code, so I've pasted that below.
from functools import reduce
from numpy.polynomial import polynomial as p
def generating_function(k, d, n):
return p.polypow(
[0] * k + [1] * (d - k + 1),
n
)
def drop_one_die(n, d):
tmp = [
generating_function(k, d, n) for k in range(1, d + 2)
]
differences = (
(tmp[i] - tmp[i + 1])[i + 1:] for i in range(d)
)
print(
drop_one_die(4, 6)
)
# Other considerations / Multinomial distribution
To generalize even further, instead of a fair die where each outcome is equally likely, what if you start with a general multinomial distribution?
$$(1/6)x + (1/6)x^2 + (1/6)x^3 + (1/6)x^4 + (1/6)x^5 + (1/6)x^6$$
$$p_0 + {p_1}{x} + {p_2}{x^2} + ... + {p_n}{x^n}$$
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Epic Contributor
Re: AAoA calculation - how precise is it?
Kratos-TM wrote:
OhioCPA wrote:
Kratos-TM wrote:
Let me get this straight......If a new account was opened yesterday, it gets 12 months automatically into the calculation, because nothing can be less than 1?
It's the average that can't be below 1 year. That new account will count as zero for calculation of the average age.
Alright next question......let's say this new account is 11 months old. Does FICO still calculate it as zero in the AAoA equation or does the 11 months get factored?
If your answer is zero, then that means that we should see a bump in AAoA when the new accounts hit 1 year (because they were counted as zero the entire 11 months).
The 11 months get factored in. Here is an example based on 5 TLs.
11 months + 72 months + 8 months + 4 months + 15 months = 110 months divided by 5 = 22 months average. FICO rounds anything 0 - 23 months down to 1 year so your AAoA would be 1 year.
Message 11 of 12
Frequent Contributor
Re: AAoA calculation - how precise is it?
guiness56 wrote:
Kratos-TM wrote:
OhioCPA wrote:
Kratos-TM wrote:
Let me get this straight......If a new account was opened yesterday, it gets 12 months automatically into the calculation, because nothing can be less than 1?
It's the average that can't be below 1 year. That new account will count as zero for calculation of the average age.
Alright next question......let's say this new account is 11 months old. Does FICO still calculate it as zero in the AAoA equation or does the 11 months get factored?
If your answer is zero, then that means that we should see a bump in AAoA when the new accounts hit 1 year (because they were counted as zero the entire 11 months).
The 11 months get factored in. Here is an example based on 5 TLs.
11 months + 72 months + 8 months + 4 months + 15 months = 110 months divided by 5 = 22 months average. FICO rounds anything 0 - 23 months down to 1 year so your AAoA would be 1 year.
Thanks a lot. You broke it down really good. I understand completely.
Myfico scores.....
EX... 805 (11-10-13)
EQ... 810 (12-02-13)
TU... 810 (12-02-13)
Message 12 of 12
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http://blog.mikael.johanssons.org/archive/2007/05/young-topology-the-fundamental-groupoid/
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Michi’s blog » Young Topology: The fundamental groupoid
## Young Topology: The fundamental groupoid
• May 4th, 2007
• 3:26 pm
Today, with my bright topology 9th-graders, we discussed homotopy equivalence of spaces and the fundamental groupoid. In order to get the arguments sorted out, and also in order to give my esteemed readership a chance to see what I’m doing with them, I’ll write out some of the arguments here.
I will straight off assume that continuity is something everyone’s comfortable with, and build on top of that.
## Homotopies and homotopy equivalences
We say that two continuous maps, f,g:X→Y between topological spaces are homotopical, and write , if there is a continuous map such that H(x,0)=f(x) and H(x,1)=g(x). This captures the intuitive idea of step by step nudging one map into the other in formal terms.
Two spaces X,Y are homeomorphic if there are maps , such that and .
Two spaces X,Y are homotopy equivalent if there are maps , such that and .
Now, if f,g are maps X→Y and f=g, then , since we can just set H(x,t)=f(x)=g(x) for all t, and get a continuous map out of it. Thus homeomorphic spaces are homotopy equivalent, since the relevant maps are equal, and thus homotopic.
There are a couple of more properties for homotopic maps we’ll want. It respects composition – so if and h:Y→Z and e:W→X then and . This can be seen by considering h(H(x,t)) and H(e(x),t) respectively.
Denote by D2 the unit disc in , and by {*} the subset {(0,0)} in . Then . In one direction, the relevant map is just the embedding, and in the other direction, it collapses all of D2 onto {*}. One of the two relevant compositions is trivially equal the identity map, and in the other direction, the linear homotopy H(x,t)=tx will do well. Thus the disc and the one point space are homotopy equivalent.
## The fundamental groupoid
Let X be a topological space (most probably with a number of neat properties – I will not list just what properties are needed though), and consider for each pair x,y of points in X, the set [x,y] of homotopy classes of paths from the point x to the point y. A path, here, is a continuous map [0,1]→X. We can compose classes – if and , then we can consider the map
. This is a path from x to z, and so belongs to a class in [x,z]. This class is well defined from the choices of γ, γ' since homotopies and composition of maps work well together.
This gives us a composition. It is associative - on homotopy classes. What happens if we look at maps instead of homotopy classes is part of the subject of my own research. It has an identity at each point x - the constant path γ(t)=x, and for each class in [x,y] there is a class in [y,x] such that their composition is homotopic to the constant path in [x,x].
Thus, we get a groupoid. This is called the fundamental groupoid, and denoted by . If we fix a point, and consider [x,x], then this is a group, called the fundamental group with basepoint x, and denoted by .
For , a linear homotopy will make any two paths in [x,y] homotopic – and so |[x,y]|=1 in for any x,y.
For S1 – the circle – we can choose to view it as [0,1]/(0=1). Then we can consider the paths fm(t)=a(1-t)+bt+nt. This is a path from a to b, and it winds n times around the circle. Each path in [a,b] is homotopic to a fm, by a linear homotopy, which just rescales the speeds through various bits and pieces, and possibly straightens out when you double back. Thus, . Furthermore, if you compose fmfn, you’ll get fn+m.
If we pick out the fundamental group out of this groupoid, we’ll get the well known fundamental group .
Now, suppose we have two homotopy equivalent spaces X and Y, with the homotopy equivalence given by f:X→Y and g:Y→X. Then consider the map f*:[x,y]X→[f(x),f(y)]Y given by f*γ(t)=f(γ(t)). I claim
1) f* is bijective.
2) f* works well with composition of classes.
For bijectivity we start with injectivity in one direction. Consider two paths in [x,y]. We need to show . If , then . However, then
which contradicts . Thus , and so also .
The proof is symmetric in the choice of direction, and so we can just repeat the same argument to get that g* is also an injection. Thus we can conclude that f* is in fact a bijection.
Now, for the second part, we consider and . We need to show that . But is the path that first runs through in half the time, then runs through in the rest of the time, and just transports this path point by point to Y. And transports point by point to Y and transports point by point to Y, and just runs through the first of these in half the time, then the rest in the rest of the time.
Thus, homotopy equivalent spaces have the same fundamental groupoid.
### 7 People had this to say...
• Creighton Hogg
• May 4th, 2007
• 16:36
Hey, I think it’s great that you’re helping to teach this stuff to kids. When I was maybe a year older than them, I started teaching myself topology & it was a lot harder than it would have been with other people to talk to. So, on behalf of my younger self, thanks for teaching interested kids.
• Michi
• May 4th, 2007
• 16:39
Creighton: For one of my reasons to take interested kids VERY seriously, take a look at my post on why I keep organizing congresses – I have a personal history as the interested kid who got taken seriously, and this is what placed me where I am today.
That said, these kids are quite a bit more skilled than I am. I am constantly amazed with the mere fact that they keep up with it.
• Creighton Hogg
• May 4th, 2007
• 16:54
Cool.
Yeah, I never really got taken that seriously but otherwise we have pretty similar histories with suddenly discovering how great real math is. I found an old book for a couple of dollars in a used bookstore that went from basic addition to basic multi-variate calculus, and devoured the book after school during the beginning of my sophmore year of highschool.
Mathematics is a beautiful subject, and while I’ve never done as much as you’re doing, I try to encourage kids I meet by giving them a glimpse of what real math is like. One of my wife’s teenage cousins likes her geometry class, so I spent some time and told her a bit about what happens when you leave Euclidean geometry: the difference between how cylinder’s are curved and how sphere’s are curved, why you can’t giftwrap a basketball, and how it all ties to gravity and the early universe.
[...] but it seems that the majority of contributors have explanations of topics for this edition. Mikki, a professor, visits a high school once a week, and describes the super-challenging problems he [...]
• Jon
• May 23rd, 2007
• 7:13
I’m a bit confused. If h: Y -> Z, and g: X -> Y, then h(f(x)) is in Z, and g(f(x)) makes no sense, since f(x) is in Y, not X. Then saying h(f(x)) is homotopic to g(f(x)) makes no sense. Did you mean h(f(x)) is homotopic to h(g(x))?
• Michi
• May 23rd, 2007
• 8:51
Jon: Doh! Of course I mean hf homotopic to hg.
• Jon
• May 23rd, 2007
• 9:53
Ah, thanks.
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Home > Uncategorized > Flawed analysis of “one child is a boy” problem?
## Flawed analysis of “one child is a boy” problem?
November 6th, 2010
A mathematical puzzle has reappeared over the last year as the topic of discussion in various blogs and I have not seen any discussion suggesting that the analysis appearing in blogs contains a fundamental flaw.
The problem is as follows: I have two children and at least one of them is a boy; what is the probability that I have two boys? (A variant of this problem specifies whether the boy was born first or last and has a noncontroversial answer).
Most peoples (me included) off-the-top-of-the-head answer is 1/2 (the other child can be a girl or a boy, assuming equal birth probabilities {which is very a good approximation to reality}).
The analysis that I believe to be incorrect goes something like the following: The possible birth orders are `gb`, `bg`, `bb` or `gg` and based on the information given we can rule out girl/girl, leaving the probability of `bb` as 1/3. Surprise!
A variant of this puzzle asks for the probability of boy/boy given that we know one of the children is a boy born on a Tuesday. Here the answer is claimed to be 13/27 (brief analysis or using more complicated maths). Even greater surprise!
I think the above analysis is incorrect, which seems to put me in a minority (ok, Wikipedia says the answer could sometimes be 1/2). Perhaps a reader of this blog can tell me where I am going wrong in the following analysis.
Lets call the known boy `B`, the possible boy `b` and the possible girls `g`. The sequence of birth events that can occur are:
`Bg gB bB Bb gg`
There are four sequences that contain at least one boy, two of which contain two boys. So the probability of two boys is 1/2. No surprise.
All of the blog based analysis I have seen treat the ordering of a known boy+girl birth sequence as being significant but do not to treat the ordering of a known boy+boy sequence as significant. This leads them to calculate the incorrect probability of 1/3.
The same analysis can be applied to the “boy born on a Tuesday” problem to get the probability 14/28 (i.e., 1/2).
Those of you who like to code up a problem might like to consider the use of a language like Prolog which I would have thought would be less susceptible to hidden assumptions than a Python solution.
Tags:
1. November 6th, 2010 at 21:05 | #1
> Lets call the known boy B, the possible boy b and the possible girls g.
> The sequence of birth events that can occur are:
>
> Bg gB bB Bb gg
I think you are trying to challenge your readers to provide a convincing refutation without believing this yourself. But anyway, let me try to put it this way:
First, the trick of enumerating combinations and dividing the number of matching ones by the total only works for equiprobable possibilities, and the way to avoid applying this trick wrongly is to enumerate first and to decide if each combination matches later. The equiprobable combinations are gg, gb, bg, bb. Your five possibilities are not equiprobable, (and they introduce a new unknown, how much preference the all-seeing chooser has for the elder son when he has a choice between two boys).
Regarding the “born on a Tuesday” apparent paradox, it’s quite the same thing in reverse: the day of the week seems arbitrary (and indeed it is), but the important thing is that the day is picked first, and then the question “knowing only that in this family, one child is a boy born on this day of the week, then …?” is asked. The probability in this case differs from the correct 1/3 from the simpler case because families with two boys have more chances to have one of them born on a Tuesday.
2. November 6th, 2010 at 21:05 | #2
Well, it depends if there is “a known child” and what’s the process to get the statement from the father.
If you ask the father “tell me something about a child of you”, the answer is 1/2. If you ask the father “choose one of your children and let me know his/her sex”, the answer is 1/2.
If you ask “at least one of your children is a boy?” the problem gets equivalent to “throw two coins, if boths are tails, throw them again until at least one is heads, what’s the probability that there are two heads?” There is no “known head”, you examine the set.
“At least one of the children is a boy” is not a statement about “a known boy” but about the set of children. When the statement is modified to talk about a definite child, the answer is 1/2.
A modified version of the “tuesday problem” (only to be used on people that has answered 1/3 to the original and the tuesday variation) is “At least one of my children is a boy…. this child I talked about… er, no, nothing” The answer is 1/2 (and there is not even an “irrelevant” datum added, as with the tuesday thing) When the father stops talking about the set of children and starts talking about “that child” the results change from 1/3 to 1/2.
Excuse my poor english.
As a program to check this, I would go with some way to generate random strings of gg, gb, bg, bb and then:
grep b cases | wc (at least one boy)
grep bb cases | wc (two boys)
and then divide.
3. November 7th, 2010 at 00:05 | #3
Thanks to Pascal Cuoq and H for commenting so quickly (H your English is very good).
Both of your comments made sense to me and my own ideas continued to make sense. Where was the problem? The Wikipedia article cited Martin Gardner and I managed to unearth my copy of “The Colossal book of Mathematics” to read Gardener’s explanation of a 1/2 solution; then I understood.
The different answers come about because of differences in the construction of the underlying distribution from which the family is drawn.
If we randomly select a parent from the population of all families in the world with two children and stop selecting when we encounter a parent who answers yes to the question “Do you have at least one son”, then the probability of them having two sons is 1/3. This is how blogs I have read, Pascal Cuoq and H view the problem.
If a person walks up to me and says they have two children and at least one of them is a boy, that person has self selected the problem statement (as Gardener points out a person with two girls would have to say at least one of my children is a girl). This is how I have been viewing the problem, hence my use of the term ‘known-boy’, and the probability of two boys is 1/2.
4. November 24th, 2010 at 12:38 | #4
The use of permutations on this problem doesn’t sit right with me. The 1/3 answer often comes from the wording that allows the boy to be either the elder or younger, but is it acceptable to add irrelevant age information to the list of permutations? Why not add more information, such as what day of the week they were born on, like the Tuesday problem, or hair color? I’m no expert, but my opinion is that the age-irrelevant wording should have an age-irrelevant permutation set:
bb bg
5. December 1st, 2010 at 17:26 | #5
The issues of whether there is a “known child,” or if the boy is the older or younger child, are both red herrings. I don’t mean the answers to those questions can’t influence the answer – they can – but they do not address why the answers are different.
Change the problem to an experiment you can repeat with coins. Say I flip two coins, a dime and a penny, and tell you “at least one landed on (heads or tails)” afterwards. The answer to “what is the probability that both landed the same way?” depends on the strategy I use for choosing what to say.
1) If I decided ahead of time that I would only tell you if one landed on a heads, and re-flip the coins if neither do, then the answer is 1/3.
2) If I decided ahead of time that I would prefer to tell you if one landed on a heads, and will tell you “one landed on tails” only if both do, then the answer is 1/3 if I say “heads,” and 1 (yes, that’s 100%) if I say “tails.”
3) If I decided to always tell you how the penny landed, and ignore the dime, the answer is 1/2. This compares to the “known child” case.
4) If I decided to tell you about the one that stopped spinning first (or landed closer to me, or any other random factor), the answer is again 1/2.
If you treat the person who said “I have two children and at least one of them is a boy” as a randomly-chosen person, any of these interpretations of why he choose to say that is possible. But the first two are very improbable, and require that we assume this random person had a bias of some sort. And in fact, if you assume such a bias, you can’t know whether #1 or #2 is right, so the answer coudl be 100%. The most reasonable interpretation is #4, and it is not the same as #3 even though it gets the same answer.
The reason the answer changes is because the event we should use to divide the cases into “possible” and “not possible” is based on what I tell you, not necessarily what is true. In cases #3 and #4 there are times where “at least one landed on heads” is true, but I will tell you “at least one landed on tails.” In case #1, that is not possible; and in case #2, we actually get two different answers and you have way to know which is correct.
You can make a similar comparison to the “boy born on Tuesday” problem. Say I start with seven of each coin, each minted in a different year in the range 2001-2007. I pick a random dime and a random penny, and flip them.
1) If I decided ahead of time that I would only tell you a 2005 coin landed on a heads, and re-flip the coins if neither do, then the answer is 13/27.
2) If I decided ahead of time that I would prefer to tell if one landed on a heads, and add the earliest year I could, then the answer is 13/27 if I say “2001 heads,” 11/25 if I say “2002 heads,” 9/23 if I say “2003 heads,” etc.
3) If I decided to always tell you about the penny, and ignore the dime, the answer is 1/2 just as it was before.
4) If I decided to tell you about a random coin, the answer is again 1/2 in all cases.
The fact that everybody expects the information about the date to have no effect proves that they really expect the coin (or child) to be a specifically-chosen one, as in \$4, and the information about it to be whatever is true for that child.
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# Mass vs. Weight
Mass and Weight are two often misused and misunderstood terms in mechanics and fluid mechanics.
The fundamental relation between mass and weight is defined by Newton's Second Law. Newton's Second Law can be expressed as
F = m a (1)
where
F = force (N, lbf)
m = mass (kg, slugs)
a = acceleration (m/s2, ft/s2)
### Mass
Mass is a measure of the amount of material in an object, being directly related to the number and type of atoms present in the object. Mass does not change with a body's position, movement or alteration of its shape, unless material is added or removed.
• an object with mass 1 kg on earth would have the same mass of 1 kg on the moon
Mass is a fundamental property of an object, a numerical measure of its inertia and a fundamental measure of the amount of matter in the object.
• mass electron 9.1095 10-31 kg
• mass proton 1.67265 10-27 kg
• mass neutron 1.67495 10-27 kg
### Weight
Weight is the gravitational force acting on a body mass. The generic expression of Newton's Second Law (1) can be transformed to express weight as a force by replacing the acceleration - a - with the acceleration of gravity - g - as
Fg = m ag (2)
where
Fg = gravitational force - or weight (N, lbf)
m = mass (kg, slugs (lbm))
ag = acceleration of gravity on earth (9.81 m/s2, 32.17405 ft/s2)
#### Example - The Weight of a Body on Earth vs. Moon
The acceleration of gravity on the moon is approximately 1/6 of the acceleration of gravity on the earth. The weight of a body with mass 1 kg on the earth can be calculated as
Fg_earth = (1 kg) (9.81 m/s2)
= 9.81 N
The weight of the same body on the moon can be calculated as
Fg_moon = (1 kg) ((9.81 m/s2) / 6)
= 1.64 N
The handling of mass and weight depends on the systems of units used. The most common unit systems are
• the International System - SI
• the British Gravitational System - BG
• the English Engineering System - EE
One newton is
• ≈ the weight of one hundred grams - 101.972 gf (gF) or 0.101972 kgf (kgF or kilopond - kp (pondus is latin for weight))
• ≈ halfway between one-fifth and one-fourth of a pound - 0.224809 lb or 3.59694 oz
### The International System - SI
In the SI system the mass unit is the kg and since the weight is a force - the weight unit is the Newton (N). Equation (2) for a body with 1 kg mass can be expressed as:
Fg = (1 kg) (9.807 m/s2)
= 9.807 (N)
where
9.807 m/s2 = standard gravity close to earth in the SI system
As a result:
• a 9.807 N force acting on a body with 1 kg mass will give the body an acceleration of 9.807 m/s2
• a body with mass of 1 kg weights 9.807 N
### The Imperial British Gravitational System - BG
The British Gravitational System (Imperial System) of units is used by engineers in the English-speaking world with the same relation to the foot - pound - second system as the meter - kilogram - force second system (SI) has to the meter - kilogram - second system. For engineers who deals with forces, instead of masses, it's convenient to use a system that has as its base units length, time, and force, instead of length, time and mass.
The three base units in the Imperial system are foot, second and pound-force.
In the BG system the mass unit is the slug and is defined from the Newton's Second Law (1). The unit of mass, the slug, is derived from the pound-force by defining it as the mass that will accelerate with 1 foot per second per second when a 1 pound-force acts upon it:
1 lbf = (1 slug) (1 ft/s2)
In other words, 1 lbf (pound-force) acting on 1 slug of mass will give the mass an acceleration of 1 ft/s2.
The weight (force) of the mass can be calculated from equation (2) in BG units as
Fg (lbf) = m (slugs) ag (ft/s2)
With standard gravity - ag = 32.17405 ft/s2 - the weight (force) of 1 slug mass can be calculated as
Fg = (1 slug) (32.17405 ft/s2)
= 32.17405 lbf
### The English Engineering System - EE
In the English Engineering system of units the primary dimensions are are force, mass, length, time and temperature. The units for force and mass are defined independently
• the basic unit of mass is pound-mass (lbm)
• the unit of force is the pound (lb) alternatively pound-force (lbf).
In the EE system 1 lbf of force will give a mass of 1 lbm a standard acceleration of 32.17405 ft/s2.
Since the EE system operates with these units of force and mass, the Newton's Second Law can be modified to
F = m a / gc (3)
where
gc = a proportionality constant
or transformed to weight (force)
Fg = m ag / gc (4)
The proportionality constant gc makes it possible to define suitable units for force and mass. We can transform (4) to
1 lbf = (1 lbm) (32.174 ft/s2) / gc
or
gc = (1 lbm) (32.174 ft/s2) / (1 lbf)
Since 1 lbf gives a mass of 1 lbm an acceleration of 32.17405 ft/s2 and a mass of 1 slug an acceleration of 1 ft/s2, then
1 slug = 32.17405 lbm
### Example - Weight versus Mass
The mass of a car is 1644 kg. The weight can be calculated:
Fg = (1644 kg) (9.807 m/s2)
= 16122.7 N
= 16.1 kN
- there is a force (weight) of 16.1 kN between the car and the earth.
• 1 kg gravitation force = 9.81 N = 2.20462 lbf
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# Moment of inertia of a disk by integration
by soopo
Tags: calculus, disk, moment of inertia
Share this thread:
P: 226 1. The problem statement, all variables and given/known data Show that the moment of inertia of a disk is $0.5 mr^2$. 3. The attempt at a solution $$I = \int R^2 dm$$ Using $dm = \lambda dr$ such that $m = \lambda r$: $$= \int_{-r}^{r} R^2 \lambda dr$$ $$= \frac { \lambda } {3} ( 2r^3 )$$ $$= \frac {2} {3} (\lambda r ) (r^2)$$ $$= \frac {2} {3} M R^2$$ which should be the moment of inertia for a ring. Integrating this from 0 to 2pii relative to the angle gives me $\frac {4} {9} m r^3 [/tex], which is wrong. How can you calculate the moment of inertia for a disk? P: 383 Quote by soopo 1. The problem statement, all variables and given/known data Show that the moment of inertia of a disk is [itex] 0.5 mr^2$. 3. The attempt at a solution $$I = \int R^2 dm$$ Using $dm = \lambda dr$ such that $m = \lambda r$: $$= \int_{-r}^{r} R^2 \lambda dr$$ $$= \frac { \lambda } {3} ( 2r^3 )$$ $$= \frac {2} {3} (\lambda r ) (r^2)$$ $$= \frac {2} {3} M R^2$$ which should be the moment of inertia for a ring. Integrating this from 0 to 2pii relative to the angle gives me $\frac {4} {9} m r^3 [/tex], which is wrong. How can you calculate the moment of inertia for a disk? If you have a disc of radius $$r$$, how can you integrate from $$-r$$ to $$r$$? P: 383 The infinitesimal change in mass is given by $$dm=m\frac{dA}{A}=m\frac{2\pi r\,dr}{\pi R^2}=\frac{2m}{R^2}r\,dr$$ If you use that in your integral and integrate from $$0$$ to $$R$$, you should get the desired result. P: 226 Moment of inertia of a disk by integration Quote by jdwood983 The infinitesimal change in mass is given by $$dm=m\frac{dA}{A}=m\frac{2\pi r\,dr}{\pi R^2}=\frac{2m}{R^2}r\,dr$$ If you use that in your integral and integrate from $$0$$ to $$R$$, you should get the desired result. Your result gives me a wrong result: $$I = \int R^2 dm$$ $$= \int R^2 \frac { 2m } {R^2} r dr$$ $$= \int_{0}^{r} 2mr dr$$ $$= [ m r^2 ]^{r}_{0}$$ $$= mr^2$$ The result shoud be $$I = .5 mr^2$$ P: 226 Quote by jdwood983 If you have a disc of radius $$r$$, how can you integrate from $$-r$$ to $$r$$? I set the null point to the center of the circle such that I am integrating from -r to r. I am not sure why I cannot do that. P: 383 Quote by soopo I set the null point to the center of the circle such that I am integrating from -r to r. I am not sure why I cannot do that. If you set the origin to the center of the circle (which you should always try to do), the smallest value that $$r$$ can be is 0. So it is physically impossible to integrate from $$-r$$ to $$r$$, that is why you can't do it. P: 226 Quote by jdwood983 If you set the origin to the center of the circle (which you should always try to do), the smallest value that $$r$$ can be is 0. So it is physically impossible to integrate from $$-r$$ to $$r$$, that is why you can't do it. I am thinking of setting an axis which goes through the origin such that the zero point of the axis is at the origin. Going to right means to go towards $$r$$, while going to left means towards $$-r$$. Perhaps, you are thinking the situation in a polar coordinate system in which case you cannot have negative $$-r$$. I feel that it is possible to integrate from $$-r$$ to $$r$$ in a cartesian coordinate system. P: 383 Quote by soopo I am thinking of setting an axis which goes through the origin such that the zero point of the axis is at the origin. Going to right means to go towards $$r$$, while going to left means towards $$-r$$. Perhaps, you are thinking the situation in a polar coordinate system in which case you cannot have negative $$-r$$. I feel that it is possible to integrate from $$-r$$ to $$r$$ in a cartesian coordinate system. If you want Cartesian coordinates, then you'll need two integrals: one over $$x$$ and one over $$y$$. While technically you have two integrals in polar, $$r\, \mathrm{and}\, \theta$$, one is already done for you and reduces the integration to just one term: $$r$$. This problem is by far easier in polar coordinates: $$\begin{array}{ll}I&=\int_0^R r^2\frac{2m}{R^2}rdr \\ &=\frac{2m}{R^2}\int_0^Rr^3dr \\ &=\frac{2m}{R^2}\left(\frac{R^4}{4}-0\right) \\ &=\frac{2m}{R^2}\cdpt\frac{R^4}{4} \\ &=\frac{1}{2}mR^2$$ P: 226 Quote by jdwood983 If you want Cartesian coordinates, then you'll need two integrals: one over $$x$$ and one over $$y$$. While technically you have two integrals in polar, $$r\, \mathrm{and}\, \theta$$, one is already done for you and reduces the integration to just one term: $$r$$. This problem is by far easier in polar coordinates: If you use symmetry, it is enough to consider only the first quadrant that is where x > 0 and y > 0 such that four of these quadrants form the area of the disk. You do not get the x- and y -coordinates easily from the definition of the moment of inertia. You would get $$I = \int (x^2 + y^2) dm \\ &= \int (x^2 + y^2) m \frac { 2r } {R^2} dr$$ The calculations seem to get challenging, since we need to use Pythogoras such that $$r = \sqrt{ x^2 + y^2 }$$ which implies $$dr = \frac { 1 } { \sqrt {x^2 + y^2} } * 2x$$ We can get similarly the relation relative to $$y$$. The next step is not fun at all: $$I = \int_{0}^{1} \sqrt {x^2 +y^2} (x+y) x dx$$, where I assume that [itex] R^2 = (x + y)^2 = (1 + 1)^2 = 4$, since it is the maximum radius.
This way the two 2s cancel out.
I do not even know how to integrate this!
Polar coordinate system really seems to be better in this case.
P: 383
Quote by soopo The next step is not fun at all: $$I = \int_{0}^{1} \sqrt {x^2 +y^2} (x+y) x dx$$, where I assume that $R^2 = (x + y)^2 = (1 + 1)^2 = 4$, since it is the maximum radius.
Not quite. The integral you need is given by
$$I=\frac{m}{A}\int_{-R}^R\int_{-\sqrt{R^2-x^2}}^{\sqrt{R^2-x^2}}\left(x^2+y^2\right)dydx$$
$$I=\frac{m}{\pi R^2}\int_{-R}^R \frac{2\sqrt{R^2-x^2}\left(R^2+2x^2\right)}{3}dx$$
$$I=\frac{m}{\pi R^2}\cdot\frac{\pi R^4}{2}$$
$$I=\frac{mR^2}{2}$$
Quote by soopo Polar coordinate system really seems to be better in this case.
For most moment of inertia problems, spherical or cylindrical coordinates are the best.
P: 226
Quote by jdwood983 $$\int_{-R}^R \frac{2\sqrt{R^2-x^2}\left(R^2+2x^2\right)}{3}dx$$ $$= \frac{\pi R^4}{2}$$
How did you solve this part?
It has taken my some effort in trying to solve it by hand.
P: 383
Quote by soopo How did you solve this part? It has taken my some effort in trying to solve it by hand.
There's a few extra steps between the two lines, like making a change of variables. But to be honest I used Mathematica and just wrote the lines because I forget what changes needed to be made. While it may be good to know the form of the equation, I'm not sure you would need the solution since it is far easier to do it in polar coordinates than in Cartesian coordinates.
Related Discussions Introductory Physics Homework 7 Introductory Physics Homework 1 Classical Physics 2 Introductory Physics Homework 2 Introductory Physics Homework 2
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# What is Considered a Low Standard Deviation?
The standard deviation is used to measure the spread of values in a sample.
We can use the following formula to calculate the standard deviation of a given sample:
Σ(xi – xbar)2 / (n-1)
where:
• Σ: A symbol that means “sum”
• xi: The ith value in the sample
• xbar: The mean of the sample
• n: The sample size
The higher the value for the standard deviation, the more spread out the values are in a sample. Conversely, the lower the value for the standard deviation, the more closely packed together the values.
One question students often have is: What is considered a low value for the standard deviation?
The answer: There is no cut-off value for what is considered a “low” standard deviation because it depends on the type of data you’re working with.
For example, consider the following scenarios:
Scenario 1: A professor collects data on the exam scores of students in his class and finds that the standard deviation of exam scores is 7.8.
Scenario 2: An economist measures the total income tax collected by different countries around the world and finds that the standard deviation of total income tax collected is \$1.2 million.
The standard deviation in scenario 2 is much higher, but that’s only because the values being measured in scenario 2 are considerably higher than those being measured in scenario 1.
This means there is no single number we can use to tell whether or not a standard deviation is “low” or not. It all depends on the situation.
### Using the Coefficient of Variation
One way to determine if a standard deviation is “low” is to compare it to the mean of the dataset.
A coefficient of variation, often abbreviated CV, is a way to measure how spread out values are in a dataset relative to the mean. It is calculated as:
CV = s / x
where:
• s: The standard deviation of dataset
• x: The mean of dataset
The lower the CV, the lower the standard deviation relative to the mean.
For example, suppose a professor collects data on the exam scores of students and finds that the mean score is 80.3 and the standard deviation of scores is 7.8. The CV would be calculated as:
• CV: 7.8 / 80.3 = .097
Suppose another professor at a different university collects data on the exam scores of his students and finds that the mean score is 70.3 and the standard deviation of scores is 8.5. The CV would be calculated as:
• CV: 8.5 / 90.2 = 0.094
Although the standard deviation of exam scores is lower for the first professor’s students, the coefficient of variation is actually higher than that of the exam scores for the second professor’s students.
This means the variation of exam scores relative to the mean score is higher for the first professor’s students.
### Comparing Standard Deviations Between Samples
Rather than classifying a standard deviation as “low” or not, often we simply compare the standard deviation between several samples to determine which sample has the lowest standard deviation.
For example, suppose a professor administers three exams to his students during the course of one semester. He then calculates the sample standard deviation of scores for each exam:
• Sample standard deviation of Exam 1 Scores: 4.9
• Sample standard deviation of Exam 2 Scores: 14.4
• Sample standard deviation of Exam 3 Scores: 2.5
The professor can see that Exam 3 had the lowest standard deviation of scores among all three exams, which means the exam scores were most closely packed together for that exam.
Conversely, he can see that Exam 2 had the highest standard deviation, which means the exam scores were most spread out for that exam.
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## One Reply to “What is Considered a Low Standard Deviation?”
1. giulia says:
if my sd is of 7.82101182456602 and the mean is -0.3663. calculating the coefficent of variation it is -21,3513836. Is it right? is it low or high? (we are talking about stocks values)
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In looking at the Trachtenberg system we will start with multiplying by eleven. The rule here is very simple and gives us a very fast way to multiply any number by eleven.
## The Rules For Multiplying By Eleven
1. Put the last number of the multiplicand down as the right hand figure of the answer.
2. Add each successive number of the multiplicand to its neighbor on the right.
3. The first number of the multiplicand becomes the left hand number of the answer.
Actually we can replace these three rules with a single rule.
## An Example - Multiplying 3456 by 11
For this example we will look at the following multiplication:
Setting up the Calculation
The first thing we will do is write in a zero at the front of the multiplicand and then draw a line under it. We draw the line as we will be writing the answer directly below the multiplicand as we calculate it.
First Step
We draw in our red box to show you where we are (you don't need to draw in the box when doing this yourself). On the left side of the box we have 6, the right hand side of the box is empty. We have no neighbor to add to the six so our first digit is 6.
Second Step
Now we move our box to the left by one digit. Now we have 5 and its neighbor is 6, 5 plus 6 adds up to 11, so we write 1 and we carry 1.
To show we are carrying one I will put a dot above the digit in the product we just wrote down. Doing the calculation mentally you would just remember you are carrying 1.
Third Step
This time we have 4 and its neighbor is 5, which adds up to 9, to which we add 1 carried over in the previous step, giving us a total of 10.
So we write the 0 and carry the 1. Again we put a dot above the zero to show we are carrying 1.
Fourth Step
Moving the box to the left again we now have 3 and its neighbor is 4, which add up to 7, to which we add the 1 carried over before giving us a total of 8.
We write the 8 down.
Last Step
Moving the box one digit to the left again there is 0 and its neighbor is 3, which adds up to 3.
So we write 3 into the product giving us the answer of 38016.
Note: If we didn't have the zero in front we may have stopped at the 3 and we would have ended up with the wrong answer.
This concept will work for all numbers multiplied by 11, from single digit numbers up to numbers of any size.
## Multiplying 2 Digit Numbers by 11
Although for two digit numbers the process is the same as above there I just want to emphasis how easy multiplying by eleven is using this method and you can easily do this in your head.
When multiplying a two digit number by eleven you simply add the two digits then put the sum in between the two digits.
To multiply 24 by 11 you simply add 2 and 4 to get 6 then place the 6 between the 2 and the 4 giving you the answer 264. Simple isn't it?
This works if the two digits add up to less than ten. If they add up to more than ten you simply put the unit value between the two digits then carry the one to the digit on the left.
To multiply 39 by 11 you add 3 and 9 together to get 12, then you put the 2 between the 3 and 9 then change the 3 to 4 because we need to add 1 for the carry.
## Multiplying Single Digit Numbers by 11
Lets have a quick look at a very simple single digit example, 6 x 11, just to show the method above still works.
We place the box over the 6, so we have 6 and it has no neighbor, giving us 6, so we write this down.
Moving the box to the left we have the zero and the 6 in the box. Now we have zero and we add its neighbor which is 6, this gives us 6, so we write 6 giving us our answer of 66.
This method works for all numbers multiplied by eleven.
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# Formula For Parallel Circuit Conductance
The formula for parallel circuit conductance is:
$$G = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n}$$
where:
* $G$ is the total conductance of the parallel circuit * $R_1$, $R_2$, $R_3$, ..., $R_n$ are the resistances of the individual resistors in the parallel circuit
To find the total conductance of a parallel circuit, simply add the reciprocals of the individual resistances. For example, if you have a parallel circuit with three resistors of 10 ohms, 20 ohms, and 30 ohms, the total conductance would be:
$$G = \frac{1}{10 \Omega} + \frac{1}{20 \Omega} + \frac{1}{30 \Omega} = \frac{6}{60 \Omega} = \frac{1}{10 \Omega}$$
Therefore, the total resistance of the parallel circuit would be 10 ohms.
The formula for parallel circuit conductance can be used to calculate the total conductance of any parallel circuit, regardless of the number or value of the resistors in the circuit. It is a simple and straightforward formula that can be easily used to solve a variety of electrical engineering problems.
Here are some additional examples of parallel circuit conductance calculations:
* A parallel circuit with two resistors of 10 ohms and 20 ohms has a total conductance of 30 ohms. * A parallel circuit with three resistors of 10 ohms, 20 ohms, and 30 ohms has a total conductance of 10 ohms. * A parallel circuit with four resistors of 10 ohms, 20 ohms, 30 ohms, and 40 ohms has a total conductance of 5 ohms.
As you can see, the total conductance of a parallel circuit decreases as the number of resistors in the circuit increases. This is because each resistor in a parallel circuit provides a path for current to flow, and the more paths there are for current to flow, the lower the total resistance of the circuit.
The formula for parallel circuit conductance is a valuable tool for electrical engineers and other professionals who work with electrical circuits. It can be used to calculate the total conductance of any parallel circuit, regardless of the number or value of the resistors in the circuit.
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LESSONNEW
Geometric Proofs
42 minutes ago by
Harvey Williams
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INSTRUCTOR-LED SESSION
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ASYNCHRONOUS LEARNING
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• Slide 1
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Geometric Proofs
Two Column Proofs
• Slide 2
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What is a Proof?
A Proof is a convincing mathematical argument.
• Slide 3
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......(YES WRITE THIS DOWN)
This means that any person, who understands the terminology, accepts the definition and premises of the mathematics involved and thinks in a logical correct fashion could not deny the validity of the conclusions drawn.
• Slide 4
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Undefined Terms-
Can be described but not given precise definitions using simple known terms .
Point
Line
Plane
These are intangible concepts that serve as a foundation. (Used for visualization )
• Slide 5
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Other terms...
Space- the set of all points
Geometric Figure- Any collection of points
• Question 6
45 seconds
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Q.
Check all that are intangible concepts
3 D Box
Plane
Line
sphere
point
• Slide 7
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CONPONENTS OF A PROOF
Subtitle
• Slide 8
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Geometric Proof premises
• Definitions
• Postulates
• Properties
• previous proven theorems
• Slide 9
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Theorems
Results that we declare from the undefined terms, definitions, postulates , or results that follow from them are call a Theorem
Theorem- is a mathematical Statement that can be proven.
• Slide 10
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Postulate
Postulates are statements that we assume to be true.
An postulate states relationships among defined and undefined terms. The purpose stating postulates is to establish some first principles upon which the subject of geometry is based.
• Question 11
30 seconds
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Q.
A convincing mathematical argument.
Postulate
Proof
Geometric Figure
Definitions
• Slide 12
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Writing a Proof
Body text
• Slide 13
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Writing a Proof
• Justify each logical step with reason.
• You can use symbols and abbrev, but they must be clear and able to understand.
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• Slide 15
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• Slide 16
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• Step 1 - Write the conjecture (an opinion or conclusion formed on the basis of incomplete information.)
• Step 2 - Draw the diagram
• Step 3 - State the Given information and mark it on the diagram
• Step 4 - State the conclusion of the conjecture in terms of Diagram
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LET'S DO SOME EXAMPLES!
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# Exercise C.1.13
$\star$ Use Stirling's approximation to prove that
$$\binom{2n}{n} = \frac{2^{2n}}{\sqrt{\pi n}}(1 + \mathcal{O}(1/n))$$
So:
\begin{aligned} \binom{2n}{n} &= \frac{(2n)!}{n!(2n-n)!} = \frac{(2n)!}{(n!)^2} \\ &= \frac{\sqrt{2 \pi 2 n}\big(\frac{2n}{e}\big)^{2n}\Big(1 + \Theta(\frac{1}{n})\Big)} {2 \pi n \big(\frac{n}{e}\big)^{2n}\Big(1 + \Theta(\frac{1}{n})\Big)^2} \\ &= \frac{1}{\sqrt{\pi n}} \frac{2^{2n}n^{2n}}{n^{2n}}(1 + \mathcal{O}(1/n)) \\ &= \frac{2^{2n}}{\sqrt{\pi n}}(1 + \mathcal{O}(1/n)) \end{aligned}
There is a little hand-waving at the end, but it is good enough.
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The first six rows of Pascal's Triangle look like:
SUM 1 n = 0 1 = 20 1 1 n = 1 2 = 21 1 2 1 n = 2 4 = 22 1 3 3 1 n = 3 8 = 23 1 4 6 4 1 n = 4 16 = 24 1 5 10 10 5 1 n = 5 32 = 25
There is an obvious pattern in this arrangement. Each row begins and ends with 1. The "middle" entries are obtained by adding the two numbers in the row above it between which it falls.
The row corresponding to n = 6 is
1 6 15 20 15 6 1
and the sum of the entries is 64 = 26.
Another interesting and useful feature is that the sum of the entries in each row is equal to 2n
REFERENCES
[1] Musser, Gary L. and William F. Burger, Mathematics for Elementary Teachers: A Contemporary Approach, Fourth Edition, Prentice-Hall, 1997.
[2] Rose, Israel H., A Modern Introduction to College Mathematics, John Wiley & Sons, Inc., 1959.
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# How to Solve Multi-step Equations & Cheats?
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0
# How can you find the factors of 72?
Updated: 12/20/2022
Wiki User
12y ago
Divide by two and three.
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Wiki User
12y ago
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Q: How can you find the factors of 72?
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### Are all the factors of 72 are multiples of the prime factors of 72?
Apart from 1, all of the other factors of 72 are multiples of prime factors.
### What are the prime factors for 72?
The prime factors for 72 are: 2, 3
### Find the number of factors of 72-?
They are: 1 2 3 4 6 8 9 12 18 24 36 and 72
1*72 = 72
### What is the lcm of 72 and 147 using prime numbers?
The first step in using primes for finding LCM is to find the factors of the given numbers. The LCM of the given numbers is the product of all the prime factors to their greatest power.Prime factors of 72 are 23 x 32Prime factors of 147 are 3 x 72The LCM of 72 and 147 is 23 x 32 x 72 = 8 x 9 x 49 = 3528
Related questions
### list all factors of 72?
find all factors of 72
90
### Common factors of 72 and 45?
The factors of 45 are: 1, 3, 5, 9, 15, 45The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72The common factors are: 1, 3, 9everyone knows that you have to find the factors of 45 and then 72 and then find what they have in commen and then find the highest now work it out ya self
### Can you find all the factors of 24 that are also factors of 72?
Every factor of 24 is a factor of 72, since 24 is itself a factor of 72. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
### How do you determine factors of 72?
A factor is any number that will divide 72 evenly. The brute-force way to find the factors of 72 is to try every number starting with 1. While this sounds awful, it's not really so bad because you only have to go up to 9:72 / 1 = 72 (72 and 1 are factors)72 / 2 = 36 (26 and 2 are factors)72 / 3 = 24 (24 and 3 are factors)72 / 4 = 18 (18 and 4 are factors)72 / 5 = 14 remainder 2 (5 is not a factor)72 / 6 = 12 (12 and 6 are factors)72 / 7 = 10 remainder 2 (7 is not a factor)72 / 8 = 9 (9 and 8 are factors)72 / 9 = 9 (8 and 9 are factors)
### How are the factors 36 and 72 related?
Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. All of the factors of 36 are also factors of 72.
### How are the factors of 36 and 72 alike?
Since 36 is a factor of 72, the factors of 36 are included in the factors of 72.
### Are all the factors of 72 are multiples of the prime factors of 72?
Apart from 1, all of the other factors of 72 are multiples of prime factors.
### How can you use a factor string with two factors to find a factor string with three factors?
By breaking down one of the factors. 8 x 9 = 72 8 x 3 x 3 = 72
### How are factors of 36 related to the factors of 72?
Since 36 is a factor of 72, all of its factors are included in the factors of 72.
### Is 72 the lowest number with 14 factors?
No. 72 has 12 factors.
### What are the prime factors for 72?
The prime factors for 72 are: 2, 3
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1. ## rate
1. (a) A car traveling at a speed of v can brake to an emergency stop in a distance x. Assuming all other driving conditions arte all similar, if traveling speed of the car doubles, the stopping distance will be (1) 2x, or (2) 2x, or (3) 4x: (b) a driver traveling at 40.0 km/h in a school zone can brake to an emergency stop in 3.00 m. What would be the braking distance if the car were traveling at 60.0 km/h?
2. A student drops a ball from the top of a building; it takes 2.8 s for the ball to reach the ground. (a) What is the ball's speed just before hitting the ground? (b) What is the height of the building?
3. A photographer in a helicopter ascending vertically at a constant rate of 12.5 m/s accidentally drops a camera out the window when the helicopter is 60 m above the ground. (a) How long will it take the camera to reach the ground? (b) What will it's speed be when it hits?
2. I'm sure this requires nothing more than some formulas.
3. Hello, John!
For the last two, we are expected to know the "free fall" formula: . $y \; = \; h_o + v_ot - gt^2$
. . where: . $\begin{array}{ccc}h_o & = & \text{initial height} \\ v_o & = &\text{initial velocity} \\ g & = & \text{gravitational constant} \\ y & = & \text{height of object} \end{array}$
Note: $g$ is usually 16 ft/secē or 4.9 m/secē.
2. A student drops a ball from the top of a building;
it takes 2.8 s for the ball to reach the ground.
(a) What is the ball's speed just before hitting the ground?
(b) What is the height of the building?
We have: . $v_o = 0$ . (the ball is dropped, not thrown).
The equation is: . $y \;=\;h_o - 16t^2$
(a) The velocity is given by the derivative: . $v(t) \:=\:y' \:=\:-32t$
Then: . $v(2.8) \:=\:-32(2.8) \:=\:-89.6$
The ball's speed on impact is 89.6 feet per second (downward, of course).
(b) When the ball strikes the ground, its height is zero.
. . We have: . $0 \;=\;h_o - 16\cdot2.8^2\quad\Rightarrow\quad h_o \:=\:125.44$
The height of the building is 125.44 feet.
3. A photographer in a helicopter ascending vertically at a constant rate of 12.5 m/s
accidentally drops a camera out the window when the helicopter is 60 m above the ground.
(a) How long will it take the camera to reach the ground?
(b) What will its speed be when it hits?
We are given: . $h_o = 60,\;v_o = 12.5$
The equation is: . $y \;=\;60 + 12.5t - 4.9t^2$
(a) "reach the ground" means: $y = 0$
We have: . $60 + 12.5t - 4.9t^2 \:=\:0$
. . which factors: . $(t - 5)(4.9t + 12) \:=\:0$
. . and has the positive root: . $t = 5$
Therefore, it takes the camera 5 seconds to crash to the ground.
(b) The velocity is: . $v(t) \;=\;y' \;=\;12.5 - 9.8t$
. . Hence: . $v(5)\;=\;12.5 - 9.8(5) \;=\;-36.5$
Therefore, its speed on impact is 36.5 meters per second.
4. aaaand...
Problem 1,a
We are told that the square of velocity is proportional to the stopping distance. (look at topsquark's post for its justification)
$v^2 \propto x$ so $(2v)^2=4v \propto 4x$.
4x
Problem 1,b
$40^2=1600\frac{km}{h} \propto 3.00m$
Using this information, we can find the stopping distance of the same car at 60.0 km/h by using a proportion and solving for x.
$\frac{3}{1600} = \frac{x}{3600} \Longleftrightarrow x=6.75m$
These veterans are just too fast...
5. Originally Posted by rualin
These veterans are just too fast...
Soroban is not the fastest so you haven't seen anything yet lol j/m. Soroban is one of the most cunning and interesting.
6. Originally Posted by rualin
Problem 1,a
We are told that velocity is directly proportional to the stopping distance.
No we weren't. Try this programme:
Find the acceleration for when it starts at speed v.
The acceleration will be the same as this for when it starts at 2v.
In both cases, apply the formula:
$v^2 = v_0^2 + 2a(x - x_0)$
-Dan
7. Thanks, topsquark! I realize my mistake and have corrected it... I hope.
,
,
### A photographer in a helicopter ascending vertically at a constant rate of 12.5 m/s accidentally drops a camera out the window when the helicopter is 60.0 m above the ground.
Click on a term to search for related topics.
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# How to do a math problem
There are also many YouTube videos that can show you How to do a math problem. We will also look at some example problems and how to approach them.
## How can we do a math problem
This can help the student to understand the problem and How to do a math problem. In the algebra stage, I think the most important thing is to develop your mathematical modeling ability. Solving algebraic equations is just a step-by-step and practice makes perfect operation. However, how to transform practical problems into algebraic solvable abstract problems is a more challenging and practical task. Mathematical modeling is not necessarily very important to the exam (because most of the math exam focuses on solving), but in real life, especially in scientific research of science and engineering, mathematical modeling ability is a hard core ability. If a simpler and more accurate mathematical model can be established, it will almost be the final victory, and the rest of the solution will be left to the computer.
Cleaning and pollution separation. 2. Reasonable configuration of commercial kitchen equipment: commercial kitchen equipment of the same type. Supplementary equipment shall be reasonably matched and set together to facilitate coordination, coordination and application. If the equipment requiring smoke exhaust and exhaust are placed together, the smoke hood shall be used for centralized smoke exhaust; Similar stoves shall be of type and model It refers to the production operation sequence in which operators can efficiently produce qualified products that meet customer needs, and it is also an important guarantee for achieving high efficiency.
The accuracy of the cross roller bearing can reach P2 level. Therefore, it is suitable for joints and rotating parts of industrial robots, rotating tables of machining centers, robot rotating parts, precision rotating tables, medical machines, calculators, IC manufacturing devices and other equipment. 2. Calculate the monthly income of families with high purchasing power according to the average monthly salary - take the average monthly salary as the starting point to calculate the average monthly income of urban families: average monthly family income = average monthly salary * 2; The monthly income of urban high purchasing power families is calculated from the average monthly income of urban families: the monthly income of high purchasing power families = the average monthly income of urban families * 1.2 ~ 1.5; Taking the monthly income of high-income families as the starting point, calculate the family's limit monthly payment: the family's limit monthly payment = the monthly income of high purchasing power families * 0.7; Using the family's maximum monthly payment as the starting point to reverse the total amount of housing loans: the formula is very complex.
Exclusion is a method of finding inconsistent options by excluding options that are consistent with the question stem, or finding options that are consistent with the question stem by excluding inconsistent options, and then solving the answer. The essence of the exclusion method is to find the options that are not involved in the question stem as the answer by excluding the options that are already involved in the question stem, or to find the options that are consistent with the question stem as the answer by excluding the options that are not involved in the question stem. In fact, you can try to use the exclusion method when solving every logic test. Problem d is an integer 0-1 programming problem.
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# NCERT Class 6 Solutions: Whole Numbers (Chapter 2) Exercise 2.1–Part 2
Q-5 Write the successor of:
Solution:
1. : The successor of 24,40,701 is
2. : The successor of 1,00,199 is
3. : The successor of 10,99,999 is
4. : The successor of 23,45,670 is
Q-6 Write the predecessor of:
Solution:
1. : The predecessor of 94 is
2. : The predecessor of 10000 is
3. : The predecessor of 208090 is
4. : The predecessor of 7654321 is
Q-7 In each of the following pairs of numbers, state which whole number is on the left of the other number on the number line? Also write them with the appropriate sign between them.
Solution:
• Draw a line, mark a point on it and label it 0. Mark out points to the right of 0, at equal intervals.
• Label them as 1, 2, 3... Thus, we have a number line with the whole numbers represented on it.
• Observe that, out of any two whole numbers, the number on the right is the greater number. [3 are to the right of 2. So 3 is the greater number and 2 is smaller]
• The number line can be used to represent larger numbers.
• The whole number 503 is on the left of the whole number 530 on the number line
• So, or
• The whole number 307 is on the left of the whole number 370 on the number line
• So, or
• The whole number 56789 is on the left of the whole number 98765 on the number line
• So, or
• The whole number 9830415 is on the left of the whole number 10023001 on the number line
• So,
Explore Solutions for Mathematics
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# Interest Rate Swaps
Hello All,
Example – Bank A enters into a \$1M quarterly-pay plain vanilla interest rate swap as the fixed rate payer of 6% based on a 360-day year. The floating-rate payer agrees to pay 90-day LIBOR plus a 1% margin. 90-day LIBOR is currently at 4%. Calculate amounts Bank A pays or receives 270 days from now.
90-day libor rates are :
• 4.5% 90 days from now.
• 5% 180 days from now.
• 5.5% 270 days from now.
OA = \$0
Here’s what I did :
270 days libor = 5.5%; Add Margin => 6.5%
Therefore, outstanding payment = {(fixed rate) * 270/360} - {6.5% * 270/360 } = -0.0375%*1M =-3750, which is wrong unfortunately.
Question #2- What’s the difference between these two terms? “90-day LIBOR is currently at 4%.” and “90-day libor rate is 4.5% 90 days from now.” Why are these two rates different (i.e. 4% and 4.5% – aren’t these 90-day Libor)? Can someone please also talk about this? I would appreciate any help.
I understand that if 90-day Libor is say 5%, effective rate = 5% * (90/360). Now, I don’t know what’s the difference between the two terms above.
Question 1: The LIBOR to be used is the 180 days LIBOR, I.e., 5%+ the 1% margin, that’s 6%, hence, the 0 net pay. The LIBOR rate used is usually that of a period before the actual period, why? I’m not sure. I think the rate is usually set in advance. Question 2: I don’t know
Question 1: that is right adekunle, for the floating side the rate is reset at the beginning of each period, with payments at the end of the period. so they both pay 6% for that 3 months, so on a net basis, the payment is zero.
Question 2: the difference is that the first term is a spot rate for 3-month libor (90 day libor is right now…) the second term is a forward rate for 3-month libor (what it will cost to borrow for 3-months, starting in 3-months).
Thanks for your response. I still didn’t understand why we chose 180-day LIBOR for this one. Is this a convention? Do you mind explaining this a bit.
Swap floating rates are set in advance but paid in arrears; i.e., the floating rate that is paid at time t is the current rate at time t – 1. For a quarterly-pay swap, time is measured in 90-day chunks, so the current floating rate on day 0 is paid on day 90, the current floating rate on day 90 is paid on day 180, the current floating rate on day 180 is paid on day 270, and so on.
By the way, make sure that you’re using the terminology correctly (it’ll help solidify your understanding). You’re not using _ 180-day LIBOR _: 180-day LIBOR is an interest rate for a 180-day loan. You’re using _ 90-day LIBOR _, but you’re using the 90-day rate that is current (i.e., is the market rate) 180 days after the start of the swap.
What I don’t understand is, if I know my current fixed rate exposure is less than the LIBOR exposure, why enter the swap when I know I’ll have to pay in the end? What’s the incentive? My thoughts have always been “it is unknown, you don’t know if you’ll gain or lose, so you just take a stab at it” and with that understanding, I’ve gotten a fair number of questions right.
The incentive is that you expect in the future to receive higher payments than you’re paying, so that the expected present value is zero.
Hello S2000magician,
Thank you so much for your response. I have two follow-up questions. How do we know the length of the chunk? For instance, in the example above, is it “The floating-rate payer agrees to pay 90-day LIBOR plus a 1% margin”? However, here 90-day LIBOR is a spot rate. I am not sure whether I can equate chunks to spot rate. The question doesn’t mention the frequency of payment. I could be wrong in questioning the question. However, please help me.
Thank you for correcting me, S2000magician. I believe 90-day LIBOR is a spot rate, and 90-day LIBOR 180 days from now is a future rate. I was a bit confused when I wrote that post. I hope I am correct now. Please let me know your thoughts.
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You are on page 1of 33
# hemical Engineering Process Principl
CHAPTER 1
Basic Concepts
## In designing a new process or analyzing an
existing one, calculation of amounts and
properties of raw materials and products is
crucial.
This chapter presents the calculation
techniques of expressing the values of
process variables.
Topic Outcomes
At the end of Chapter 1, you should:
Convert one set of units in a function or
equation into another equivalent set for
mass, length, area, volume, time, energy and
force using conversion factor tables.
Identify the units commonly used to express
both mass and weight.
Identify the number of significant figures in a
given value and state the precision with
which the value is known.
## What are in this chapter?
Introduction to
Engineering
Calculations
Units and
Dimensions
Conversion of
Units
Systems of
Units
Units
Example: 2.05kg
Units
+3.56kg (yes)
Subtracted
Example: 5m 3m
(yes)
7hr 2min (no)
Multiplied
Divided
Example : 4m
12 m2
x 3m =
Conversion of Units
To convert a quantity expressed in terms of one unit to
equivalent in terms of another unit, multiply the given
quantity by the conversion factor.
Conversion factor
## a ratio of equivalent values
of a quantity expressed in different units.
## Let say to convert 36 mg to gram.
36 mg
1g
1000 mg
0.036 g
Conversion
factor
Dimensional Equation
Convert 1 cm/s2 to km/yr2
1. Write the given quantity and units on left.
2. Write the units of conversion factors that cancel the old unit and
replace them with the desired unit.
1 cm
s2
h2
day2
km
s2
h2
day2
yr2
cm
1 cm
36002 s2
s2
12 h2
12 day2
km
100 x 1000
yr2
12 yr2
=
1m
1 km
100 cm
1000 m
Systems of Units
8
Base Units
Base Units
Quantity
Length
SI
Symbol
American
Symbol
CGS
Symbol
meter
foot
ft
centimeter
cm
lbm
gram
Mass
kilogram
kg
pound
mass
Moles
grammole
mole
pound mole
lbmole
gram-mole
mole
Time
second
second
second
Temperature
Kelvin
Rankine
Kelvin
Multiple Units
10
-Example :
Years
Days
Hours
Minutes
Seconds
-Example :
## Multiple Unit Preferences
tera (T) = 10
12
centi (c) = 10
-2
giga (G) = 10
milli (m) = 10
-3
mega (M) = 10
micro () = 10
-6
nano (n) = 10
-9
kilo (k) = 10
12
Derivatives SI Units
Derived SI Units
Quantity
Unit
Volume
Liter
## 0.001m3 = 1000 cm3
Force
Newton
(SI)
Dyne
(CGS)
1 kg.m/s2
1 g.cm/s2
Pressure
Pascal
Pa
1 N/m2
Energy/
Work
Joule
Calorie
J
cal
1 N.m = 1 kg.m2/s2
4.184 J =4.184 kg.m2/s2
Power
Watt
1 J/s = 1 kg.m2/s3
Systems of Units
3 systems of unit:
a) SI system
b) American engineering system
c) CGS system
Derived unit for velocity in the SI System? The CGS
System? The American Engineering System?
EXERCISE
## Convert 1 miles per hour to meter per
second
Length
1 m = 100 cm = 1000 mm
=106 microns = 1010
angstrom
= 39.37 in = 3.2808 ft
= 1.0936 yd =
0.0006214 mile
1 ft = 12 in = 1/3 yd =
0.3048 m
= 30.48 cm
mi
mile
1
m
1 hr
m
1 1
0.447
hr
hr 0.0006214 mile 3600 s
s
14
EXERCISE
Convert 23 Ibm.ft/min
kg.cm/s2
Mass
1 kg = 1000 g = 0.001
metric tonne
= 2.20462 Ibm =
35.2739 oz
1 Ibm = 16 oz = 5 x 10-4 ton
= 453.593 g =
0.453593 kg
to its equivalent
Length
1 m = 100 cm = 1000 mm
=106 microns = 1010
angstrom
= 39.37 in = 3.2808 ft
= 1.0936 yd =
0.0006214 mile
1 ft = 12 in = 1/3 yd =
0.3048 m
= 30.48
cm2
100 cm
12 min
kg.cm
Ibm. ft
Ibm. ft 0.453593 kg
23
23
2 2
2
2
min
min
1 Ibm
3.2808 ft 60 s
15
0.088
s2
16
## Force & Weight
Force is proportional to product of mass and acceleration.
Usually defined using derived units ;
1 Newton (N)
=
1 kg.m/s2
1 dyne
=
1 g.cm/s2
1 Ibf
= 32.174 Ibm.ft/s2
Weight of an object is force exerted on the object by
gravitational attraction of the earth i.e. force of gravity, g.
Value of gravitational acceleration:
g
= 9.8066 m/s2 = 980.66 cm/s2
= 32.174 ft/s2
## Force & Weight
gc is used to denote the conversion factor from a natural
## force unit to a derived force unit.
gc
1 kg.m/s2
1N
= 32.174 lbm.ft/s2
1 lbf
## Weight & Mass
See this example:
Given the density of 2 ft3 water is 62.4 lbm/ft3. At the sea level,
the gravitational acceleration is 32.174 ft/s2.(Refer to page 13)
The mass of water is
lbm
M 62.4 3 2 ft 3 124.8lbm
ft
## The weight of water is
lbm
W 62.4 3 2 ft 3 32.174 ft / s 2
ft
Conversion factor
1lb f
2
32.174lbm ft / s
124.8lb f
Dimensions
## BASE UNITS DIMENSIONS
Quantity
SI
Unit
Dimension
Kilogra
m
Meter
Temperature
Time
Mass
Length
21
22
EXAMPLE
Spaghetti Recipe
Ingredients:
20 ml of cooking oil
100 gram of minced meat
15 cm spaghetti sticks
Value
Unit
Dimension
20
milliliter
LENGTH
[L]
100
gram
MASS
[M]
15
centimete
r
LENGTH
[L]
23
Dimensional Homogeneity
Quantities can be added/subtracted if ONLY their units
are same.
Unit same, the DIMENSION of each term must be the
same.
Eg. : VELOCITY = LENGTH / TIME
(L) / (T)
(L) / (T)
(m/s)
(m/s)
24
## Every valid equation must be
dimensionally homogeneous:
all additive terms on both sides of
the equation must have same
dimensions
25
Examples:
F = ma
## where F = Force (N = kg.m/s2)
m = mass (kg) = ( M )
a = acceleration (m/s2)= ( L) / ( T )2
Unit:
## kg.m/s2 = (kg )(m/s2)
Dimension:
(M)(L) =(M) x (L)
( T )2
( T )2
(M)(L) =(M)(L)
( T )2
( T )2
26
LEFT
= RIGHT
DIMENSIONALLY
HOMOGENEOUS
(CONSISTENT )
Dimensional Analysis
This is a very important tool to check your work
Eg. : Doing a problem you get the answer distance
d = v t2 (velocity x time2)
Units on left side = ( L )
Units on right side = ( L )/( T ) x ( T )2 = ( L ) .( T )
## Left units and right units dont match, so
27
The period P of a swinging pendulum depends only
on the length of the pendulum d and the
acceleration of gravity g.
Which of the following formulas for P could be
correct ?
Given :
d = units of length ( L )
g = units of ( L / T 2).
(a)
P 2 dg
(b)
P 2
(c
d
g
28
P 2
d
g
## Identify unit of P. P has units of time (T )
Make sure Left & Right unit dimensionally
homogeneous (consistent).
(a) P
(a)
(a)
2 dg
2
L L
L 2 4 T
T T
Not
Right !!
(b) P 2
d
g
L
T2 T
L
T2
Not
Right !!
29
(c) P 2
d
g
L
T2 T
L
T2
Correct units!!
If an equation is dimensionally
homogeneous but its additive terms have
inconsistent unit, the terms may be made
consistent by applying conversion factors
Example:
V (m/s) = Vo (m/s) + g (m/s2) t
(min)
Apply the conversion factor
## V (m/s) = Vo (m/s) + g (m/s2) t
(min)
V = Vo + 60 g t
(60s/min)
Free Template from www.brainybetty.com
30
## An equation is only VALID when it is dimensionally
Homogeneous
= consistent in UNITS!!!
## Free Template from www.brainybetty.com
31
Dimensionless Quantities
Can be a pure number
Eg. : 2, 1.3 ,5/2
a multiplicative combination of variables with no net
dimensions
Eg. :
ud
Re
= (g/cm3) , u = (cm/s),
d = (cm), = (g/cm.s)
32
DIMENSIONLESS
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Suggested languages for you:
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Found in: Page 775
### College Physics (Urone)
Book edition 1st Edition
Author(s) Paul Peter Urone
Pages 1272 pages
ISBN 9781938168000
# (a) What is the resistance of ten ${\mathbf{275}}{\mathbf{-}}{\mathbf{\Omega }}$resistors connected in series? (b) In parallel?
(a)The resistance of ten $275\Omega$resistors connected in series is $2750\Omega$.
(b)The resistance of ten $275\Omega$ resistors connected in parallel is $27.50\Omega$.
See the step by step solution
## Step 1: Definition of resistance in series and parallel
An unwillingness to accept anything, such as a change or a new thought, is known as resistance.
Resistance in series: When resistors are daisy-chained together in a single line and a common current flows through them, they are said to be connected in series.
Resistance in parallel: In a parallel circuit, the total resistance is always less than any of the branch resistances. The total resistance in the circuit decreases when more parallel resistances are added to the lines.
## Step 2: Finding resistance of resistors connected in series
(a)
The equivalent resistance of the n resistors with each of resistance and connected in series can be expressed as,
${\mathrm{R}}_{\mathrm{s}}=\mathrm{nR}$
Therefore, substituting the given data, we will get,
${R}_{s}=10×275\phantom{\rule{0ex}{0ex}}=2750\Omega$
Thus, the resistance of ten $275\Omega$ resistors connected in series is $2750\Omega$.
## Step 3: Finding resistance of resistors connected in parallel
(b)
The equivalent resistance of the n resistors with each of resistance and connected in parallel can be expressed as,
${\mathrm{R}}_{\mathrm{P}}=\frac{\mathrm{R}}{\mathrm{n}}$
Therefore, substituting the given data, we will get,
${\mathrm{R}}_{\mathrm{P}}=\frac{275\mathrm{\Omega }}{10}\phantom{\rule{0ex}{0ex}}=27.50\mathrm{\Omega }$
Hence, the resistance of ten$275\Omega$resistors connected in parallel is$27.50\Omega$.
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# Binet's Formula for Fibonacci
## Recursion Algorithms: Factorial and Fibonacci Series
Recursion is a powerful concept in computer programming that involves a function calling itself. It allows us to break down complex problems into simpler, more manageable parts. In this tutorial, we will explore two classic examples of recursion algorithms: factorial and Fibonacci series.
### The Factorial Algorithm
Let's start with the factorial algorithm, which calculates the product of an integer and all the positive integers below it. To express this algorithm recursively, we can define the factorial of n (denoted as n!) as follows:
• If n is 0 or 1, the factorial is 1.
• Otherwise, the factorial of n is n multiplied by the factorial of n minus 1.
``````def factorial(n):
if n == 0 or n == 1:
return 1
else:
return n * factorial(n - 1)
``````
In this code snippet, we first check if the input `n` is equal to 0 or 1. If so, we return 1 as the base case. Otherwise, we multiply `n` by the factorial of `n - 1`, which is the recursive step. This recursive call keeps repeating until we reach the base case.
### The Fibonacci Series Algorithm
Next, let's dive into the Fibonacci series algorithm, which generates a sequence of numbers in which each number is the sum of the two preceding ones. The Fibonacci sequence typically starts with 0 and 1.
To express this algorithm recursively, we can define the Fibonacci number at index n (denoted as F(n)) as follows:
• If n is 0, the Fibonacci number is 0.
• If n is 1, the Fibonacci number is 1.
• Otherwise, the Fibonacci number at index n is the sum of the Fibonacci numbers at indices n minus 1 and n minus 2.
``````def fibonacci(n):
if n == 0:
return 0
elif n == 1:
return 1
else:
return fibonacci(n - 1) + fibonacci(n - 2)
``````
In this code snippet, we first check if the input `n` is equal to 0 or 1. If so, we return the respective Fibonacci number as the base cases. Otherwise, we recursively call the `fibonacci()` function with `n - 1` and `n - 2` as arguments, then add the results together.
### Applying Binet's Formula to the Fibonacci Series
While the recursive Fibonacci algorithm is straightforward, it can be computationally expensive for large values of n. Fortunately, there is an alternative formula called Binet's Formula, which allows us to directly calculate the Fibonacci number at any index.
Binet's Formula states that the nth Fibonacci number, denoted as F(n), can be computed using the following equation:
F(n) = [φ^n - (1-φ)^n] / √5
where φ is the golden ratio, approximately equal to 1.61803.
``````import math
def fibonacci_binet(n):
phi = (1 + math.sqrt(5)) / 2
return int((phi**n - (1 - phi)**n) / math.sqrt(5))
``````
In this code snippet, we first import the `math` module to access the square root function. Then, we define the `fibonacci_binet()` function, which calculates the Fibonacci number at index `n` using Binet's Formula. By directly computing the Fibonacci number, we can avoid the recursive overhead and improve performance.
### Conclusion
Recursion is a powerful technique in programming, and understanding recursion algorithms opens the door to solving a wide range of problems efficiently. In this tutorial, we explored two popular recursion algorithms: factorial and Fibonacci series. We learned how to express these algorithms recursively, provided code examples, and even introduced Binet's Formula to directly compute Fibonacci numbers. Armed with this knowledge, you can confidently tackle recursive problems and utilize these algorithms in your projects.
Remember, practice is key to mastering recursion. Challenge yourself with additional exercises and explore other recursion algorithms to deepen your understanding. Happy coding!
Please note that the above blog post is written in Markdown format and can be easily converted into HTML or any other desired format using a Markdown converter tool.
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It is currently 26 Jun 2017, 16:13
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# Last year the price per share of Stock X increased by k
Author Message
Director
Joined: 17 Oct 2005
Posts: 928
Last year the price per share of Stock X increased by k [#permalink]
### Show Tags
18 Mar 2006, 21:45
This topic is locked. If you want to discuss this question please re-post it in the respective forum.
Last year the price per share of Stock X increased by k percent and the earnings per share
of Stock X increased by m percent, where k is greater than m. By what percent did the
ratio of price per share to earnings per share increase, in terms of k and m?
A. k/m %
B. (k-m) %
C. [100(k-m)]/(100+k) %
D. [100(k-m)]/(100+m) %
E. [100(k-m)]/(100+k+m) %
SVP
Joined: 05 Apr 2005
Posts: 1710
### Show Tags
18 Mar 2006, 21:58
p/e ratio before = p/e
p/e ratio after = [p (100+k)]/[e(100+x)]
increament = [{p (100+k)}/{e(100+x)} -p/e] 100
should come [100(k-m)]/(100+m)%.
SVP
Joined: 05 Apr 2005
Posts: 1710
### Show Tags
18 Mar 2006, 22:00
HIMALAYA wrote:
p/e ratio before = p/e
p/e ratio after = [p (100+k)]/[e(100+x)]
increament = [{p (100+k)}/{e(100+x)} -p/e] 100
should come [100(k-m)]/(100+m)%.
oh, i did it: http://www.gmatclub.com/phpbb/viewtopic ... t=earnings
Director
Joined: 17 Oct 2005
Posts: 928
### Show Tags
18 Mar 2006, 23:24
thanks himalya, nice to see you here again
18 Mar 2006, 23:24
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• Study Resource
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Transcript
```Comparing Two Groups
Statistics 2126
So far..
• We have been able to compare a sample
mean to a population mean
– z test
– t test
• Often times though we have two groups to
compare
• Is Group 1 different from Group 2
Matched pairs or correlated t test
• AKA dependent sample t test
• When subjects are matched on a variable
or are used as their own controls, a sort of
before and after thing if you will
• Be very careful with this
• But it is way powerful and easy to do
Back to our mythical IQ course..
Before
After
Difference
103
98
-5
100
107
7
111
119
8
97
100
3
133
134
1
106
111
5
87
85
-2
A couple of summary statistics
d 27
d
d
n
27
d
7
d 3.86
sd 3.48
Now it is a simple t test
d
t
sd / n
3.86 0
t
3.48 / 7
3.86
t
3.48 / 2.65
3.86
t
1.31
t 2.95
And now for the decision
•
•
•
•
t(6) = 2.447
tobt = 2.95
Reject H0
Our IQ course works!!
Two sample problems
• While is is useful to know how to compare
a sample mean to a population mean and
check for significance it is not all that
common
• We rarely know μ
– Sometimes we do
• IQ
• Differences
• Theoretical values
The much more common question
is…
• Does one group differ from another?
• Let’s say we had two classes with different
teaching methods
• Is there an effect of teaching method?
Statistic
Class 1
Class 2
Mean
77
71
Standard
deviation
6.2
6.7
Number of
students
49
52
Our hypotheses
•
•
•
•
•
•
Are the two classes different?
H 0 μ1 = μ2
H A μ1 ≠ μ2
Or we could restate them like this:
H 0 μ1 - μ2 = 0
H A μ1 ≠ μ2 ≠ 0
Let’s go back to the original t
formula
Statistic
H0
↓
↓
x
t
s/ n
← Error
Figure it out
( x1 x2 ) ( 1 2 )
t
error
practicall y....
( x1 x2 )
t
error
the values of s for
each group
• They must be
weighted
2
1
2
2
s
s
n1 n2
6.2
2
49
6.7
2
52
So the formula is
t
x1 x2
2
1
2
2
s
s
n1 n1
Degrees of freedom
• With a one sample t test we lose one
degree of freedom
• Because we calculated one standard
deviation
• Here was have calculated 2
• So we lose 2 df
• In our case we have 99 df
Sub in the values
t
77 71
2
2
6.2 6.7
49
52
6
t
1.65
t 4.67
rejectH0
conclusions
• All t tests are based on the same formula
• Keep the assumptions in mind
– SRS
– Homogeneity of variance
– Independence of observations
```
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### UNIT 2 : FUNCTIONS & TRANSFORMATIONS
LESSON 4: REFLECTIONS
1. Vertical Reflections (reflections in the x-axis):
Example 1:
The graphs of y = x2 and y = -x2 are given below.
Note that the graphs are congruent. The graph of y = -x2 (red) is a reflection (mirror image) in the x-axis of the graph of y = x2 (blue).
The transformation may be depicted in mapping form. Each point (x, y) on the base curve y = x2 has been transformed as follows:
(x, y) -------------------------------à (x, -y) [reflection in the x-axis]
For example, the point (1, 1) on the curve y = x2 reflects into the point (1, -1) on the image curve y = - x2
Note that the x-coordinate stays the same and the y-coordinate changes sign.
Other points on the image curve may be obtained using this mapping as follows.
(x, y) -------------------------------à (x, -y)
(-2, 4) ---------------------à (-2 , -4)
(-1, 1) ---------------------à (-1 , -1)
(0,0) ---------------------à (0, 0)
(1, 1) ---------------------à (1, -1)
(2, 4) ---------------------à (2, -4) yielding the graph as shown above in (red)
(x, y) -------------------------------à (x, -y)
(x, y) -------------------------------à(x, - y)
(0, 0) --------------------------à(0, 0)
(1, 1) --------------------------à(1, – 1)
(4, 2) --------------------------à(4, -2)
(9, 3) --------------------------à(9, - 3)
(16, 4) -------------------------à(16, -4) yielding the graph at left (red)
In general, the graph of y = -f(x) is a reflection in the x-axis of the graph of y = f(x).
Example 3: Given the graph of y = f(x) as shown, draw the graph of y = -f(x).
The required graph will be a reflection in the x-axis of the graph of y = f(x) and may be represented in mapping form:
(x, y) ---------------------------(x, -y)
Now take key points on the graph of y = f(x) -- {(-5, 1), (-3, 1), (-1, 3),(0,1.5), (1, 0), (3, 2)} and change the sign of the y-coordinates using the mapping.
(x, y) ---------------------------------à(x, -y)
(-5, 1) ---------------------------à(-5, -1)
(-3, 1) ---------------------------à(-3, -1)
(-1, 3) ---------------------------à(-1, -3)
(1, 0) ----------------------------à(1, 0)
(3, 2) ----------------------------à(3, -2)
2. Horizontal Reflections (reflections in the y-axis):
Example 4: Given the function f(x) = 3x – 1, find the equation of f(-x) and draw its graph.
To find f(-x), simply substitute (-x) for x in the equation
f(-x) = 3(-x) – 1 = -3x – 1 or y = -3x - 1
Draw the graph of both functions by completing the tables of values below.
For y = 3x – 1
If x = -2, y = 3(-2) –1 = -7
If x = -1, y = 3(-1) – 1= -4, etc. giving
x -2 -1 0 1 2 y -7 -4 -1 2 5
For y = -3x – 1
If x = -2, y = -3(-2) –1 = 5
If x = -1, y = -3(-1) – 1= 2, etc. giving
x -2 -1 0 1 2 y 5 2 -1 -4 -7
Note that the graph of y = f(-x) = -3x – 1 (red) is a reflection (mirror image) of the graph of y = f(x) = 3x – 1 (blue) in the y – axis.
For example, the point (2, 5) on y = 3x – 1 is reflected into its image point (-2, 5) on the graph of y = -3x –1.
Notice the x-values change sign and the y-values stay the same. This reflection could be put in mapping form as follows;
(x, y) --------------------------------à (-x, y) [reflection in the y-axis]
(x, y) --------------------------à(-x, y)
(0, 0) --------------------------à(0, 0)
(1, 1) --------------------------à(-1, 1 )
(4, 2) --------------------------à(-4, 2 )
(9, 3) --------------------------à(-9, 3)
(16, 4) -------------------------à(-16, 4)
.
(x, y) -------------------------------à(-x - 3, y)
(0, 0) --------------------------à(-3, 0)
(1, 1) --------------------------à(-4, 1 )
(4, 2) --------------------------à(-7, 2 )
(9, 3) --------------------------à(-12, 3)
(16, 4) -------------------------à(-19, 4)
Example 7: Given f(x) = x2 + 4x, find the equation for f(-x) and draw both graphs.
Solution: By completing the square, we can rewrite f(x) as follows:
f(x) = x2 + 4x + 4 – 4 ** Recall – divide the coefficient of x by 2 and square it [ (4/2)2] = 4
= (x2 + 4x + 4) – 4
= (x + 2)2 - 4 ** trinomial x2 + 4x + 4 gets factored as (x+2)(x+2) = (x + 2)2
The graph of f(x) (blue) will be a translation (shift) left 2 and down 4 relative to the basic function y = x2 . In mapping form we have
(x, y) ---------------------à (x - 2, y - 4) and using the key points for the function y = x2, we obtain the points of the transformed function.
(x, y) ---------------------à (x - 2, y - 4)
(-2, 4) --------------------à (-2-2 , 4-4) = (-4, 0)
(-1, 1) --------------------à (-1-2 , 1-4) = (-3, -3)
(0,0) ----------------------à (-2, -4)
(1, 1) ---------------------à (-1, -3)
(2, 4) ---------------------à (0, 0) yielding the graph as shown with vertex at (-2, -4)
Now determine f(-x) = (-x)2 + 4(-x)
= x2 – 4x
The graph of f(-x) (red) will be a reflection in the y-axis of the above parabola (blue) with vertex at (-2, -4),
Take the points on this parabola and use the reflection mapping: (x, y) -------------------------------à (-x, y).
(x, y) ---------------------------à (-x, y) [reflection in the y-axis]
(-4, 0) --------------------------à (4, 0)
(-3, -3) -------------------------à(3, -3)
(-2, -4) -------------------------à(2, -4)
(-1, -3) -------------------------à (1, -3)
(0, 0) --------------------------à (0, 0)
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Rhombus Calculator
Scroll down for Instructions and Definitions
Click on the data you know: Side and Angle a Side and Angle b Side and Altitude Long Diagonal AD & Short Diagonal CB
First, click on one of the 5 choices, enter the data in the appropriate input boxes, then click "CALCULATE".
Rhombus Facts
All sides are equal.
Lines AD and CB are called diagonals and always meet at right angles.
Line AD is the long diagonal and line CB is the short diagonal.
The diagonals bisect the vertex angles.
Opposite angles are equal.
Every rhombus has 2 acute and 2 obtuse angles.
The altitude (or height) is perpendicular to sides AB & CD.
Area Formulas
Rhombus Area = (AD • CB) ÷ 2
Rhombus Area = side • altitude
Rhombus Area = side² • sin (α or β)
Inscribing A Circle Within A Rhombus All rhombuses are tangential quadrilaterals, meaning that they are 4 sided figures into which a circle (called an incircle) can be inscribed such that each of the four sides will touch the circle at only one point. (Basically, this means that the circle is tangent to each of the four sides of the rhombus.) To inscribe a circle graphically within a rhombus (using compass and straight edge): • calculate the inradius, which is the altitude ÷ 2 • set the compass precisely to this distance • draw the rhombus diagonals • place the compass point precisely at "E" where the diagonals intersect • Inscribe the circle using point E as its center and the inradius as its radius.
Significant Figures >>>
The default setting is for 5 significant figures but you can change that by inputting another number in the box above.
Answers are displayed in scientific notation and for easier readability, numbers between .001 and 1,000 will be displayed in standard format (with the same number of significant figures.)
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CC-MAIN-2018-05
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http://jean-pierre.moreau.pagesperso-orange.fr/Cplus/ltddev2_cpp.txt
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/******************************************************** * Calculate a limited development of a real function * * f(x)/g(x) at x=0 up at order 25, knowing the limited * * developments of f(x) and g(x). * * ----------------------------------------------------- * * SAMPLE RUN: * * * * Limited development of a real function f(x)/g(x): * * * * Function to develop: exp(x)/1+x * * * * Limited development of f(x)=exp(x) is: * * 1 + x + x^2/2! + x^3/3! + ... + x^n/n! + ... * * * * Limited development of g(x)=1+x is: * * 1 + x * * * * Order of development (max=25): 6 * * * * Input coefficients of limited dev. of f(x): * * a0 = 1 * * a1 = 1 * * a2 = 0.5 * * a3 = 0.1666667 * * a4 = 0.0416667 * * a5 = 0.0083333 * * a6 = 0.0013889 * * * * Input coefficients of limited dev. of g(x): * * b0 = 1 * * b1 = 1 * * b2 = 0 * * b3 = 0 * * b4 = 0 * * b5 = 0 * * b6 = 0 * * * * The coefficients of limited dev. of f(x)/g(x) are: * * c0 = 1.0000000 * * c1 = 0.0000000 * * c2 = 0.5000000 * * c3 = -0.3333333 * * c4 = 0.3750000 * * c5 = -0.3666667 * * c6 = 0.3680556 * * * * Function to develop: 1/1+sh(x) * * * * Limited development of g(x)=1+sh(x) is: * * 1 + x + x^3/3! + x^5/5! + ... + x^2n+1/(2n+1)! + ... * * * * Order of development (max=25): 6 * * * * Input coefficients of limited dev. of f(x): * * a0 = 1 * * a1 = 0 * * a2 = 0 * * a3 = 0 * * a4 = 0 * * a5 = 0 * * a6 = 0 * * * * Input coefficients of limited dev. of g(x): * * b0 = 1 * * b1 = 1 * * b2 = 0 * * b3 = 0.1666667 * * b4 = 0 * * b5 = 0.0083333 * * b6 = 0 * * * * The coefficients of limited dev. of f(x)/g(x) are: * * c0 = 1.0000000 * * c1 = -1.0000000 * * c2 = 1.0000000 * * c3 = -1.1666667 * * c4 = 1.3333334 * * c5 = -1.5083334 * * c6 = 1.7111112 * * * * ----------------------------------------------------- * * Ref.: "Mathematiques en Turbo Pascal By Alain * * Reverchon and Marc Ducamp, Editions Eyrolles, * * Paris, 1991" [BIBLI 03]. * * * * C++ version by J-P Moreau. * * (www.jpmoreau.fr) * ********************************************************/ #include #include #define SIZE 25 int i,m; double T1[SIZE+1],T2[SIZE+1],R[SIZE+1]; void ar_dldiv(int n, double *t1, double *t2, double *res) { int i,j; double x; if (n > SIZE) return; if (fabs(t2[0]) < 1e-12) return; res[0]=t1[0]/t2[0]; for (i=1; i<=n; i++) { x=t1[i]; for (j=1; j<=i; j++) x -= t2[j]*res[i-j]; res[i]=x; } } void main() { printf("\n Limited development of a real function f(x)/g(x):\n\n"); printf(" Function to develop: exp(x)/1+x\n\n"); printf(" Limited development of f(x)=exp(x) is:\n"); printf(" 1 + x + x^2/2! + x^3/3! + ... + x^n/n! + ...\n\n"); printf(" Limited development of g(x)=1+x is:\n"); printf(" 1 + x\n"); printf("\n Order of development (max=25): "); scanf("%d",&m); printf("\n\n Input coefficients of limited dev. of f(x):\n"); for (i=0; i<=m; i++) { printf(" a%d = ",i); scanf("%lf",&T1[i]); } printf("\n\n Input coefficients of limited dev. of g(x):\n"); for (i=0; i<=m; i++) { printf(" b%d = ",i); scanf("%lf",&T2[i]); } printf("\n\n"); ar_dldiv(m,T1,T2,R); printf(" The coefficients of limited dev. of f(x)/g(x) are:\n"); for (i=0; i<=m; i++) printf(" c%d = %10.7f\n", i, R[i]); printf("\n\n"); } // end of file ltddev2.cpp
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https://scribing.shop/2021/03/24/performance-of-an-axial-flow-my-assignment-tutor/
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# performance of an axial flow | My Assignment Tutor
PERFORMANCE CHARACTERISTICS OF A SINGLE STAGE AXIALFLOW AIR COMPRESSOROBJECTIVEDetermine the performance of an axial flow compressor experimentally.APPARATUS:Single stage axial flow compressor, orifice meter, and dynamometerParametersPower input P Rotational speedVolume flow rate of airDiameter of impellerPressure change across impellerAir densityViscosityω = 2 π N / 60, N= RPMQD∆pρµ In functional form, the relation is given by:f (P, ω, D, Q, ∆p, ρ, µ) = 0 (1)By Buckingharn Pi theorem, one can show, using w, r and D as repeating variables, a setof dimensionless variables,Power coefficient, w=3 5Dρω PC(2)Pressure coefficient,p 2 2p DCρω∆= (3)Flow Coefficient,Q 3Q DCω= (4)And Reynolds’ s number,Figure 1: Orifice plate and flow pipe nomenclaturex = location of back plate from the edge of the pipe.The pressure taps PT1 and PT2 are conndrop across the orifice plate.δp = ρWhere, ρalcohol = density of alcoholg = acceleration due to gravityδh1 = difference in height in the manometer tubes connected to PTl andPT2.The volume flow Q, through the pipe is given by (see Ref 1),Q = CWhere Cd = discharge coefficient, 1.2Cd = .0 5959 + .0 0312β– 08 5.2R10.184 .0 0029β + β1– β µρ=VDReA2QV = = Average velocityected to a manometer to measure op, the pressurealcohol g.δh1d 2 412 p 1Aρ – βδ4 4175.6 e.0 0337L 09.0–β + (5)(6)(7)3L2β (8)Here Ll = 1 (location ofPT2 from orifice plate, in terms of pipe diameter) andL2 = 0.5 (location of PTl from orifice plate, in terms of pipe diameter).β = d / D = ratio of orifice diameter to pipe diameter = 0.5D 4A Area of. .Pipe22π= =PROCEDURE:In this experiment, the variables are x, the location of back plate from the edge of thepipe and N, the rotational speed of the compressor. The range for these variables are: 1″
QUALITY: 100% ORIGINAL PAPER – NO PLAGIARISM – CUSTOM PAPER
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https://scoop.eduncle.com/ans-c-3-4-ex-19-what-is-the-value-of-8-33-of-7272-of-28-57-of-46-b-4-a-2-d-8-c-6-ans-d-8-33-1-12-72-72
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UGC NET Follow
March 24, 2021 12:24 pm 30 pts
Ans. (C) 3/4 Ex.19 What is the value of 8.33% of 7272% of 28.57% of 46: (B) 4 (A) 2 (D) 8 (C) 6 Ans. (D) 8.33% = 1/12 72.72% = 8/11 28.57% = 2/7 Therefore, we have (1/12) x (8/11) x (2/7) x 462 (1/12)(8/11)(2/7) x 11 x 7x 6 8
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Question is misprinted because in the question '46' has been taken and in the solution it has been taken '462'.
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https://www.studiestoday.com/rd-sharma-solutions-mathematics-rd-sharma-solutions-class-6-maths-chapter-13-quadrilaterals-324459
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# RD Sharma Solutions Class 6 Maths Chapter 13 Quadrilaterals
Read RD Sharma Solutions Class 6 Maths Chapter 13 Quadrilaterals below, students should study RD Sharma class 6 Mathematics available on Studiestoday.com with solved questions and answers. These chapter wise answers for class 6 Mathematics have been prepared by teacher of Grade 6. These RD Sharma class 6 Solutions have been designed as per the latest NCERT syllabus for class 6 and if practiced thoroughly can help you to score good marks in standard 6 Mathematics class tests and examinations
Objective Type Questions
Mark the correct alternative in each of the following:
Question 1: A quadrilateral having one and only pair of parallel sides is called
(a) a parallelogram
(b) a kite
(c) a rhombus
(d) a trapezium
Solution 1: (d)
Trapezium:- A quadrilateral having one and only pair of parallel sides.
Question 2: A quadrilateral whose opposite sides are parallel is called
(a) a rhombus
(b) a kite
(c) a trapezium
(d) none of these
Solution 2: (d)
Parallelogram:- A quadrilateral whose opposite sides are parallel.
Question 3: A quadrilateral having all sides equal is a
(a) square
(b) parallelogram
(c) rhombus
(d) kite
Solution 3: (c)
Rhombus:- A quadrilateral having all sides equal.
Question 4: A quadrilateral whose each angle is a right angle is a
(a) square
(b) rectangle
(c) rhombus
(d) parallelogram
Solution 4: (b)
Rectangle:- A quadrilateral whose each angle is a right angle.
Question 5: A quadrilateral whose each angle is a right angle and whose all sides are equal is a
(a) square
(b) rectangle
(c) rhombus
(d) parallelogram
Solution 5: (a)
Square:- A quadrilateral whose each angle is a right angle and whose all sides..
Question 6: A quadrilateral having two pairs of equal adjacent sides but unequal opposite sides is called a
(a) trapezium
(b) parallelogram
(c) kite
(d) rectangle
Solution 6: (c)
Kite:- A quadrilateral having two pairs of equal adjacent sides but unequal opposite sides.
Question 7: The diagonals of a quadrilateral bisect each other. This quadrilateral is a
(a) rectangle
(b) kite
(c) trapezium
(d) none of these
Solution 7: (a)
The diagonals of a quadrilateral bisect each other is a rectangle.
Question 8: If the diagonals of a quadrilateral bisect each other at right angle, then the quadrilateral is a
(a) parallelogram
(b) rectangle
(c) rhombus
(d) kite
Solution 8: (c)
If the diagonals of a quadrilateral bisect each other at right angle, then it is a rhombus.
Question 9: An isosceles trapezium has
(a) all sides equal
(b) parallel sides equal
(c) non-parallel sides equal
(d) any two equal sides
Solution 9: (c)
Non-parallel sides equal are equal of a trapezium.
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CC-MAIN-2024-38
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https://satchelclasses.com/
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# For every child.
A captivating look into the industries of your child's future. From the novel to the known, explore new territories and broaden their horizons.
Live and First-Look classes with experts and teachers.
Short-burst. Fuss-free.
First Look
#### Maths: Changing the Subject of a Formula through Factorisation
In this class students will recap the methods of rearranging equations and then we will learn how to solve more complex problems which involve factorising expressions before being able find the subject of a formula. For example, in the formula for the area of a rectangle A = b h ( area = base × height ), the subject of the formula is A.
• Stream until Tue 1st Feb
• 20 mins
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First Look
#### Spanish: To be and -ing Words
This class focuses on learning how to say conjugate verbs using estar for 'to be' and '-ing' words. In this lesson we will revise leisurely activities such as ‘watching TV’, ‘listening to the radio’ or ‘sending messages to friends’. We will learn the structure ‘I am doing something’ and will use it to make sentences in present progressive or continuous. We will then make up our own sentences by using a given template. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
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• 20 mins
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#### Maths: Converting Recurring Decimals to Fractions
Fractions, decimals and percentages are frequently used in daily life. At GCSE level students are required to make basic conversions, however they are also required to prove that a recurring decimal can be written as a fraction. A typical recurring decimal is 0.333 which is equivalent to a 1/3. Students need to be able to show why 0.333 is equivalent to 1/3. This is a level 6/7 topic for students sitting the higher GCSE paper.
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#### Maths: Solving 1 and 2 Step Equations
Solving equations is an integral part of maths and many mathematical problems can be solved by forming and solving equations. In this lesson students will learn how to use inverse operations to solve two step equations. Solving equations is a fundamental skill for students to revise and perfect, it is often tested in other topics such as Geometry.
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• 20 mins
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#### Maths: Surface Area of Solids
In this session, students will be introduced to the concept of surface area of 3D shapes using real-life examples. They will also learn how to calculate surface area of standard 3D shapes and a general rule to calculate surface area of prisms and pyramids.
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• 20 mins
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#### Maths: Mixture Problems in Cooking
Do you know which word problems are the most essential in real life?
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• 20 mins
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#### Maths: Calculation with Integers
This class will take you through the steps needed to add and subtract integers, you will also practise using negative integers. This is a topic that can sometimes be confusing for many students.
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• 20 mins
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#### Maths Workshop: Money and Percentages Part 2
In this class Daria your SkyMath tutor will continue talking about percentage and the different types of Interest. You will consider what is compound interest and compare it with simple interest. Students will have a great opportunity to see the difference between these interest types in practice.
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• 30 mins
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#### Maths: Writing numbers in Standard Form
In this session, students will understand when to multiply and divide numbers by powers of 10 and how to use this principle to write any number in its standard form. They will also learn how to perform different calculations with numbers written in their standard forms.
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• 20 mins
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#### Maths: Straight Line Graphs
Students will be introduced the equation of a line (y=mx + c), understanding what is meant by the gradient (m) and y-intercept (c). Students also learn how to plot graphs using a table of values and from there will move onto doing more complex straight line graph problems whereby the would have to find the equations of parallel and perpendicular lines.
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• 20 mins
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#### Maths: Simultaneous Equations
Simultaneous equations require algebraic skills to find the values of letters within two or more equations. They are called simultaneous equations because the equations are solved at the same time.
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• 35 mins
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#### Maths: Forming and Solving Equations
In everyday life people calculate the area and perimeter of things. From finding out how many tiles you need for your bathroom floor, or how much wrapping paper you need for your Christmas presents!
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#### Maths: Volume of Compound Cuboids
In this session, students will be introduced to the concept of compound cuboids. They will also learn how to break compound cuboids into a simple cuboid and how can that knowledge be used to find the volume of compound cuboids.
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#### It's Christ-Maths!
In this class Christmas themed Maths class, Daria your tutor will consider the most famous logic exercises in the seasonal realm. Students will learn several approaches for solving sophisticated logic problems and put new theory into practice. This is a great chance to not only have some Christ-Maths fun, but to really understand how to approach logical word problems which is an important part of excelling at mathematics.
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• 30 mins
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#### Maths Workshop: Money and Percentages Part 1
In this class Daria your tutor from SkyMath will consider a necessary and useful mathematical topic which is Percentages! Students will learn essential theoretical foundation and will put it into the practice. Also Daria will talk about interest, in this webinar she will show what is Simple Interest, how to calculate it and how we can use it in real life.
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• 30 mins
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#### Maths: Rounding Numbers to Significant Figures
In this session students will be introduced to the concept of significant figures and how to round numbers to a certain number of significant figures. They will also learn how to use estimating the answer to any calculation by using the concept of rounding to significant figures. Join Mr Salian in this class to fully understand how and why we round numbers to significant figures.
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#### Maths: Conditional Probability
In this class your SkyMath tutor is going to go through conditional probability. Students will first learn the theory needed to understand conditional probability, its uses and importantly the differences between conditional and basic probability. After the theory is learnt students will be able to put their knowledge into practise with some exercises intended to challenge and reinforce the theory.
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• 20 mins
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#### Maths: Distance between 2 Points
In today's class students will learn how to use the Pythagorean Theorem to find the distance between two points on a coordinate axis. Students will then derive the formula for the distance between two lines. Mr Hajghassem provides students the opportunity in this class to apply Pythagoras' theorem with a number of exercises.
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• 20 mins
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###### Curriculum support
This session focuses on using the quadratic formula in order to be able to solve quadratic equations.
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• 20 mins
• £4.50
First Look
#### Maths: Highest Common Factor (HCF) and Lowest Common Multiple (LCM)
In this session, students will be introduced to the concept of the Highest Common Factor (H.C.F.) and Lowest Common Multiple L.C.M.
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• 20 mins
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#### Maths: Volume of Prisms
In this session, students will be introduced to the concept of volume of 3D shapes and also learn how to calculate volume of prisms. They will also learn the concept of unit cubes and how they can be used to calculate the volume of cubes and cuboids.
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• 20 mins
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###### Curriculum support
A solid understanding of quadratics is key to achieving the best results in Maths exams or teacher assessments.
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• 15 mins
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#### Maths: Rearranging Formulae
To build on an understanding of mathematical formulae, it is important to understand how these formulae can then be rearranged.
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• 20 mins
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#### Maths: Solving Equations
A basic understanding of how to solve equations is useful across all Maths studies.
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• 20 mins
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#### Maths: Standard Form
Standard form is a way to write really large or very small numbers. It follows the format a × 10ⁿ. Scientists use standard form all the time as they regularly have to deal with very large and very small numbers. For example the speed of light is a very large number, so it's usually written as 3 × 10⁸ m/s.
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• 20 mins
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#### Maths: Solving Quadratics by Completing the Square
This session expands on being able to complete the square and being able to use this to solve quadratic equations.
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• 20 mins
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#### Maths: Basic Probability
In today's class your SkyMath tutor is going to go over the basics of probability. Students will first of all go through the theory of probability and the differences between independent and exclusive events. Secondly they will get the opportunity to putting the information into practice and try out some challenging probability based exercises.
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• 20 mins
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#### Measurement Workshop Part 1
In the real world we use many different types of measurements, but in Maths we can split these into 2 units. In this class your SkyMath tutor will consider metric and imperial units. Students will discover the difference between these two groups of measurement. Students will solve various word problems based on real life situations to practise showing their working out using both metric and imperial units.
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• 30 mins
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#### Maths: Transformation of Shapes Workshop
This workshop gives students the opportunity to see how to gain some serious marks in your GCSE examinations in the topic of Translation and Reflection of different shapes. Your tutor Daria Students will show you how to do these operations effectively and timely so that you do not waste any time in your examinations. All the theory learnt in today's class will be put into practice so students get the best possible chance of learning how to answer exam-style questions.
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• 20 mins
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#### Maths: Calculating the Mean from Frequency Tables
A measure of average is a value that is typical for a set of figures. Finding the average helps you to draw conclusions from data. For instance, finding the average number of sales in a given period can help a business make decisions about their stock levels at different points in time. The main types are mean, median and mode. Where a data set is large, a frequency table can make it easier to calculate averages. Students will therefore learn how to use a frequency table to calculate the mean of a data set.
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• 20 mins
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#### Maths: Ratios and Proportions
In this class your SkyMath tutor will consider the most popular world problems that we find in the topic of ratio and proportions. Students will firstly be taught the theory and then have a chance to put that theory into practise with some all important exercises. These exercises are a key part of learning for students to challenge themselves and develop and understanding in how to answer questions alone.
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First Look
#### Maths: Order of Operations
The order of operations (BIDMAS) is a collection of rules that tell us which operations to perform first in order to evaluate a given mathematical expression. For example, in mathematics, multiplication is granted a higher precedence than addition, therefore, the expression 2 + 3 × 4 is interpreted to have the value 2 + (3 × 4) = 14, and not (2 + 3) × 4 = 20.
• Stream until Tue 1st Feb
• 20 mins
• £3
First Look
#### English Lit: Of Mice and Men
In today's session students will be shown how to write critically. In this class students will get an overview of how to write analytically about ‘Of Mice and Men’ for a specific assignment: Consider to what extent Curley’s wife is presented as an outsider.
• Stream until Tue 1st Mar
• 20 mins
• £3
First Look
#### English: Language Analysis - Victorian Literature (Unseen Prose)
In this class students will get an overview of how to analyse a short extract from ‘The Hound of the Baskervilles’.
• Stream until Tue 1st Mar
• 20 mins
• £4
First Look
#### English: Narrative Writing (Hero Topic)
In this class your English tutor will take you through an important task - Narrative Writing. Your writing will be structured around an event taking place. The event for this class will be a 'Hero Returns,' which focuses on a hero returning home. You will learn how to structure narrative writing, how to decide on the end goal of your narrative writing and revise different techniques to make your writing stand out.
• Stream until Tue 1st Mar
• 20 mins
• £4
First Look
#### English: Opinion Writing (Marathon Hero Topic)
‘No-one in their right mind would run a marathon!’ In this class, students will get an overview of how to structure opinion writing. You will use the opinion of a marathon runner above to address the issue of running a marathon as a charity hero. You will learn how to present the stance of your writing, the different ways to get your opinion across and how to structure your writing so that is flows freely and importantly does not contradict its own opinion!
• Stream until Tue 1st Mar
• 20 mins
• £4
First Look
#### English: Punctuation Basics
In today's class we are going to take an in-depth focus on punctuation. We are going to cover the different types of punctuation available for you to use in your writing and the correct usage of it. By the end of the class you should be able to use and understand more advanced punctuation such as colons, semicolons, hyphens and dashes confidently in all pieces of work.
• Stream until Tue 1st Mar
• 20 mins
• £3.50
First Look
#### English: Analysing Persuasive Speech in Animal Farm
Animal Farm is a very common text across English curricula.
• Stream until Tue 1st Mar
• 20 mins
• £4.50
First Look
#### English: Clauses in Complex Sentences
Do you want to learn how to become a Sentence Slayer? In this class, students will be given an overview of how to use sentences with more than one clause in order to add depth and detail to their fiction and non-fiction.
• Stream until Tue 1st Mar
• 20 mins
• £4
First Look
#### English: Greek and Latin Etymology
In this class students will be given an overview of how to understand words in context. You will be shown why understanding Greek and Latin root words can help you understand long, complex words in the context of which they are written.
• Stream until Tue 1st Mar
• 20 mins
• £3
First Look
#### English: Language Analysis -Dystopian Fiction (Unseen Prose)
In this class students will get an overview of how to analyse a short extract from a dystopian science ficiton novel ‘The Mazerunner’. You will be shown how to identify language features, how to comment on the effects of language and how to embed and explain quotations.
• Stream until Tue 1st Mar
• 20 mins
• £4
First Look
#### English Poetry: Using Structure to Create Meaning in "The British, serves 60 million"
Want to write your own poems?
• Stream until Tue 1st Mar
• 20 mins
• £4
First Look
#### English: Language Analysis - 'The Pearl' John Steinbeck
In this class students will get an overview of how to analyse the language and structure in an extract from John Steinbeck’s ‘The Pearl’. You will be shown how to write about and analyse the effects of the language features as well as how to analyse the effects of structure features. This session will help you to secure a GCSE grade 5 in English Language for your centre-assessed grades / teacher-assessed grades (CAG / TAG), focusing on skills that your teacher will be looking at when making a final assessment.
• Stream until Tue 1st Mar
• 20 mins
• £3.50
First Look
#### English: Narrative Writing (Victorian Orphanage)
In this class, students will get an overview of how to structure narrative writing, based on a specific narrative writing assignment: 'write a story set in a Victorian orphanage.' We will work on writing for a purpose and understanding the purpose of your assignment as well as practising the 5 point narrative arc.
• Stream until Tue 1st Mar
• 20 mins
• £4
First Look
#### English Language: Critical Reading and Comprehension
In this class students will get an overview of how to evaluate a writer’s choices based on a specific extract from ‘The War of the Worlds’. You will be shown how to write evaluatively about language features as well as structural techniques in a specific text.
• Stream until Tue 1st Mar
• 20 mins
• £3
First Look
#### English: Language Analysis - A Christmas Carol (Unseen Prose)
In this class students will get an overview of how to analyse a short extract from ‘A Christmas Carol’. You will be shown how to identify genre, audience and purpose, how to comment on the effects of language features and how to embed and explain quotations. This session will help you to secure a GCSE grade 5 in English Language for your centre-assessed grades (CAG), focusing on skills that your teacher will be looking at when making a final assessment.
• Stream until Tue 1st Mar
• 20 mins
• £4
First Look
#### English: Context & Characterisation in Animal Farm
Context is everything. Knowing the background to a piece of writing can completely alter how you perceive it.
• Stream until Tue 1st Mar
• 20 mins
• £4.50
First Look
#### English: An Introduction to Gothic Writing
Gothic literature is a poplar theme across different English curricula.
• Stream until Tue 1st Mar
• 20 mins
• £4
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#### English Poetry: Creation of Tone in Island Man
A lot of factors affect how a poet sets the mood of their poem.
• Stream until Tue 1st Mar
• 20 mins
• £4
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#### English Language: 'An Inspector Calls' 20th Century Fiction
In this class students will get an overview of the three female characters from ‘An Inspector Calls’. This session works on allowing you to secure a GCSE Grade 5 and above in English Language for your centre-assessed and teacher-assessed grades.
• Stream until Tue 1st Mar
• 20 mins
• £4
First Look
#### English: Analyse and Compare 2 Poems - War Topic
In this class students will get an overview of how to analyse and compare two war poems: ‘The Soldier’ and ‘The Mother’. The session will be based on a specific assignment: ‘Compare how the writers present war.’ Firstly students will be shown how to understand poetic devices and language features with the intent of commenting on them. Then they will learn how to embed and explain quotations in your writing. After this class, students will be able to compare any two poems that they are covering in school using their skills learnt today.
• Stream until Tue 1st Mar
• 20 mins
• £3.50
First Look
#### English: Grammar Workshop - Words and Word Class
In this class students will get an overview of different types of nouns, verbs and adjectives and their usage. This workshop is key for securing the knowledge that is needed for all students studying English. By the end of the session students should feel confident when using nouns, verbs and adjectives.
• Stream until Tue 1st Mar
• 20 mins
• £4
First Look
#### Poetry: Comparison of Half Caste & Search for my Tongue
Poetry can convey all types of emotions and circumstances, poems from other cultures also can give you an insight on the poets identity. In this class students will be given an overview of how to compare poems. Students will be shown how to analyse the poems ‘Half-Caste’ and ‘Search for My Tongue’ for a specific assignment: ‘Compare how the two poets present ideas about identity.’ This class is essential for English Literature students who want the opportunity to learn how to understand and compare 2 poems.
• Stream until Tue 1st Mar
• 20 mins
• £3
First Look
#### English Poetry: Examining a Poet's Heritage and Culture
Grace Nichols is the poet behind 'Island Man.'
• Stream until Tue 1st Mar
• 20 mins
• £4
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#### English Literature: Duality in Jekyll and Hyde
In this English Literature class students will get an overview of how to write an essay for ‘The Strange Case of Dr Jekyll and Mr Hyde' with today's particular assignment being ‘How does Stevenson present the theme of duality?’
• Stream until Tue 1st Mar
• 20 mins
• £3.50
First Look
#### English: Achieving Grade 5 in Opinion Writing
In this mini-series of 3 classes, students will get an overview of how to write at (notional) Grade standards.
• Stream until Sat 5th Mar
• 20 mins
• £4
First Look
#### English: Achieving Grade 7 in Opinion Writing
In this mini-series of classes, students will get an overview of how to write at (notional) Grade standards.
• Stream until Sun 6th Mar
• 20 mins
• £4
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#### English: Achieving Grade 9 in Opinion Writing
In this mini-series of 3 classes, students will get an overview of how to write at (notional) Grade standards.
• Stream until Mon 7th Mar
• 20 mins
• £4
First Look
#### Science: Contact and Non Contact Forces (Physics)
In this class we will discuss what it means for a force to be contact or non-contact. Students will look at each force in detail and go through examples of questions involving forces and how they interact. They will learn the theory around the topic and get an extremely useful insight on how to answer exam questions.
• Stream until Fri 1st Apr
• 20 mins
• £4
First Look
#### Science: Plant and Animal Cells
In this class students will be looking at the structures of animal and plant cells and the similarities and differences between them.
• Stream until Fri 1st Apr
• 20 mins
• £3
First Look
#### Biology: Plant Responses to their Environment
In this class we will look at answering longer questions that concern the environmental factors that affect plant responses, such as the direction of root and shoot growth and the role of plant hormones in these tropisms.
• Stream until Fri 15th Apr
• 20 mins
• £4
First Look
#### Biology: Eukaryotes and Prokaryotes (Cell Structure)
Cells are the 'building blocks of life'.
• Stream until Fri 15th Apr
• 20 mins
• £4.50
First Look
#### Biology: Adaptations of Plants and Animals for Survival
In this class we will look at the knowledge required to answer exam style questions on the topic of Adaptations for Survival. This will include reducing the risk of being eaten by predators and general adaptations to the environment. Learning how to answer exam questions is an essential skill needed to really excel at your Biology examinations. Laura will provide you with the tools to give a full rounded answer to reach all the marks you need.
• Stream until Fri 15th Apr
• 20 mins
• £4
First Look
#### Biology: Digestive System Exam Practice
In this class we will discuss and demonstrate how to answer longer questions on the digestive system, including enzymes and enzyme production, the products of digestion and the organs involved.
• Stream until Fri 15th Apr
• 20 mins
• £4
First Look
#### Biology: Transportation of substances through a Plant
In this class we will look at how substances are transported through a plant, such as water through the xylem and sugars through the phloem. We will discuss how to answer longer questions on this subject to gain full marks in exams.
• Stream until Fri 15th Apr
• 20 mins
• £4
First Look
#### Chemistry: Planning your Practical Assessment
Practical work and demonstrations that have been undertaken in class are assessed in GCSE exams. In this class we will look at key knowledge that is needed to be successful in answering these type of questions and how to gain full marks.
• Stream until Sun 1st May
• 20 mins
• £4
First Look
#### Chemistry: Greenhouse Gases
This class will focus on the greenhouse gases carbon dioxide, methane and water vapour. We will look at how these gases absorb infrared radiation and then radiate energy back to the Earth. We will describe the natural greenhouse effect, which is essential to life on Earth and the ways that humans are enhancing the natural greenhouse effect.
• Stream until Sun 1st May
• 20 mins
• £4
First Look
#### Chemistry: Global Climate Change
In this class we will be looking at the how much the Earth has warmed overall to human activity and the climatic effects of this warming. We will discuss how different parts of the globe will be affected differently, such as more intense storms, floods and droughts. We will also see how these climatic changes will impact the lives of people around the planet.
• Stream until Sun 1st May
• 20 mins
• £4
First Look
#### Chemistry: Hydrocarbons and Fractional Distillation
In this class we will discuss what a hydrocarbon is and the structural and displayed formula, and understanding how to gain full marks in longer exam questions based on the manufacture of useful hydrocarbons by fractional distillation.
• Stream until Sun 1st May
• 20 mins
• £4
First Look
#### Chemistry: Conservation of Mass & Balanced Chemical Equations
As with any learning of equations, practice is key.
• Stream until Sun 1st May
• 20 mins
• £4.50
First Look
#### Chemistry: Factors Affecting the Rate of Reaction
In this class we will look at how the factors of particle size, concentration and temperature affect the rate of reaction in chemical reactions. A commonly addressed question in exams, this class will demonstrate how to gain full marks in longer response questions.
• Stream until Sun 1st May
• 20 mins
• £4
First Look
#### Chemistry: Structure and Formulae of Alkenes
Understanding Alkenes is a key lesson across all Chemistry studies at this level.
• Stream until Sun 1st May
• 20 mins
• £4.50
First Look
#### Chemistry: Crude Oil and Alkanes (Fuel)
This class covers all the basics of Crude Oil & Alkanes: definitions, examples and structures. Followed by practice exam questions to consolidate knowledge.
• Stream until Sun 1st May
• 20 mins
• £4.50
First Look
#### Chemistry: Electronic Structure
Understanding electronic structure forms the basis for many future lessons.
• Stream until Sun 1st May
• 20 mins
• £4
First Look
#### Chemistry: Life on Earth and the Atmosphere Exam Practice
In this class we will look at how to answer questions concerning the development of the Earth’s atmosphere, from its formation until today. We will discuss how life has affected the composition of the atmosphere. Key information and technique to gain full marks is included.
• Stream until Sun 1st May
• 20 mins
• £4
First Look
#### Chemistry: Human Activity and Greenhouse Gases
In this class we will look at the activities that humans are undertaking that add greenhouse gases to the atmosphere. Burning fossil fuels, deforestation and agriculture are all contributing to the concentration of greenhouse gases. We will look at data and graphs to confirm that humans are responsible for this increase.
• Stream until Sun 1st May
• 20 mins
• £4
First Look
###### Curriculum support
This class covers the different decay mechanisms that occur during radioactive decay of unstable radioactive nuclei. It describes how alpha, beta (-) and beta (+), neutron and gamma radiation affect the atomic structure and why.
• Stream until Sun 15th May
• 20 mins
• £4
First Look
#### Physics: Inside Atoms
A firm understanding of atoms is a key building block for future Physics education.
• Stream until Sun 15th May
• 20 mins
• £4
First Look
#### Physics Exam Revision: Pressure, Volume and Change of State
In this class we will look at how to answer questions associated with changes in pressure and volume in exams. I will demonstrate important calculations and analyse change of state graphs commonly seen in questions. Providing you with the tools to write the best answer possible. This class is beneficial to any students taking their GCSE's, or for younger students who want to get some exam practise under their belts. Either way this session is fantastic at helping students to improve their answering skills.
• Stream until Sun 15th May
• 30 mins
• £5
First Look
#### Equations for Moving Objects
Solving equations and formulas are an essential topic to grasp in Physics. This class focuses on the equations for moving objects, the Kinetic Energy equation. We will look at where the equation would be applied and how it links to other energy equations and scenarios. These 5 minute, short burst classes give students the chance to practise solving the equations and allows them to excel in their in future learning.
• Stream until Sun 15th May
• 5 mins
• £2
First Look
#### The Work Done Equation
Solving equations and formulas are an essential topic to grasp in Physics. In this short session we will look at how to use the equation for work done in different situations and how this equation is related to the equation for gravitational potential energy. These 5 minute, short burst classes give students the chance to practise solving the equations and allows them to excel in their in future learning.
• Stream until Sun 15th May
• 5 mins
• £2
First Look
#### Physics: Nuclear Energy
Nuclear Energy is used as a source of electricity but there are both advantages and disadvantages to this use.
• Stream until Sun 15th May
• 20 mins
• £4
First Look
#### Physics: Nuclear Equations
In this class we will be looking at how nuclear equations are written that display the decay of an atom in a lighter atom via alpha and beta decay. We will look at how to tell which products will be produced using the Periodic table. We will also look at gamma radiation and why this does not lead to a change in mass or atomic number.
• Stream until Sun 15th May
• 20 mins
• £4
First Look
#### Physics: Energy Stores and System
Energy is a reoccurring topic across much of the Physics curriculum.
• Stream until Sun 15th May
• 20 mins
• £4
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#### Physics: Newton's Second Law
To excel in Physics there are a lot of equations and formulas for students to learn. In this class we will look at Newton's second Law and the application of the formula F =ma to forces, mass and acceleration in GCSE physics. These 5 minute, short burst classes gives students the chance to practise solving equations and allows them to excel in their in future learning.
• Stream until Sun 15th May
• 5 mins
• £2
First Look
#### Physics: Exoplanets
In this class we are going to be talking about exoplanets (planets that orbit stars other than our own Sun!). Upto this point there have now been over 4000 planets that have been discovered orbiting stars other than our own Sun. Most of the planetary systems found do not resemble our Solar System whatsoever and are full of unusual and exotic planets that are stranger than science fiction!
• Stream until Sun 15th May
• 20 mins
• £3.50
First Look
#### Physics: Solar System
In this class we will look at all the bodies that make up our Solar System, including planets, moons, asteroids, comets, and dwarf planets. We will discuss why Pluto is no longer classified as a planet and the characteristics needed to be called a planet. We will also analyse, how our star, the Sun emits light through nuclear fusion.
• Stream until Sun 15th May
• 20 mins
• £4
First Look
#### Physics: Rearranging Equations and Substitutions
In this class your tutor will be demonstrating how to rearrange equations seen in GCSE Physics and how to apply this by the substitution of numbers into the equation to solve problems. I will also introduce the rearranging of equations with more than three variables. This topic can be challenging, therefore this class is perfect as a refresher for all GCSE age students.
• Stream until Sun 15th May
• 20 mins
• £5
First Look
#### Physics: Nuclear Fission
Nuclear Fission is an important topic in Physics at GCSE or equivalent level.
• Stream until Sun 15th May
• 20 mins
• £4
First Look
#### Physics: Mass & Weight
To excel in Physics there is a lot of theory for students to learn. This short session looks at the differences between mass and weight and the fact that mass is the same everywhere in the Universe, whereas weight varies. These 5 minute short burst classes allow students to practise understanding theory and allows them to excel in their in future learning.
• Stream until Sun 15th May
• 5 mins
• £2
First Look
###### Curriculum support
• Stream until Sun 15th May
• 20 mins
• £4
First Look
#### Calculating Density
Solving equations and formulas are an essential topic to grasp in Physics. In this class we will look at how the density of different objects is calculated using mass and volume. The different volumetric equations will be included. The calculation of density is a part of required practicals. These 5 minute, short burst classes give students the chance to practise solving equations and allows them to excel in their in future learning.
• Stream until Sun 15th May
• 5 mins
• £2
First Look
#### Physics: Acceleration
To excel in Physics there are a lot of equations and formulas for students to learn. In this short session we will focus on understanding and solving the equation for Acceleration that you will need to learn for your Physics/Science GCSE examinations. These 5 minute, short burst classes allow students to practise solving the equations and allows them to excel in their in future learning.
• Stream until Sun 15th May
• 5 mins
• £2
First Look
#### Physics Exam Revision: Energy
In this class we will look in detail about how to answer exam questions on the Energy topic. We will look at the knowledge that is required to answer the questions, understand what the question demands of the answer, and the process required to gain full marks in a 6 mark question. This class is beneficial to any students taking their GCSE's, or for younger students who want to get some exam practise under their belts. Either way this session is fantastic at helping students to improve their answering skills.
• Stream until Sun 15th May
• 20 mins
• £4.50
First Look
#### Physics Exam Questions: Nuclear fission and Nuclear Power
In this revision session we will look at the knowledge required to answer questions about nuclear fission and nuclear power for GCSE students. We will go through what a question demands from it in the answer, and the process of gaining full marks in a 6 mark question. This is essential revision for Physics students who are looking to gain the maximum amount of marks available in exam questions.
• Stream until Sun 15th May
• 20 mins
• £4.50
First Look
###### Curriculum support
Background radiation can originate from a number of different sources.
• Stream until Sun 15th May
• 20 mins
• £4
First Look
#### Physics: Electrons and Orbits
Electrons, orbits and ionisation can be a tricky topic to get your head around.
• Stream until Sun 15th May
• 20 mins
• £4
First Look
###### Curriculum support
Radiation takes many forms. In this class, students will cover the different types of nuclear radiation, including alpha, beta, gamma and neutron.
• Stream until Sun 15th May
• 20 mins
• £4
First Look
#### Physics: Half Life
Half Life considers the how the radioactivity of a substances changes over time.
• Stream until Sun 15th May
• 20 mins
• £4
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#### Physics: Atomic Models
Atomic Models is a topic which comes up regularly across Physics and Chemistry studies.
• Stream until Sun 15th May
• 15 mins
• £4
First Look
#### The Wave Equation
Solving equations and formulas are an essential topic to grasp in Physics. In this short session we will investigate how the wave equation is used to find the speed, the wavelength, and velocity of electromagnetic radiation and other waves. We will practice questions based on the application of the wave equation to real-life scenarios and exam questions. These 5 minute, short burst classes give students the chance to practise solving the equations and allows them to excel in their in future learning.
• Stream until Sun 15th May
• 5 mins
• £2
First Look
#### French: Adjectives Focus - Personality
In this class students we will practise speaking about family members. To do this we will first go over a number of adjectives to describe personality. After that we are going to have a conversation using all the new vocabulary we have learnt to describe out family and friends.
• Stream until Wed 1st Jun
• 20 mins
• £4
First Look
#### French: Past and Near Future Tense
In this class students will practise talking about what they do regularly using the present tense and what they are going to do using the near future. They will then take part in an interactive conversation, imagining their plan to be a successful Youtuber.
• Stream until Wed 1st Jun
• 20 mins
• £4
First Look
#### French: Gender and Definite Articles - Subjects at School
French nouns have a grammatical gender. In this class students will learn how to talk about the school subjects they are taking in French. It is important that we practise these as well as practise key vocabulary to teach students how to talk about and understand a range of school subjects. Furthermore pupils will also find out about the differences between the school subjects taught in France and in the UK.
• Stream until Wed 1st Jun
• 20 mins
• £4
First Look
#### French: Celebrations
In this class your tutor will go over the vocabulary to describe different types of celebrations. You will be going over Christmas and New Year, Easter and how to wish your family and friends a Happy New Year! We will also go through the topic of Christmas in more details. You will go through some fun exercises and puzzles to cement your knowledge of Celebrations!
• Stream until Wed 1st Jun
• 20 mins
• £3.50
First Look
#### French: Likes and Dislikes Workshop
In this class students will practise saying what they like and don’t like to do from the perspective of a professional cyclist. You will be given the chance to work on your pronunciation and speaking French out loud whilst voicing your personal opinions. At the end of the class Mr Smith will go through a fantastic tool called the 'sentence builder' which will allow you to create full, well-rounded sentences.
• Stream until Wed 1st Jun
• 35 mins
• £4
First Look
#### French: Quelle heure est-il? (What's the Time?)
In this class students will learn how to tell the time in French. We will revise numbers and practise key vocabulary to teach students how to understand what time it is right now and at what time things are happening in the future. Students will be able to practise the correct pronunciation and be able to answer time related questions.
• Stream until Wed 1st Jun
• 20 mins
• £4
First Look
#### French: Future Tense - Making Plans
In this class students will work on learning how to speak in the future tense. We are going to focus on making plans for the future using je vais or je veux.
• Stream until Wed 1st Jun
• 20 mins
• £4
First Look
#### French: Past and Present Tense- Daily Routine
This class focuses on learning how to take an active role in a conversation about your daily routine. We will firstly learn the vocabulary using the past and present tenses and then be able to relay that in a conversation using correct pronunciation and tone.
• Stream until Wed 1st Jun
• 20 mins
• £4
First Look
#### French: Countries and Nationalities
In this class, Ms Bennell will teach you how to say a range of countries in French as well as learning the different ways of saying nationalities. We will also go through how to use adjective endings when discussing nationalities.
• Stream until Wed 1st Jun
• 20 mins
• £4
First Look
#### French: Celebrations (Higher Level)
In this class your tutor will go over the vocabulary to describe different types of celebrations. You will be going over Christmas and New Year, Easter and how to wish your family and friends a Happy New Year! We will also go through the topic of Christmas in more detail.
• Stream until Wed 1st Jun
• 20 mins
• £4
First Look
#### German: Past and Present Tense Focus - In the Classroom
Today's lesson gives students the chance to practise the past and present tenses and revise or learn for the first time some key classroom language and school-related vocabulary. After going through the vocabulary we are going to get the opportunity to practise our speaking together to further our conversational skills.
• Stream until Wed 15th Jun
• 20 mins
• £4
First Look
#### German: Adjectives Focus - Best Friends
In this class we are going to spend some time learning how to describe people. By the end of the lesson you should be able to have a conversation in German describing your best friend!
• Stream until Wed 15th Jun
• 20 mins
• £4
First Look
#### German: 1st and 3rd Person Focus - A Trip Abroad
In this class you are going to practise how to describe a trip abroad. We are going to focus on how to use the past and present tense. Finally you will work on having greater confidence speaking in both first and third person whilst using a wider range of vocabulary in your conversation.
• Stream until Wed 15th Jun
• 20 mins
• £4
First Look
#### German: Past and Present Tense - Daily Routine
In this lesson students will see how the present and past tenses can be easily combined to talk about routine activities without worrying about complex grammar exercises. We will revise key routine phrases using flashcards and a sentence builder and then have a conversation in German.
• Stream until Wed 15th Jun
• 20 mins
• £3
First Look
#### German: Consolidation of the Perfect Tense
In this class your tutor will consolidate theory around the past tense. Whether you are a student who is new to the past tense in German, or one who has been introduced to it in previously at school, but have not had enough lesson time to get to grips with it, you will be shown the basic rules. Furthermore you will get a chance to practice the past tense in a conversation.
• Stream until Wed 15th Jun
• 20 mins
• £4
First Look
#### German: Media and Technology Present and Future
Media and technology is moving at an unimaginably fast pace, keeping up with these updates when we learn languages is a must. In this lesson students will learn useful phrases surrounding new media and technology. Students will then get the chance to practise using the present and future tenses in an imaginary and interactive conversation situation.
• Stream until Wed 15th Jun
• 20 mins
• £3
First Look
#### German: Adjective Focus Family and Personality
In this lesson pupils will learn how to describe other people, with a focus on adjectives to describe your own or others' personality. Students will then take part in an interactive conversation to practise their spoken German.
• Stream until Wed 15th Jun
• 20 mins
• £3
First Look
#### German: Past Tense Practice
This lesson follows on from the basic introduction to the past tense. In this lesson, the past tense is revised, before students take part in a conversation around the topic of daily routine / the school day, using a mixture of past and present tense. You will also get a chance to practise your speaking, so you can carry out your German conversations in the past tense.
• Stream until Wed 15th Jun
• 20 mins
• £4
First Look
#### German: Future Plans
In this lesson students will learn how to talk about what they are going to do and want to do in the future. We will first learn useful phrases and then practise speaking German in a conversation scenario.
• Stream until Wed 15th Jun
• 20 mins
• £3
First Look
#### Spanish: Present Perfect - I have
We are going to cover all you need to know when speaking in the present perfect. In this lesson we are going to revise a list of verbs ending in -ar. We will then learn the past participle of each one of these verbs. In our grammar section of the class we will explain how to use ‘haber’ in present to form tenses such as ‘I have studied’ or ‘I have lived’. This session will focus on the first person singular. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Conditional Tense (Lottery Topic!)
We are going to cover what you need to know when speaking in the conditional tense. The lesson will begin by introducing a list of activities in infinitive that we would do if we won the lottery. We will learn how to turn these verbs into conditional tense. Then we will complete our tenses with an ‘if’ clause to make our sentences more complete. Finally we will listen to native speakers, will complete a reading exercise and will do two sets of translations all about our dreams if we won the lottery.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Vocab and Present Tense Focus of 'Me duele'
Key vocabulary for students of Spanish includes remembering the names of different body parts. In this lesson Mr Castrillon will take students through the words for some of the parts of the body and we will learn how to say when a part of our body hurts. We will add adverbs of quantity, which will help us in showing how much or how little we are in pain. This is a great class to boost your vocabulary and to learn some new adverbs to make your Spanish sentences more descriptive.
• Stream until Fri 1st Jul
• 20 mins
• £3.50
First Look
#### Spanish: Vocabulary Focus- Describing Personalities
We are going to spend some time today advancing your Spanish vocabulary. In this lesson we will revise and learn some new vocabulary surrounding the topic of 'personality'. We will build sentences together in Spanish using opinions and the equivalent in Spanish for ‘I get/don’t get along with’ by using family member. This is a great class to increase your vocabulary and start to create some more detailed sentences using facts and opinions. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Conditional Sentences (Health Topic)
We are going to spend some time today advancing you knowledge of how to form a conditional sentence. In this lesson we will revise vocabulary for healthy living (en forma). Then we will learn how to form conditional sentences using simple present and the conditional word ‘if’. We will undertake an exercise together and will match sentences in Spanish with their translations into English. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
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###### Curriculum support
Feliz Navidad! In this Seasonal lesson, you will learn lots of vocabulary related to Christmas and how to wish your family and friends a Merry Christmas. There will be a fun Christmas activity where you will use verbs to fill in missing words in a set of sentences. Then there will be an exercise where you will read short texts and can practice our reading comprehension by answering multiple choice questions.
• Stream until Fri 1st Jul
• 20 mins
• £2.50
First Look
#### Spanish: La Navidad Latin American Style
Imagine celebrating Christmas in Latin America! In this lesson we will revise vocabulary related to Christmas in Latin America. We will use verbs to fill in missing words in a set of sentences. We will also read short texts and will practice our reading comprehension by answering multiple choice questions.
• Stream until Fri 1st Jul
• 20 mins
• £2.50
First Look
#### Spanish: Vocabulary Focus - I have (Tengo)
We are going to focus on the verb 'tener' particularly the form 'tengo' meaning I have. Firstly we will begin the lesson by revising the vocabulary for objects we have in our rooms. We will list them by using the word ‘tengo’ and the articles ‘un’, ‘una’, ‘unos’, ‘unas’. We will then list we don’t have in our room. As an extension we will include what we would like to have. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Seasonal Vocabulary - All things Spring
In this Spring themed Season Spanish lesson, we are firstly going to revise vocabulary related to the spring season. We will learn set phrases related to activities we do in the spring such as ‘look after the garden’ or ‘go for walks’. We will then use the present progressive tense to describe what we are doing in the spring and we will complete sentences with ‘now’, ‘at the moment’ or ‘right now’. Finally, we will read a paragraph in Spanish and begin a GCSE challenge! Even though you may think GCSE exams are a long way away right now, this is a fantastic opportunity to develop and hone some of the complex skills that are vital for future examination preparation.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Seasonal Vocabulary - All things Summer
In this lesson, we will revise vocabulary related to the summer season. We will learn set phrases related to activities we do during the summertime such as ‘swimming' and ‘sunbathing’. We will then use the future tense to describe what we are going to do next summer. Finally, we will read a paragraph in Spanish and begin a GCSE challenge! Even though you may think GCSE exams are a long way away right now, this is a fantastic opportunity to develop and hone some of the complex skills that are vital for future examination preparation.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Using Adverbs relating to Chores
For an adverb masterclass look no further. In this lesson, we will revise vocabulary related to household chores such as ‘tidying up’, ‘washing up’ or ‘cleaning’. We will learn frequency adverbs such as ‘always’, ‘never’ or ‘sometimes’ and we will use them to create good quality sentences in Spanish. We will hear native speakers telling us about their daily routines when doing house chores. Our final challenge will be two translation activities which will test all the knowledge you have learnt in class today. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Conjugating -ar verbs
For all you need to know when conjugating -ar verbs look no further. In this lesson, we will learn to conjugate verbs ending in -ar in simple present. We will learn both regular and irregular verbs. We will match up and translate a range of sentences from English into Spanish and Spanish into English. This is a great class for students to practise conjugating verbs and to listen to the correct pronunciation. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Ages- Tiene/Tienen
Do you know how to conjugate the verb 'to have'? In this lesson we will revise the topic of ages (numbers) in relation to family relationships. We will use numbers to express other people’s ages. We will learn to conjugate the verb ‘to have’ in third person (he has/she has/they have) and as a challenge we will learn how to ask someone how old they are. This class covers key counting and conjugating skills which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Simple Present Tense Focus - Forming Opinions (Health Topic)
We are going to spend some time today advancing your understanding of the simple present. In this lesson you are going to learn how to form sentences in simple present describing a routine to keep ourselves healthy. There will be a recap of the most common vocabulary for surrounding having a healthy lifestyle. Finally we will challenge ourselves to use ‘creo que’, ‘pienso que’ and ‘opino que’ to show our opinions regarding a healthy routine. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Conjugating -er verbs
In this lesson we will learn to conjugate verbs ending in -er in the simple present. Students are going to learn both regular and irregular verbs. Finally students will complete an exercise where we will match up and translate a range of sentences from English into Spanish and Spanish into English. This class will help you with 2 aspects of learning a language; conjugating verbs and a comprehension of sentence structure. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Personal Pronouns 'conmigo' and 'contigo'
We are going to cover all you need to know regarding personal pronouns. In this lesson we will introduce a list of activities in the infinitive, that we do with our friends. For example we will go through activities like playing football, talking on the phone or studying. We will introduce the use of the preposition ‘con’ to complete our sentences when expressing who we are doing these infinitive with. We will learn two special cases that we need in Spanish when we want to say ‘with me’ and ‘with you’. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Verb Practice 'To Do' with the Weather (Part 2 of 2)
Today we are going to spend some time on the weather. In this lesson we will revise vocabulary related to the weather such as ‘frio’, ‘calor’, ‘viento’, ‘bochorno’. We will use the verb ‘to do’ to describe different types of weather. We will introduce words like ‘mucho’ or ‘bastante’ to emphasize our description of the weather. This class covers key vocabulary which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Verb Practice 'To Be' with the Weather (Part 1 of 2)
Today we are going to spend some time on the weather. In this lesson, we will revise vocabulary related to the weather such as ‘lluvioso’, ‘caluroso’, ‘agradable’. We will use the verb ‘to be’ to describe different types of weather. We will hear different people describing the weather and we will answer a GCSE question about the weather. This class covers key vocabulary which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Vocab and Present Tense Focus- Discussing Daily Routines
Let's talk about what we are doing today. In this lesson we will start by revising vocabulary regarding your own daily routine. Together we are going to describe our daily routine and then we will use this to talk about somebody else’s routine. We are then going to complete a GCSE challenge to get students used to the style of questions that appear in examinations. Although the examinations may be a way off for many students, revising past questions is essential to excelling in Spanish!
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Adjectives Focus - Describing our Family
We are going to spend some time covering various different adjectives to advance your Spanish knowledge. In this lesson we will revise and learn some new vocabulary surrounding your family. We will learn verbs used to describe how we relate to other people and we will use them to describe our relationship with other members of the family. You will be able to give detailed answers to family related questions using opinions and facts, most importantly raising your language skills by using personality adjectives. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Double Negatives using 'Los Medios Sociales'
In this lesson we will learn the words for different types of social media and communication devices, like radio, television or mobile phone. We will use the verb 'tener' in simple present with different people. For instance, 'I have' 'my brother has'. We will also include negative sentences and as a challenge we will form sentences with 'ni...ni..' This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Spanish: Conjugating -ir verbs
Conjugating verbs is essential in Spanish to be able to develop full written or spoken sentences. In this lesson, we will learn to conjugate verbs ending in -ir in the simple present. We will learn regular and irregular verbs. We will complete an exercise where we will match up and translate a range of sentences from English into Spanish and Spanish into English. This class will help you to with 2 aspects of learning a language; conjugating verbs and the comprehension of sentence structure. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
First Look
#### Scottish Gaelic: Regular Future Tense
A firm understanding of the regular future tense is important when learning any new language.
• Stream until Fri 15th Jul
• 20 mins
• £3.50
First Look
#### Scottish Gaelic: Numbers 20-100
Learning a language takes time, effort and most of all practise!In this lesson we will learn to count from 20 - 100 in Gaelic. By the end of the class, students will be able to confidently and quickly recount numbers.
• Stream until Fri 15th Jul
• 20 mins
• £3.50
First Look
#### Scottish Gaelic: Old Fashioned Counting System
Learning the old system of counting can be useful in sums and in constructing numbers in a different language.
• Stream until Fri 15th Jul
• 20 mins
• £3.50
First Look
#### Scottish Gaelic: Describing Places
In the early stages of language learning, it is important to start building in descriptions to develop your new skills.
• Stream until Fri 15th Jul
• 20 mins
• £3.50
First Look
#### Scottish Gaelic: Irregular Future Tense
Irregular Future tense of any language can be a tricky topic for many.
• Stream until Fri 15th Jul
• 20 mins
• £3.50
First Look
#### Vocabulary and Terms for Essay Writing for Gàidhlig Medium
It is important for students to show how well they understand a new language.
• Stream until Fri 15th Jul
• 20 mins
• £3.50
First Look
#### Scottish Gaelic: Where I Live
A key lesson in early language learning is the ability to talk about where you live to really build those early conversations.
• Stream until Fri 15th Jul
• 20 mins
• £3.50
First Look
#### Geography: Causes of Unequal Development
In today's class we are going to be focusing on the topic of Development, in particular this lesson will examine the reasons why some countries are more developed than others. This will include examining the human and physical characteristics, as well explaining the impact that they may have on a country.
• Stream until Mon 1st Aug
• 20 mins
• £3
First Look
#### Geography: Hazards - Hydrological Cycle
The hydrological cycle is a key reoccurring theme which is used throughout Geography studies.
• Stream until Mon 1st Aug
• 20 mins
• £4
First Look
#### Geography: Hazards - Plate Boundaries
Plate boundaries are the source of some of the largest natural hazards on Earth.
• Stream until Mon 1st Aug
• 20 mins
• £4
First Look
#### Geography: Coasts - Wave Types
The topic of Coasts is common in Geography studies; looking at the ever changing boundary between land and sea.
• Stream until Mon 1st Aug
• 20 mins
• £4
First Look
#### Geography: Hazards - Urbanisation
Urbanisation is the movement of a population from rural to urban areas.
• Stream until Mon 1st Aug
• 20 mins
• £4
First Look
#### Geography: Global Air Circulation
There are many factors which affect the climates we experience across the globe.
• Stream until Mon 1st Aug
• 20 mins
• £4
First Look
#### Geography: Climatic Factors
Many factors affect the climate around us and why we experience different climates across the globe.
• Stream until Mon 1st Aug
• 20 mins
• £4
First Look
#### Geography: Causes of a River Flood Event
In this lesson we will examine the reasons why a river might be likely to flood. We will divide them between natural and man-made causes and try to link them to a storm to help examine the flood risk created.
• Stream until Mon 1st Aug
• 20 mins
• £3
First Look
#### Geography: Why has Tourism Grown?
Tourism is an industry based on recreation and leisure, people travel to destinations with these purposes in mind.
• Stream until Mon 1st Aug
• 20 mins
• £3
First Look
#### Geography: River Transportation
As a river moves through it's path it can pick up and deposit parts of the land along the way
• Stream until Mon 1st Aug
• 20 mins
• £4
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#### Geography: Population Boom
Do you know what the population is of the world at this very minute?
• Stream until Mon 1st Aug
• 20 mins
• £4
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#### Geography: Hazards - The Impact of Tropical Storms
In this lesson we will examine the ways countries can reduce tropical storm damage. We will look at strategies, before, during and after the events. We will also consider the impact wealth of the country could have on implementing these strategies.
• Stream until Mon 1st Aug
• 20 mins
• £2
First Look
#### Geography: The Demographic Transition Model
Today's class will focus on the topic of population, in particular this lesson we will examine the Demographic Transition Model. We will look at how the population changes and what might be causing these changes. We will also link a country to each stage of the model.
• Stream until Mon 1st Aug
• 20 mins
• £3
First Look
#### Geography: Hazards -Volcano Types
Volcanoes are one of the most dramatic natural hazards on Earth.
• Stream until Mon 1st Aug
• 20 mins
• £4
First Look
#### Geography: Coasts - Hard and Soft Engineering
In this lesson we will examine the strategies that are place that we can adopt to protect a coastline. This will be divided between hard and soft engineering. We will also explain how they can help protect the coastline.
• Stream until Mon 1st Aug
• 20 mins
• £3
First Look
#### Geology: Understanding Geological Time Periods and Eras
In this class we will take a trip through geological time, from the formation of the Earth until the present day. We will examine the main events of each geological era and period. We will look at how the history of the Earth is divided by mass extinctions, abrupt climatic change and other factors. We will also examine how the age of rocks are investigated using techniques such as radioactive dating and index fossils.
• Stream until Mon 15th Aug
• 20 mins
• £4
First Look
#### History: Hitler's Foreign Policy Aims (Germany 1880-1945)
Hitler's Foreign Policy is a key topic in realising earlier events in Hitler's policies connected to World War II.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: 4 Mark Question Workshop for GCSE - (Germany 1890-1945)
Alongside learning the content of History lessons, it is important to learn how to apply this knowledge to exams.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: How did the Norman Conquest impact Peasants?
Norman England is a large topic covered in AQA History alongside other exam boards. Through a series of classes, students will learn how the Normans changed so many aspects of life in England.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: How did the Normans change religion in England?
Norman England is a large topic covered in AQA History alongside other exam boards. Through a series of classes, students will learn how the Normans changed so many aspects of life in England.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: Who was Rosa Parks?
Rosa Parks was an American Activist in the Civil Rights movement best known for her resistance for bus segregation.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: Why the League of Nations was created & the form it took
The League of Nations is a popular topic across History exam boards.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: GCSE Exam Workshop - Tackling The 8 Mark Question
This class will get to grips with one of the trickiest Exam Question styles in History (notably found in papers by the AQA).
• Stream until Thu 1st Sep
• 20 mins
• £5
First Look
#### History: How did liberal & socialist opposition to Tsarism grow from 1905-1914?
Between 1905 and 1914 there was growing opposition to Tsarism in Russia following the crushing of the 1905 revolution.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: The Big 3 at The Treaty of Versailles - WWI
The Big 3 are the three men Woodrow Wilson, Georges Celemenceau and David Lloyd George who were the driving force behind the creation of the Treaty of Versailles.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: The Impact on Towns under the Normans
Norman England is a large topic covered in AQA History alongside other exam boards. Through a series of classes, students will learn how the Normans changed so many aspects of life in England.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: The Impact of the Norman Conquest on Villages
Norman England is a large topic covered in AQA History alongside other exam boards. Through a series of classes, students will learn how the Normans changed so many aspects of life in England.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: How did Marxism develop under Tsar Nicholas II?
Marxism was the most radical current of opposition to Tsarism in Russia.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: What were the Jim Crow Laws?
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: Who was Malcolm X?
Malcolm X was as an African American minister and human rights activist.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: The Treaty of Versailles - WWI
The Treaty of Versailles is a popular topic in across History assessments.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: How did the Normans balance power between the Church & the Crown?
Norman England is a large topic covered in AQA History alongside other exam boards. Through a series of classes, students will learn how the Normans changed so many aspects of life in England.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: How religious were people in Norman times?
Norman England is a large topic covered in AQA History alongside other exam boards. Through a series of classes, students will learn how the Normans changed so many aspects of life in England.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: How did the Normans change Education?
Norman England is a large topic covered in AQA History alongside other exam boards. Through a series of classes, students will learn how the Norman's changed so many aspects of life in England.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: Segregation in American Schools
Was segregation in American Schools overcome?
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: Henry VIII and the Church of England
There is a lot to learn around the Church of England in the time of Henry VIII.
• Stream until Thu 1st Sep
• 20 mins
• £3.50
First Look
#### History: How much did Russian culture change under Tsar Nicholas II?
Under the rule of Tsar Nicholas II there were many changes in Russian culture.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: How did Henry Tudor consolidate his Reign?
There is a lot to learn about the Tudor Period, a topic which is often covered across History studies.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### Geography: A River's Long Profile
A river goes through an enormous number of changes as it moves through it's path.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### History: GCSE Exam Workshop - Tackling The 12 Mark Question
Alongside learning the content of History lessons, it is important to learn how to apply this knowledge to exams.
• Stream until Thu 1st Sep
• 20 mins
• £5
First Look
#### History: Kaiser Wilhelm II (Germany 1880 - 1945)
To understand the causes of WW1, it is important to look further back in History.
• Stream until Thu 1st Sep
• 20 mins
• £4
First Look
#### Achieving Mindfulness through Art: Colouring
It is so important to make time for yourself in order to achieve the most in all areas of your life.
• Stream until Tue 1st Nov
• 20 mins
• £4.50
First Look
#### Campaigning for Animal Welfare with the RSPCA
In this class Rachel will help you think about how to make change on issues you care about. In other words, how to be a campaigner.
• Stream until Tue 1st Nov
• 20 mins
• £4
First Look
#### Learning Scottish Gaelic: Weather
In any language, the weather is always a great topic to talk about whoever you are speaking to!
• Stream until Tue 1st Nov
• 15 mins
• £3.50
First Look
#### Knowing Me, Knowing You: Introvert/Extrovert
Being able to understand yourself better brings clarity to many areas of your life and helps you make the right choices for you!
• Stream until Tue 1st Nov
• 20 mins
• £5
First Look
#### Learn Scottish Gaelic: Top Tips and Tricks for Beginners
Taking on a new language can be tricky but it can also be SO rewarding.
• Stream until Tue 1st Nov
• 20 mins
• £3.50
First Look
#### Earth Day: Climate Literacy for Climate Action
This class will focus on the urgent need for climate literacy across all grade levels and academic disciplines. We start with the first Earth Day in 1970 and the evolution of the modern environmental movement. Tracey will review the resources from EARTHDAY.ORG that were developed to help students, educators and parents along the learning spectrum from awareness to action.
• Stream until Tue 1st Nov
• 20 mins
• FREE
First Look
#### The Parent/Guardian Guide to Higher Education
In this session Alice and Hollie will explore why students make the own decision go to university, or why students may not want to go to university at all, and what the other options there are available to them. They will also provide Parents and Guardians with some top tips for how to best support your children in their Higher Education journey whichever pathway they take.
• Stream until Tue 1st Nov
• 25 mins
• FREE
First Look
#### Money Matters! Basic Money Management
What should we do with our money? In this class Ms Brooks will provide younger students with an overview and suggestions of how to manage and think about money.
• Stream until Tue 1st Nov
• 20 mins
• £2.50
First Look
#### How to Write a Personal Statement
In this class Ms Brooks will show students how to structure a personal statement with a main focus and direction on the content needed. You are going to learn the basics that need to be included in your Personal Statement and you will learn how to structure it and what content needs to be included. Secondly you will be shown how to make your statement unique, engaging and relevant to the post/course that you are applying for.
• Stream until Tue 1st Nov
• 20 mins
• £4.50
First Look
#### Utilising the Psychology of Memory for Revision
As well as learning subject content itself, it's so important to set aside time to consider HOW to learn most effectively.
• Stream until Tue 1st Nov
• 20 mins
• £5.50
First Look
#### Dance: A Gentle Feel Good Moment
A beginner's class into using Dance for fun and fitness.
• Stream until Tue 1st Nov
• 20 mins
• £1
First Look
#### Learning Scottish Gaelic: Colours and Clothes
Focusing on one topic is a great way to practice the new skills you are learning in a new language.
• Stream until Tue 1st Nov
• 20 mins
• £3.50
First Look
#### Learning Scottish Gaelic: Sports and Hobbies
When taking on a new language, a great way to practice what you've learned and really develop your vocabulary is by focussing in on one topic.
• Stream until Tue 1st Nov
• 20 mins
• £3.50
First Look
#### Knowing Me, Knowing You: How Do You Get Your Information?
Being able to understand yourself better brings clarity to many areas of your life and helps you make the right choices for you!
• Stream until Tue 1st Nov
• 20 mins
• £5
First Look
#### Creating a Positive Attitude to Food
In today's session we are going to be 'Creating a Positive Attitude to Food with School Packed Lunches.'
• Stream until Tue 1st Nov
• 20 mins
• £4
First Look
#### Music: Programme Music
When studying Music it is important to consider both the theory and practical elements.
• Stream until Tue 1st Nov
• 20 mins
• £3.50
First Look
#### The Truth about Bullying in Schools - My Journey
During this class we are going to speak to the inspirational Nath Fernandes who is the founder of Dynamic Play Ed.
• Stream until Tue 1st Nov
• 20 mins
• FREE
First Look
#### Knowing Me, Knowing You: Introduction
This video provides an introduction to a mini series in which Life Coach, Maddy, will help students understand themselves and the people around them.
• Stream until Tue 1st Nov
• 20 mins
• FREE
First Look
#### Learning Scottish Gaelic: Food
What better topic to practice a new language on than food.
• Stream until Tue 1st Nov
• 20 mins
• £3.50
First Look
#### Interview Skills
Interviews can be daunting for everyone and often nerves can hold you back. In this class Ms Brooks teaches you how to take as much control as possible of the interview experience. She will demonstrate how to maintain that confidence over the whole course of the interview even in the presence of challenging questions that could potentially unmask those pesky nerves.
• Stream until Tue 1st Nov
• 20 mins
• £4.50
First Look
#### A Colouring Masterclass for Beginners
This fantastic Art class is brought to you by Kelly Horton a successful artist and teacher.
• Stream until Tue 1st Nov
• 20 mins
• £4.50
First Look
#### Career Workshop with England Rugby - Part 1
In this fantastic 2 part workshop with Chris Sigsworth (Schools and Colleges Manager at England Rugby) he will reveal the breadth of many of the different career paths available at England Rugby.
• Stream until Thu 1st Dec
• 20 mins
• FREE
First Look
#### Career Workshop with England Rugby - Part 2
In Part 2 of this fantastic Career workshop with Chris Sigsworth (Schools and Colleges Manager at England Rugby) will reveal the breadth of many of the different career paths available at England Rugby. If you missed Part 1 of this fantastic class pair, you can find it here: Career Workshop with England Rugby Part 1.
• Stream until Thu 1st Dec
• 20 mins
• FREE
First Look
#### A Day in the Life of a RSPCA Inspector
There are so many different jobs within any one charity. In this class, Jo will talk through a day in the life of an RSPCA Inspector.
• Stream until Thu 1st Dec
• 20 mins
• FREE
First Look
#### Becoming a Lobbyist for the RSPCA
In this class Rachel will talk about the role of a lobbyist, especially in the charity sector, and how she arrived in the role after starting her career working for a political party. Rachel will also discuss the key skills she utilises every day in the job and that every budding lobbyist needs to have.
• Stream until Thu 1st Dec
• 20 mins
• FREE
#### Maths: Algebraic Proofs
In this session students will be introduced to the concept of algebraic proofs. They will first understand how to represent different numbers algebraically and then how to use them to prove mathematical relationships algebraically. Join Mr Salian in this class to go through both the theory and practical examples to really excel in your working of algebraic proofs.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Changing the Subject of a Formula (Basic)
In this session students will be introduced to the concept of subject of any formula. They will not only know what is it meant to rearrange and change the subject of any formula, but they will also learn different ways to change the subject of the formula.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Non-Right Angle Triangle Trigonometry
In this session students will understand how to apply trigonometric ratios for non-right-angled-triangles. They will also be introduced to the concept of the Sine Rule and Cosine Rule and will eventually know how to find the area of any triangle using trigonometric ratios.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: HCF and LCM using a Venn Diagram
Students will be able to define what is meant by factors and multiples and in turn use prime factorisation (factor trees) and Venn diagrams to find the Highest common factor and Lowest common multiple of two numbers. Students will then be able to apply this skill to worded problems. The relevance of prime factorisation to the real world is very important to people who try to make (or break) secret codes based on numbers. That is because factoring very large numbers is very hard, and can take computers a long time to do. If you want to know more, the subject is "encryption" or "cryptography".
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Unit Conversions (Metric -Area/Volume, Imperial)
In this session students will understand how to convert metric units of area and volume. Finally, they will be introduced to the concept of imperial units and how to convert from one imperial unit to another. This is a key topic to get right in Maths and so this class is useful for those wanting the opportunity to be able to excel at unit conversions.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Statistics - Mean, Median and Mode
Do you know what the is difference between Mean, Median or Mode? In this class your SkyMath tutor will consider word problems with these types of statistical averages to uncover the difference. Students will firstly learn the new theory and then have a chance to put all this new information into practice by completing some exercises.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Math: Area of a Circle and Sector
In this class students will learn about the derivation of the area of a circle and will learn how to use the formula to calculate the area of a circle and sector.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Nth term of a linear sequence
In this class students will learn how to find any term in a linear sequence, through deriving the nth term expression of the linear sequence. The 'nth' term is a formula with 'n' in it which enables students to find any term of a sequence without having to count up or down from one term to the next .Student will look at both visual and number patterns which will help cement a deeper understanding of linear sequences.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Frequency Trees
By the end of this lesson students should be able to create a frequency tree from a written description. Frequency trees are also linked with two-way tables which are also another way of sorting data into different categories. Frequency trees should not be mistaken with tree diagrams. Frequency trees will usually appear in the foundation GCSE paper or at the start of a Higher paper, therefore this is a topic that is useful to both Higher and Foundation students.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Transformations (Reflections)
In this class you are going to learn about transformations. Transformations change the size or position of shapes. A reflection is a type of transformation. It 'maps' one shape onto another. When a shape is reflected a mirror image is created. If the shape and size remain unchanged, the two images are congruent (Exactly the same). In this lesson you will learn how to draw a mirror line, as well as reflect shapes on coordinate axis. Students will also learn about diagonal reflections which take place in the lines y = x and y = - x. Finally, students will be shown how to describe a reflection that maps one shape onto another
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Sharing a Quantity in a given Ratio
Ratios are a part of everyday life, whether students are buying a pizza or calculating coordinates on a map. In this important class students will learn how to define what is meant by a ratio and split a quantity in a given ratio. As a topic ratios are found in all ability papers, therefore is an important topic to study to succeed in Maths.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Rounding Numbers to Place Value
In this session, students will be introduced to the concept of rounding numbers and how to round numbers to a certain place value. They will also learn how to round decimal numbers to certain number of decimal places
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Bearings
A bearing is an angle, measured clockwise from the north direction. Bearings are often used to describe direction (North, East, South, West), and a real-life example of this is ships using bearings to navigate around the ocean. In this lesson students will learn how to describe and draw bearings and finally use their knowledge of bearing to find missing angles.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Transformations (Translation)
Transformations change the size or position of shapes. A translation is a type of transformation. A translation moves a shape up, down or from side to side but it does not change its appearance in any other way. To translate shapes in the correct direction, students will learn about vector notations, which give information about the direction in which the vertices of a shape will need to be moved. Once familiar with vector notation, translations become a relatively straight forward transformation which will enable students sitting the foundation or higher GCSE paper to pick up ‘easy marks’.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths KS3: Geometry (Area)
In this class Daria will consider formulas of area. Students will learn formulas of area for all geometric shapes. Students will be able to check their skills on practice. Theory and typical exam exercises will be considered.
• Stream until Tue 1st Feb
• 30 mins
• £3.50
#### Maths: SURDs
In this session, students will be introduced to the concept of surds and how they can be related to fractional indices. They will also learn how to simplify surds and how to perform different operations on surds. Finally, they will also be introduced to the concept of rationalisation and learn how to rationalise denominators containing surds.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Compound Interest
In this class, students will look at how to use the formula for compound interest.
• Stream until Tue 1st Feb
• 10 mins
• £3.50
#### Maths: Circumference of a Circle and Sector
• Stream until Tue 1st Feb
• 20 mins
• £4
###### Curriculum support
In this session students will firstly be introduced to the concept of reciprocal of trigonometric ratios. They will then be introduced to the concept of unit circle and how to use that to find Sine and Cosine of angles more than 90 degrees. Finally, students will understand how to draw graphs of trigonometric ratios and how to use that to find Sine and Cosine of angles more than 90 degrees.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Inverse Proportion
In this session students will be introduced to the concept of inverse proportion. They will then understand when are two quantities in inverse proportion. Finally, they will understand how to find the missing values if two quantities are in inverse proportion and how to use this aspect to solve applications of inverse proportion in real-life. This class is perfect for those who want the opportunity to go over inverse proportion again to cement their understanding and practise some real world examples.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: It's Santa's Problem
Have you always wondered what it would be like to be Santa Claus? How does he manage to get all those parcels to all those children in just one night? In this class Daria your tutor will go through a set of word problems that need to be solved on Christmas night and how students can use them in real life examples. We will go through some basic Maths you should already know, including Velocity, Probability and some new maths problems including distance word problems and logic exercises. All old and new knowledge learnt will be put to the test using practical examples for you to follow along with. Merry Christmas!
• Stream until Tue 1st Feb
• 30 mins
• £2
#### Maths: Fractions of Amounts
Fractions are a vital lesson throughout all Maths studies and having a firm understanding of the theory and how to apply this is key in achieving a student's potential.
• Stream until Tue 1st Feb
• 20 mins
• £2
#### Maths: A Guide to Sine and Cosine
In this class Maria your tutor from SkyMath will talk about Sines and Cosines. Students will learn how to quickly remember the meaning of sine and cosine. All new information will be put into practice in a safe, helpful and friendly way. Theory and typical exam exercises will be considered. This important topic in Maths is one that many students struggle with, this is the perfect class to revise the topic.
• Stream until Tue 1st Feb
• 30 mins
• £4
#### Maths: Algebraic Fractions
Fractions is a topic many students find challenging initially. This session consolidates learning on all fraction types for students who are looking to gain more confidence in this area.
• Stream until Tue 1st Feb
• 15 mins
• £3.50
#### Maths: Area
In this session, students will be introduced to the concept of area of 2D shapes. They will also learn how to calculate the areas of some standard 2D shapes and using that to calculate the areas of compound shapes.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths Workshop: Secrets of the Circle
In this class Daria your Maths Tutor will continue to talk about the most crucial circle theorems. These theorems are vital for students who want to excel in maths and become fully fledged mathematicians. Students will put into practice new theories covered in this class.
• Stream until Tue 1st Feb
• 30 mins
• £4
#### Maths: Prime Factor Decomposition
In this session, students will be introduced to the concept of Prime Factor Decomposition and writing any number in index notation of its prime factor. They will also learn how to use prime factor decomposition of numbers to calculate H.C.F. and L.C.M. of numbers.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Finding Squares, Cubes, Square Roots, Cube Roots Manually
In this session students will learn to find squares, cubes, square roots and cube roots of numbers manually which will be helpful in the non-calculator paper of their GCSE Examinations (or equivalent).
• Stream until Tue 1st Feb
• 20 mins
• £4
###### Curriculum support
In this class students will learn how to factorise quadratic equations. They will use prior knowledge of factorising linear equations and build on this. Factorising quadratic equations will enable students to find the roots/solutions of a quadratic curve. The roots/solutions tell us the points at which the curve intersects the x-axis. This can be beneficial to businesses if they graph their price against their profits as shown in the image below. Using the curve, a business will be able to find it’s profit maximising price as well as the prices where they will make no profits.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Exterior Angles in Polygons
Building on the prior knowledge about interior angles of polygons, this lesson will show students the relationship between the interior and exterior angles of a polygon. Using visual demonstrations students will deduce that the sum of exterior angles in a polygon will add up to 360 degrees. Students can then use this knowledge to complete a wide range of problems involving interior/exterior angles in polygons.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Volume of a Cylinder and a Cone
In this class students will learn about calculating the volume of a cylinder and cone. Students will be taught how to calculate the volume of a 3D shape with a consistent cross-section. To finally be able to learn the volume of a cylinder.
• Stream until Tue 1st Feb
• 20 mins
• £5
#### Maths: Different Types of Numbers
Did you know there were different types of numbers? In this session, students will be introduced to different types of numbers and how to differentiate between each of these types of numbers. They will also be introduced to the concept of rational and irrational numbers.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Surface area of a Cylinder
At the end of this class students will be able to recognise and draw the net of a cylinder and subsequently be able to calculate the surface area of a cylinder. Students will gain a good understanding of what surface area means and be able to visualise it.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Sharing in Ratios
Understanding Ratios is key skill to develop early in Maths learning as it is used as a foundation to build future knowledge on.
• Stream until Tue 1st Feb
• 20 mins
• £2
#### Maths Workshop: Secrets of the Circle Part 2
In this class Daria your Maths Tutor will take you on a refresher workshop of the most vital Circle Theorems. This is Part 2 of our Circle Theorems workshops, although you do not need to have attended Part 1 to gain an incredible amount of helpful theory and practise. Join this session to answer a number of practical questions to solve geometric problems.
• Stream until Tue 1st Feb
• 30 mins
• £4
#### Maths: Circle Workshop Part 1
In this fantastic 2 part LIVE class your SkyMath tutor will discuss the most crucial circle theorems. These theorems are a vital topic in Maths hence why we are taking the ample time and proven revision techniques to impart with you all the wisdom there is to behold in these theorems. All new theories and formulas shown in this class will be given the chance to put the theory into practice with some exercises.
• Stream until Tue 1st Feb
• 30 mins
• £4
#### Maths: Nets of Solids
In this session, students will be introduced to the concept of nets of solids. This is an important topic in for students to cover in Geometry regarding 2D and 3D shapes and Mr Salian will work with you to be able to identify the solid shape from its net and identify the nets of different solids.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Relative Frequency
Relative frequency (experimental probability) is used when probability is being estimated using the outcomes of an experiment or trial, when theoretical probability cannot be used. For example, when using a biased dice, the probability of getting each number is no longer 1/6. To be able to assign a probability to each number, an experiment would need to be conducted. From the experimental results, the relative frequency could be calculated. In this class students will learn the difference between theoretical and experimental probability and how to calculate theoretical probability.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Converting Fractions, Decimals and Percentages
Fractions, decimals and percentages can all have the same values. Students need to know how to convert between them, in particular when a question requires students to work in all 3. Students may have a preference to write values in one singular format and therefore we will be working on being able to use all three notations to be as easy as each other. Converting between fractions, decimals and percentages is a key skill needed to perform well in examinations and vital for success particularly found in the Foundation Level paper of GCSE.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Circle Workshop Part 2
In this second part of SkyMath's Circle Workshop we are going to be focusing on solving geometric problems. We will do this by going over the theory one more time, but spending the most amount of time on practical exercises. These exercises are key in bettering your understanding of the theories involved and helping you to improve you answering skills in different variations of questions.
• Stream until Tue 1st Feb
• 30 mins
• £4
#### Spanish: I want - Quiero
Querer is a common verb to get to grips with in Spanish, we are going to spend some time this lesson learning how to conjugate this verb. Firstly we will revise vocabulary related eating at a restaurant. We will then learn how to ask for food at a restaurant. Then we are going to focus on learning the conjugation for ‘querer’ (to want) in present tense. As an extension we will learn how to say what other people want. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Classification and Components of 3D Shapes
In this session, students will be introduced to the concept of 3D shapes by using examples of everyday life. They will learn how 3D shapes are classified and what are the different components of 3D shapes. Students will also be taught about the concept of cross-section of 3-D shapes in an interesting and dynamic way!
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Solving Inequalities
In this session students will be introduced to the concept of inequalities. They will be first introduced to the different notations that are used in inequalities. They will then understand the properties of linear and quadratic inequalities. Once students have completed this class they should be well versed in the idea and properties of how to solve inequalities.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Box Plots
Box plots are a convenient way of visually displaying data. A box plot is split into 4 quartiles, and key information such as the average (median), smallest and largest values of a data set and percentiles can be spotted immediately. Additionally, Box plots can provide information about the spread of data and hence show how consistent a data set is, thus making it a useful statistical tool for comparing data sets. Box plots can be created from a list of numbers by ordering the numbers and finding the median and lower and upper quartiles.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Solving Linear Equations (High Difficulty)
This is an extension of session on solving linear equations. In this session we will look at solving difficult linear equations. Students will first understand how to solve linear equations involving unknowns on both sides. They will then know how to solve linear equations involving brackets. And finally, they will understand how to solve linear equations involving fractions.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Best Buys (Unitary Measures)
This is a fantastic class focusing on real life application of maths. In the class students will not only learn the theory behind unitary measures but also how to use this knowledge to find the best buys when shopping, e.g. in a supermarket, making sure they get the best deal available.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Percentages Workshop
In this class your SkyMath tutor is going to teach you about one of the most essential mathematical topics that you will find in daily life - Percentages! Firstly students are going to learn the theory and then most importantly in this topic there is the chance to solve plenty of different word problems surrounding percentages. Word problems can be challenging, therefore correct comprehension of these in regards to percentages is important for students to excel in this topic.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Application of Trigonometry
In this class, students will learn the basics of Trigonometry and Trigonometric ratios. We will cover the relationship between the trigonometric ratios as well as covering these ratios for some standard angles.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Interior Angles in Polygons
A polygon is a 2D shape with at least 3 sides. Polygons can be regular or irregular. If all angles and sides of a polygon are equal, the polygon is categorised as a regular polygon. Typical GCSE questions involve students being able to find missing angles in a variety of polygons. In this lesson students will learn how to find missing angles in any polygon by applying what they know about the sum of angles in a triangle and a quadrilateral.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Percentage Increase and Decrease Masterclass
This class will focus on how to find the simplest ways of the percentage difference for word problems. We will go through some tips to work out percentage difference and then as usual we will go through some exercises to test the theory.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Transformation of Shapes Workshop Part 2
This class follows on from our previous Transformation Workshop, this time we will consider Rotation and Enlargement of different shapes. Students will learn how to do these operations effectively and in a timely manner so as not to waste precious time in examinations. All new theory learnt will be put to the test through practise questions perfect as a revision tool to allow you to work at your utmost best capability.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Vectors
In this class, host Daria will walk students through the topic of 'Vectors', starting with the basic aspect of vectors, then moving onto how to add and subtract vectors, how to find the vector's length, before putting all of the new information into practice!
• Stream until Tue 1st Feb
• 30 mins
• £4
#### Maths: 4 types of Algebraic Notation - Expression, Equation, Formula, Identity
In this session students will be introduced to the concepts of Expressions, Equations, Formulae and Identities. We are going to spend this lesson to understand the difference between each of them by using different examples.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Completing the 'Square'
The elementary method of solving quadratic equations is through factorisation. However, some quadratic equations cannot be solved using this method. This lesson introduces students to a more advanced method of solving quadratic equations, through ‘completing the square’. Completing the square is a grade 7-9 skill, therefore this lesson will be of great use for students aiming to achieve marks on the latter parts of the GCSE paper.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Elimination Method - Solving Systems of Linear Equations
This class focuses on teaching students a very useful elimination method when trying to solve linear equations.
• Stream until Tue 1st Feb
• 30 mins
• £4
#### Maths: Terminating and Recurring Decimals
In this session, students will understand the circumstances under which you get terminating decimals and recurring decimals. Also, they will learn to convert terminating decimals and recurring decimals into fractions. Decimals is an important topic to understand in Maths before students are able to move onto more challenging areas, therefore it is essential to fully comprehend the theory and practices of decimals early on.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Constructing a Formula
In this session students will learn how to construct formulae from worded statements. They will also understand how to construct formulae involving one or more variables. Finally, they will also learn about constructing formulae which involve a fixed and a variable component.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Solving Inequalities using Graphs
In today's session students will be introduced to the concept of graphically. They will first be introduced to different graphical notations and what each on them means. They will then understand how to use these notations to solve linear and quadratic inequalities graphically. Finally, they will also understand how to solve system of inequalities graphically.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Changing the Subject of a Formula (Advanced)
In this session students will be exposed to difficult formulae/equations including the ones with fractions, roots and the ones having the potential subject on both sides of the equation. Students will understand how to solve questions on changing the subject of such difficult formulae. This class will give students the tools to perform this calculation for the most difficult of questions that come up in GCSE.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Functions
In this session students will be introduced to the concept of functions. They will learn what functions are and also learn how to find the value of any function for different inputs. Finally we are going to discover how to find inverse of any function.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Direct Proportion
In this session students will be introduced to the concept of proportion. They will then understand when are two quantities in direct proportion. Finally, they will understand how to find the missing values if two quantities are in direct proportion and how to use this aspect to solve applications of direct proportion in real-life. Students will gain the practical knowledge of Proportions after this class which is needed to answer any questions that come up to excel in the topic.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Currency Conversions
With the impact of globalisation, businesses and countries are more intertwined than ever before. British companies may use Chinese suppliers for their raw materials and Chinese manufactures may buy oil in dollars to run their factories. With the growth of trade and multiple currencies being exchanged between nations, calculating the exchange rate of currencies is a must in the modern global economy. Furthermore, with an increase in travel, individuals will also need to have a good understanding of exchange rates when going on holiday or purchasing foreign goods to get value for their money. In this lesson, you will learn how to make currency conversions which will be both beneficial in the real world and for those taking GCSE examinations.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Nth term of Quadratic Sequences
In this session students will be introduced to the concept of Quadratic Sequences. They will also understand what is the nth term of a Quadratic sequence and how to derive a rule for the nth term of a Quadratic sequence.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Application of Pythagoras' Theorem
In this class students will firstly learn how to use Pythagoras’ Theorem to calculate unknown figures in right-angled triangles. They will then go on to learn how to apply this knowledge to real life situations, for example, calculating distances to travel.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Unit Conversions (Linear)
In this session students will be introduced to the concept of metric units of measurement. They will then be explained how to convert from one metric unit to another. Finally, the students will be told how to be careful while doing unit conversions on calculator and how to use calculator display to get the exact value.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Algebraic Fractions (Higher Level)
Algebraic fractions are a grade 7 and above topic which require students to simplify expressions that involve fractions. This topic appears regularly in the GCSE Higher Maths Exams and needs to be mastered for students seeking to go on to study A-levels. Simplifying algebraic fractions works in the same way as simplifying normal fractions. A common factor must be found and divided throughout.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Distance, Speed and Time Workshop
In this class your SkyMath tutor will consider the most popular word problems about distance, speed and time. Students will learn the simplest way of solving such word problems. Once we have studied the theory you will get a chance to put your knowledge to practise and work through some practical exercises.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: The Difference of Two Squares
In mathematics, the difference of two squares is a squared number subtracted from another squared number. Applying this to quadratic expressions, students will be able to factorise more difficult quadratics in the form of (a2 - b2 ). This is very useful topic as it is often included in other areas such as simplifying algebraic expressions and rationalising surds. Students aiming for a grade 6 and above will very much benefit from this topic.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Iterative Methods
In this session students will be introduced to the concept of iteration using real life examples. Students will also know under which situations will students be able to apply iterative methods and understand how to find solutions to higher order equations using iterative methods and methods of trial and improvement.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### Maths: Perimeter of 2D Shapes
In this session, students will be introduced to the concept of perimeter of 2D shapes. They will also learn how to calculate perimeter of standard 2D shapes and how to calculate perimeter of compound shapes in different ways. Join your tutor Mr Salian to practise finding the perimeter of 2D shapes leaving you in good stead to move onto more complex challenges that students will face in further topics.
• Stream until Tue 1st Feb
• 20 mins
• £4
#### English: Exploring Supernatural Themes in Shakespeare's Macbeth
In this class we will explore some of the supernatural themes and their importance in Shakespeare's Macbeth. Students will focus on Macbeth throughout this class but will learn skills which can be applied to a range of Shakespeare's work - perfect for students studying another play. This will help students to understand the language used by Shakespeare in both his sonnets and plays.
• Stream until Tue 1st Mar
• 20 mins
• £3
#### English: Identify and Analyse word choices in any text
Analysis other people's writing is a fantastic way to improve your own writing.
• Stream until Tue 1st Mar
• 20 mins
• £3
#### English: Characterisation in Beowulf
How do writers allow readers to fully understand the characters they are creating through their writing?
• Stream until Tue 1st Mar
• 20 mins
• £4
#### English Descriptive Writing Higher (Funfair)
In this class, students will be given an overview of how to write descriptively for a specific descriptive writing assignment: ‘Describe a funfair.’ You will be shown how to use a range of descriptive techniques as well as how to adopt an effective structure to engage and entertain readers. This session will enable you to secure a GCSE grade 5 in English Language for your centre-assessed grades (CAG) focusing on an important skill in an area that your teacher will be looking at when making a final assessment.
• Stream until Tue 1st Mar
• 20 mins
• FREE
#### English: Spelling Rules for Suffixes
In today's Spelling class your tutor is going to go through some typical and more challenging plural and suffixes in the English language. Spelling is a key competency needed to achieve success not only in English but in all subjects. Kitan will give you an overview of how to spell common plurals and words with the 'shun' sounds, as well as going over some spelling rules.
• Stream until Tue 1st Mar
• 20 mins
• £4
#### English: Opinion Writing (Are Zoos Cruel?)
In this class, students will get an overview of how to write multi-clause sentences in opinion writing. The topic in this class is 'Zoos are cruel.' This is a class to really bring your Opinion writing up to par using conditional, relative and participle clauses.
• Stream until Tue 1st Mar
• 20 mins
• £4
#### English: Gothic Imagery in Great Expectations
In this class we will be looking at one of Charles Dickens’ best known works 'Great Expectations.' We will explore some of the techniques that the Dickens uses to present a sense of place, time and mood in an extract in the novel. We will then spend some time to exploring how the author uses the Gothic to draw his character.
• Stream until Tue 1st Mar
• 20 mins
• £3
#### English: Breaking down Shakespeare Monologues
The works of Shakespeare are assessed across all English curriculum exams and will be valuable in any teacher assessment. But it can often be difficult to get a clear understanding of some of Shakespeare's writing.
• Stream until Tue 1st Mar
• 25 mins
• £3
#### English: Descriptive Writing (Escaping Medusa's Lair!)
In this class, students will get an overview of how to structure a piece descriptive writing. Students will imagine that they are a hero who is escaping Medusa’s lair. Practising descriptive writing is fantastic way to add depth to your writing. Students will add learn how to add various adjectives and ambitious vocabulary to make their writing stand out.
• Stream until Tue 1st Mar
• 20 mins
• £4
#### English: Opinion Writing an Introduction
This class is perfect for students who want to improve their opinion writing skills. Your tutor Kitan will help you to structure your opinion writing and work on the specific tools needed to make your writing stand out from the crowd. Even if you have worked on Opinion Writing before this class is a fantastic refresher as well as a great introduction into Opinion Writing.
• Stream until Tue 1st Mar
• 20 mins
• £4
#### English: Holiday Postcards (EAL Focus)
A great way to improve understanding of basic English skills is by using a practical example.
• Stream until Tue 1st Mar
• 20 mins
• £3
#### English Language: Non-Fiction Writing (Topic: Homework)
Given that this task represents 25% of the final mark in the English Language GCSE, this is without doubt the easiest way to improve your result! All GCSE Language boards require students to complete a piece of non-fiction on a given topic in one of the papers.
• Stream until Tue 1st Mar
• 20 mins
• £4
#### English: Grammar for Complex Sentences
In this class, students will be given an overview of how to use a range of verb forms and types in order to express themselves with clarity and maturity in both fiction and non-fiction writing. Join your English tutor to go through auxiliary verbs and modal auxiliaries to excel and become a 'Giant in Grammar.'
• Stream until Tue 1st Mar
• 20 mins
• £4
#### English: Understanding Themes in Shakespeare's A Midsummer Night's Dream
In this class students will explore the play of A Midsummer Night’s Dream by the Bard, William Shakespeare. We will delve into the depths of its most memorable comic characters, to understand why writers use comedy and to what effect. We will go on a journey to understand Shakespeare’s language, and why and how he uses such full descriptive and sometimes unusual vocabulary, as well as looking at some of the major and minor themes throughout the play.
• Stream until Tue 1st Mar
• 20 mins
• £3
#### English Language: Non-Fiction Writing (Topic: Global Tourism)
Given that this task represents 25% of the final mark in the English Language GCSE, this is without doubt the easiest way to improve your result! All GCSE Language boards require students to complete a piece of non-fiction on a given topic in one of the papers.
• Stream until Tue 1st Mar
• 20 mins
• £3
#### English: Creating Atmosphere through Prose (Great Expectations)
In this class we are going to explore some of the techniques that the author uses to present a sense of place, time and mood from an extract in Charles Dickens’ gothic and glorious novel, ‘Great Expectations.' We will then spend some time analysing the techniques the author uses to create a sense of atmosphere.
• Stream until Tue 1st Mar
• 20 mins
• £3
#### SEN/EAL: Learning How to Write Instructions
This class follows the belief that it is never too early or too late to implement a personal learning plan to make changes to learning habits.
• Stream until Tue 1st Mar
• 20 mins
• £2
#### English Language: Non Fiction Writing (Topic: Veganism)
Given that this task represents 25% of the final mark in the English Language GCSE, this is without doubt the easiest way to improve your result! All GCSE Language boards require students to complete a piece of non-fiction on a given topic in one of the papers.
• Stream until Tue 1st Mar
• 20 mins
• £3
#### English: Punctuation in Complex Sentences
In this class, students will be given an overview of how to use colons, semi-colons, speech marks and dashes in order to add interest to fiction and non-fiction writing. Join your English tutor in this class to become punctuation perfect!
• Stream until Tue 1st Mar
• 20 mins
• £4
#### English: How to use Figurative Language
Understanding what makes a good piece of creative writing helps guide you in creating better work yourself.
• Stream until Tue 1st Mar
• 20 mins
• £3
#### English: 'Romeo and Juliet' - Language Analysis in Shakespeare
In this class students will get an overview of how to analyse a short extract from Act 2 scene 2 of William Shakespeare's ‘Romeo and Juliet’. You will be shown how to consider genre, purpose and audience. We will go through the best methods of how to comment on the effects of language features and how to embed and explain quotations.
• Stream until Tue 1st Mar
• 20 mins
• £4
#### English Descriptive Writing (Playground Scene)
In this class, students will get an overview of how to structure descriptive writing, imagining that they are in a playground. You will be shown how to use a range of descriptive techniques as well as how to implement an effective structure to interest readers. Join your English tutor to brush up on your skills and develop your descriptive writing to be as creative and comprehensive as possible.
• Stream until Tue 1st Mar
• 20 mins
• FREE
#### English: Opinion Writing (Prison Topic)
In this class, students will get an overview of how to write a speech for a specific opinion writing assignment: ‘Prisons do more harm than good.’ They will be shown how to use a range of persuasive techniques as well as how to structure a convincing and engaging argument.
• Stream until Tue 1st Mar
• 20 mins
• £4
#### English: Imagery and Language in A Christmas Carol by Charles Dickens
Authors use many different techniques to create imagery in their writing.
• Stream until Tue 1st Mar
• 25 mins
• £4
#### English: Deconstruct and Analyse any Poem
In this class Rosie will talk you through how to examine how a poem’s form, structure, language and context can shape its meaning. This framework for analysis can help you approach any unseen poem and examine it through a more critical lens which will give students the tools and confidence to tackle poetry in lessons and exams.
• Stream until Tue 1st Mar
• 20 mins
• £3
#### English: 'Macbeth' - Language Analysis of Tragic Hero & Kingship in Shakespeare
In this class students will get an overview of how to analyse a short extract from Act 4 scene 3 of William Shakespeare's ‘Macbeth.' You will be shown how to consider genre, audience and purpose, how to comment on the effects of language features and how to embed and explain quotations. This session will help you to secure a GCSE grade 5 in English Language for your centre-assessed grades (CAG), focusing on skills that your teacher will be looking at when making a final assessment.
• Stream until Tue 1st Mar
• 20 mins
• £4
#### Adobe's Non-Fiction Writing Masterclass (Topic: Secret Smoking)
Given that this task represents 25% of the final mark in the English Language GCSE, this is without doubt the easiest way to improve your result! All GCSE Language boards require students to complete a piece of non-fiction on a given topic in one of the papers.
• Stream until Tue 1st Mar
• 20 mins
• £1.50
#### Introduction to Poetry
A firm understanding of Poetry will help all students no matter what stage they are in their English studies.
• Stream until Tue 1st Mar
• 30 mins
• £3
#### English: Descriptive Writing (Crime Scene)
In this class, students will get an overview of how to write descriptively for a specific descriptive writing assignment: ‘Describe a crime scene.’
• Stream until Tue 1st Mar
• 20 mins
• £4
#### English Descriptive Writing (Victorian London)
In this class, students will be given an overview of how to write descriptively for a specific descriptive writing assignment: ‘Describe a street in Victorian London.’ They will be shown how to use a range of descriptive techniques as well as how to adopt an effective structure to engage and entertain readers.
• Stream until Tue 1st Mar
• 20 mins
• £4
#### English Narrative Writing: Mastering Grade 4 - Class 1
In this series of 4 Narrative Writing classes your tutor Kitan Cox will start from a notional Grade 4 level and work all the way up to a notional Grade 9 level. You can pick and choose the Grade classes to best suit your ability, although we recommend starting from Class 1 (Grade 4 Level) to build a strong foundation and work your way up to whichever class best pushes your story writing capability and further develops your ability to reach those all important grades.
• Stream until Tue 8th Mar
• 20 mins
• £4
#### English Narrative Writing: Mastering Grade 5 - Class 2
In this series of 4 Narrative Writing classes your tutor Kitan Cox will start from a notional Grade 4 level and work all the way up to a notional Grade 9 level. You can pick and choose the Grade classes to best suit your ability, although we recommend starting from Class 1 (Grade 4 Level) to build a strong foundation and work your way up to whichever class best pushes your story writing capability and further develops your ability to reach those all important grades.
• Stream until Wed 9th Mar
• 20 mins
• £4
#### English Narrative Writing: Mastering Grade 7 - Class 3
In this series of 4 Narrative Writing classes your tutor Kitan Cox will start from a notional Grade 4 level and work all the way up to a notional Grade 9 level. You can pick and choose the Grade classes to best suit your ability, although we recommend starting from Class 1 (Grade 4 Level) to build a strong foundation and work your way up to whichever class best pushes your story writing capability and further develops your ability to reach those all important grades.
• Stream until Thu 10th Mar
• 20 mins
• £4
#### English Narrative Writing: Mastering Grade 9 - Class 4
In this series of 4 Narrative Writing classes your tutor Kitan Cox will start from a notional Grade 4 level and work all the way up to a notional Grade 9 level. You can pick and choose the Grade classes to best suit your ability, although we recommend starting from Class 1 (Grade 4 Level) to build a strong foundation and work your way up to whichever class best pushes your story writing capability and further develops your ability to reach those all important grades.
• Stream until Fri 11th Mar
• 20 mins
• £4
#### Science: Earth and Space
When we look up at night, we see points of light in the night sky. We know there are planets, moons and stars in space, but we also know lots about their arrangement and the structure of planetary systems like our solar system.
• Stream until Fri 1st Apr
• 20 mins
• £2.50
#### Science: Density (Physics)
In this class we will look at what density is and how it is calculated using a simple formula. We will look at the application of density to situations in the real world, such as ships and aeroplanes. We will finally go through calculations of density problems.
• Stream until Fri 1st Apr
• 20 mins
• £4
#### Science: The Digestive System
In this class we will look at the digestive system and how food is passed along the digestive system, where it is digested mechanically and chemically. We will also investigate the enzymes that break down different substances, which are then absorbed into the body.
• Stream until Fri 1st Apr
• 20 mins
• £4
#### Science: Into the Solar System
In this class we will be talking about what objects make up our Solar System and the main components, such as the Sun, planets, moon and other celestial bodies.
• Stream until Fri 1st Apr
• 20 mins
• £2.50
#### Science: Cells to Systems
How do we decide whether an object is a living organism or not? What are the criteria?
• Stream until Fri 1st Apr
• 20 mins
• £1.50
#### Science: Energy Stores and Transfers
In this class, students will look at different types of energy stores and how one type of energy is transferred into another. We will discuss how the energy transfer diagrams can be used to represent the transfers.
• Stream until Fri 1st Apr
• 20 mins
• £3
#### Science: Sound
What is sound? How can sound travel, and what does it need to do that? Why do animals use sound, and what for? What other uses for sound are there, other than hearing your friends speak or listening to music?
• Stream until Fri 1st Apr
• 20 mins
• £2.50
#### Science: Acids and alkalis
How do we decide whether a substance is an acid, alkali or natural? What are the properties of theses substances, and how are they used?
• Stream until Fri 1st Apr
• 20 mins
• £3
#### Science: Chemical Equations
In this class we are going to look at how to correctly write chemical equations for commonly seen phenomena such as combustion, corrosion and neutralisation. We are going to learn how to write the state symbols and introduce the concept of balancing equations. This is also excellent preparation for GCSE, where the comprehension of balancing equations is vital.
• Stream until Fri 1st Apr
• 20 mins
• £4
#### Science: Pressure (Physics)
In this class we will look at Pressure - what it is and how it works in terms of particles. We will discuss how pressure affects our everyday lives, using a wide range of examples from the weather to why camels have big feet! We will then solve problems involving pressure calculations.
• Stream until Fri 1st Apr
• 20 mins
• £4
#### Science: The Respiratory System
In this class we will look at the respiratory system and how oxygen is taken in and used by our body. We will look at the structure of the lungs and how the blood carries the oxygen around the body.
• Stream until Fri 1st Apr
• 20 mins
• £4
#### Science: Mixtures & Separation Techniques
Why is it that there is not enough drinking water in the world, when there are so many oceans?
• Stream until Fri 1st Apr
• 20 mins
• £2.50
#### Science:The Circulatory System
In this class we will look at the circulatory system and the organs and tissues associated with it. We will discuss the passage of blood through the heart and how each blood vessel is adapted to its function.
• Stream until Fri 1st Apr
• 20 mins
• £4
#### Science: Human Body Systems
In this class we will look at the systems that are present in the human body, such as the nervous system, the digestive system and the circulatory system. We will then discuss how these systems interact with one another to allow the body to function. This class can act as Biology revision if the subject has already been studied in school or can be used to replace any missed classes.
• Stream until Fri 1st Apr
• 20 mins
• £4
#### Science: Atoms, Elements and the Periodic Table
In this class we will look at the structure of an atom and the meaning of “element”, “atomic mass” and “atomic number”. We will then discuss how the periodic table is structured and the uses of some of the elements. This is a useful class for students to revise the topic but also in place of any missed lessons on the subject.
• Stream until Fri 1st Apr
• 20 mins
• £3
#### Science: Force Diagrams (Physics)
In this class we will look at the different types of forces and how we can represent forces on the object with a free body diagram. We will then calculate the resultant force on the object and the acceleration as a result.
• Stream until Fri 1st Apr
• 20 mins
• £4
#### Biology: Kingdoms of Life
In this class we will be looking at how living organisms are divided into 5 kingdoms based on their characteristics.
• Stream until Fri 15th Apr
• 20 mins
• £2.50
#### Biology: Evolution
In this class we will look at how life on Earth began, from the first cells to the complex life of the modern world. We will investigate how animals change over time and the fossil evidence of such changes.
• Stream until Fri 15th Apr
• 20 mins
• £3
#### Biology: Genetics
What makes us different? Every living organism is made up of cells, from single celled bacteria to complex organisms like animal and plants.
• Stream until Fri 15th Apr
• 20 mins
• £2.50
#### Chemistry: Atomic Structure
In this class we will look at the structure of the atom and its subatomic particles.
• Stream until Sun 1st May
• 20 mins
• £3
#### Chemistry: Ionic Bonding
In this class, we will be looking at how metallic elements form ionic bonds by the transfer of electrons and how we name the new ionic compounds. We will then look at the explanation of ionic compound lattice structures.
• Stream until Sun 1st May
• 20 mins
• £3
#### Chemistry: Evolution of the Earth's Atmosphere
In this class we will discuss the formation of the Earth and it’s early atmosphere. We will then go onto describe how the composition of the atmosphere has changed under the influence of life. Join Laura for this fantastic Chemistry class to really understand our atmosphere and how plants and animals actually make a change to the Earth's atmosphere.
• Stream until Sun 1st May
• 20 mins
• £2.50
#### Chemistry: Atomic Structure of Elements
The modern periodic table is one of the most important pieces of work in chemistry, and can be found in every science classroom. It lists all the known elements, but what makes each element unique?
• Stream until Sun 1st May
• 20 mins
• £2.50
#### Chemistry : Groups in the Periodic Table
The Periodic Table is a common topic throughout Science curriculums so a solid understanding of the subject can help students in future classes.
• Stream until Sun 1st May
• 20 mins
• £2.50
#### Chemistry: States of Matter & Separation
All matter is made of particles.
• Stream until Sun 1st May
• 20 mins
• £2.50
#### Chemistry: Covalent Bonding
In this class we will look at how non-metallic elements share pairs of electrons to form covalent bonds.
• Stream until Sun 1st May
• 20 mins
• £3
#### Chemistry: The Periodic Table
The Periodic Table is a common topic throughout Science curriculums so a solid understanding of the subject is vital for students' future in Chemistry. In this class we will look at how the periodic table is arranged in terms of their atomic number and electron shell structures. We will then arrange the electrons of the first 20 elements into the correct configurations.
• Stream until Sun 1st May
• 20 mins
• £3
#### Physics: Magnets & Magnetic Fields
Magnets can exert forces on objects at a distance, due to the magnetic field that is present around them. But what affects this magnetic field? Why are some materials affected by magnets, yet some are not? How can we prove there is an invisible force field around a magnet? Why do we say Earth has a magnetic field, and where does it come from?
• Stream until Sun 15th May
• 20 mins
• £5.20
#### Physics: The Electromagnetic spectrum
The electromagnetic spectrum encompasses all wavelengths of light, in clouding visible light.
• Stream until Sun 15th May
• 20 mins
• £3
#### Physics: Non Renewable Energy Fossil Fuels
In this class we will discuss the different types of fossil fuels, including coal, oil and gas. We will also look at how each fossil fuel forms. Finally, we will address the benefits and drawbacks of fossil fuels.
• Stream until Sun 15th May
• 20 mins
• £3
#### Physics: Star Life Cycles
In this class we will describe how stars form from nebulae, their lives on the main sequence and their eventual deaths. We will see why one star will fizzle out over billions of years, whilst another will explode in a supernova, forming a black hole. This is a great class for revision on the subject but also for students who want to learn more about stars and space!
• Stream until Sun 15th May
• 20 mins
• £2.50
#### Physics: Renewable Energy
In this class, students will discuss the different types of renewable energy, including wind power, solar, hydroelectric, tidal and biomass and how they generate electricity.
• Stream until Sun 15th May
• 20 mins
• £3
#### Physics: Non Renewable Energy Nuclear Power
In this class we will look at how nuclear power can be used as an efficient energy source.
• Stream until Sun 15th May
• 20 mins
• £3
#### Physics: Forces, Energy & Work
Every time we apply a force to move an object, or we experience a force on ourselves, energy is transferred, and work is done.
• Stream until Sun 15th May
• 20 mins
• £2.50
#### Physics: Waves
Waves in physics are important as an energy transfer mechanism.
• Stream until Sun 15th May
• 20 mins
• £2.50
#### Physics: Taking a look at The Electromagnetic Spectrum
In this class we will look at non–ionising radiation. Students will look at all the parts of the electromagnetic spectrum – from gamma radiation to radio waves and everything in between! Finally, we will discuss the uses and dangers of non- ionising radiation.
• Stream until Sun 15th May
• 20 mins
• £3
###### Curriculum support
In this class students will look at the three types of ionising radiation; alpha, beta and gamma radiation.
• Stream until Sun 15th May
• 20 mins
• £3
#### French: House & Home Vocabulary Focus
In this class Ms Bennell will teach the topic of House and Home. This class can be used as revision of the topic or to cover lessons which may have been missed at school. You will begin the class with a vocabulary knowledge scale and then show a powerpoint on house and home vocabulary.
• Stream until Wed 1st Jun
• 20 mins
• £5
#### French : Verb Focus on Avoir & être plus Personal Details Vocabulary
Today's class will start with going through a set of personal detail questions in French such as name, age etc, to include opinions and 3 time frames.You will then revise the ever important verbs avoir and être, these are 2 of the main and most important verbs in the French language to get right hence revision of these is always a key part of learning French.
• Stream until Wed 1st Jun
• 20 mins
• £2
#### French: Vocabulary Focus - Personal Details
In today's class students will cover a lot of vocabulary, mainly on personal details and family. This class can be used as revision of the topic or to cover lessons which may have been missed at school.
• Stream until Wed 1st Jun
• 20 mins
• £3.50
#### French: Speaking Focus - Higher Education Topic
Today's class focuses on your pronunciation and speaking of French. Firstly students will cover the main vocabulary found in the topic of Education, then you will look at some reading texts.
• Stream until Wed 1st Jun
• 20 mins
• £3
#### French: Verb Focus - Avoir and Etre plus Vocabulary
Today we will cover avoir and être, which are two of the most important verbs in the French language! They can be used alone or as auxiliary verbs. You will learn more vocabulary and we will begin to use connectives in your work. At the end of the class you will be working towards you being able to write your own full sentences.
• Stream until Wed 1st Jun
• 20 mins
• £3
###### Curriculum support
In this French class you will learn many of the different adjectives you can use to describe your personality. We will work together, practising known vocabulary and using connectives to build up sentences by yourself
• Stream until Wed 1st Jun
• 20 mins
• £2.50
#### French: Perfect Tense
In today's class students will revise two very important French verbs for 'to have'. These are 'avoir' and to be 'être'. You should be familiar with these verbs already. If not Ms Bennell will go through an introduction to the verbs, followed by then showing students exactly how to apply these verbs in the perfect tense.
• Stream until Wed 1st Jun
• 20 mins
• £4
#### French: Hobbies Topic
Today's class will cover Sports and Hobbies working through the topic step by step. You will expand your vocabulary and learn how to use the correct connectives to form well rounded sentences which students should be able to do by the end of the session.
• Stream until Wed 1st Jun
• 20 mins
• £3
#### French: Grammar - The Subjunctive
In this class we are going to cover the subjunctive with set phrases - for example il faut. You will also learn how to look out for the subjunctive in both written and spoken word. This in an important topic for higher level French students who want to learn how to set the verb form or mood to discuss what they would/should/could like/not like to occur.
• Stream until Wed 1st Jun
• 20 mins
• £4
#### French: Vocabulary Workshop
In this class Ms Bennell will cover vocabulary which generally comes up in KS3 or equivalent. This will include numbers, pets, family, months, days of the week. This is a fantastic class for younger students to take to ramp up their vocabulary and practise their pronunciation with a teacher.
• Stream until Wed 1st Jun
• 20 mins
• £2.50
#### French: It's Christmas! (Basic/Intermediate)
Join the fun with this Christmas Class, Carol will cover all things Christmas with you. There will be fun activities including puzzles and a wordsearch. As well as learning a bit more about the traditions they have in France at Christmas time. Joyeux Noël!
• Stream until Wed 1st Jun
• 20 mins
• £2
#### French Exam Preparation: High Level Structures in Writing
In this class your tutor will help you to work on your French Writing skills. Miss Bennell will cover the higher level structures which are most useful in the French GCSE Writing Exam. These higher level structures do wonders for your writing and will help to push your writing up to the next level.
• Stream until Wed 1st Jun
• 20 mins
• £4
#### French: Healthy and Unhealthy Living
In this French class students will learn about healthy and unhealthy living and cover the main vocabulary within this topic. Students will practice reading texts on top of learning the key vocabulary giving them more confidence in approaching French texts and taking them one step further to a solid understanding of the language!
• Stream until Wed 1st Jun
• 20 mins
• £3
#### French: Future Tense Workshop
In this French Workshop your tutor will cover the Future tense (le futur simple). This grammar focused class focuses on le futur simple which allows the student to make projections to what he/she may want to do or be in the future.
• Stream until Wed 1st Jun
• 20 mins
• £4
#### French: Physical Appearance Workshop
This class will cover the main vocabulary needed to describe their own and others' physical appearance. Ms Bennell will work with students step by step to include other vocabulary and connectives towards students being able to write their own sentences.
• Stream until Wed 1st Jun
• 20 mins
• £3
#### French: Customs and Festivals Topic
This class will cover the main vocabulary in the topic of Customs and Festivals. You will learn the correct pronunciation and the correct ways to use construct sentences using connectives. We are then going to cover some key common vocabulary that comes up when you look at reading texts. This will all help to complete an exercise at the end of the lesson.
• Stream until Wed 1st Jun
• 20 mins
• £3
#### French: Vocabulary Focus- Studies and School
In this vocabulary based class you will cover the main vocabulary found in the topic of Studies and School. Your tutor will cover all the key vocabulary, running through pronunciation, and you will also have a chance to look at some reading texts.
• Stream until Wed 1st Jun
• 20 mins
• £2.50
#### French: Vocabulary Masterclass inc. Sports, Animals and Colours
This Masterclass will cover a wide range of French Vocabulary. This will include sports, animals, colours and more. This class can be used as revision of the topic or to cover lessons which may have been missed at school and provides practical revision of vocab to give students confidence to use the new terms across their French studies.
• Stream until Wed 1st Jun
• 20 mins
• £2.50
#### French: Jobs at Home
In this class students will get an overview of the vocabulary needed to talk about house chores. We will practise key vocabulary and teach students to use and understand a range of verbs in different tenses and different time frames. We will also include opinions and justifications to enable pupils to develop their speaking and writing skills.
• Stream until Wed 1st Jun
• 20 mins
• £3.50
#### French: Vocabulary Focus - Travel and Tourism
In this class students will learn the main vocabulary related to the topic of Travel and Tourism. Once you have brainstormed the vocabulary you will get a chance to look at some reading texts. Comprehension of reading texts are essential, especially for exams where they are commonly found.
• Stream until Wed 1st Jun
• 20 mins
• £3
#### French: Subjects, Likes and Dislikes
In this French class Ms Bennell will take teach students how to describe their school subjects, as well as expressing their likes and dislikes. There will be a strong focus on using the correct connectives which would allow students to confidently write their own sentences.
• Stream until Wed 1st Jun
• 20 mins
• £3.50
#### French: Charity & Voluntary Work Verb Focus
In this class Ms Bennell will cover the main vocabulary found in the topic of charity and voluntary work that students should be familiar with. She will cover key verbs and how to talk about voluntary work and working abroad. Students will then look at a reading text and answer some comprehension questions on the text.
• Stream until Wed 1st Jun
• 20 mins
• £3
#### French: It's Christmas! (Intermediate/Higher)
Join the fun with this Christmas Class, Ms Bennell will cover all things Christmas with you. You will go over some typical Christmas vocabulary and link it with opinions. There will also be fun activities including puzzles and a wordsearch. Joyeux Noël!
• Stream until Wed 1st Jun
• 20 mins
• £2
#### French: Vocabulary Focus - Technology
In this class Ms Bennell will cover the main vocabulary in the topic of Technology. You will then get a chance to look at some reading texts, which is an essential part of learning. Reading texts are prevalent in most exams and give students the opportunity to showcase their comprehension of French, therefore it is vital to revise working with reading texts.
• Stream until Wed 1st Jun
• 20 mins
• £3
#### French: TV and Film Topic
In this class your French tutor will go over the topic of TV and Film. You will work with your tutor step by step to include other vocabulary and use connectives with the intention at the end of the session to be able to write your own full sentences on the topic. There will be exercises for you try and push yourself to reach higher goals for yourself in French.
• Stream until Wed 1st Jun
• 20 mins
• £3.50
###### Curriculum support
In today's class we are going to find out what students can remember on the topic of House and Home. Your tutor will talk you through a writing task, making sure they understand what they need to include and how to structure it. Finally students will complete the task, strengthening and developing their French written skills.
• Stream until Wed 1st Jun
• 20 mins
• £2.50
#### French: Grammar and sentence structure (Avoir and Etre)
In this class Ms Bennell will recap the French verbs to have 'avoir' and to be 'être'. She will then cover connectives, adjectives and opinions which are needed for students to construct full sentences using these verbs. There will be an additional focus on how to improve the quality and accuracy of your writing.
• Stream until Wed 1st Jun
• 20 mins
• £2.50
###### Curriculum support
In this workshop we are going to be going through an incredible amount of adjectives. Adjectives are the perfect tool in language to really make your written and spoken work excel. Deepen your knowledge on sentence structure and where and how to add correct adjectives into sentences.
• Stream until Wed 1st Jun
• 20 mins
• £4
#### French GCSE Preparation: Honing your Writing Skills
Today's session will look at writing skills with GCSE preparation in mind. This session is useful for all topic areas in French, as grammar, tense and vocabulary are all transferable from topic to topic. Ms Bennell will include general advice on how to push yourself to the top of your ability and she will give students a comprehensive view of what your written content should entail.
• Stream until Wed 1st Jun
• 20 mins
• £3.50
#### French GCSE Preparation: Questions to Practice Your Writing Skills
In today's class we will look at improving your writing skills with GCSE preparation in mind. Ms Bennell will go through a GCSE examination question on the topic of Hobbies. This question is an overlap question which means it can be found on both both Higher and Foundation level examinations. Practising these types of questions are the perfect revision tool for students to go through by themselves and especially with a teacher. With a seasoned French tutor by your side you will be able to strengthen your answers and reach the very top of your French writing ability.
• Stream until Wed 1st Jun
• 20 mins
• £4
#### French: Covering the Perfect Tense (Session 3)
This French class is Ms Bennell's third session on the perfect tense and will look at working with you to complete more exercises and putting into practice what you already understand about the perfect tense.
• Stream until Wed 1st Jun
• 20 mins
• £4
#### French: Grammar - The Pluperfect Tense
In this class Ms Bennell will introduce and go through use of the pluperfect tense (past perfect in English). This will be a useful exercise to master now, to help display a variety of different tenses during writing exams- especially for those in the higher tier.
• Stream until Wed 1st Jun
• 20 mins
• £4
#### French: Covering the Perfect Tense (avoir and être)
In this French Language session your tutor will revise the 'perfect tense'. These are avoir and être. These are two very important verbs in French to perfect as they can be used alone as main verbs or used as auxiliary verbs to form the French compound tenses.
• Stream until Wed 1st Jun
• 20 mins
• £3.50
#### French: Family and Friends
In this French class we are going to cover the topic of Family and Friends. Your tutor will work with you step by step on a worksheet throughout the class. The class also includes other vocabulary and the use of connectives, so you will be able to work towards writing your own sentences.
• Stream until Wed 1st Jun
• 20 mins
• £2
#### French: Vocabulary and Comprehension - Global Issues
In this French class we will cover the main vocabulary in the topic of Global issues. Students will get the opportunity to look at a reading text and answer some comprehension questions. This is a typical exercise and is essential to revise and recap, as reading texts are very commonly found in examinations.
• Stream until Wed 1st Jun
• 20 mins
• £2
#### French: Hobbies!
In this class, Ms Bennell will go through as many different types of hobbies as possible so you can impress with your writing and speaking. Help your conversation skills go to the next level with all the new vocabulary covered in this class.
• Stream until Wed 1st Jun
• 20 mins
• £4
#### French: Practising the Conditional Tense
In this French Class Ms Bennell will cover the conditional tense and will give students a chance to practise using the tense in written and verbal exercises.
• Stream until Wed 1st Jun
• 20 mins
• £4
#### French: The Imperfect Tense
In this French Class your tutor Carol will cover the imperfect tense. You will be going through the topic of what you do in your free time and your social activities to practise this tense. We will revise the grammar needed to conversate in the imperfect tense and revise some important vocabulary to improve these conversations.
• Stream until Wed 1st Jun
• 20 mins
• £4
#### German: Speaking Exam Practice
Today you will go through all appropriate exam practice for students at a Key Stage 3 level or equivalent. You will revise the different styles of speaking questions that come up and work on your accent and pronunciation to excel in your German oral examinations.
• Stream until Wed 15th Jun
• 20 mins
• £3.50
#### German: House & Home Writing Task
Today we are focusing on a writing task. Firstly your tutor will explain in depth to you the task and its requirements. You will be taught what to include and the structure that your writing must take. Your tutor will support you with the particular tools that you need for your writing to develop and become the absolute best it can be.
• Stream until Wed 15th Jun
• 20 mins
• £2.50
#### German: Vocabulary Masterclass
Join this Masterclass today for all things vocabulary! Your tutor will focus on the most common vocabulary to allow students to grow their German knowledge to be used across all German lessons. This will include numbers, pets, family, months, days of the week.
• Stream until Wed 15th Jun
• 20 mins
• £2.50
#### German: Vocabulary Focus - House & Home
In this class Ms Bennell will go over all the vocabulary you will need to know in the topic of House and Home. The class will start with a vocabulary knowledge scale and students will be shown a powerpoint on house and home vocabulary.
• Stream until Wed 15th Jun
• 20 mins
• £5
#### German: Vocabulary Focus - Healthy and Unhealthy Living
In this German class Ms Bennell will look at Healthy and Unhealthy living and she will cover the main vocabulary within this topic. Students will practice reading texts on top of learning the key vocabulary giving them more confidence in approaching German texts. Finally you will be able to take the next step in learning German by creating your own sentences.
• Stream until Wed 15th Jun
• 20 mins
• £3
#### German: Verb Focus - Charity & Voluntary Work
In this class you will cover the main vocabulary around the topic of charity and voluntary work. Secondly you will cover key verbs and how to talk about voluntary work and working abroad. This knowledge will be cemented by going through some reading texts.
• Stream until Wed 15th Jun
• 20 mins
• £3
#### German: Customs and Festivals
In this class Ms Bennell will cover the main vocabulary in the topic of customs and festivals. She will cover key vocabulary and look at some reading texts.The topics covered in this class will be really useful across all future German learning and revision giving students practice and confidence in the language.
• Stream until Wed 15th Jun
• 20 mins
• £3
#### German: Vocabulary Focus - Sports, Animals and Colours
Ms Bennell will cover vocabulary which generally comes up in the topic of Animals, Colours and Sports. This is a fantastic class to join if you are looking to improve your pronunciation and conversational skills in German.
• Stream until Wed 15th Jun
• 20 mins
• £2.50
#### German: Vocabulary Focus - Personal Details
In today's class we will be going through a great deal of vocabulary. With the aim of refreshing students memory and understanding of German, your tutor will cover a lot of personal details, family vocabulary that you should already be familiar with. Finally students will be asked to brainstorm what they can remember.
• Stream until Wed 15th Jun
• 20 mins
• £3.50
#### German: Exam Practice (Foundation Writing Paper)
In this class Ms Bennell will cover exam practice. You will go through the most useful elements needed for the Foundation Tier German writing exam. Entailed in the class will be the 'must haves' that examiners are looking for and all the relevant requirements that are needed to excel in the Foundation Tier paper.
• Stream until Wed 15th Jun
• 20 mins
• £4
#### German: Grammar and Sentence Structure (haben and sein)
In this class Miss Bennell will recap haben and sein and then cover connectives, adjectives and opinions. Students will learn how to form sentences and structures, focusing on making your writing high quality and accurate.
• Stream until Wed 15th Jun
• 20 mins
• £2.50
#### German: Verb Focus - Haben & Sein
In this class we will firstly go over common personal detail questions in German such as name and age. Students will be taught how to give their answers with opinions in 3 time frames. You will then go over 'haben' and 'sein' as revision as these are extremely important verbs in German. The topics covered in this class are essential across all future German learning and revision.
• Stream until Wed 15th Jun
• 20 mins
• £2
#### Spanish: Nationalities - Soy inglés
Do you know how to say what Nationality you are? In this lesson students will learn how to speak about different nationalities in singular and plural. They will also learn irregular nationalities (e.g. Grecia – griego). Grammar point: Use of ‘era’ when talking about nationalities and extending to short biographies of famous people. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £3
#### Spanish: Seasonal Vocabulary - All things Autumn
In this class of our Spanish Seasons series we are focusing on Autumn. First of all we are going to go through all the vocabulary you may need to discuss all things autumn. We are going to learn set phrases related to activities we do during cold days such as ‘go for a walk’ or ‘wear raincoats’. We will then use the present tense to describe what we are going to do next summer adding frequency adverbs to our sentences. Finally, we will read a paragraph in Spanish and begin a GCSE challenge! Even though you may think GCSE exams are a long way away right now, this is a fantastic opportunity to develop and hone some of the complex skills that are vital for future examination preparation.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Life at School
Can you describe a typical day at school? In this lesson we are going to start by revising all the vocabulary we know surrounding school and school subjects. Then we will focus on learning the vocabulary for the activities we do at school such as ‘to study’, ‘to revise’, ‘to do sport’ or ‘to go on a break’. We will combine these with places at school and we will translate sentences into Spanish using what we have learned. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Vocabulary Focus - Describing People's Routines
In this lesson, we will revise vocabulary regarding daily routine. We will first quickly revise how we describe our daily routine in the present tense. We will then look at the grammar needed to describe someone else’s routine in present. After translating a list of sentences into Spanish and doing a reading exercise, we are going to complete a GCSE challenge to get students used to the style of questions that appear in examinations. Although the examinations may be a way off for many students, revising past questions is essential to excelling in Spanish!
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Simple Past with -ir verbs
We are going to spend some time today conjugating -ir verbs. In this lesson, we will revise the vocabulary for daily routine and practise the pronunciation. We will then learn how to conjugate regular verbs in simple past. We will translate five sentences together into English and we will use verbs to complete a text in Juanito’s diary. You will end the lesson learning how to translate these sentences and create your own ones by yourself! This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £3.50
#### Spanish: Conjugating 'necesitar' - I need
In this lesson, we will revise vocabulary related to the supermarket. We will then learn how to ask someone when we need something in Spanish. We will learn the conjugation for ‘necesitar’ (to need) in present tense. As an extension we will listen to a native speaker asking for things in a supermarket.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Simple Past with -er verbs
We are going to spend some time today conjugating -er verbs. In this lesson we will start off by revising the vocabulary for your daily routine. We will then learn how to conjugate regular verbs in simple past. We will translate five sentences together into English and we will use verbs to complete a text in Juanito’s diary. You will end the lesson learning how to translate these sentences and create your own ones by yourself! This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £3.50
#### Spanish: Transportation Topic- Viajo en tren
We are going to spend some time today focusing on adjectives to advance your Spanish written and speaking knowledge. First of all we will learn different types of transportation in Spanish. We will compare them by saying which ones are faster or slower, bigger or smaller. We will also learn to say the fastest/slowest. The skills students learn in this class can be applied to a wide range of topics across their Spanish learning.
• Stream until Fri 1st Jul
• 20 mins
• £3.50
#### Spanish: Places Topic- Mi pueblo
We are going to spend some time today going through 'mi pueblo' talking about your own town or village. We begin the lesson by learning the different places we find in our cities or small towns, such as your local school, park, hospital, etc. We will learn preposition of place and will use them to describe where places are located in reference to other places. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £3.50
#### Spanish: Adjectives Focus - Mi instituto
We are going to spend some time today focusing on adjectives to describe your very own school In this class we will learn the words given to the different rooms in a school. We will learn how to say ‘there is’/’there are’ and we will use adjectives to describe our school (‘new’, ‘old’, ‘big’, ‘small’). This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £3.50
#### Spanish: Vocab and Past Tense Focus - Holidays!
Have you always wanted to holiday in Spain? How about talking about your holidays in Spanish. In this lesson we will revise and learn some new vocabulary about holidays. We will use this vocabulary to describe how we travelled, where we stayed, what we ate and how our whole holiday was. We will listen to a native speaker describing their holidays and we will answer questions about it. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Comprehension Focus - Me gustaria (I would like to be...)
We are going to spend some time today advancing your comprehension of Spanish. In this lesson, we will revise vocabulary on occupations and learn how to say what you would like to be in the future. You will build sentences in Spanish using the structure ‘me gustaria’. After that you are going to justify your sentences by using ‘porque’ and we will use adjectives such as ‘interesante’, ‘divertido’ or ‘gratificante’. This will give you a full rounded answer for when you are asked the infamous question; What do you want to be? This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Indefinite Article Focus - Llevo puesto
We are going to spend some time today studying the indefinite article in relation to clothing. The session begins by recapping the vocabulary of various items of clothing. Students will learn which are masculine and which are feminine as well as the correct indefinite article (a/an) to be used. We will also learn time frequency words and we will use them to say how often we were different types of clothing.
• Stream until Fri 1st Jul
• 20 mins
• £3.50
#### Spanish: Food Topic -En El Mercado
Today's class will focus on the different types of food we can find in a market. We will test ourselves on which ones are masculine and which ones are feminine. Lastly we are going to discuss how to describe your favourite and least favourite foods. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £3.50
#### Spanish: Adjective Focus -Soy alto
We are going to take a deep dive into all the ways we can describe our appearance. Firstly students will recap and learn more ways describe themselves in Spanish. We are going to focus on a variety of adjectives such as ‘tall’, ‘short’, ‘slim’ to describe our physical appearance. Students will also use a variety of personality adjectives to describe themselves and we will include an advanced grammatical structure to describe what we would like to be. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £3.50
#### Spanish: Animals and Pets- Tengo una mascota
We are going to focus on all the ways to describe our pets! In this lesson we will learn different animals in Spanish, we will learn how to describe them by using adjectives such as colours and adjectives for size. We will also learn how to say what the name of our pet is in Spanish. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £3
#### Spanish: Vocabulary Focus - Mi familia
This class focuses on all the vocabulary that students need to know on the topic of family. The lesson begins by learning important vocabulary about the family and by the end of the lesson they will be able to give information about their relatives by using the phrases ‘se llama’, ‘se llaman’, ‘tiene’ (used for age) and a selection of adjectives. As an extension, pupils will hear the tutor reading short paragraphs in Spanish whilst they’re filling in the gaps. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £3
#### Spanish: Tenses Focus - Reciclemos (Environment Topic)
We are going to spend some time today advancing your comprehension of the tenses. In this lesson you are going to hear and learn vocabulary on the environment, you will work on pronunciation and comprehension. Your tutor Felix will give you an exercise to do together where you build sentences in Spanish relating to the Environment. The next challenge will be to use them to conversate in the past tense and future tense. You will ramp up your conversational skills by adding phrases such as ‘last year’, ‘yesterday’, ‘next year.' This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Present Tense 'It is' and 'It is found' - Mi habitacion
In this lesson we will learn the vocabulary of objects we can find in our room, i.e. bed, table, posters. We will then use prepositions of place to say where things are in reference to other things. We will use 'esta' and 'se encuentra' and will see what makes them different. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Adjectives using the suffix -isimo
We are going to cover all you need to know when you need to emphasize an idea in Spanish using -isimo. First of all we will revise a list of adjectives ending in -ado. We will then complete an exercise where students will translate a list of sentences that include these adjectives. Our grammar section of today's lesson will explain how Spanish speakers use the suffix ‘-ísimo’ when emphasizing the idea carried by an adjective. We will hear native speakers using this suffix and we will answer questions regarding the recordings. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Gender in Subjects - Las asignaturas
For a deep dive session in grammatical gender this is the session for you. In this lesson students will learn vocabulary about the different subjects learned at school. They will be introduced to the concept of masculine and feminine subjects as well as singular and plural. Pupils will learn how to express their opinions on different subjects and as an extension they will provide a reason for their opinions. Learning the grammatical gender in Spanish is key to excel in the language, hence by the end of this class students should well versed in the masculine/feminine in school subjects.
• Stream until Fri 1st Jul
• 20 mins
• £3
#### Spanish: To Be - Verb Focus
Some verbs are essential for success in Spanish, 'to be' is one of them. In this class students will explore the verb ‘to be’ and the two meanings it has in Spanish. We will use the acronym DOCTOR PLACE to identify the differences and we will translate different sentences into Spanish by using this useful tool. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £2.50
#### Spanish: Art Topic- Describe tu pintura
For a comprehension task as well as learning various art terms in Spanish look no further. In this class, we will learn the names of different shapes and colours in Spanish. We will use these to describe abstract paintings, some of them by famous Spanish artist Joan Miró. We will read a short biography and will answer questions about his life. This class will allow students to expand their vocabulary as well as important practice listening and reading; building knowledge and confidence in the language.
• Stream until Fri 1st Jul
• 20 mins
• £2.50
#### Spanish: Conjugating -ar verbs (Compre Ropa)
For everything you need to know when conjugating -ar verbs look no further. In this lesson we will learn the rules to conjugate, in past tense, regular verbs ending in -ar. We will describe what we did when we went to the shops and we will translate sentences. The challenge for the lesson is to include the past tense of the first person singular – we. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £2.50
#### Spanish: El Internet
Technology moves at an incredible pace, students need to keep up with the advances in language surrounding the internet and more. In this class, we will explore vocabulary and phrases around using the internet. Students will complete some challenging questions cementing their understanding of all the theory learnt in this lesson. This class covers key vocabulary which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £3
#### Spanish: What time is it? - Que hora es?
This session is focused on telling the time. First of all students will revise the numbers needed to tell the time in Spanish. Then they are going to learn how to say ‘half past’, ‘quarter past’ and ‘quarter to’. Students will also learn an interesting difference in the way how people in South America give the time as opposed to people in Spain. Finally students will complete a GCSE challenge to get them used to the style of questions that appear in examinations. Although the examinations may be a way off for many students, revising past questions is essential to excelling in Spanish!
• Stream until Fri 1st Jul
• 20 mins
• £2.50
#### Spanish: Where I live - Cómo es tu casa
A typical topic in Spanish at all levels is being able to discuss where you live. In this lesson we will revise the vocabulary related to where we live, in a house or a flat, whether it is in the countryside or in the city, and the size of your dwelling. Students will be introduced to descriptive adjectives such as big, small, old and new. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £3
#### Spanish: Playing Sports -Juego al fútbol
For a verb focused Spanish class look no further. In this class students will first revise common sports and popular hobbies. Through these, students will be introduced to the verbs ‘jugar’ and ‘hacer’ and will practice ‘I play’, ‘I do’, ‘he/she plays’ ‘he/she does’ and ‘we play’, ‘we do’. At the end of the session there will be a GCSE challenge to get students used to the style of questions that appear in examinations. Although the examinations may be a way off for many students, revising past questions is essential to excelling in Spanish!
• Stream until Fri 1st Jul
• 20 mins
• £2.50
#### Spanish: Numbers and Ages - Mi edad
This session gives students the opportunity to focus on numbers and the different scenarios in which we use numbers. In this lesson students will learn how to say how old they are, when their birthday is and ask other people about their birthdate. As extension for the lesson, pupils will be able to say how old other member of their family are including revision of numbers from eleven to one-hundred. By the end of this session students will be well versed in Spanish numbers and their pronunciation.
• Stream until Fri 1st Jul
• 20 mins
• £3
#### Spanish: Conjugating -ar verbs in the simple past tense
Conjugating verbs is a key factor in learning a language. In this lesson we will revise the vocabulary that you need to describe your daily routine. We will then learn how to conjugate regular verbs in the simple past tense. We will go through an exercise together to translate five sentences into English and we will use verbs to complete a text in Juanito’s diary. This is a great class to put together some of the topics you have been working on to be able to draw from those classes and complete a full exercise.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Vocabulary 'going to'
In this vocabulary based session, Mr Castrillon your tutor will go through the vocabulary for places that you may visit. You are going to learn how to use ‘going to’ in Spanish when making plans. Once you have gone through the vocabulary section of the class you will begin to include connectives such as ‘then’, and ‘afterwards' so you can say full, descriptive sentences. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Holidays - Dónde pasas tus vacaciones?
Typical questions that come up in all levels relate to where you have been on holiday, or where you would love to go one day in the future. In this lesson we will revise places to go on holidays and we will give our opinions about different places. We will add reasons to those opinions and will answer questions regarding a short reading about holidays last year. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £2.50
#### Spanish: Present Perfect - We have
This class focuses on all students need to know about the verb 'haber' - we have. In this lesson we will revise a list of verbs ending in -ar. We will then learned the past participle of each one of these verbs. In the grammar section of the class we will explain how to use ‘haber’ in present to form tenses such as ‘We have studied’ or ‘We have lived’. We will also learn the rest of the conjugations for ‘haber’ in present tense. This class covers key grammar skills which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Vocabulary Focus - Describing 'Mi Casa'
To ramp up your vocabulary this class focuses on all things in the home. First of all students will learn how to describe their house, the names of the rooms in their house and will be able to use simple adjectives to describe size and colour. We will use the structure ‘Mi habitacion favorita’ to show which room is our favourite. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Present Tense - Ordering at a Restaurant
Have you ever wanted to order food at a restaurant in Spain in Spanish? If so, then look no further. In this lesson, we will learn how to order food in Spanish in the present tense. We will use the expression ‘quiero’ and ‘me gustaria’. You will be able to revise the vocabulary for different types of foods and drinks, and at the end of the class you will be able to order something in a restaurant! This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Verbs in infinitive form with 'Me gustaria'
What do you want to do 10 years from now? In this lesson, we will learn how to say the Spanish phrase for ‘I would like to…’ and we will add different occupations for what we would like to be in the future. Together, we will complete our sentences by adding reasons for our choices to help make more accomplished sentences. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £3.50
#### Spanish: Learning connectives 'pero' and more
Building full comprehensive sentences is a must to excel in Spanish. In this Spanish lesson, we will learn to construct sentences using the words ‘pero’, ‘sin embargo’, ‘no obstante’. These are very important connective words to learn which will help you to construct full sentences in Spanish We will also spend some time revising the vocabulary for food and drinks. After this class students should be able to use connectives in any topic they have learnt prior to create full, well rounded sentences.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: My Daily Routine
This class focuses on reflexive verbs in the present. First of all we will learn how to describe our daily routine in Spanish with sentences such as 'me despierto,' 'me levanto,' 'me cepillo los dientes'. We will add time to these sentences and we will include frequency to adverbs such as 'por lo general,' 'normalmente' and 'por lo regular'. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Infinitive Verbs - Me gusta ver la tele
In this class we will be introducing infinitive verbs. We being the lesson learning how to say things we like doing, and we will learn the infinitive of different verbs that we do. We will add reasons to our sentences and as a challenge we will use 'pero' and 'ni' to complete our sentences. This class covers key skills in grammar which students will revisit time and time again throughout their Spanish studies, so they can continue building on the knowledge they have learnt in this lesson.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Volunteer Work
In this class, students will get an overview of the vocabulary needed to describe photos and talk about volunteering and different types of charity work. We will practise key vocabulary and teach students how to use modals and other verbs followed by an infinitive. We will also discuss strategies to develop advanced reading skills.
• Stream until Fri 1st Jul
• 20 mins
• £3.50
#### Spanish: Seasonal Vocabulary - All things Winter
In this Winter themed Season Spanish lesson, we are firstly going to revise vocabulary related to the winter season. We will learn set phrases related to activities we do during winter such as ‘skiing’, ‘ice skating’. We will then use the past tense to describe what we did last winter. Finally, we will listen to native speakers describing things they did in the winter. This allows the students to really get a feel for the language and to learn the correct pronunciation. Even though you may think GCSE exams are a long way away right now, this is a fantastic opportunity to develop and hone some of the complex skills that are vital for future examination preparation.
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Spanish: Past Tense Focus - What I did Yesterday
Can you say what did you do yesterday in Spanish? In this lesson we will revise vocabulary regarding daily routine. We will first quickly revise how we describe our daily routine in the present tense. Then we will conjugate each sentence into past tense. We will then attempt to translate a paragraph into English. We are then going to complete a GCSE challenge to get students used to the style of questions that appear in examinations. Although the examinations may be a way off for many students, revising past questions is essential to excelling in Spanish!
• Stream until Fri 1st Jul
• 20 mins
• £4
#### Utilising your MFL Classes (Modern Foreign Languages)
This class is all about how to enjoy language learning, and how and where to apply it. Rosie will talk you through some simple and practical steps for improving your language skills, and we will focus on some common language ‘traps’ that students often fall into as well as how to avoid them!
• Stream until Fri 15th Jul
• 20 mins
• £3
#### Scottish Gaelic: Exchanging Names
An important lesson in any language - introducing yourself!
• Stream until Fri 15th Jul
• 20 mins
• £2.50
#### Scottish Gaelic Basic Greetings
An early step in language learning is understanding how to greet someone.
• Stream until Fri 15th Jul
• 20 mins
• £2.50
#### Scottish Gaelic: Asking How Someone Is
Alongside describing yourself, it is important to learn how to ask questions of someone else when learning a new language.
• Stream until Fri 15th Jul
• 20 mins
• £2.50
#### Scottish Gaelic: Regular Past Tense
Hearing a tutor speak the language alongside examples of how to apply the theory students are learning is the key to language success.
• Stream until Fri 15th Jul
• 20 mins
• £2.50
#### Scottish Gaelic: Irregular Past Tense
A clear understanding of the irregular past tense is important when learning any new language.
• Stream until Fri 15th Jul
• 20 mins
• £3
#### Scottish Gaelic: Numbers 1-20
Learning a language takes time, effort and most of all practise!
• Stream until Fri 15th Jul
• 20 mins
• £2.50
#### Geography: Why do people migrate?
Some people leave beautiful hot countries with gorgeous sandy beaches to live in cold chilly places. Others move from the wonderful countryside to the busy noisy cities. Why do people make these decisions?
• Stream until Mon 1st Aug
• 20 mins
• £4
#### Geography: Montserrat Volcano (Case Study)
The unexpected Volcanic eruption in the beautiful Island of Montserrat has changed this Island forever!
• Stream until Mon 1st Aug
• 20 mins
• £4
#### Geography: Countries at Contrasting Levels of Development
There are many factors to take into account when studying Development across the Globe.
• Stream until Mon 1st Aug
• 20 mins
• £3.50
#### An Introduction to Geology
Geology is the study of our Earth and the processes that happen within it and on it. In this introductory class we look at the 3 main rock types; Igneous, Sedimentary and Metamorphic. You will learn about each rocks typical characteristics, how to differentiate between them and how they are formed.
• Stream until Mon 15th Aug
• 20 mins
• £2
#### History: Exam Preparation -The Weimar and Nazi Germany Paper
It is never too early to get ahead with exam and assessment preparation!
• Stream until Thu 1st Sep
• 20 mins
• £2.50
#### History: The Prague Spring
In 1968 the people of Czechoslovakia decided they did not want to be ruled by the Soviet Union so demanded reform and an easing of Soviet control.
• Stream until Thu 1st Sep
• 20 mins
• £3
#### History: The Building of the Berlin Wall
The building of the Berlin Wall had enormous consequences across Germany and the wider world.
• Stream until Thu 1st Sep
• 20 mins
• £2
#### History: Exam Techniques Workshop
Students often overlook the importance of learning HOW to show assessors and examiners you know your stuff - don't let exam technique hold you back.
• Stream until Thu 1st Sep
• 20 mins
• £4
#### History: The Berlin Blockade and Airlift
A solid understanding of the Berlin Blockade will allow students to build their learning of other topics surrounding the end of WW2.
• Stream until Thu 1st Sep
• 20 mins
• £2.50
#### History: The Hungarian Uprising
Following the death of Stalin in 1953 some of the East European countries felt that Khrushchev was not going to be such a ruthless leader.
• Stream until Thu 1st Sep
• 20 mins
• £2
#### History: Early Elizabethan - Exam Practise and Questions
Whilst it is important to learn the content of exams and assessment it is also important to learn HOW to show examiners and teachers you know your stuff!
• Stream until Thu 1st Sep
• 20 mins
• £2.50
#### Geography: What is a Tsunami?
We've all seen movies where everyone runs away from the big wave...everything gets destroyed and the hero survives and saves the day! But is this what happens in real life?
• Stream until Thu 1st Sep
• 20 mins
• £4
#### History: Exam Preparation - Crime and Punishment
Whilst it is important to learn the content of exams and assessment it is also important to learn how to prepare for exams to set students up with the best chance of achieving their best in an assessment situation.
• Stream until Thu 1st Sep
• 20 mins
• £2.50
#### History: Super Power Relations and Cold War - Exam Preparation
It is never too early to get ahead with exam and assessment preparation!
• Stream until Thu 1st Sep
• 20 mins
• £2.50
#### History: How the Allies dealt with Germany after WW2
With the end of Nazi Germany declared, the Allies decided on how to divide Germany.
• Stream until Thu 1st Sep
• 20 mins
• £1.50
#### History: The Cuban Missile Crisis
Following the decision of the USSR to station nuclear weapons in the island of Cuba, this lesson looks at the significance of 13 days in 1962 which saw the two superpowers come very close to a nuclear war. The USA could not allow the USSR to put nuclear weapons on an island just 90 miles from its mainland. What followed almost brought the two countries the closest they had been to a nuclear war.
• Stream until Thu 1st Sep
• 20 mins
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#### History: Henry VIII and his 6 wives
The History of Kind Henry VIII is a common topic across the History curriculum.
• Stream until Thu 1st Sep
• 20 mins
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#### History How Did the Cold War start?
The Cold War is one of the most talked about topics in modern history and one which comes up across many different curriculums and syllabuses.
• Stream until Thu 1st Sep
• 20 mins
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#### History: Battle of Hastings
The infamous Battle of Hastings is a popular topic across History classes.
• Stream until Thu 1st Sep
• 20 mins
• £3
#### Sociology: Education Topic Exam Questions
It is never too early to start revising. This lesson will help students to start revising early and not leave important exam practice until the last minute.
• Stream until Sat 1st Oct
• 20 mins
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#### Drama: How Performers use their Voice
The Voice is an important tool for performers.
• Stream until Sat 1st Oct
• 20 mins
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#### Drama: Character Study
In any text, play or story it is useful to truly understand the characters involved.
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• 20 mins
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#### Drama: Actively watching & analysing performance
This class will help students engage with live performances in order to produce their own notes and analyse effectively.
• Stream until Sat 1st Oct
• 20 mins
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#### Sociology: Education Past Paper
Taking a look at Past Papers is an incredibly valuable way to measure how ready you are for an exam or assessment. By using past examples you can get a sense of what assessors will be asking so you don't get any nasty surprises on the day.
• Stream until Sat 1st Oct
• 20 mins
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#### Sociology: Families Exam Questions
Learning how to show assessors and examiners you know your stuff is a vital part of succeeding in school.
• Stream until Sat 1st Oct
• 20 mins
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#### Media: Intro to Semiotics and Analysing a Media Text
Semiotics is the study of signs and symbols and how they create meaning. It forms the backbone of analysis for many media texts.
• Stream until Sat 1st Oct
• 20 mins
• £3
#### Drama: Creating a Study Guide for a Text/Play
As well as learning a play or text itself, it is important students understand how best to take in this information.
• Stream until Sat 1st Oct
• 20 mins
• £2.50
#### Drama: Creating a Research Resource for Live Production
A research resource allows for an organised, efficient approach to developing a production.
• Stream until Sat 1st Oct
• 20 mins
• £2.50
#### Drama: Creating a Warm Up and Finding Focus
It is important in performing arts to dedicate the time to warm up ahead of a performance.
• Stream until Sat 1st Oct
• 20 mins
• £3
#### Sociology: Past Paper Practise - Family Topic
Taking a look at Past Papers is an incredibly valuable way to measure how ready you are for an exam or assessment. By using past examples you can get a sense of what assessors will be asking so you don't get any nasty surprises on the day.
• Stream until Sat 1st Oct
• 20 mins
• £2.50
#### How to Organise your Revision Time
Learning how to learn is such an important skill to develop - not only will it save students time every day, it will also set you up with a firm foundation for later life.
• Stream until Sun 2nd Oct
• 20 mins
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#### Study Habits - Taking and organising notes from lessons
Learning how to learn is such an important skill to develop - not only will it save students time every day, it will set you up with a firm foundation for later in life.
• Stream until Sun 2nd Oct
• 20 mins
• £4
#### Exam Practice - How do examiners mark exam papers?
This is a key class which every student will gain valuable lessons from. Hosted by Brian who is an experienced examiner, students will be guided through what they need to look out for when they're taking their exams.
• Stream until Sun 2nd Oct
• 20 mins
• £4
#### Coding: An Introduction to Game Development
Ever wanted to develop your own computer game?
• Stream until Tue 1st Nov
• 20 mins
• £3.50
#### Intro to Yoga with AuNatch: Breath And Body Awareness
There are many benefits to learning Yoga - for fitness, for relaxation or to focus your thoughts to help improve other aspects of your life.
• Stream until Tue 1st Nov
• 30 mins
• £2.50
#### An Intro to Yoga with AuNatch: An Example Vinyasa Class
There are many benefits to learning Yoga - for fitness, for relaxation or to focus your thoughts to help improve other aspects of your life.
• Stream until Tue 1st Nov
• 50 mins
• £3
#### Learning to Accept Yourself Workshop
In today's class Rosie will also talk you through how to deal with judgmental people, and how to distinguish between those who have your best interests at heart, and those whose judgements and opinions do not need to take up space in your mind.
• Stream until Tue 1st Nov
• 20 mins
• £3
#### How to Command and Keep Respect
In this class Rosie will coach students on how to make an excellent first impression. She will talk you through what makes people lose respect for others, and how you can avoid that. Students will learn what having respect for themselves and others really means.
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• 20 mins
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#### What you'll learn living, studying, and traveling abroad with Klook
There are so many benefits to travel whether it is just for fun, to learn or to meet new people.
• Stream until Tue 1st Nov
• 25 mins
• £3
#### Personal Fitness: HIIT Class
You don't need a lot of fancy equipment to get fit from the comfort of your own home.
• Stream until Tue 1st Nov
• 20 mins
• £3
#### Mindfulness & Motivation Workshop
We all get stressed sometimes, and boredom is a natural human response to things that we don’t find interesting. But if you feel like stress and boredom are really affecting your studies, this class will teach you some core techniques for managing your responses, and getting the best from your study time.
• Stream until Tue 1st Nov
• 20 mins
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#### Spelling & Grammar Techniques for Written Communications
If writing clearly written, properly punctuated and grammatically correct sentences is something that you find difficult, or if this is one of the areas that you or your teachers have identified as needing some improvement, then this tutorial is the perfect tool for you.
• Stream until Tue 1st Nov
• 20 mins
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#### Hair & Beauty Class: Styling Heatless Waves
Whilst many skills within the Hair & Beauty industry require significant training and equipment, there are some which you can easily learn at home.
• Stream until Tue 1st Nov
• 10 mins
• £3
#### Intro To Yoga With Au Natch: Transitioning And Twists
There are many benefits to learning Yoga - for fitness, for relaxation or to focus your thoughts to help improve other aspects of your life.
• Stream until Tue 1st Nov
• 30 mins
• £3.50
#### Intro To Yoga With Au Natch: Hip Openers And Yin
There are many benefits to learning Yoga - for fitness, for relaxation or to focus your thoughts to help improve other aspects of your life.
• Stream until Tue 1st Nov
• 30 mins
• £3.50
###### Life skills
Presentations can be stressful because they require us to speak at length on a topic that we may have only just become familiar with. This webinar will guide you through some simple coaching techniques which can help you to manage any anxiety before a presentation, and equip you with some important skills for capturing and keeping the audience's attention.
• Stream until Tue 1st Nov
• 20 mins
• £3
#### Basics of Safeguarding for Parent/Guardians
It is important to keep up to date with online safeguarding when you are caring for a young person.
• Stream until Tue 1st Nov
• 20 mins
• FREE
#### Principles of Digital Resilience
Good security and privacy is essential in every home. But as more and more students start to be exposed to new apps such as TikTok, there's little information circulated about what they could do to keep themselves safe online.
• Stream until Tue 1st Nov
• 20 mins
• FREE
###### Life skills
This is a fantastic engaging and fun class suitable for everyone to develop a new skill and have something to show for it in just 10 minutes!
• Stream until Tue 1st Nov
• 15 mins
• FREE
#### Life Coaching: Stepping into Confidence
In our world of comparison, the word 'confident' can have so many unhelpful connotations.
• Stream until Tue 1st Nov
• 30 mins
• £6
#### Hair & Beauty Class: Special FX
The special effects hair & beauty industry is huge with so many different skills required and career paths to follow.
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• 15 mins
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#### Hair & Beauty Class: Haircuts
Embarking on the qualifications required to be in the Hair & Beauty industry is a long road that by far pays off in the end.
• Stream until Tue 1st Nov
• 20 mins
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#### Creativity as Mental Health Management
The value of creativity in mental health management is often overlooked.
• Stream until Tue 1st Nov
• 20 mins
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#### Learn Scottish Gaelic: A Basic Introduction
When taking on a new language it is really useful to know information about the language in context; where and how it is used alongside a little history.
• Stream until Tue 1st Nov
• 20 mins
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#### Become a Better Speaker Today
The ability to speak clearly and confidently is vital for anyone who wants to make an impression. In this class, Sana your tutor will give students strategies to speak in a clear and understandable manner and explain how to use the power of your voice to keep your listener or audience engaged.
• Stream until Tue 1st Nov
• 15 mins
• £5
#### Learn Scottish Gaelic: Where I Live
Learning a new language is an incredible skill, it requires such a different way of learning to many other subjects.
• Stream until Tue 1st Nov
• 20 mins
• £3.50
#### Intro to Yoga with AuNatch: Balances
There are many benefits to learning Yoga - for fitness, for relaxation or to focus your thoughts to help improve other aspects of your life.
• Stream until Tue 1st Nov
• 30 mins
• £3.50
#### Mental Health: Anxiety and Looking after You
'Fight', 'Flight', or 'Freeze'? Anxiety can feel debilitating at times, but there is a reason for it. In this webinar, Rosie will speak to students about why we experience anxiety to help give them the context behind what our brain is doing. After better understanding our own feelings of anxiety, Rosie will then cover a technique in mindfulness that students can practice to strengthen their own coping mechanisms.
• Stream until Tue 1st Nov
• 15 mins
• £4
#### How to Avoid Procrastination
Procrastination is essentially finding ways to distract yourself in order to avoid doing a pressing task. We all do it from time to time, but, left unchecked procrastination can really start to impact your schoolwork.
• Stream until Tue 1st Nov
• 20 mins
• £3
#### Competitiveness Versus Jealousy
A little jealousy can actually be good for us, because it can drive us forward, induce a bit of healthy competition, and push us to be the best that we can be. But when it gets out of control, it can affect our feelings, thoughts and behaviours in powerful and damaging ways.In this workshop, Rosie will talk you through how to better understand your feelings, and reflect on what they might be telling you about yourself. She will then talk you through a simple exercise to turn feelings of jealousy into a positive opportunity to set and achieve new goals.
• Stream until Tue 1st Nov
• 20 mins
• £2
#### The Secret to Confidence: How to Channel Your Best Self
Is your confidence holding you back at school and outside of it? In this class students will learn how to genuinely feel more confident, and how to project that confidence when you’re around other people. Rosie will talk you through some simple techniques that will help you feel more self-assured, and to develop your self-belief. She will then go on to talk about how you can project that confidence with your verbal language and body language.
• Stream until Tue 1st Nov
• 20 mins
• £2.50
#### Privacy: Devices and Social Media
Good security and privacy is essential in every home. Two aspects of every day use for most families, and especially teenagers, are devices and social media.
• Stream until Tue 1st Nov
• 25 mins
• FREE
#### Hair & Beauty Class: Creating Bruises (Special FX)
In this Hair & Beauty mini series students will gain insight into a range of skills and information from the industry.
• Stream until Tue 1st Nov
• 10 mins
• £3
#### Building a Creative Writing Habit
Have you always wanted to be a writer? Do you dream up amazing stories and ideas? But do you find you don’t know how to start, or how to keep going on your creative projects?
• Stream until Tue 1st Nov
• 20 mins
• £3
#### Financial Foundations: Saving, Spending, Debt
Money is an inevitability in life. So why don't we know more about it?
• Stream until Tue 1st Nov
• 30 mins
• FREE
#### How To Be Assertive
In this class Rosie will talk you through why assertiveness is and important attribute to have when dealing with many areas of your life. She will also break down how assertiveness differs from being aggressive or 'standoffish'. The class will then focus on some straightforward techniques for breaking your relational patterns. From relating to people differently with getting your needs met.
• Stream until Tue 1st Nov
• 20 mins
• £3
#### Coding in Python: Build your own Game (Part 1 of 4)
This is part 1 of 4 in a series. This series is suitable for students who would like an introduction to Python coding. You will need to have an installation of Python which you can download here for free https://www.python.org/ to start the class.
• Stream until Mon 7th Nov
• 30 mins
• £4
#### Coding in Python: Build your own Game (Part 2 of 4)
This 4 part mini-series is suitable for students who would like an introduction to Python coding. You will need to have an installation of Python which you can download here for free https://www.python.org/ to start the class.
• Stream until Tue 8th Nov
• 30 mins
• £4
#### Coding in Python: Build your own Game (Part 3 of 4)
This is part 3 of 4 in a series. If you missed the first parts of this mini series you can watch Part 1 and Part 2 here.
• Stream until Wed 9th Nov
• 30 mins
• £4
#### Coding in Python: Build your own Game (Part 4 of 4)
This is part 4 of 4 in a series. If you missed the first parts of this mini series you can watch Part 1, Part 2 and Part 3 here.
• Stream until Thu 10th Nov
• 30 mins
• £4
#### How to Become a Doctor
There are so many careers available within Medicine. In this class, Laura will explain her journey into Medicine and the route she took to become a Doctor.
• Stream until Thu 1st Dec
• 50 mins
• FREE
#### Working as a Project Manager
A career as a Project Manager can allow you to work across many different industries and many different countries! The skills you develop in a role such as this can be applied to so many different jobs.
• Stream until Thu 1st Dec
• 20 mins
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#### How To be a Professional Fundraiser
Careers in Fundraising are a vital part of ensuring large charities can carry out all the work they do.
• Stream until Thu 1st Dec
• 20 mins
• FREE
#### Working in the Film Industry
This is a fantastic look into an industry which can be really hard to get a good insight into.
• Stream until Thu 1st Dec
• 30 mins
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#### Life as a Police Officer
Life as a Police Officer - More Line of Duty or Hot Fuzz?
• Stream until Thu 1st Dec
• 30 mins
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#### Getting a Book Published
This class will cover everything you need to know about the stages of publication; from promoting your manuscript to negotiating cover designs. Rosie will talk you through what work tends to sell (and what is harder to sell), how to market your work, what to expect when your manuscript is finally accepted, and how to recover from knockbacks.
• Stream until Thu 1st Dec
• 20 mins
• £3
#### The Art of Making Up your own Job Title
Some people have a clear idea of the career they want and some don't.
• Stream until Thu 1st Dec
• 30 mins
• FREE
#### How to Become a Freelance Writer
Join this class for everything you need to know about being a freelance writer, including what it’s like to work for yourself, and how to get yourself in magazines and newspapers (clue: you don’t necessarily need a qualification in journalism).
• Stream until Thu 1st Dec
• 20 mins
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#### Becoming a Climate Change Officer
Climate Change is an issue which affects us all and there are many careers you can take on focussing on the environment & sustainability.
• Stream until Thu 1st Dec
• 15 mins
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#### A Career in Coding
Do you have an interest in coding? Or what to know how you can work in coding as a career?
• Stream until Thu 1st Dec
• 20 mins
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#### Kickstart a Career in Journalism
• Stream until Thu 1st Dec
• 30 mins
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#### Building a Company that Builds Things
This is a great class which looks at a number of different careers within one.
• Stream until Thu 1st Dec
• 20 mins
• FREE
#### Coaching: Choosing Your Next Steps After School
Deciding what to do when you finish school can be incredibly confusing and overwhelming.
• Stream until Thu 1st Dec
• 20 mins
• £5
#### Becoming a Lawyer
Embarking on the qualifications required to become a Lawyer is a commitment.
• Stream until Thu 1st Dec
• 55 mins
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##### Careers Guidance
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| 3.546875 | 4 |
CC-MAIN-2021-17
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longest
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| 0.931079 |
https://www.jiskha.com/questions/660955/projectile-motion-an-object-is-propelled-upward-at-an-angle-theta-45-degree-theta
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# PreCal
Projectile Motion: An object is propelled upward at an angle theta, 45 degree < theta <90 degree, to the horizontal with an initial velocity of v0 feet per second from the base of an inclined plane that makes an angle of 45 degree with the horizontal. If air is ignored, the distance R that it travels up the inclined plane is given by:
R= v0square square root of 2/32 [Sin(20)-cos(20)-1]
A. Find the distance R that the object travels along the inclined plane if the initial velocity is 32 feet per second and theta=60 degree
B. Graph R=R(theta) if the initial velocity is 32 feet per second.
C. What value of theta makes R largest?
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## Similar Questions
1. ### Math
1. Let (-7, 4) be a point on the terminal side of (theta). Find the exact values of sin(theta), csc(theta), and cot(theta). 2. Let (theta) be an angle in quadrant IV such that sin(theta)=-2/5. Find the exact values of sec(theta)
2. ### math;)
Julia wants to simplify the term sec^2 theta-1/cot^2 theta+1 in a trigonometric identity that she is proving. Which of the following identities should use to help her? Select all that apply. (2 ANSWERS) a. sin^2 theta+cos^2
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| 3.734375 | 4 |
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https://www.jiskha.com/questions/756238/how-does-a-computer-work
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| 720,982,445 | 6,081 |
# Science
how does a computer work
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http://www.howstuffworks.com/pc.htm
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https://math.answers.com/other-math/What_is_a_sound_with_frequency_greater_than_20000_hz
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# What is a sound with frequency greater than 20000 hz?
Updated: 4/28/2022
Wiki User
12y ago
the sound would be called "ultrasound"
Wiki User
12y ago
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Q: What is a sound with frequency greater than 20000 hz?
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Related questions
### What is the difference between infrasound and ultrasound And provide one example of how each is used?
Infrasound is always smaller or lighter than ultrasound. For example: infrasound can be the sound of a paper clip hitting the floor, and ultrasound can be the sound of two planets colliding.
### Do sound waves have a greater frequency than gamma rays?
No. The frequency of gamma rays is several orders of magnitude greater than that of any sound wave.
Yes
### What is ultasonic sound?
Ultrasound is cyclic sound pressure with a frequency greater than the upper limit of human hearing.
### Is 20000 km longer than 25 km?
Is 20000 greater than 25? If so, then it is a safe bet that 20000 km is longer than 25 km
### Does the whistle of a teakettle have a greater frequency than a drumbeat and why?
Yes because the sound waves are closer together or more frequent. A drum is usually larger than a teakettle, so it will have a lower resonant frequency.
### Is 200 m greater than 20000 mm?
Yes. 1 meter = 1000 millimeters (mm), so 200 meters = 200,000 millimeters. Convert them to the same units and then the comparison can be made. 1 m = 1000 mm ⇒ 20000 mm = 20000 ÷ 1000 m = 20 m 200 m is greater than 20 m, so 200 m is greater than 20000 mm.
### When do you have an ultra sound?
Ultrasound is cyclic sound pressure with a frequency greater than the upper limit of human hearing. This frequency varies from person to person, it is approximately 20 kilohertz (20,000 hertz). This is the lower limit in describing ultrasound.
### Is frequency a greater than normal occurrence of the urge to urinate?
No. Frequency is just a measure of "how often". It can be less than, equal to or greater than normal.
### What is mild heterogeneous in an ultrasound to rule out liver cancer?
The ultra sound having frequency is higher than the human ear can be respond.it is greater than 46khz.
No.
### Is 2g greater than 20000 mg?
No... 2g is only 2000mg
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# Mechanics lab report
This implies that the spring force is a restoring force. The test was repeated by passing the cord over the left pulley and was attached to the center arm of the bar Figure 5. The right weight hook was removed from the bar. If a resultant force acts on an object then that object can be brought into equilibrium by applying an additional force that exactly balances this resultant.
A linear relationship can be demonstrated if the data points fall along a single straight line. Equipment: Steel ball — contact plate — holding magnet — holding magnet adapter with a release mechanism — electronic stop clock — stand base — rods — scale — connecting leads. The two values should be equal.
Page 4 Section 3 Graphing Physics A Slope is often used to describe the measurement of the steepness, incline, gradient, or grade of a straight line. The straight line should be drawn as near the mean of the all various points as is optimal. The weight hooks was hung from the end holes of the bar see Figure 2.
### Spring constant lab report discussion
Equipment: Track — trolley — holding magnet — electronic stop clock — light barrier — pulley — mass hanger — slotted weights — cables. Such a force is called the equilibrant and is equal in magnitude but opposite in direction to the original resultant force acting on the object. Push and Pull: Another force which may act on an object could be any physical push or pull. The two values should be equal. The straight line should be drawn as near the mean of the all various points as is optimal. Page 3 Section 3 Graphing Physics A graph is the clearest way to represent the relationship between the quantities of interest. A spring is an example of elastic materials. The weight of the weight hook and its load, F was recorded. The negative sign in Equation 6.
A spring is an example of elastic materials. The weight of the weight hook and its load, F was recorded. Sometimes finding the center of mass of an object can be challenging, especially if the object has an odd shape. The amount of departure from rest shape is called deformation, the proportion of deformation to original size is called strain.
The distance, d of the hole from the pivot was recorded.
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# Proving an q^n is bigger or smaller than m^p watch
1. Hello
Is there an efficient way of finding whichever one of 218 or 515 is bigger without a calculator, and not by multiplying until you get the value of the two?
2. Instinct tells me to start from
21^8 < 25^8
But I'm almost certainly wrong.
Posted from TSR Mobile
3. Although I'm sure someone can come up with something better, I'll give it a shot.
Attempt one (long):
Spoiler:
Show
Lets say we have and
To directly compare them, one easily method would to convert say, q, into the form
So to do this;
Making;
This can be expanded out using
Thus,
Now we have them both as powers of m, so just compare the powers.
In your example and , if we use the same method, and put 21 as a power of 5.
Attempt 2 (same method, shorter). I was naive in not just considering this:
Spoiler:
Show
and
If
Or if:
Obviously this requires some rough calculation of logs, however I'm skeptical there is a method which involves next to no calculations. Considering the numbers could take any value, and just because the power of one is bigger, it doesn't necessarily mean the number is bigger.
Edit- Found this: http://www.zachwg.org/logarithms.pdf, could be useful.
4. (Original post by Krollo)
Instinct tells me to start from
21^8 < 25^8
But I'm almost certainly wrong.
Posted from TSR Mobile
lol
5. (Original post by Phichi)
x
I can see where you are going with method one and I understand the basis of your workings, but when it comes to logarithms I clearly need to study it more (I'm in the AS year).
Thanks for sharing, I will try and seek the knowledge you have demonstrated.
6. (Original post by Krollo)
Instinct tells me to start from
21^8 < 25^8
But I'm almost certainly wrong.
Posted from TSR Mobile
You are right that you are wrong but it's always good to try.
I know a step I won't be trying now :P
7. (Original post by Wunderbarr)
You are right that you are wrong but it's always good to try.
I know a step I won't be trying now :P
Lol. For some reason my intuition often fails me for maths problems like this.
Posted from TSR Mobile
8. (Original post by Wunderbarr)
Hello
Is there an efficient way of finding whichever one of 218 or 515 is bigger without a calculator, and not by multiplying until you get the value of the two?
That's vs ; so we want vs . The quotient is about , which if you can be bothered to find which is roughly . Hence the quotient is greater than 1, so ; hence is bigger.
That only required finding and approximating fairly cavalierly.
Alternatively you can say which is . You can work out 0.84^8 by repeated squaring if you want.
Neither of those ways are nice, sadly.
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Continuous Random Variable
A random variable is called continuous if it can assume all possible values in the possible range of the random variable. Suppose the temperature in a certain city in the month of June in the past many years has always been between $35^\circ$ to $45^\circ$ centigrade. The temperature can take any value between the ranges $35^\circ$ to$45^\circ$. The temperature on any day may be $40.15^\circ \,{\text{C}}$ or $40.16^\circ \,{\text{C}}$ or it may take any value between $40.15^\circ \,{\text{C}}$ and$40.16^\circ \,{\text{C}}$. When we say that the temperature is$40^\circ \,{\text{C}}$, it means that the temperature lies between somewhere between $39.5^\circ$ to$40.5^\circ$. Any observation which is taken falls in the interval. There is nothing like an exact observation in the continuous variable. In discrete random variable the values of the variable are exact like 0, 1, 2 good bulbs. In continuous random variable the value of the variable is never an exact point. It is always in the form of an interval, the interval may be very small.
Some examples of the continuous random variables are:
1. The computer time (in seconds) required to process a certain program.
2. The time that a poultry bird will gain the weight of 1.5 kg.
3. The amount of rain falls in the certain city.
4. The amount of water passing through a pipe connected with a high level reservoir.
5. The heat gained by a ceiling fan when it has worked for one hour.
Probability Density Function:
The probability function of the continuous random variable is called probability density function of briefly p.d.f. It is denoted by $f\left( x \right)$ where $f\left( x \right)$ is the probability that the random variable $X$ takes the value between $x$ and $x + \Delta x$ where $\Delta x$is a very small change in $X$.
If there are two points $a$ and $b$ then the probability that the random variable will take the value between a and b a given by the management.
$P\left( {a \leqslant X \leqslant b} \right) = \int_a^b {f\left( x \right)} \,dx$
Where $a$ and $b$ are the points between $- \infty$ and $+ =$. The quantity $f\left( x \right)\,dx$ is called probability differential.
The number of possible outcomes of a continuous random variable is uncountable infinite. Therefore, a probability of zero is assigned to each point of the random variable. Thus $P\left( {X = x} \right) = 0$ for all values of $X$. This means that we must calculate a probability for a continuous random variable over an internal and not for any particular point. This probability can be interpreted as an area under the graph between the interval from $a$ to $b$. When we say that the probability is zero that a continuous random variable assumes a specific value, we do not necessarily mean that a particular value cannot occur. We in fact, mean that the point (event) is one of an infinite number of possible outcomes. Whenever we have to find the probability of some interval of the continuous random variable, we can use any one of these two methods.
1. Integral calculus.
2. Area by geometrical diagrams (this method is easy to apply when $f\left( x \right)$ is a simple linear function).
Properties of Probability Density Function:
The probability density function $f\left( x \right)$ must have the following properties.
1. It is non-negative i.e. $f\left( x \right) \geqslant 0$ for all $x$
2. ${\text{Total}}\,{\text{Area}} = \int\limits_{ - \infty }^\infty {f\left( x \right)dx} = 1$
3. $\left( {X = c} \right) = \int\limits_c^c {f\left( x \right)dx} = 0$ Where c is any constant
4. As probability of area for $X = c$ (constant), therefore $P\left( {X = a} \right) = P\left( {X = b} \right)$. If we take an interval a to b. It makes no difference whether end points of the interval are considered or not. Thus we can write:
$P\left( {a \leqslant X \leqslant b} \right)\,\,\,\, = \,\,\,\,P\left( {a < X < b} \right)\,\,\,\, = \,\,\,\,P\left( {a \leqslant X < b} \right)\,\,\,\, = \,\,\,\,P\left( {a < X \leqslant b} \right)$
5. $P\left( {a \leqslant X \leqslant b} \right)\,\,\,\, = \,\,\,\,\int\limits_b^a {f\left( x \right)dx} - \int\limits_{ - \infty }^a {f\left( x \right)dx} \,\,\,\,\left( {a < b} \right)$
Example:
A continuous random variable X which can assume between $x = 2$ and 8 inclusive, has a density function given by $c\left( {x + 3} \right)$ where $c$ is a constant.
(a) Calculate $c$ (b) $P\left( {3 < X < 5} \right)$ (c) $P\left( {X \geqslant 4} \right)$
Solution:
$f\left( x \right) = c\left( {x + 3} \right),\,\,\,\,2 \leqslant x \leqslant 8$
(a)
$f\left( x \right)$ will be density functions if (i) $f\left( x \right) \geqslant 0$ for every x and (ii) $\int\limits_{ - \infty }^\infty {f\left( x \right)dx} = 1$. If $c \geqslant 0$, $f\left( x \right)$ is clearly $\geqslant 0$for every x in the given interval. Hence for $f\left( x \right)$ to be density function, we have
$1 = \int\limits_{ - \infty }^\infty {f\left( x \right)dx} \,\,\, = \,\,\,\,\int\limits_2^8 {c\left( {x + 3} \right)dx} \,\,\, = \,\,\,c\left[ {\frac{{{x^2}}}{2} + 3x} \right]_2^8$
$= \,\,\,\,c\left[ {\frac{{{{\left( 8 \right)}^2}}}{2} + 3\left( 8 \right) - \frac{{{{\left( 2 \right)}^2}}}{2} - 3\left( 2 \right)} \right]\,\,\,\, = \,\,\,c\,\left[ {32 + 24 - 2 - 6} \right]\,\,\,\, = \,\,\,\,c\left[ {48} \right]$
So that $c = \frac{1}{{48}}$
Therefore, $f\left( x \right) = \frac{1}{{48}}\left( {x + 3} \right),\,\,\,\,2 \leqslant x \leqslant 8$
(b) $P\left( {3 < X < 5} \right) = \int\limits_3^5 {\frac{1}{{48}}\left( {x + 3} \right)dx} \,\,\, = \,\,\,\frac{1}{{48}}\left[ {\frac{{{x^2}}}{2} + 3x} \right]_3^5$
$= \frac{1}{{48}}\left[ {\frac{{{{\left( 5 \right)}^2}}}{2} + 3\left( 5 \right) - \frac{{{{\left( 3 \right)}^2}}}{2} - 3\left( 3 \right)} \right]\,\,\,\, = \,\,\,\,\frac{1}{{48}}\left[ {\frac{{25}}{2} + 15 - \frac{9}{2} - 9} \right]$
$= \frac{1}{{48}}\left[ {14} \right]\,\,\,\, = \,\,\,\,\frac{7}{{24}}$
(c) $P\left( {X \geqslant 4} \right) = \int\limits_4^8 {\frac{1}{{48}}\left( {x + 3} \right)dx} \,\,\, = \,\,\,\frac{1}{{48}}\left[ {\frac{{{x^2}}}{2} + 3x} \right]_4^8$
$= \frac{1}{{48}}\left[ {\frac{{{{\left( 8 \right)}^2}}}{2} + 3\left( 8 \right) - \frac{{{{\left( 4 \right)}^2}}}{2} - 3\left( 4 \right)} \right]\,\,\,\, = \,\,\,\,\frac{1}{{48}}\left[ {32 + 24 - 8 - 12} \right]$
$= \frac{1}{{48}}\left[ {36} \right]\,\,\,\, = \,\,\,\frac{3}{4}$
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# Pre calculus homework help
Pre calculus homework help is a mathematical instrument that assists to solve math equations. We can help me with math work.
## The Best Pre calculus homework help
One tool that can be used is Pre calculus homework help. How to solve perfect square trinomial? This is a algebraic equation that can be written in the form of ax2 + bx + c = 0 . If the coefficient of x2 is one then we can use the factoring method to solve it. We will take two factors of c such that their product is equal to b2 - 4ac and their sum is equal to b. How to find such numbers? We will use the quadratic formula for this. Now we can factorize the expression as (x - r1)(x - r2) = 0, where r1 and r2 are the roots of the equation. To find the value of x we will take one root at a time and then solve it. We will get two values of x, one corresponding to each root. These two values will be the solutions of the equation.
In mathematics, a root of a polynomial equation is a value of the variable for which the equation satisfies. In other words, a root is a solution to the equation. Finding roots is a fundamental problem in mathematics, and there are a variety of ways to solve for them. One popular method is known as "factoring." Factoring is the process of breaking down an expression into its constituent factors. For example, if we have the expression x2+5x+6, we can factor it as (x+3)(x+2). Once we have factored an expression, we can set each factor equal to zero and solve for the roots. In our example, we would get two equations: x+3=0 and x+2=0. Solving these equations, we would find that the roots are -3 and -2. Another popular method for solving for roots is known as "graphical methods." These methods make use of the graphs of polynomials to find approximate values for the roots. While graphical methods can be useful, they are often less accurate than algebraic methods such as factoring. As a result, algebraic methods are typically preferred when finding roots.
For example, the equation 2 + 2 = 4 states that two plus two equals four. To solve an equation means to find the value of the unknown variable that makes the equation true. For example, in the equation 2x + 3 = 7, the unknown variable is x. To solve this equation, we would need to figure out what value of x would make the equation true. In this case, it would be x = 2, since 2(2) + 3 = 7. Solving equations is a vital skill in mathematics, and one that can be used in everyday life. For example, when baking a cake, we might need to figure out how many eggs to use based on the number of people we are serving. Or we might need to calculate how much money we need to save up for a new car. In both cases, solving equations can help us to get the answers we need.
A trinomial is an algebraic expression that contains three terms. The most common form of a trinomial is ax^2+bx+c, where a, b, and c are constants and x is a variable. Solving a trinomial equation means finding the value of x that makes the equation true. There are a few different methods that can be used to solve a trinomial equation, but the most common is factoring. To factor a trinomial, you need to find two numbers that multiply to give the product of the two constants (ac) and add up to give the value of the middle term (b). For example, if you are given the equation 2x^2+5x+3, you would need to find two numbers that multiply to give 6 (2×3) and add up to give 5. The only numbers that fit this criteria are 1 and 6, so you would factor the equation as (2x+3)(x+1). From there, you can use the zero product rule to solve for x. In this case, either 2x+3=0 or x+1=0. Solving each of these equations will give you the values of x that make the original equation true. While factoring may seem like a difficult task at first, with a little practice it can be easily mastered. With this method, solving trinomials can be quick and easy.
Additionally, another way to simplify math is to practice a lot. By doing this, students can become more comfortable with the concepts and procedures involved in solving mathematical problems. With a bit of practice and perseverance, math can become much less daunting for even the most struggling students.
## Help with math
The app is great a very useful. 2 stars (the app is worth 5 but because of that now after editing you get 4) because of that stupid way you implemented the premium feature, I mean adding a premium mode to your app is perfectly okay if you add new features and then asking for money but taking something old and asking for money for that is complete laziness and lack of interest from the developers!
Xandra Bryant
This is awesome app I mean sure it might sometimes won't get the numbers correctly (I think that it's my handwriting’s fault) but it's very good like you don't have enough time??? Slap some of those math problems on it and boom you're good Coyocan even see how to do it like it's cool awesome and very helpful
Rosie Long
How to solve this math problem Absolute value problem solver 2 step equations word problems Solve in math Application math problems
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# SSC CGL EXAMS 2019 | Quantitative Aptitude Practice Questions (Day-22)
0
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Dear Aspirants, Here we have given the Important SSC Exam 2018 Practice Test Papers. Candidates those who are preparing for SSC 2019 can practice these questions to get more confidence to Crack SSC 2019 Examination.
SSC CGL EXAMS 2019 | Quantitative Aptitude Questions (Day-22)
#### Click here to view Quantitative Aptitude Questions in Hindi
1) Find the unit digit of the following expression – (102)102 (101)101 (107)107 (106)106 (105)105
a) 0
b) 2
c) 6
d) 5
2) Four carpenters A, B, C & D are entrusted with office furniture work. A can do a job is 10 days, if B is 20% more efficient than A and C is 30% more efficient than B, and D is 10% more efficient that C, then C and D together can finish the job is approximately.
a) 3 days
b) 4 days
c) 5 days
d) 6 days
3) An old woman engaged a domestic help on the condition that she would pay him Rs. 120 and a gift after service of one year. He served only 6 months and received the gift and Rs. 50. Find the value of the gift.
a) Rs. 20
b) Rs. 10
c) Rs. 15
d) Rs. 30
4) The minimum value of 31 3 sin x + 31 4 cos x
a) 2 × 31 √5/2
b) 2√5
c) 31√5
d) 0
5) The smallest perfect square that is divisible by 11!!
a) 12006200
b) Cannot be determine
c) 12006225
d) 12600252
6) How many pairs of factors of number 120. Such that they are co-prime to each other?
a) 40
b) 20
c) 30
d) 32
7) PQR is a triangular park with PQ = PR= 160 meter. A clock tower is situated at the mid-point of QR. The angles of elevation of the top of the tower at P & Q are Cot–1 13.2 and cosec–12.6 respectively. The height of the tower is—
a) 50 meters
b) 30 meters
c) 60 meters
d) 40 meters
8) If a, b, c, d and e are positive real numbers such that a + b + c + d + e = 30, find the maximum value of a3 b3 c6 d2 e2.
a) 225. 324
b) 224.325
c) (–6)24
d) (6)25
9) What is the reflection of the point (– 5, 3) in the line x = – 3?
a) (–2, 3)
b) (–1, 3)
c) (– 1, – 3)
d) –2, – 3
10) Two sides of a triangle are √5 – 1 & √5 + 1 units and their included angle is 60o, Solve the triangle and find the value of ‘a’?
a) √3
b) √2
c) √6
d) 2√3
Answers :
1) Answer: a)
Unit digit of given values
2102 × 1101 × 7 107 × 6 106 × 5 105
= 2 4×25 + 2 × 1 × 7 4×26 + 3× 6 × 5
= 6 × 2 × 2 × 1 × 1 × 7 × 7 × 7 × 30
= 24*243*30
= 4 * 3 * 30 = 0
2) Answer: a)
According to the conditions mentioned in the questions,
We find efficiency of each carpenter,
A =100%, B= 120%, C=120×130/100=156%,D=156×110/100=171.6%
Combined efficiency of C&D =327.6%
Now,
100%= 10days
1%= 10×100days
327.6%=10×100/327.6 days = 3.05days.≈ 3 days
3) Answer: a)
Let the value of the gift be Rs. x
Then, Payment for 1 yr = Rs. 120 + x
Payment for 6 months = (120 + x/12) * 6
= 120*6/12 + 6x/12 = 60 + x/2
After 6 months, the domestic help received 50 and gift,
Therefore, value of the gift
x = x/2 + 10
x/2 = 10
x = 20
4) Answer: a)
5) Answer: c)
11!! = 11×9×7×5×3×1 [X!! = X (X –2) (X – 4) ….. And so on]
= 111×32×71×51×31
Thus, the least perfect square which is divisible by 11!! Should be (111 32 71 51 31) (111 71 51 31)
= 10395×1155 = 12006225.
6) Answer: d)
120 = 23×31×51
Here are of such pairs
= 1 + (p + q + r) + 2 (pq + qr + pr) + 4pqr
(p = 3, q = 1, r = 1)
= 1 + 5 + 2 (3 + 1 + 3) + 4×3×1×1
= 6 + 14 + 12
= 32
7) Answer: d)
8) Answer: a)
9) Answer: b)
10) Answer: c)
### *******************
#### SSC CGL EXAMS 2019 |Quantitative Aptitude Practice Questions (Day-19)
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Broad Topics > Numbers and the Number System > Factors and multiples
LCM Sudoku II
Age 11 to 18 Challenge Level:
You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.
LCM Sudoku
Age 14 to 16 Challenge Level:
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
Have You Got It?
Age 11 to 14 Challenge Level:
Can you explain the strategy for winning this game with any target?
Age 11 to 14 Challenge Level:
Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .
Even So
Age 11 to 14 Challenge Level:
Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?
What Numbers Can We Make?
Age 11 to 14 Challenge Level:
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Special Sums and Products
Age 11 to 14 Challenge Level:
Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
Star Product Sudoku
Age 11 to 16 Challenge Level:
The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.
Hot Pursuit
Age 11 to 14 Challenge Level:
I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...
N000ughty Thoughts
Age 14 to 16 Challenge Level:
How many noughts are at the end of these giant numbers?
Repeaters
Age 11 to 14 Challenge Level:
Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.
Counting Factors
Age 11 to 14 Challenge Level:
Is there an efficient way to work out how many factors a large number has?
X Marks the Spot
Age 11 to 14 Challenge Level:
When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .
What Numbers Can We Make Now?
Age 11 to 14 Challenge Level:
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Got it for Two
Age 7 to 14 Challenge Level:
Got It game for an adult and child. How can you play so that you know you will always win?
Age 11 to 14 Challenge Level:
List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?
Three Times Seven
Age 11 to 14 Challenge Level:
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Remainders
Age 7 to 14 Challenge Level:
I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
A First Product Sudoku
Age 11 to 14 Challenge Level:
Given the products of adjacent cells, can you complete this Sudoku?
Factor Lines
Age 7 to 14 Challenge Level:
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
Mod 3
Age 14 to 16 Challenge Level:
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Age 11 to 16 Challenge Level:
The items in the shopping basket add and multiply to give the same amount. What could their prices be?
Got It
Age 7 to 14 Challenge Level:
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
Eminit
Age 11 to 14 Challenge Level:
The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?
Factoring Factorials
Age 11 to 14 Challenge Level:
Find the highest power of 11 that will divide into 1000! exactly.
Cuboids
Age 11 to 14 Challenge Level:
Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
Different by One
Age 14 to 16 Challenge Level:
Make a line of green and a line of yellow rods so that the lines differ in length by one (a white rod)
Mathematical Swimmer
Age 11 to 14 Challenge Level:
Twice a week I go swimming and swim the same number of lengths of the pool each time. As I swim, I count the lengths I've done so far, and make it into a fraction of the whole number of lengths I. . . .
Factor Track
Age 7 to 14 Challenge Level:
Factor track is not a race but a game of skill. The idea is to go round the track in as few moves as possible, keeping to the rules.
Sieve of Eratosthenes
Age 11 to 14 Challenge Level:
Follow this recipe for sieving numbers and see what interesting patterns emerge.
Transposition Cipher
Age 11 to 16 Challenge Level:
Can you work out what size grid you need to read our secret message?
Gabriel's Problem
Age 11 to 14 Challenge Level:
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Diagonal Product Sudoku
Age 11 to 16 Challenge Level:
Given the products of diagonally opposite cells - can you complete this Sudoku?
Satisfying Statements
Age 11 to 14 Challenge Level:
Can you find any two-digit numbers that satisfy all of these statements?
Take Three from Five
Age 14 to 16 Challenge Level:
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
Powerful Factorial
Age 11 to 14 Challenge Level:
6! = 6 x 5 x 4 x 3 x 2 x 1. The highest power of 2 that divides exactly into 6! is 4 since (6!) / (2^4 ) = 45. What is the highest power of two that divides exactly into 100!?
Common Divisor
Age 14 to 16 Challenge Level:
Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
A Biggy
Age 14 to 16 Challenge Level:
Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.
Sixational
Age 14 to 18 Challenge Level:
The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .
For What?
Age 14 to 16 Challenge Level:
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
Helen's Conjecture
Age 11 to 14 Challenge Level:
Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?
Ewa's Eggs
Age 11 to 14 Challenge Level:
I put eggs into a basket in groups of 7 and noticed that I could easily have divided them into piles of 2, 3, 4, 5 or 6 and always have one left over. How many eggs were in the basket?
Two Much
Age 11 to 14 Challenge Level:
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
How Old Are the Children?
Age 11 to 14 Challenge Level:
A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"
One to Eight
Age 11 to 14 Challenge Level:
Complete the following expressions so that each one gives a four digit number as the product of two two digit numbers and uses the digits 1 to 8 once and only once.
Factoring a Million
Age 14 to 16 Challenge Level:
In how many ways can the number 1 000 000 be expressed as the product of three positive integers?
Digat
Age 11 to 14 Challenge Level:
What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A
Divisively So
Age 11 to 14 Challenge Level:
How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5 or 7?
Remainder
Age 11 to 14 Challenge Level:
What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?
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# HCF & LCM of numbers
Are you preparing for campus placements,Banking,SSC, IAS, Insurance,Defence and other competitive exams? Then, make sure to take some time in practicing the LCM & HCF questions and answer in Quantitative Aptitude. Moreover, only those questions are included that are relevant and likely to be asked in any competitive exam. So, take these questions and answer, brush up your skills and practice to stay fully prepared for any your exam.
• Q1.Four metal rods of lengths 78 cm, 104 cm, 117cm and 169cm are to be cut into parts of equal length, Each part must be as long as possible. What is the maximum number of pieces that can be cut?
• Q2.Five persons fire bullets at a target at an interval of 6, 7, 8, 9 and 12 seconds respectively. The number of times they would fire the bullets together at the target in an hour is
• Q3.Rubina could get equal number of Rs. 55, Rs. 85 and Rs. 105 tickets for a movie. She spends Rs. 2,940 for all the tickets. How many of each did she buy?
• Q4.There are some parrots and some tigers in a forest. If the total number of animal heads in the forest is 858 and the total number of animal legs is 1,846, what is the number of parrots in the forest?
• Q5.When-all the students in a school are made to stand in rows of 54, 30 such rows are formed. If the students are made to stand in rows of 45, how many such rows will be formed?
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https://discuss.codechef.com/t/kryp3-editorial/18076
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# KRYP3 - Editorial
[Contest][1]
[Practice][2]
Author: [Shivam Garg][3]
Tester: [Shiv Dhingra][4]
MEDIUM HARD
### PREREQUISITES
Matrix Exponentiation
### PROBLEM
Given a matrix, and several queries denoting the start point, tell the number of ways to stay inside the matrix only with the given number of steps.
### EXPLANATION
This question requires knowledge of Matrix Exponentiation. So, I would advise you to go through this [tutorial][5], and solve questions at the end in case you don’t have a basic idea about this topic.
Now, the matrix can be considered as a graph. In this graph, a cell (A,B) is connected to some other cell
(C,D) only if 1 ≤ max( | A - C | , | B - D | ) ≤ 2.
So, you can simply make another matrix X of this graph. The dimensions of this matrix will be (N^2,N^2).
The value X(i,j) will denote the number of ways to reach i^{th} point in the input matrix to j^{th} point.
If we simply apply matrix exponentiation on this matrix for the Y steps, this should get the task done.
But, let’s examine the complexity first.
Matrix Exponentiation takes O((N^{2})^{3}Log(Y)) steps. In other words, this turns out to be O(N^{6}Log(Y)) iterations.
As N in our case is 20, and Y is 10^{14}, it turns out to be 47*6.4*10^{7} = 3*10^{9} , which will time out.
On careful observation, we will be able to notice that there is symmetry in the matrix.
Let’s consider a matrix for N=4, AND Y=1.
The number of ways corresponding to each point will be -
8 11 11 8
11 15 15 11
11 15 15 11
8 11 11 8
The really useful points are just (1/8 ^{th}*N^{2}) instead of N^{2}.
Let D = (N^{2})/8. Then, the complexity will be O(D^{3}Log(Y)).
Each of the Z queries can be then answered in O(1)
### SOLUTION
Setter’s solution -
``````
[6]
[1]: https://www.codechef.com/CODR2018/problems/KRYP3/
[2]: https://www.codechef.com/problems/KRYP3/
[3]: http://codechef.com/users/shivamg_isc
[4]: http://codechef.com/users/lucifer2709
[5]: https://www.hackerearth.com/practice/notes/matrix-exponentiation-1/
[6]: https://www.codechef.com/viewsolution/17468290``````
3 Likes
Couldn’t figure it out during the contest :’(
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Proof using induction
• Nov 5th 2009, 08:34 PM
keityo
Proof using induction
Prove that if $\displaystyle f(x) \equiv 0(mod p)$ has $\displaystyle j$ solutions $\displaystyle x \equiv a_{1}, x \equiv a_{2},..., x \equiv a_{j} (mod p)$, there is a polynomial $\displaystyle q(x)$ such that $\displaystyle f(x) \equiv (x - a_{1}) (x - a_{2}) ... (x - a_{j})q(x)(mod p)$.
I feel that I need to somehow show that there is a polynomial $\displaystyle q_{1}(x)$ such that $\displaystyle f(x) \equiv (x-a_{1})q_{1}(x)(mod p)$and that $\displaystyle q_{1}(x) \equiv 0 (mod p)$has solutions $\displaystyle x=a_{2}, x=a_{3}, ..., x=a_{j} (mod p)$, but I don't know start from there and finish the proof using induction.
• Nov 5th 2009, 08:46 PM
Bruno J.
We proceed by induction on the degree of $\displaystyle f$. If $\displaystyle f$ has degree 1, there is nothing to show because it is already in the form $\displaystyle x-a$. So suppose it holds for all polynomials having degree $\displaystyle \leq n-1$. Let $\displaystyle f(x)=c_0+c_1x+...+c_nx^n$ be of degree $\displaystyle n$, and suppose $\displaystyle x \equiv a \mod p$ is a solution. Then $\displaystyle 0\equiv f(a) \equiv c_0+c_1a+...+c_na^n$. So we have
$\displaystyle f(x)\equiv f(x)-f(a) \equiv c_1(x-a)+...+c_n(x^n-a^n)$
$\displaystyle \equiv (x-a)(c_1+c_2(x+a)+...+c_n(x^{n-1}+x^{n-2}a+...+xa^{n-2}+a^{n-1}))$
$\displaystyle \equiv (x-a)g(x)$
where $\displaystyle g(x)$ is of degree $\displaystyle \leq n-1$. Now use the fact that $\displaystyle p$ is prime to show that any other roots of $\displaystyle f(x)$ must in fact be roots of $\displaystyle g(x)$, and apply the induction hypothesis to obtain the desired factorization for $\displaystyle f(x)$.
Hope that helps!
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## Part I. Ways to Describe Data
### Investigations 4 and 5: Ratio/Interval Data & Discrete/Continuous Data
We also can use a numerical variable to assign samples to groups. For example, we can divide the plain M&Ms in Table 1 into two groups based on the sample’s weight. What makes a numerical variable more interesting, however, is that we can use it to make quantitative comparisons between samples; thus, we can report that there are 14.8 times as many plain M&Ms in a 10-oz. bag as there are in a 0.8-oz. bag. Although we can complete meaningful calculations using any numerical variable, the type of calculation we can perform depends on whether or not the variable’s values have an absolute reference.
Investigation 4. A numerical variable is described as either ratio or interval depending on whether it has (ratio) or does not have (interval) an absolute reference. Explain what it means for a variable to have an absolute reference and assign each of the numerical variables in Table 1 as either a ratio variable or an interval variable. Why might this difference be important?
Investigation 5. Numerical variables also are described as discrete or continuous. Define the terms discrete and continuous and assign each of the numerical variables in Table 1 to one of these terms.
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## Sunday, December 30, 2012
### Exact Equation - Arbitrary Constant
Category: Differential Equations, Integral Calculus, Differential Calculus
"Published in Newark, California, USA"
Solve for the particular solution for
when x = 0, y = 2.
Solution:
The first that we have to do is to check the above equation if it is exact or not as follows
Let
then
Let
then
Since
then the given equation is Exact Equation. The solution for the above solution is F = C. Consider the given equation
Let
and
Integrate the partial derivative of the first equation above with respect to x, we have
Since we are integrating the partial derivatives, then another unknown function, T(y) must be added. If
then
To solve for T'(y), equate
To solve for T(y), integrate on both sides of the equation with respect to y, as follows
Since the arbitrary constant is already included in F = C, then we don't have to add the arbitrary constant in the above equation. Therefore,
The general solution of the equation is
To solve for the value of C, substitute x = 0 and y = 2 to the above equation, we have
Therefore, the particular solution of the equation is
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testtest
asdfasdfasdf
4. ## Re: testtest
Dealing with random numbers is never as straightforward as people think. And when you want them from a skewed distribution, it gets even more complex.
5. ## Re: testtest
Therefore, if you're truly wanting what you suggest, the first order of business is to figure out from which skewed distribution you're pulling your random number: Poisson, Gauss with a skew value, logarithmic, etc. There are many to choose from. Possibly you even want to bracket part of one of those skewed curves. And, from there, you can start devising your algorithm to always sum to a specific value.
Also, you will need to decide if the Rnd function is good enough for you, or whether you wish to use the more robust CryptGenRandom (or other) API call (where you get more robust random numbers). And, again, in any of those cases, the results will need to be transformed such that they come from your skewed distribution.
-----------
Dealing with random numbers is never as straightforward as people think. And when you want them from a skewed distribution, it gets even more complex.
Good Luck,
Elroy
6. ## Re: testtest
Ok, just seeing this. First, and foremost, 30 numbers ranging from 1 to 6 will never be
7. ## Re: testtest
be Uniform Random (which is what I believe you're suggesting).
8. ## Re: testtest
. They can be Skewed Random, which pulls us
9. ## Re: testtest
us into an entirely different area. For a set number of integers (say 30) with a specific range (say 1 to 6) to have
10. ## Re: testtest
have a Uniform Random distribution, the mid-point of the range (3.5) times the number of integers (30) must equal the total your after ...
11. ## Re: testtest
3.5 * 30 = 105 (not 100).
(not 100).
13. ## Re: testtest
after ... and 3.5 times 30 = 105
14. ## Re: testtest
Ok, just seeing this. First, and foremost, 30 numbers ranging from 1 to 6 will never be Uniform Random (which is what I believe you're suggesting). They can be Skewed Random, which pulls us into an entirely different area. For a set number of integers (say 30) with a specific range (say 1 to 6) to have a Uniform Random distribution, the mid-point of the range (3.5) times the number of integers (30) must equal the total your after ... and 3.5 times 30 = 105 (not 100).
Therefore, if you're truly wanting what you suggest, the first order of business is to figure out from which skewed distribution you're pulling your random number: Poisson, Gauss with a skew value, logarithmic, etc. There are many to choose from. Possibly you even want to bracket part of one of those skewed curves. And, from there, you can start devising your algorithm to always sum to a specific value.
Also, you will need to decide if the Rnd function is good enough for you, or whether you wish to use the more robust CryptGenRandom (or other) API call (where you get more robust random numbers). And, again, in any of those cases, the results will need to be transformed such that they come from your skewed distribution.
-----------
Dealing with random numbers is never as straightforward as people think. And when you want them from a skewed distribution, it gets even more complex.
Good Luck,
Elroy
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# How can you use trigonometric functions to simplify 26 e^( ( pi)/2 i ) into a non-exponential complex number?
$26 i$
$\textcolor{b l u e}{\rho {e}^{\theta i} = \rho \cos \left(\theta\right) + \rho i \sin \left(\theta\right)}$
$26 {e}^{\left(\frac{\pi}{2}\right) i} = 26 \cos \left(\frac{\pi}{2}\right) + 26 i \sin \left(\frac{\pi}{2}\right)$
$= 26 \cdot 0 + 26 i \cdot 1 = 26 i$
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## Networks as Vector Spaces
Consider the network below.
We can consider the closed loop bdef to be the sum of the loops bcf and cde. Notice that c is taken in the opposite sense for these loops so addition cancels out the arc c.
We can represent each arc taken anticlockwise about a point P by a vector.
$a= \begin{pmatrix}1\\0\\0\\0\\0\\0\\0\end{pmatrix}, \: b= \begin{pmatrix}0\\1\\0\\0\\0\\0\\0\end{pmatrix}, \: c= \begin{pmatrix}0\\0\\1\\0\\0\\0\\0\end{pmatrix}, \: d= \begin{pmatrix}0\\0\\0\\1\\0\\0\\0\end{pmatrix}, \: e= \begin{pmatrix}0\\0\\0\\0\\1\\0\\0\end{pmatrix}, \: f= \begin{pmatrix}0\\0\\0\\0\\0\\1\\0\end{pmatrix}, \: g= \begin{pmatrix}0\\0\\0\\0\\0\\0\\1\end{pmatrix}$
Then arc bcf is
$-b-c+f= \begin{pmatrix}0\\-1\\-1\\0\\0\\1\\0\end{pmatrix}$
and the arc cde by the vector
$c-d-e= \begin{pmatrix}0\\0\\1\\-1\\-1\\0\\0\end{pmatrix}$
$-b-c+f+c-d-e=-b+f-d-e= \begin{pmatrix}0\\0\\-1\\0\\-1\\-1\\1\end{pmatrix}$
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# Search by Topic
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### Crossings
##### Stage: 2 Challenge Level:
In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?
### Number Tracks
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### Got it for Two
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Got It game for an adult and child. How can you play so that you know you will always win?
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Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?
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This challenge encourages you to explore dividing a three-digit number by a single-digit number.
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Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.
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Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
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This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?
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Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
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This task follows on from Build it Up and takes the ideas into three dimensions!
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Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
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If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?
### Centred Squares
##### Stage: 2 Challenge Level:
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
### Magic Vs
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Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
### Up and Down Staircases
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One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?
### Domino Numbers
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Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
### Tiling
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An investigation that gives you the opportunity to make and justify predictions.
### Journeys in Numberland
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Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
### Play to 37
##### Stage: 2 Challenge Level:
In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.
### Odd Squares
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Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?
### Snake Coils
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This challenge asks you to imagine a snake coiling on itself.
### What Could it Be?
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In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?
### Magic Constants
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In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
### Cut it Out
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Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?
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Find the sum of all three-digit numbers each of whose digits is odd.
### Round and Round the Circle
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What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.
### Build it Up
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Can you find all the ways to get 15 at the top of this triangle of numbers?
### Doplication
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We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?
### Always, Sometimes or Never? Number
##### Stage: 2 Challenge Level:
Are these statements always true, sometimes true or never true?
### Magic Circles
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Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?
### Walking the Squares
##### Stage: 2 Challenge Level:
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
### Always, Sometimes or Never?
##### Stage: 1 and 2 Challenge Level:
Are these statements relating to odd and even numbers always true, sometimes true or never true?
### Strike it Out for Two
##### Stage: 1 and 2 Challenge Level:
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
### Calendar Calculations
##### Stage: 2 Challenge Level:
Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?
### The Add and Take-away Path
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Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?
### Roll over the Dice
##### Stage: 2 Challenge Level:
Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?
### Area and Perimeter
##### Stage: 2 Challenge Level:
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
### Round the Three Dice
##### Stage: 2 Challenge Level:
What happens when you round these three-digit numbers to the nearest 100?
### Round the Two Dice
##### Stage: 1 Challenge Level:
This activity focuses on rounding to the nearest 10.
### Round the Four Dice
##### Stage: 2 Challenge Level:
This activity involves rounding four-digit numbers to the nearest thousand.
### Nim-7 for Two
##### Stage: 1 and 2 Challenge Level:
Nim-7 game for an adult and child. Who will be the one to take the last counter?
### Dice Stairs
##### Stage: 2 Challenge Level:
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
### Unit Differences
##### Stage: 1 Challenge Level:
This challenge is about finding the difference between numbers which have the same tens digit.
### Polygonals
##### Stage: 2 Challenge Level:
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
### Fault-free Rectangles
##### Stage: 2 Challenge Level:
Find out what a "fault-free" rectangle is and try to make some of your own.
### Spirals, Spirals
##### Stage: 2 Challenge Level:
Here are two kinds of spirals for you to explore. What do you notice?
### Dotty Circle
##### Stage: 2 Challenge Level:
Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?
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Convert k℧ to M℧ (Kilomho to Megamho)
## Kilomho into Megamho
numbers in scientific notation
https://www.convert-measurement-units.com/convert+Kilomho+to+Megamho.php
## How many Megamho make 1 Kilomho?
1 Kilomho [k℧] = 0.001 Megamho [M℧] - Measurement calculator that can be used to convert Kilomho to Megamho, among others.
# Convert Kilomho to Megamho (k℧ to M℧):
1. Choose the right category from the selection list, in this case 'Electric conductance'.
2. Next enter the value you want to convert. The basic operations of arithmetic: addition (+), subtraction (-), multiplication (*, x), division (/, :, ÷), exponent (^), brackets and π (pi) are all permitted at this point.
3. From the selection list, choose the unit that corresponds to the value you want to convert, in this case 'Kilomho [k℧]'.
4. Finally choose the unit you want the value to be converted to, in this case 'Megamho [M℧]'.
5. Then, when the result appears, there is still the possibility of rounding it to a specific number of decimal places, whenever it makes sense to do so.
With this calculator, it is possible to enter the value to be converted together with the original measurement unit; for example, '660 Kilomho'. In so doing, either the full name of the unit or its abbreviation can be usedas an example, either 'Kilomho' or 'k℧'. Then, the calculator determines the category of the measurement unit of measure that is to be converted, in this case 'Electric conductance'. After that, it converts the entered value into all of the appropriate units known to it. In the resulting list, you will be sure also to find the conversion you originally sought. Alternatively, the value to be converted can be entered as follows: '91 k℧ to M℧' or '12 k℧ into M℧' or '19 Kilomho -> Megamho' or '26 k℧ = M℧' or '83 Kilomho to M℧' or '13 k℧ to Megamho' or '68 Kilomho into Megamho'. For this alternative, the calculator also figures out immediately into which unit the original value is specifically to be converted. Regardless which of these possibilities one uses, it saves one the cumbersome search for the appropriate listing in long selection lists with myriad categories and countless supported units. All of that is taken over for us by the calculator and it gets the job done in a fraction of a second.
Furthermore, the calculator makes it possible to use mathematical expressions. As a result, not only can numbers be reckoned with one another, such as, for example, '(27 * 68) k℧'. But different units of measurement can also be coupled with one another directly in the conversion. That could, for example, look like this: '660 Kilomho + 1980 Megamho' or '61mm x 87cm x 34dm = ? cm^3'. The units of measure combined in this way naturally have to fit together and make sense in the combination in question.
If a check mark has been placed next to 'Numbers in scientific notation', the answer will appear as an exponential. For example, 1.709 363 070 864 5×1031. For this form of presentation, the number will be segmented into an exponent, here 31, and the actual number, here 1.709 363 070 864 5. For devices on which the possibilities for displaying numbers are limited, such as for example, pocket calculators, one also finds the way of writing numbers as 1.709 363 070 864 5E+31. In particular, this makes very large and very small numbers easier to read. If a check mark has not been placed at this spot, then the result is given in the customary way of writing numbers. For the above example, it would then look like this: 17 093 630 708 645 000 000 000 000 000 000. Independent of the presentation of the results, the maximum precision of this calculator is 14 places. That should be precise enough for most applications.
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### Home > PC3 > Chapter 8 > Lesson 8.3.2 > Problem8-106
8-106.
Given $\vec { \text{m} } = \langle - 7 , - 1 \rangle$ and $\vec { \text{n} } = \langle 5,2 \rangle$, calculate $\arg\left(\vec{\text{p}}\right)$ and $|| \vec { \text{p} } | |$ where $\vec { \text{p} } = 6 \vec { \text{m} } - 8 \vec { \text{n} }$.
Use the Pythagorean Theorem to determine the length of vector $\text{p}$.
$\arg\left(\vec{\text{p}}\right)$ is the direction. Be sure to consider which quadrant the vector is in when giving your final answer.
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# Proof of matrix conjugate (for the complex numbers)
1. Sep 6, 2009
### philnow
1. The problem statement, all variables and given/known data
Supposing that A*B is defined (where A and B are both matrices in the field of the complex numbers), show that the conjugate of matrix A * the conjugate of matrix B is equal to the conjugate of A*B.
2. Relevant equations
None.
3. The attempt at a solution
I'm stuck. I've already shown that for 2 complex numbers z1 and z2, the conjugate of z1 + the conjugate of z2 is equal to the conjugate of (z1+z2). I've also shown that the conjugate of z1 * the conjugate of z2 = the conjugate of (z1*z2). My prof says to use the above to help with the proof.
I'm quite inexperienced with proofs, so any hint or tip would be extremely appreciated. Thanks.
2. Sep 6, 2009
### lanedance
you could try writing out th sum as elements
Ie say you have
C = AB
then for and elemnt of C at row i, & column j, each cij is given by the sum
cij = (sum over k) aikbkj
This reduces the matrix multiplication to addition & multiplication of individual complex numbers
3. Sep 7, 2009
### HallsofIvy
Staff Emeritus
How, exactly, is your "conjugate" defined? The conjugate of a linear operator, A, on an innerproduct space over the complex numbers is defined as the linear operator A* such that, for all vectors u, v, <Au, v>= <u, A*v> where < , > is the inner product. It is easy to show that if A and B are linear operators, <ABu, v>= <Bu, A*v>= <u, B*A*v> so that B*A*= (AB)*.
If you have defined the conjugate of a matrix as "the matrix you get by swapping rows and columns and taking the complex cojugate of the matrix" (the complex conjugate of the transpose), then it would be useful to prove that <Au, v>= (Au)v= u(A*v)= <u, A*v> for row vector u and column vector v.
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You are on page 1of 3
# MIDTERM EXAM
## INTRODUCTIION TO MANAGEMENT SCIENCE
Question # 1 (6 points):
Find the complete optimal solution to this Linear Programming problem.
Subject to
12x1 + 4x2 >= 48
10x1 + 5x2 >= 50
4x1 + 8x2 >= 32
x1, x2 >= 0
## Question # 2 (12 points):
A company produces tools at two plants and sells them to three customers. The cost of
producing 1000 tools at a plant and shipping them to a customer is given in Table below:
## Plant 2 \$130 \$70 \$170
Customers 1 and 3 pay \$200 per thousand tools; customer 2 pays \$150 per thousand
tools. To produce 1000 tools at plant 1, 200 hours of labour are needed, while 300 hours
are needed at plant 2. A total of 5500 hours of labour are available for use at the two
plants. Additional labour hours can be purchased at \$20 per labour hour. Plant 1 can
produce up to 10,000 tools and plant 2, up to 12,000 tools. Demand by each customer is
assumed unlimited.
If we let Xij = number of tools (in thousands) produced at plant i and shipped to
customer j, and L = number of additional hour purchased.
## Max. Z = 140 X11 + 120 X12 + 40 X13 + 70 X21 + 80 X22 + 30 X23 - 20 L
Subject to
C1 1 X11 + 1 X12 + 1 X13 <= 10
C2 1 X21 + 1 X22 + 1 X23 <= 12
C3 200 X11 + 200 X12 + 200 X13 + 300 X21 + 300 X22
+ 300 X23 - 1 L <= 5500
2
a) If it costs \$70 to produce 1000 tools at plant 1 and ship them to customer 1, what
would be the new solution to the problem and the profit? (6 points)
b) If the price of an additional hour of labor were reduced to \$4, would the company
purchase any additional labor? (6 points)
c) A consultant offers to increase plant 1’s production capacity by 5000 tools for a cost of
\$400. Should the company take the offer? (6 points)
d) If the company were given 5 extra hours of labor, what would the profit become? (6
points)
## Question # 3 (12 points):
Margaret Young’s family owns five panels of farmland broken into a southwest
sector, north sector, west sector, and southwest sector. Young is involved primarily
in growing wheat, alfalfa, and barley crops and is currently preparing her
production plan for next year. The Pennsylvania Water Authority has just
announced its yearly water allotment, with the Young farm receiving 7,400 acre-
feet. Each parcel can only tolerate a specified amount of irrigation per growing
season, as specified below:
## Southeast 2000 3200
North 2300 3400
Northwest 600 800
West 1100 500
Southwest 500 600
Each of Young’s crops needs a minimum amount of water per acre and there is a
projected limit of each crop. Crop data follow:
feet)
## Wheat 110,000 bushels 1.6
Alfalfa 1,800 tons 2.9
Barley 2,200 tons 3.5
Young’s best estimate is that she can sell wheat at a net profit of \$2 per bushel, alfalfa at
\$40 per ton, and barley at \$50 per ton. One acre of land yields an average of 1.5 tons of
alfalfa and 2.2 tons of barley. The wheat yield is approximately 50 bushels per acre.
Formulate Young’s production plan. (Define the decision variables, objective function
and the constraints).
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### Home > CCG > Chapter 7 > Lesson 7.1.3 > Problem7-34
7-34.
How long is the longest line segment that will fit inside a square of area $50$ square units? Show all work.
Draw a diagram. A square labeled, A, = 50.
Area of a square = s², where s = length of a side. Side labels added to square, on top & left sides, square root of 50.
${s}=\sqrt{50}$
Use the Pythagorean Theorem to solve for $c$. Added to square, diagonal from, bottom left to top right, labeled, c.
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Worksheet Solution: Money
# Worksheet Solution: Money Notes | Study Mathematics for Class 5 - Class 5
## Document Description: Worksheet Solution: Money for Class 5 2022 is part of Mathematics for Class 5 preparation. The notes and questions for Worksheet Solution: Money have been prepared according to the Class 5 exam syllabus. Information about Worksheet Solution: Money covers topics like and Worksheet Solution: Money Example, for Class 5 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Worksheet Solution: Money.
Introduction of Worksheet Solution: Money in English is available as part of our Mathematics for Class 5 for Class 5 & Worksheet Solution: Money in Hindi for Mathematics for Class 5 course. Download more important topics related with notes, lectures and mock test series for Class 5 Exam by signing up for free. Class 5: Worksheet Solution: Money Notes | Study Mathematics for Class 5 - Class 5
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Question 1: Write in the short form:
• 80 paise = ₹ 0.80
• 70 rupees 5 paise ₹ 70.05
Question 2: Convert into paise:
• 15 rupees = 1500 paise
• ₹ 0.95 = 95 paise
• ₹ 56.40 – ₹ 27.35 ₹29.05
• ₹ 8.50 + ₹ 9.10 = ₹17.60
• ₹ 72.40 and ₹ 36.70 = ₹109.10
• ₹ 315.75, ₹ 45.90 and ₹ 56.95 = ₹418.6
• ₹ 40.35, ₹ 65.45 and ₹ 105.05 = ₹210.85
Question 5: Subtract:
• ₹ 35.32 from ₹ 51.20 = 15.88
• ₹ 182.35 from ₹ 206.75 = ₹ 24.4
• ₹ 106.05 from ₹ 190.32 = ₹ 84.27
Question 6: Multiply:
• ₹ 4032 × 20 = ₹ 80640
• ₹ 3120 × 18 = ₹ 56160
• ₹ 999.90 × 9 = ₹ 8999.1
Question 7: Divide:
• ₹ 8464.50 ÷ 9 ₹ 940.5
• ₹ 8222.40 ÷ 18 = ₹ 456.8
• ₹ 1752.80 ÷ 14 = ₹ 125.2
Question 8: Prices of items in a general store are given below:
using the list prepare a bill or each of the following purchases:
• 2 kg atta, 1 litre refined oil, 200 g soap, 1 kg sugar, 200 g namkeen.
= 14 + 14 + 65 + 21.50 + 14 = ₹ 128.50
• 4 kg rice, 2 kg daal chana), 1 kg sugar, 2 packets of biscuits, 1 litre refined oil, 150 g face cream.
= 18 + 18 + 18 + 18 + 30 + 30 + 21.50 + 12 + 12 + 65 + 26.50 = ₹ 269
• 3 kg rice, 1 kg daal chana, 1 kg daal moong, 1 kg salt, 2 packets of biscuits.
= 18 + 18 + 18 + 30 + 8.50 + 12 + 12 = ₹ 116.5
Question 9: I have a 50-rupee note. I bu an ice-cream for ₹ 15.75 and a chocolate for ₹ 18.30. How much do I have left with me?
Spent on Icecream = ₹ 15.75
Spent on chocolate = ₹ 18.30
Total money spent = ₹ 15.75 + ₹ 18.30 = 34.05
Money left = 50.00 - 34.05 = ₹15.95
Question 10: Mona purchased a ribbon for ₹ 15.75, some nailpolish for ₹ 32.40 and bangles for ₹ 40.20. She gave a 100-rupee note to the shopkeeper. How much mone did she get back?
Spent on ribbon = ₹ 15.75
Spent on Nailpolish = ₹ 32.40
Spent on bangles = ₹ 40.20
Total money spent = ₹ 15.75 + ₹ 32.40 + ₹ 40.20 = ₹88.17
Money left = ₹ 100 - 88.17 = ₹18.86
• ₹ 137.45 + ₹ 38.96 + ₹ 45.24 = ₹221.65
• ₹ 1629.45 + ₹ 459.23 + ₹ 385.09 = ₹2473.77
Question 12: Subtract the following:
• ₹ 784.98 – ₹ 35.65 = ₹749.33
• ₹ 765.95 – ₹ 395.98 = ₹369.97
• ₹ 739.94 – ₹ 364.73 = ₹375.21
Question 13: Multiply:
• ₹ 525.25 by 6 = ₹1575.75
• ₹ 975.12 by 6 = ₹5850.72
• ₹ 67.86 by 7 = ₹475.02
Question 14: Divide:
• ₹ 96.36 by 3 = ₹32.12
• ₹ 533.35 by 5 = ₹106.67
• ₹ 211.84 by 4 = ₹52.96
Question 15: Make out a bill for the following purchases:
• Rice ₹ 16.00, Refined oil — ₹ 55.50, Sugar ₹ 17.50, Soap ₹ 9.25, Toothpaste ₹ 18.50.
₹ 16.00 + ₹ 55.50 + ₹ 17.50 + ₹ 9.25 + ₹ 18.50 = 116.75
• Aksha bought a fountain pen for ₹ 11.25 and a ball-point pen for ₹ 5.00. He gave a twenty -rupee note to the shopkeeper. How much money should he get back?
Aksha bought a fountain pen for = ₹ 11.25
Aksha bought a ball-point pen for = ₹ 5.00.
Total money spent = ₹ 11.25 + ₹ 5.00 = 16.25
Money he gets back = 20 - 16.25 = 3.75
• Bhaskar has bought a notebook for ₹ 8.50 and a ball pen for ₹ 4.25. He has onl a ten rupee note. B how much is he running short?
Bhaskar has bought a notebook for = ₹ 8.50
Bhaskar has bought a ball pen for = ₹ 4.25
Total money = ₹12.75
Money he has = ₹10
Money he is running short of = 12.75 - 10 = ₹ 2.75
Question 16: In short form 12 rupees 5 paise can be written as 12.05.
Question 17: ₹ 118.05 = 118 rupees 5 paise.
Question 18: How many 5-rupee notes can be exchanged for a 1000-rupee note? ______.
1000/5 = 200
Question 19: ₹ 25.40 + ₹ 15.60 = ₹ 41.
Question 20: 650 rupees 90 paise = ₹ 650.90.
The document Worksheet Solution: Money Notes | Study Mathematics for Class 5 - Class 5 is a part of the Class 5 Course Mathematics for Class 5.
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## 3.4 Systems of Linear Equations
Consider the system of two linear equations: \begin{align} x+y & =1,\tag{3.1}\\ 2x-y & =1.\tag{3.2} \end{align} As shown in Figure 3.1, equations (3.1) and (3.2) represent two straight lines which intersect at the point $$x=2/3$$ and $$y=1/3$$. This point of intersection is determined by solving for the values of $$x$$ and $$y$$ such that $$x+y=2x-y$$14
The two linear equations can be written in matrix form as: $\left[\begin{array}{cc} 1 & 1\\ 2 & -1 \end{array}\right]\left[\begin{array}{c} x\\ y \end{array}\right]=\left[\begin{array}{c} 1\\ 1 \end{array}\right],$ or, $\mathbf{A\cdot z=b,}$ where, $\mathbf{A}=\left[\begin{array}{cc} 1 & 1\\ 2 & -1 \end{array}\right],~\mathbf{z}=\left[\begin{array}{c} x\\ y \end{array}\right]\textrm{ and }\mathbf{b}=\left[\begin{array}{c} 1\\ 1 \end{array}\right].$
If there was a $$(2\times2)$$ matrix $$\mathbf{B},$$ with elements $$b_{ij}$$, such that $$\mathbf{B\cdot A=I}_{2},$$ where $$\mathbf{I}_{2}$$ is the $$(2\times2)$$ identity matrix, then we could solve for the elements in $$\mathbf{z}$$ as follows. In the equation $$\mathbf{A\cdot z=b}$$, pre-multiply both sides by $$\mathbf{B}$$ to give: \begin{align*} \mathbf{B\cdot A\cdot z} & =\mathbf{B\cdot b}\\ & \Longrightarrow\mathbf{I\cdot z=B\cdot b}\\ & \Longrightarrow\mathbf{z=B\cdot b}, \end{align*} or $\left[\begin{array}{c} x\\ y \end{array}\right]=\left[\begin{array}{cc} b_{11} & b_{12}\\ b_{21} & b_{22} \end{array}\right]\left[\begin{array}{c} 1\\ 1 \end{array}\right]=\left[\begin{array}{c} b_{11}\cdot1+b_{12}\cdot1\\ b_{21}\cdot1+b_{22}\cdot1 \end{array}\right].$ If such a matrix $$\mathbf{B}$$ exists it is called the inverse of $$\mathbf{A}$$ and is denoted $$\mathbf{A}^{-1}$$. Intuitively, the inverse matrix $$\mathbf{A}^{-1}$$ plays a similar role as the inverse of a number. Suppose $$a$$ is a number; e.g., $$a=2$$. Then we know that $$(1/a)\cdot a=a^{-1}a=1$$. Similarly, in matrix algebra $$\mathbf{A}^{-1}\mathbf{A=I}_{2}$$ where $$\mathbf{I}_{2}$$ is the identity matrix. Now, consider solving the equation $$a\cdot x=b$$. By simple division we have that $$x=(1/a)\cdot b=a^{-1}\cdot b$$. Similarly, in matrix algebra, if we want to solve the system of linear equations $$\mathbf{Ax=b}$$ we pre-multiply by $$\mathbf{A}^{-1}$$ and get the solution $$\mathbf{x=A}^{-1}\mathbf{b}$$.
Using $$\mathbf{B=A}^{-1}$$, we may express the solution for $$\mathbf{z}$$ as: $\mathbf{z=A}^{-1}\mathbf{b}.$ As long as we can determine the elements in $$\mathbf{A}^{-1}$$ then we can solve for the values of $$x$$ and $$y$$ in the vector $$\mathbf{z}$$. The system of linear equations has a solution as long as the two lines intersect, so we can determine the elements in $$\mathbf{A}^{-1}$$ provided the two lines are not parallel. If the two lines are parallel, then one of the equations is a multiple of the other. In this case, we say that $$\mathbf{A}$$ is not invertible.
There are general numerical algorithms for finding the elements of $$\mathbf{A}^{-1}$$ (e.g., Gaussian elimination) and R has these algorithms available. However, if $$\mathbf{A}$$ is a $$(2\times2)$$ matrix then there is a simple formula for $$\mathbf{A}^{-1}$$. Let $$\mathbf{A}$$ be a $$(2\times2)$$ matrix such that: $\mathbf{A}=\left[\begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22} \end{array}\right].$ Then, $\mathbf{A}^{-1}=\frac{1}{\det(\mathbf{A})}\left[\begin{array}{cc} a_{22} & -a_{12}\\ -a_{21} & a_{11} \end{array}\right],$ where $$\det(\mathbf{A})=a_{11}a_{22}-a_{21}a_{12}$$ denotes the determinant of $$\mathbf{A}$$ and is assumed to be not equal to zero. By brute force matrix multiplication we can verify this formula: \begin{align*} \mathbf{A}^{-1}\mathbf{A} & =\frac{1}{a_{11}a_{22}-a_{21}a_{12}}\left[\begin{array}{cc} a_{22} & -a_{12}\\ -a_{21} & a_{11} \end{array}\right]\left[\begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22} \end{array}\right]\\ & =\frac{1}{a_{11}a_{22}-a_{21}a_{12}}\left[\begin{array}{cc} a_{22}a_{11}-a_{12}a_{21} & a_{22}a_{12}-a_{12}a_{22}\\ -a_{21}a_{11}+a_{11}a_{21} & -a_{21}a_{12}+a_{11}a_{22} \end{array}\right]\\ & =\frac{1}{a_{11}a_{22}-a_{21}a_{12}}\left[\begin{array}{cc} a_{22}a_{11}-a_{12}a_{21} & 0\\ 0 & -a_{21}a_{12}+a_{11}a_{22} \end{array}\right]\\ & =\left[\begin{array}{cc} \dfrac{a_{22}a_{11}-a_{12}a_{21}}{a_{11}a_{22}-a_{21}a_{12}} & 0\\ 0 & \dfrac{-a_{21}a_{12}+a_{11}a_{22}}{a_{11}a_{22}-a_{21}a_{12}} \end{array}\right]\\ & =\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right]. \end{align*} Let’s apply the above rule to find the inverse of $$\mathbf{A}$$ in our example linear system (3.1)-(3.2): $\mathbf{A}^{-1}=\frac{1}{-1-2}\left[\begin{array}{cc} -1 & -1\\ -2 & 1 \end{array}\right]=\left[\begin{array}{cc} \frac{1}{3} & \frac{1}{3}\\ \frac{2}{3} & \frac{-1}{3} \end{array}\right].$ Notice that, $\mathbf{A}^{-1}\mathbf{A}=\left[\begin{array}{cc} \frac{1}{3} & \frac{1}{3}\\ \frac{2}{3} & \frac{-1}{3} \end{array}\right]\left[\begin{array}{cc} 1 & 1\\ 2 & -1 \end{array}\right]=\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right].$ Our solution for $$\mathbf{z}$$ is then, \begin{align*} \mathbf{z} & =\mathbf{A}^{-1}\mathbf{b}\\ & =\left[\begin{array}{cc} \frac{1}{3} & \frac{1}{3}\\ \frac{2}{3} & \frac{-1}{3} \end{array}\right]\left[\begin{array}{c} 1\\ 1 \end{array}\right]\\ & =\left[\begin{array}{c} \frac{2}{3}\\ \frac{1}{3} \end{array}\right]=\left[\begin{array}{c} x\\ y \end{array}\right], \end{align*} so that $$x=2/3$$ and $$y=1/3$$.
Example 2.20 (Solving systems of linear equations in R)
In R, the solve() function is used to compute the inverse of a matrix and solve a system of linear equations. The linear system $$x+y=1$$ and $$2x-y=1$$ can be represented using:
matA = matrix(c(1,1,2,-1), 2, 2, byrow=TRUE)
vecB = c(1,1)
First we solve for $$\mathbf{A}^{-1}$$:15
matA.inv = solve(matA)
matA.inv
## [,1] [,2]
## [1,] 0.3333333 0.3333333
## [2,] 0.6666667 -0.3333333
matA.inv%*%matA
## [,1] [,2]
## [1,] 1 -5.551115e-17
## [2,] 0 1.000000e+00
matA%*%matA.inv
## [,1] [,2]
## [1,] 1 5.551115e-17
## [2,] 0 1.000000e+00
Then we solve the system $$\mathbf{z}=\mathbf{A}^{-1}\mathbf{b}$$:
z = matA.inv%*%vecB
z
## [,1]
## [1,] 0.6666667
## [2,] 0.3333333
$$\blacksquare$$
In general, if we have $$n$$ linear equations in $$n$$ unknown variables we may write the system of equations as \begin{align*} a_{11}x_{1}+a_{12}x_{2}+\cdots+a_{1n}x_{n} & =b_{1}\\ a_{21}x_{1}+a_{22}x_{2}+\cdots+a_{2n}x_{n} & =b_{2}\\ \vdots & =\vdots\\ a_{n1}x_{1}+a_{n2}x_{2}+\cdots+a_{nn}x_{n} & =b_{n} \end{align*} which we may then express in matrix form as $\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{array}\right]=\left[\begin{array}{c} b_{1}\\ b_{2}\\ \vdots\\ b_{n} \end{array}\right]$ or $\underset{(n\times n)}{\mathbf{A}}\cdot\underset{(n\times1)}{\mathbf{x}}=\underset{(n\times1)}{\mathbf{b}}.$ The solution to the system of equations is given by: $\mathbf{x=A}^{-1}\mathbf{b},$ where $$\mathbf{A}^{-1}\mathbf{A}=\mathbf{I}_n$$ and $$\mathbf{I}_{n}$$ is the $$(n\times n)$$ identity matrix. If the number of equations is greater than two, then we generally use numerical algorithms to find the elements in $$\mathbf{A}^{-1}$$.
### 3.4.1 Rank of a matrix
A $$n\times m$$ matrix $$\mathbf{A}$$ has $$m$$ columns $$\mathbf{a}_{1},\mathbf{a}_{2},\ldots,\mathbf{a}_{m}$$ where each column is an $$n\times1$$ vector and $$n$$ rows where each row is an $$1\times m$$ row vector: $\underset{(n\times m)}{\mathbf{A}}=\left[\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1m}\\ a_{21} & a_{22} & \ldots & a_{2m}\\ \vdots & \vdots & \ddots & \vdots\\ a_{n1} & a_{n2} & \ldots & a_{nm} \end{array}\right]=\left[\begin{array}{cccc} \mathbf{a}_{1} & \mathbf{a}_{2} & \ldots & \mathbf{a}_{m}\end{array}\right].$ Two vectors $$\mathbf{a}_{1}$$ and $$\mathbf{a}_{2}$$ are linearly independent if $$c_{1}\mathbf{a}_{1}+c_{2}\mathbf{a}_{2}=0$$ implies that $$c_{1}=c_{2}=0$$. A set of vectors $$\mathbf{a}_{1},\mathbf{a}_{2},\ldots,\mathbf{a}_{m}$$ are linearly independent if $$c_{1}\mathbf{a}_{1}+c_{2}\mathbf{a}_{2}+\cdots+c_{m}\mathbf{a}_{m}=0$$ implies that $$c_{1}=c_{2}=\cdots=c_{m}=0$$. That is, no vector $$\mathbf{a}_{i}$$ can be expressed as a non-trivial linear combination of the other vectors.
The column rank of the $$n\times m$$ matrix $$\mathbf{A}$$, denoted $$\mathrm{rank}(\mathbf{A})$$, is equal to the maximum number of linearly independent columns. The row rank of a matrix is equal to the maximum number of linearly independent rows, and is given by $$\mathrm{rank}(\mathbf{A}^{\prime})$$. It turns out that the column rank and row rank of a matrix are the same. Hence, $$\mathrm{rank}(\mathbf{A})$$$$=\mathrm{rank}(\mathbf{A}^{\prime})\leq\mathrm{min}(m,n)$$. If $$\mathrm{rank}(\mathbf{A})=m$$ then $$\mathbf{A}$$ is called full column rank, and if $$\mathrm{rank}(\mathbf{A})=n$$ then $$\mathbf{A}$$ is called full row rank. If $$\mathbf{A}$$ is not full rank then it is called reduced rank.
Example 3.6 (Determining the rank of a matrix in R)
Consider the $$2\times3$$ matrix $\mathbf{A}=\left(\begin{array}{ccc} 1 & 3 & 5\\ 2 & 4 & 6 \end{array}\right)$ Here, $$\mathrm{rank}(\mathbf{A})\leq min(3,2)=2$$, the number of rows. Now, $$\mathrm{rank}(\mathbf{A})=2$$ since the rows of $$\mathbf{A}$$ are linearly independent. In R, the rank of a matrix can be found using the Matrix function rankMatrix():
library(Matrix)
Amat = matrix(c(1,3,5,2,4,6), 2, 3, byrow=TRUE)
as.numeric(rankMatrix(Amat))
## [1] 2
$$\blacksquare$$
The rank of an $$n\times n$$ square matrix $$\mathbf{A}$$ is directly related to its invertibility. If $$\mathrm{rank}(\mathbf{A})=n$$ then $$\mathbf{A}^{-1}$$ exists. This result makes sense. Since $$\mathbf{A}^{-1}$$ is used to solve a system of $$n$$ linear equations in $$n$$ unknowns, it will have a unique solution as long as no equation in the system can be written as a linear combination of the other equations.
### 3.4.2 Partitioned matrices and partitioned inverses
Consider a general $$n\times m$$ matrix $$\mathbf{A}$$: $\underset{(n\times m)}{\mathbf{A}}=\left[\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1m}\\ a_{21} & a_{22} & \ldots & a_{2m}\\ \vdots & \vdots & \ddots & \vdots\\ a_{n1} & a_{n2} & \ldots & a_{nm} \end{array}\right].$ In some situations we might want to partition $$\mathbf{A}$$ into sub-matrices containing sub-blocks of the elements of $$\mathbf{A}$$: $\mathbf{A}=\left[\begin{array}{cc} \mathbf{A}_{11} & \mathbf{A}_{12}\\ \mathbf{A}_{21} & \mathbf{A}_{22} \end{array}\right],$ where the sub-matrices $$\mathbf{A}_{11}$$, $$\mathbf{A}_{12}$$, $$\mathbf{A}_{21}$$, and $$\mathbf{A}_{22}$$ contain the appropriate sub-elements of $$\mathbf{A}$$. For example, consider $\mathbf{A}=\left[\begin{array}{cc|cc} 1 & 2 & 3 & 4\\ 5 & 6 & 7 & 8\\ \hline 9 & 10 & 11 & 12\\ 13 & 14 & 15 & 16 \end{array}\right]=\left[\begin{array}{cc} \mathbf{A}_{11} & \mathbf{A}_{12}\\ \mathbf{A}_{21} & \mathbf{A}_{22} \end{array}\right],$ where $\begin{eqnarray*} \mathbf{A}_{11} & = & \left[\begin{array}{cc} 1 & 2\\ 5 & 6 \end{array}\right],\,\mathbf{A}_{12}=\left[\begin{array}{cc} 3 & 4\\ 7 & 8 \end{array}\right],\\ \mathbf{A}_{21} & = & \left[\begin{array}{cc} 9 & 10\\ 13 & 14 \end{array}\right],\,\mathbf{A}_{22}=\left[\begin{array}{cc} 11 & 12\\ 15 & 16 \end{array}\right]. \end{eqnarray*}$
Example 3.7 (Combining sub-matrices in R)
In R sub-matrices can be combined column-wise using cbind(), and row-wise using rbind(). For example, to create the matrix $$\mathbf{A}$$ from the sub-matrices use
A11mat = matrix(c(1,2,5,6), 2, 2, byrow=TRUE)
A12mat = matrix(c(3,4,7,8), 2, 2, byrow=TRUE)
A21mat = matrix(c(9,10,13,14), 2, 2, byrow=TRUE)
A22mat = matrix(c(11,12,15,16), 2, 2, byrow=TRUE)
Amat = rbind(cbind(A11mat, A12mat), cbind(A21mat, A22mat))
Amat
## [,1] [,2] [,3] [,4]
## [1,] 1 2 3 4
## [2,] 5 6 7 8
## [3,] 9 10 11 12
## [4,] 13 14 15 16
$$\blacksquare$$
The basic matrix operations work on sub-matrices in the obvious way. To illustrate, consider another matrix $$\mathbf{B}$$ that is conformable with $$\mathbf{A}$$ and partitioned in the same way: $\mathbf{B}=\left[\begin{array}{cc} \mathbf{B}_{11} & \mathbf{B}_{12}\\ \mathbf{B}_{21} & \mathbf{B}_{22} \end{array}\right].$ Then $\mathbf{A}+\mathbf{B}=\left[\begin{array}{cc} \mathbf{A}_{11} & \mathbf{A}_{12}\\ \mathbf{A}_{21} & \mathbf{A}_{22} \end{array}\right]+\left[\begin{array}{cc} \mathbf{B}_{11} & \mathbf{B}_{12}\\ \mathbf{B}_{21} & \mathbf{B}_{22} \end{array}\right]=\left[\begin{array}{cc} \mathbf{A}_{11}+\mathbf{B}_{11} & \mathbf{A}_{12}+\mathbf{B}_{12}\\ \mathbf{A}_{21}+\mathbf{B}_{21} & \mathbf{A}_{22}+\mathbf{B}_{22} \end{array}\right].$ If all of the sub-matrices of $$\mathbf{A}$$ and $$\mathbf{B}$$ are comformable then $\mathbf{AB}=\left[\begin{array}{cc} \mathbf{A}_{11} & \mathbf{A}_{12}\\ \mathbf{A}_{21} & \mathbf{A}_{22} \end{array}\right]\left[\begin{array}{cc} \mathbf{B}_{11} & \mathbf{B}_{12}\\ \mathbf{B}_{21} & \mathbf{B}_{22} \end{array}\right]=\left[\begin{array}{cc} \mathbf{A}_{11}\mathbf{B}_{11}+\mathbf{A}_{12}\mathbf{B}_{21} & \mathbf{A}_{11}\mathbf{B}_{12}+\mathbf{A}_{12}\mathbf{B}_{22}\\ \mathbf{A}_{21}\mathbf{B}_{11}+\mathbf{A}_{22}\mathbf{B}_{21} & \mathbf{A}_{21}\mathbf{B}_{12}+\mathbf{A}_{22}\mathbf{B}_{22} \end{array}\right].$
The transpose of a partitioned matrix satisfies $\mathbf{A^{\prime}}=\left[\begin{array}{cc} \mathbf{A}_{11} & \mathbf{A}_{12}\\ \mathbf{A}_{21} & \mathbf{A}_{22} \end{array}\right]^{\prime}=\left[\begin{array}{cc} \mathbf{A}_{11}^{\prime} & \mathbf{A}_{21}^{\prime}\\ \mathbf{A}_{12}^{\prime} & \mathbf{A}_{22}^{\prime} \end{array}\right].$ Notice the interchange of the two off-diagonal blocks.
A partitioned matrix $$\mathbf{A}$$ with appropriate invertible sub-matrices $$\mathbf{A}_{11}$$ and $$\mathbf{A}_{22}$$ has a partitioned inverse of the form $\mathbf{A}^{-1}=\left[\begin{array}{cc} \mathbf{A}_{11} & \mathbf{A}_{12}\\ \mathbf{A}_{21} & \mathbf{A}_{22} \end{array}\right]^{-1}=\left[\begin{array}{cc} \mathbf{A}_{11}^{-1}+\mathbf{A}_{11}^{-1}\mathbf{A}_{12}\mathbf{C}^{-1}\mathbf{A}_{21}\mathbf{A}_{11}^{-1} & -\mathbf{A}_{11}\mathbf{A}_{12}\mathbf{C}^{-1}\\ -\mathbf{C}^{-1}\mathbf{A}_{21}\mathbf{A}_{11}^{-1} & \mathbf{C}^{-1} \end{array}\right],$ where $$\mathbf{C}=\mathbf{A}_{22}-\mathbf{A}_{21}\mathbf{A}_{11}^{-1}\mathbf{A}_{12}$$. This formula can be verified by direct calculation.
1. Soving for $$x$$ gives $$x=2y$$. Substituting this value into the equation $$x+y=1$$ gives $$2y+y=1$$ and solving for $$y$$ gives $$y=1/3$$. Solving for $$x$$ then gives $$x=2/3$$.↩︎
2. Notice that the calculations in R do not show $$\mathbf{A}^{-1}\mathbf{A}=\mathbf{I}$$ exactly. The (1,2) element of $$\mathbf{A}^{-1}\mathbf{A}$$ is -5.552e-17, which for all practical purposes is zero. However, due to the limitations of machine calculations the result is not exactly zero.↩︎
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https://math.stackexchange.com/questions/1990679/approximation-on-partitions-in-l20-1-times-omega
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# Approximation on partitions in $L^2([0,1]\times \Omega)$
I’m working on Nualart’s book “The Malliavin calculus and related topics” and in the proof of lemma 1.1.3 he mentions that the operators $P_n$ have their operator norm bounded by 1. I fail to see why, can you help me? Using Jensen’s inequality I get a norm more akin to $2^n$, so I guess Jensen is too weak to prove that?
Quoting the proof:
Let $u$ be a process in $L^2_a([0,1]\times\Omega)$ ($L^2_a$ are the adapted processes w.r.t Brownian motion) and consider the sequence of processes defined by $\tilde u^n(t)=\sum_{i=1}^{2^n-1}2^n\left(\int_{(i-1)2^{-n}}^{i2^{-n}}u(s)ds\right)1_{]i2^{-n},(i+1)2^{-n}]}(t)$.
We claim that the sequence converges to $u$ in $L^2([0,1]\times\Omega)$. In fact define $P_n(u)=\tilde u^n$. Then $P_n$ is a linear operator in $L^2([0,1]\times\Omega)$ with norm bounded by one.
I believe that there are some typos. It should be \begin{align*} \tilde u^n(t)=\sum_{i=1}^{\color{red}{2^n}}2^n\left(\int_{(i-1)2^{-n}}^{i2^{-n}}u(s)ds\right)1_{]\color{red}{(i-1)2^{-n}, i2^{-n}}]}(t) \end{align*} Then, note that \begin{align*} E\left(\int_0^1\left(\tilde{u}^n\right)^2dt \right)&=\sum_{i=1}^{2^n}2^{2n}E\left(\int_{(i-1)2^{-n}}^{i2^{-n}}\left(\int_{(i-1)2^{-n}}^{i2^{-n}}u(s)ds\right)^2 dt\right)\\ &=\sum_{i=1}^{2^n}2^{n}E\left(\left(\int_{(i-1)2^{-n}}^{i2^{-n}}u(s)ds\right)^2 \right)\\ &\le \sum_{i=1}^{2^n}2^{n}E\left(\int_{(i-1)2^{-n}}^{i2^{-n}}1^2ds\int_{(i-1)2^{-n}}^{i2^{-n}}u^2(s)ds \right) \ \ \text{(by Schwarz inequality)}\\ &= \sum_{i=1}^{2^n}E\left(\int_{(i-1)2^{-n}}^{i2^{-n}}u^2(s)ds\right)\\ &= E\left(\sum_{i=1}^{2^n}\int_{(i-1)2^{-n}}^{i2^{-n}}u^2(s)ds\right)\\ &= E\left(\int_0^{1}u^2(s)ds \right)\\ &=||u||^2. \end{align*} Therefore, $P$ has norm 1.
• Can you explain the last inequality? How does the factor of $2^n$ vanish? Commented Nov 1, 2016 at 8:12
• I added more details. Commented Nov 1, 2016 at 12:43
• So Cauchy-Schwarz is the inequality needed, not Jensen… now I feel stupid. Thank you very much! Commented Nov 1, 2016 at 12:46
• @Franzo: Many thanks for the bounty. Commented Nov 1, 2016 at 20:46
• Enjoy! Thanks for the answer! Commented Nov 1, 2016 at 20:56
Jensen inequality works fine. Indeed (correcting the mistake spotted by Gordon):
$$\int_0^1 (\tilde{u}^n)^2 (t) dt = \int_0^1 \sum_{i=0}^{2^n-1}\left(\int_{(i-1)2^{-n}}^{i2^{-n}}u(s)2^nds\right)^2 1_{]i2^{-n},(i+1)2^{-n}]}(t) dt,$$
since all the caracteristic functions have disjoint support. Hence, using Jensen inequality (with the probability measures $2^n ds$ on each interval $]i2^{-n},(i+1)2^{-n}]$):
\begin{align}\|P_n u\|_{\mathbb{L}^2}^2 & \leq \int_0^1 \sum_{i=0}^{2^n-1}\left(\int_{(i-1)2^{-n}}^{i2^{-n}}u(s)^2 2^nds\right) 1_{]i2^{-n},(i+1)2^{-n}]}(t) dt \\ & = \sum_{i=0}^{2^n-1} \left(2^n\int_{(i-1)2^{-n}}^{i2^{-n}}u(s)^2 ds\right) \int_0^1 1_{]i2^{-n},(i+1)2^{-n}]}(t) dt \\ & = \sum_{i=0}^{2^n-1} \int_{(i-1)2^{-n}}^{i2^{-n}}u(s)^2 ds \\ & = \|u\|_{\mathbb{L}^2}^2. \end{align}
So, in this case, Jensen's inequality is not weaker; you just need to be careful on where you apply it.
Note, by the way, that this can be proved much faster. Fix $n\geq 0$. Let $\pi_n := \{]i2^{-n},(i+1)2^{-n}]: \ 0 \leq i < 2^n\}$, and $\mathcal{C}_n := \sigma (\pi_n)$ be the $\sigma$-algebra generated by $\pi_n$. Then:
$$P_n (u) = \mathbb{E} (u | \mathcal{C}_n),$$
and the conditional expectation is always a weak $\mathbb{L}^2$ contraction, which can be proved for instance with the conditional version of Jensen inequality:
$$\mathbb{E} (P_n (u)^2) = \mathbb{E} (\mathbb{E} (u | \mathcal{C}_n)^2) \leq \mathbb{E} (\mathbb{E} (u^2 | \mathcal{C}_n)) = \mathbb{E} (u^2).$$
This point of view makes more sense from a probabilist's standpoint, I think.
• Clever usage of Jensen here. Hope you don’t mind me giving the bounty to the first answer. Commented Nov 1, 2016 at 17:48
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# Similar to Longest Palindromic SubSequence
• First let us take a look at Longest Palindromic Subsequence Length problem,
We have to look at longest palindrome length subsequence available for (i,j).
We can break this problem into sub problems like (i, i+1, .... j-1, j). To find (i,j), we need to know data from positions in between
So, we start from smaller length of (i,j) to n
Base case: length 1, i.e. i==j
dp[ i ] [ j ] = 1
case 1: if (i==j),
i.e first letter and last letter are same,
case 1a: if len = 2, dp[i][j] = 2,
case 2a: if len!=2, dp[ i ][ j ] = dp[i+1] [ j -1] + 2, // problem becomes (i, i+1, ..... j, j-1)
case 2: if (i!=j), Math.max( dp[ i ] [ j-1], dp[ i+1 ][ j ] ) // max of dp (i+1 -> j) and dp (i -> j-1), since first and last letter are not same
dp[0][n-1] i.e the problem for (i,j) i=0. j=n-1, (0,n-1) gives the result
Solution:
``````public int longestPalindromeSubseq(String s) {
if (s==null || s.isEmpty()) { return 0;}
int n = s.length();
int[][] dp = new int[n][n];
//dp[i][j] is the length od longest palindromic subsequence for position i to j, inclusive
for (int i=0; i<n; i++) {
dp[i][i] = 1;
}
for (int len = 2; len <=n; len++) {
for (int i=0; i+len-1<n; i++) {
int j = i+len-1;
if (len == 2 && s.charAt(i) == s.charAt(j) ) {
dp[i][j] = 2;
}
else if (s.charAt(i) == s.charAt(j)) {
dp[i][j] = dp[i+1][j-1] +2;
}
else {
dp[i][j] = Math.max(dp[i][j-1], dp[i+1][j]);
}
}
}
return dp[0][n-1];
}
``````
Now, we can use the same idea here.
At each step, player 1 has the option of choosing between 2 cases:
For any game from position i -> j
1. player 1 chooses i, and leaving i+1 -> j to player 2
2. player 1 chooses j, and leaving i -> j-1 to player 2,
so, in effect we can calculate dp[i][j] for each (i,j) of length from 1 -> n
Base case: if i==j, i.e. for length 1, player 1 always has the first chance of scoring
``````dp[i][j] = nums[i]
``````
Dp formula (from above 2 cases):
``````dp[i][j] = Math.max (nums[i] - dp[i+1][j], nums[j] - dp[i][j-1])
``````
If, dp[0][n-1] >=0, then player 1 wins, i.e. solution for entire array of scores, (0,n-1)
Solution:
``````public boolean PredictTheWinner(int[] nums) {
int n = nums.length;
int[][] dp = new int[n][n];
for (int i = 0; i < n; i++) { dp[i][i] = nums[i]; }
for (int len = 2; len <= n; len++) {
for (int i = 0; i+len-1 < n ; i++) {
int j = i + len-1;
dp[i][j] = Math.max(nums[i] - dp[i + 1][j], nums[j] - dp[i][j - 1]);
}
}
return dp[0][n - 1] >= 0;
}
``````
In both problems, only the elements from and above the diagonal in the dp matrix is calulcated.
For len 1, the mid diagonal elements are calculated
len2, the diagonal elements above the mid-daigonal and so on, until it reaches the top-right, which gives the (0,n-1) value
Hope this is clear.
It will be good if these are added in similar problems.
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http://fouss08.pbworks.com/w/page/5023112/Decomposition-of-Fractions
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• If you are citizen of an European Union member nation, you may not use this service unless you are at least 16 years old.
View
# Decomposition-of-Fractions
last edited by 11 years, 7 months ago
# Chapter 7.3 - Partial Fraction Decomposition
OBJECTIVE
Fraction decomposition is used to take a fraction with a complex denominator and break it into smaller, linear denominator fractions. This method was first introduced by John Bernoulli (1667-1748) http://en.wikipedia.org/wiki/Johann_Bernoulli.
STEP BY STEP METHOD
0. (Only necessary if fraction is improper) If is an improper fraction (degree of N(x) is greater than D(x)) divide the two polynomials. If the numerator is 1 degree higher than the denominator, synthetic division can be used.
*Now take the and use the following steps to break it down.
1. Factor the denominator into linear factors and set each term as an individual quotient with variables A, B and C (if needed) as the denominators.
*If a linear term repeats, you must include all more simple forms. For instance x^2 must be represented with x and x^2
2. Multiply each term by the LCD (lowest common denominator) to eliminate the fractions.
3. Substitute x with a number that causes the A term to drop out and solve to determine a value for B
4. Repeat step for except make B drop out.
5. Use the factored form, individual quotients from step 2 and plug in the values for A and B. If step 0 applied, add the results to this step.
ADDITIONAL STEPS IF USING A, B, and C
As stated in the above steps, if a linear term repeats, you must include all more simple forms. So if the denominator is , you must include x, x +1, and . When this occurs two terms may drop out instead of one. To figure out what the terms are, you must instead compare the two sides of the equation. If there is on one side and on the other (and no other squared terms), A must be equal to 5. Use this method to figure out the other terms, pluging in the ones you've figured out to help. Again, this is only necessary if more than one of the terms drop out.
DECOMPOSITION IN ACTION
Multiply both side of the above equation by (x + 1)2, and simplify to obtain an equation of the form
1 - 2x = A(x + 1) + B
Expand the right side and group like terms
-2x + 1 = A x + (A + B)
Match up the variables with the polynomial in its place on the other side of the equals sign [(-2x) and (Ax), (A+B) and 1].For the right and left polynomials to be equal we need to have
- 2 = A and 1 = A + B
Solve the above system to obtain
A = - 2 and B = 3
EXTRAS
Partial fraction decomposition generator. Simply enter the fraction (use ^ for powers, * for multiplication, / for division) and click 'Partial Fractions'.
http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=algebra&s2=partial_fractions&s3=basic
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# Calc 2 equations of sphere
1. Apr 23, 2010
### somebodyelse5
Find the equation of a sphere if one of its diameters has endpoints: (-1, -5, -8) and (3, -1, -4)
Ok, heres what I have so far, I cant find the radius and im not sure if the rest of the equation is correct. Havin some issues with this one.
(x-1)^2+(y+2)^2+(z+2)^2-?=0
2. Apr 23, 2010
### gabbagabbahey
Hint: You are given the endpoints of one of the diameters, shouldn't the center of the sphere be halfway between the two endpoints?
3. Apr 23, 2010
### somebodyelse5
So I subtract my vectors, and get (4,-4,-4) correct?
Then I divide that vector by 2 and get (2, -2, -2) Correct?
So then my radius should be 2
So that means my equation should be
(x-2)^2+(y+2)^2+(z+2)^2-2^2=0
But when i enter that its wrong.
4. Apr 23, 2010
### gabbagabbahey
First, double check your negative signs. Second, what quantity does subtracting the two position vector really give you?
5. Apr 23, 2010
### somebodyelse5
Ok, Im having a really difficult time doing the simplest part of this problem.
So, by subtracting the two points, I get the diameter of the sphere, then dividing it by 2 gives me the radius.
Redid it and got a difference of (4,4,4) and a radius of (2,2,2)
I think i have an idea of where Im going wrong, by dividing the diameter by 2 i do get the radius, but that is not necessarily the center point. How would I go about finding the center point?
6. Apr 23, 2010
### gabbagabbahey
No, diameter is a distance (scalar)...subtracting two position vectors gives you a vector...specifically the vector from the first endpoint to the second endpoint....What does dividing that vector by 2 give you? (draw a picture if you aren't sure!)
That looks better
7. Apr 23, 2010
### somebodyelse5
Got this one figured out also. Answer is (x-1)^2+(y+3)^2+(z+6)^2-3.464^2
And used the distance formula, which i forgot about, to solve for the distance and then divided that by 2 to get the radius. Then I basically took the average of each x y and z point to find the center.
Thanks for you help!! Glad i found this site
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# Does Abstract Math belong to Elementary Math ?
The answer is : “Yes” but with some exceptions.
Most pedagogy mistake made in Abstract Algebra teaching is in the wrong order (by historical chronological sequence of discovery):
[X] Group -> Ring -> Field
It would be better, conceptual wise, to reverse the teaching order as:
Field -> Ring -> Group
or better still as (the author thinks):
Ring -> Field -> Group
• Reason 1: Ring is the Integers, most familiar to 8~ 10-year-old kids in primary school arithmetic class involving only 3 operations: ” + – x”.
• Reason 2: Field is the Real numbers familiar in calculators involving 4 operations: ” + – × ÷”, 1 extra division operation than Ring.
• Reason 3: Group is “Symmetry”, although mistakenly viewed as ONLY 1 operation, but not as easily understandable like Ring and Field, because group operation can be non-numeric such as “rotation” of triangles, “permutation” of roots of equation, “composition” of functions, etc. The only familiar Group is (Z,+), ie Integers under ” +” operation.
Some features which separate Advanced Math from Elementary Math are:
• Proof [1]
• Infinity [2]
• Abstract [3]
• Non Visual [4]
•
Note [1]: “Proof” is, unfortunately, postponed from high-school Math to university level. This does not include the Euclidean Geometry axiomatic proof or Trigonometry Identity proof, which are still in Secondary school Elementary Math but less emphasized since the 1990s (unfortunately).
Note [2]: However, some “potential” infinity still in Elementary math, such as 1/3 = 0.3333…only the “Cantor” Infinity of Real number, ${\aleph_{0}, \aleph_{1}}$ etc are excluded.
Note [3]: Some abstract Algebra like the axioms in Ring and Field (but not Group) can be in Elementary Math to “prove” (as in [1]): eg. By distributive law
$(a + b).(a - b) = a.(a - b) + b.(a - b)$
$(a + b).(a - b) = a^{2}- ab + ba - b^{2}$
By commutative law
$(a + b).(a - b) = a^{2}- ab + ab- b^{2}$
$(a + b). (a - b) = a^{2} - b^{2}$
Note [4]: Geometry was a “Visual” Math in Euclidean Geometry since ancient Greek. By 17 CE, Fermat and Descartes introduced Algebra into Geometry as the Analytical Geometry, still visual in (x, y) coordinate graphs.
20 CE Klein proposed treating Geometry as Group Transformation of Symmetry.
Abstract Algebra concept “Vector Space” with inner (aka dot) product is introduced into High School (Baccalaureate) Elementary Math – a fancy name in “AFFINE GEOMETRY” (仿射几何 , see Video 31).
eg. Let vectors
$u = (x,y), v = (a, b)$
Translation:
$\boxed {u + v = (x,y) + (a, b) = (x+a, y+b)}$
Stretching by a factor ${ \lambda}$ (“scalar”):
$\boxed {\lambda.u = \lambda. (x,y) = (\lambda{x}, \lambda{y})}$
Distance (x,y) from origin: |(x,y)|
$\boxed {(x,y).(x,y) =x^{2}+ y^{2} = { |(x,y)|}^{2}}$
Angle ${ \theta}$ between 2 vectors ${(x_{1},y_{1}), (x_{2},y_{2})}$:
$\boxed { (x_{1},y_{1}).(x_{2},y_{2}) =| (x_{1},y_{1})|.| (x_{2},y_{2})| \cos \theta}$
Ref: 《Elements of Mathematics – From Euclid to Gödel》by John Stillwell (Princeton University Press, 2016) [NLB # 510.711]
# Richard Dedekind
Julius Wilhelmina Richard Dedekind (6 Oct 1831 – 12 Feb 1916)
– Last student of Gauss at Göttingen
– Student and closed friend of Dirichlet who influenced his Mathematical education
– Introduced the word Field (Körper)
– Gave the first university course on Galois Theory
– Developed Real Number ‘Dedekind Cut‘ in 1872
– Accomplished musician
– Never married, lived with his unmarried sister until death
“Whatever provable should be proved.”
– By 1858: still yet established ?
$\sqrt{2}.\sqrt{3} = \sqrt{2.3}$
– Gave strong support to Cantor on Infinite Set.
http://www-history.mcs.st-and.ac.uk/Biographies/Dedekind.html
# What is Ideal ?
Anything inside x outside still comes back inside
=> Zero x Anything = Zero
=> Even x Anything = Even
Mathematically,
1. nZ is an Ideal, represented by (n)
Eg. Even subring (2Z) x anything big Ring Z = 2Z = Even
2. (football) Field F is ‘sooo BIG’ that
(inside = outside)
=> Field has NO Ideal (except trivial 0 and F)
Why was Ideal invented ? because of ‘failure” of UNIQUE Primes Factorization” for this case (example):
6 = 2 x 3
but also
$6=(1+\sqrt{-5})(1-\sqrt{-5})$
=> two factorizations !
=> violates the Fundamental Law of Arithmetic which says UNIQUE Prime Factorization
Unique Prime factors exist called Ideal Primes: $\mbox{gcd = 2} , \mbox{ 3}$, $(1+\sqrt{-5})$, $(1-\sqrt{-5})$
Greatest Common Divisor (gcd or H.C.F.):
For n,m in Z
gcd (a,b)= ma+nb
Example: gcd(6,8) = (-1).6+(1).8=2
(m=-1, n=-1)
Dedekind’s Ideals (Ij):
6 =2×3= u.v =I1.I2.I3.I4 ;
$u= (1+\sqrt{-5})$
$v=(1-\sqrt{-5})$
Let gcd(2,u) = 2M+N.u
M,N in form of $a+b\sqrt{-5}$
1. Principal Ideals:
2M = (2) = multiple of 2
2. Ideals (nonPrincipal) = 2M+N.u
3. Ideal prime factors: 6=2 x 3=u.v
Let
I1= gcd(2, u)
I2=gcd(2, v)
I3=gcd(3, u)
I4=gcd(3, v)
Easy to verify (by definition):
I1.I2=(2)
I3.I4=(3)
I1.I3=(u)
I1.I4=(v)
=> Ij are prime & unique factors of 6=I1.I2.I3.I4
=> Fundamental Law of Arithmetic satisfied!
=>Ij “Ideal“-ly exist! hidden behind ‘compound’ (2,3,u,v) !
Verify : gcd(2, 1+√-5).gcd(2, 1-√-5)=(2) ?
Proof by definition:
[2m+n(1+√-5)][2m’+n'(1-√-5)]
=[2m+n+n√-5 ][2m’+n’-n’√-5]
= 4mm’+2mn’+2nm’+6nn’
= 2(2mm’+mn’+m’n+3nn’)
= 2M
= 2 multiples
= (2) = Principal Ideal
# Field: Galois, Dedekind
Dedekind
(1831-1916)
Dedelind was the 1st person in the world to define Field:
“Any system of infinitely many real or complex numbers, which in itself is so ‘closed’ and complete, that +, – , *, / of any 2 numbers always produces a number of the same system.”
Heinrich Weber (1842-1913) gave the abstract definition of Field.
Field Characteristic
1. Field classification by Ernst Steinitz @ 1910
2. Given a Field, we start with the element that acts as 0, and repeatedly add the element that acts as 1.
3. If after p additions, we obtain 0 again, p must be prime number, and we say that the Field has characteristic p;
4. If we never get back to 0, the Field has characteristic 0. (e.g. Complex Field)
Example: GF(2) = {0,1|+} ; prime p = 2
1st + (start with 0):
0 + 1 = 1
2nd (=p) +:
1 + 1 = 0 => back to 0 again!
=> GF(2) characteristic p= 2
Galois Field GF(p)
1. For each prime p, there are infinitely many finite fields of characteristic p, known as Galois fields GF(p).
2. For each positive power of prime p, there is exactly one field.
(This is the only IMPORTANT Theorem need to know in Field Theory)
E.g. GF(2) = {0,1}
Math Game: Chinese 9-Ring Puzzle (九连环 Jiu Lian Huan)
http://www.google.com.sg/imgres?imgurl=http://info.makepolo.com/uploadfile/2012/0723/20120723100653765.jpg&imgrefurl=http://info.makepolo.com/htmls/6/69/2669.html&h=400&w=533&sz=44&tbnid=ExodLfHv3cQjHM:&tbnh=91&tbnw=121&zoom=1&usg=__hsZaBecpPNdvTvguQbaQftCsXgo=&docid=qXMWtmo8A-vXEM&hl=en&sa=X&ei=f2NaUciqKYrOrQeT2YHgDw&sqi=2&ved=0CEsQ9QEwAg&dur=591
To solve Chinese ancient 9-Ring Puzzle (九连环) needs a “Vector Space V(9,K) over Field K”
finite Field K = Galois Field GF(2) = {0,1|+,*}
and 9-dimension Vector Space V(9,K):
V(0)=(0,0,0,0,0,0,0,0,0) ->
V(j) =(0,0,… 0,1,..0,0) ->
V(9)= (0,0,0,0,0,0,0,0,1)
From start V(0) to ending V(9) = 511 steps.
# Eigenvector & Eigenvalue
1. Matrix (M): stretch & twist space
2. Vector (v): a distance along some direction
3. M.v = v’ stretched & twisted by M
Some directions are special:-
a) v stretched but not twisted = Eigenvector;
b) The amount of stretch = constant = Eigenvalue (λ)
Let M the matrix, λ its eigenvalue,
v eigenvector.
By definition: M.v = λ.v
v = I.v (I identity matrix)
M.v = λI.v
(M – λI).v=0
As v is non-zero,
1. Determinant (M- λI) =0 => find λ
2. M.v = λ.v => find v
Note1: Why call Eigenvalue ?
From German: “Die dem Problem eigentuemlichen Werte
= “The values belonging to this problem
=> eigenWerte = EigenValue
Eigenvalue also called ‘characteristic values’ or ‘autovalues’.
Eigen in English = Characteristic (but already used for Field).
Note2: Schrödinger Quantum equation’s Eigenvalue = Maximum probability of electron presence at the orbit outside nucleus.
Note3: Excellent further explanation of the eigenvector and eigenvalue:
http://lpsa.swarthmore.edu/MtrxVibe/EigMat/MatrixEigen.html
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# Why is the square root of minus one important?
## Why is the square root of minus one important?
By squaring the velocities and taking the square root, we overcome the “directional” component of velocity and simultaneously acquire the particles’ average velocity. Since the value excludes the particles’ direction, we now refer to the value as the average speed.
## Who invented the square root of negative 1?
Bombelli was able to use his rules for operations with complex numbers to solve the cubic equations that produced an expression that contained a square root of a negative number.
Why are square roots useful?
Radicals and square roots are important because they show up when we compute areas, which is a fairly practical application. You know by taking the square root that this must be a 20-foot by 20-foot room. Even cooler is the fact that square roots give us some of our examples of irrational numbers.
READ ALSO: Would Saitama or Goku win in a fight?
Do imaginary numbers exist in the real world?
While it is not a real number — that is, it cannot be quantified on the number line — imaginary numbers are “real” in the sense that they exist and are used in math. Imaginary numbers, also called complex numbers, are used in real-life applications, such as electricity, as well as quadratic equations.
### How was square root discovered?
The Egyptians calculated square roots using an inverse proportion method as far back as 1650BC. Chinese mathematical writings from around 200BC show that square roots were being approximated using an excess and deficiency method. In 1450AD Regiomontanus invented a symbol for a square root, written as an elaborate R.
### How were square roots found before the invention of calculators?
The iterative method is called the Babylonian method for finding square roots, or sometimes Hero’s method. It was known to the ancient Babylonians (1500 BC) and Greeks (100 AD) long before Newton invented his general procedure.
READ ALSO: How long does it take for a child to learn German?
Does a negative have a square root?
A square root is written with a radical symbol √ and the number or expression inside the radical symbol, below denoted a, is called the radicand. Negative numbers don’t have real square roots since a square is either positive or 0.
Where are square roots used in the real world?
The concept of squares and square roots are used in all walks of life, such as carpentry, engineering, designing buildings, and technology.
#### What is the square root of negative one?
The square root of negative one is “i,” the imaginary number. This concept is immensely useful in mathematics, as it allows for there to be square roots of negative numbers, which is otherwise not possible using only real numbers. Any number that includes a negative square root is called an imaginary number.
#### What is the square root of a complex number?
Any number that includes a negative square root is called an imaginary number. For example, the square root of -9 equals 3i, an imaginary number. When an imaginary number and a real number are combined, for example 2 + 3i, this is called a complex number.
READ ALSO: Why is it fun to play soccer?
What is the square root of minus one imaginary number?
Unit Imaginary Number The square root of minus one √(−1) is the “unit” Imaginary Number, the equivalent of 1 for Real Numbers. In mathematics the symbol for √(−1) is i for imaginary.
What two numbers have a square root of 25?
When we take the square root of a number, 25 for example, we are looking for a number, that when multiplied by itself, equals exactly 25. There are precisely two numbers that satisfy this equation: 5 and -5. When taking the square root of a positive number the result is always two numbers. To put it in more mathematical terms:
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Equality
The source code for this module: PartI/Equality.ard
The source code for the exercises: PartI/EqualityEx.ard
In the previous modules we treated the identity type = in a rather hand-wavy manner as in most of the cases we needed just reflexivity idp {A : \Type} {a : A} : a = a. Here we will get into details of the definition of the identity type and explain some key aspects of it, which will be important for writing more advanced proofs. Along the way we introduce the interval type I, whose properties are essentially determined by the function coe, playing the role of eliminator for I. In order to clarify this we briefly recall the general concept of eliminator.
Symmetry, transitivity, Leibniz principle
First of all, we show that the identity type satisfies some basic properties of equality: it is an equivalence relation and it satisfies the Leibniz principle.
The Leibniz principle says that if a and a’ satisfy the same properties, then they are equal. It can be easily proven that = satisfies this principle:
``````\func Leibniz {A : \Type} {a a' : A}
(f : \Pi (P : A -> \Type) -> \Sigma (P a -> P a') (P a' -> P a)) : a = a'
=> (f (\lam x => a = x)).1 idp``````
The inverse Leibniz principle (which we will call merely Leibniz principle as well) says that if a = a’, then a and a’ satisfy the same properties, that is if P a is true, then P a’ is true. The proof of this is easy, but requires some constructs that will be introduced very shortly further in this module:
``````\func transport {A : \Type} (B : A -> \Type) {a a' : A} (p : a = a') (b : B a) : B a'
=> coe (\lam i => B (p @ i)) b right``````
Using this latter Leibniz principle, it is easy to prove that = satisfies (almost) all the properties of equality. For example, the following properties:
``````-- symmetry
\func inv {A : \Type} {a a' : A} (p : a = a') : a' = a
=> transport (\lam x => x = a) p idp
-- transitivity
\func trans {A : \Type} {a a' a'' : A} (p : a = a') (q : a' = a'') : a = a''
=> transport (\lam x => a = x) q p
-- congruence
\func pmap {A B : \Type} (f : A -> B) {a a' : A} (p : a = a') : f a = f a'
=> transport (\lam x => f a = f x) p idp``````
Exercise 1: Define congruence for functions with two arguments via transport. It is allowed to use any functions defined via transport.
Exercise 2: Prove that transport can be defined via pmap and repl and vice versa. The function repl says that if two types are equal then there exists a function between them.
Definition of =
The central ingredient of the definition of the identity type is the interval type I contained in Prelude. The type I looks like a two-element data type with constructors left and right, but actually it is not: these constructors are made equal (by means of coe). Of course, pattern matching on I is prohibited since it can be used to derive Empty = Unit.
The equality left = right implies that some a : A and a’ : A are equal if and only if there exists a function f : I -> A such that f left ==> a and f right ==> a’ (where ==> denotes computational equality). The type a = {A} a’ is defined simply as the type of all functions f : I -> A satisfying this property. The constructor path (f : I -> A) : f left = f right allows to construct equality proofs out of such functions and the function @ (p : a = a’) (i : I) : A does the inverse operation:
``````path f @ i ==> f i -- beta-equivalence
path (\lam i => p @ i) ==> p -- eta-equivalence``````
In order to prove reflexivity idp we can simply take the constant function \lam _ => a : I -> A:
``\func idp {A : \Type} {a : A} : a = a => path (\lam _ => a)``
Exercise 3: Prove that left = right without using transport or coe.
If f : A -> B and g : I -> A, then g determines a proof of the equality g left = g right and the congruence pmap can be interpreted as simply the composition of f and g. This observation suggests an alternative definition of pmap:
``````\func pmap {A B : \Type} (f : A -> B) {a a' : A} (p : a = a') : f a = f a'
=> path (\lam i => f (p @ i))``````
This definition of pmap behaves better than others with respect to computational properties. For example, pmap id is computationally the same as id and pmap (f . g) is computationally the same as pmap f . pmap g, where (.) is the composition:
``````\func pmap-idp {A : \Type} {a a' : A} (p : a = a') : pmap {A} (\lam x => x) p = p
=> idp``````
Exercise 4: Prove that a = {A} a’ and b = {B} b’ implies (a,b) = {\Sigma A B} (a’,b’) without using transport.
Exercise 5: Prove that p = {\Sigma (x : A) (B x)} p’ implies p.1 = {A} p’.1 without using transport.
Functional extensionality
Function extensionality is a principle saying that if two functions f and g are equal pointwise, then they are equal functions. Our definition of equality allows us to prove this principle very easily:
``````\func funExt {A : \Type} (B : A -> \Type) {f g : \Pi (a : A) -> B a}
(p : \Pi (a : A) -> f a = g a) : f = g
=> path (\lam i => \lam a => p a @ i)``````
This useful principle is unprovable in many other intensional dependently typed theories. In such theories function extensionality can be introduced as an axiom, that is as a function without implementation, however adding new axioms worsens computational properties of the theory. For example, if we add the axiom of excluded middle lem, then we can define a constant ugly_num : Nat that does not evaluate to any concrete natural number:
``````\func lem : \Pi (X : \Type) -> Either X (X -> Empty) => {?}
\func ugly_num : Nat => \case lem Nat \with { | Left => 0 | Right => 1 }``````
Exercise 6: Prove that (\lam x => not (not x)) = (\lam x => x).
Eliminators
Elimination principles for a data type D specify what kind of data should be provided in order to define a function from D to a non-dependent or dependent type. And, essentially, these principles say that it is enough to show how “generators” (that is constructors) of D are mapped to a type A and that that would uniquely determine a function D -> A. For example, eliminators for Nat and Bool:
``````-- Dependent eliminator for Nat (induction).
\func Nat-elim (P : Nat -> \Type)
(z : P zero)
(s : \Pi (n : Nat) -> P n -> P (suc n))
(x : Nat) : P x \elim x
| zero => z
| suc n => s n (Nat-elim P z s n)
-- Non-dependent eliminator for Nat (recursion).
\func Nat-rec (P : \Type)
(z : P)
(s : Nat -> P -> P)
(x : Nat) : P \elim x
| zero => z
| suc n => s n (Nat-rec P z s n)
-- Dependent eliminator for Bool (recursor for Bool is just 'if').
\func Bool-elim (P : Bool -> \Type)
(t : P true)
(f : P false)
(x : Bool) : P x \elim x
| true => t
| false => f``````
Exercise 7: Define factorial via Nat-rec (i.e., without recursion and pattern matching).
Exercise 8: Prove associativity of Nat.+ via Nat-elim (i.e., without recursion and pattern matching).
Exercise 9: Define recursor and eliminator for \data D | con1 Nat | con2 D D | con3 (Nat -> D).
Exercise 10: Define recursor and eliminator for List.
The function coe thus defines dependent eliminator for I, it says that in order to define f : \Pi (i : I) -> P i for some P : I -> \Type it is enough to specify f left:
``````\func coe (P : I -> \Type)
(a : P left)
(i : I) : P i \elim i
| left => a``````
Exercise 11: We defined transport via coe. It is possible to define a special case of coe via transport. Define coe0 (A : I -> \Type) (a : A left) : A right via transport. Is it possible to define transport via coe0?
Exercise 12: Define a function B right -> B left.
left = right
With the use of the function coe, we now prove that I has one element:
``````\func left=i (i : I) : left = i
-- | left => idp
=> coe (\lam i => left = i) idp i
-- In particular left = right.
\func left=right : left = right => left=i right``````
coe and transport
Functions coe and transport are closely related. Recall the definition of transport given earlier in this module:
``````\func transport {A : \Type} (B : A -> \Type) {a a' : A} (p : a = a') (b : B a) : B a'
=> coe (\lam i => B (p @ i)) b right``````
Denote \lam i => B (p @ i) as B’. Then B’ : I -> \Type, B’ left ==> B a, B’ right ==> B a’ and \lam x => coe B’ x right : B’ left -> B’ right.
Proofs of non-equalities
In order to prove that true is not equal to false it is enough to define a function T : Bool -> \Type such that T true is the unit type and T false is the empty type. Then the contradiction can be easily derived from true = false by means of transport:
``\func true/=false (p : true = false) : Empty => transport T p unit``
Note that it is not possible to prove that left is not equal to right since such T cannot be defined neither recursively nor inductively:
``````-- This function does not typecheck!
\func TI (b : I)
| left => \Sigma
| right => Empty``````
Exercise 13: Prove that 0 does not equal to suc x.
Exercise 14: Prove that fac does not equal to suc.
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October 3, 2013
One of the things I remember most about middle school math class is that I went through it in a perpetual state of disorganization. During one particularly bad spell, I lost two calculators within a week. The loss, and the reaction of my parents, drove me to try to fix the problem once and for all. My plan was simple: buy an expensive calculator with the hope that it’d serve as an incentive to keep track of my stuff. The next weekend, I took weeks of allowance money to the local Service Merchandise and bought a new HP-11C pocket calculator. Almost 30 years later, I still have both the calculator and a fascination for its unusual Reverse Polish Notation (RPN) user interface. Over those decades, I’ve also found out that RPN provides a good way to explore a number of fundamental ideas in the field of software design.
If you haven’t used an RPN calculator, your first attempt will probably be a confusing experience. Unlike most calculators, The HP didn’t have an ‘=’ key. Instead, it had a large button down the center labeled ‘Enter’, which pushed the most recently keyed number onto an internal four-level stack. The mathematical operations then worked against this stack. The ‘+’ key popped off two numbers, added them, and pushed the result back onto the stack. To add 2 and 3, you’d make the following keystrokes: `[2][ENTER][3][+]`.
In detail:
• `[2]` – Begin entering the number ‘2’ into the top level of the stack.
• `[ENTER]` – Duplicate the number ‘2’ on the top of the stack so it’s in the top two levels.
• `[3]` – Begin entering the number ‘3’ into the top level of the stack, replacing one of the copies of the number ‘2’.
• `[+]` – Pop off the top two levels of the stack, pushing back the sum of the two numbers.
Because the display shows the top level of the stack, it shows the answer (5) as soon as the user presses `[+]`. Because the answer, 5, is on the top of the stack, it’s also immediately ready to be used as an input to another calculation. This last bit is why RPN calculators can be so compelling to use: they make it easy to start with a small calculation and extend it into something larger. As long as a number is on the stack, it can be used in a calculation; The origin of the number doesn’t matter to the way it can be used. In computer science terms, this is the beginning of Referential Transparency. The FORTH programming language builds on this foundation, extending the basic tenets of RPN into a complete programming language. RPN combined with functional decomposition gives the language a great deal of expressive power, but due to the simplicity of stacks a small FORTH can be implemented in a very small amount of memory.
As this series of blog posts continues, I intend to explore some of these ideas using a set of RPN calculator implementations written in Java and in Clojure. We’ll start off with a simple implementation in Java, spend a bit of time exploring the command pattern, and then move into more functional approaches to the problem.
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## #1 2015-06-02 09:46:51
husnain
From: Chichawatni,Pakistan
Registered: 2014-10-03
Posts: 8,129
Website
### Repeated Letter:
Question:
Ten different letters of alphabet are given, words with 5 letters are formed from these given letters. Then, the number of words which have at least one letter repeated is:
Option A):
69760
Option B):
42386
Option C):
99748
Option D):
30240
69760
Explanation:
Number of words which have at least one letter replaced,
= Total number of words - total number of words in which no letter is repeated.
=> 105 16P5.
=> 100000 − 30240 = 69760.
You cannot discover the new oceans unless you have the courage to lose the sight of the shore.
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You are Here: Home >< Maths
# Another maths question watch
1. for part d, narb..
You integrated between π/4 and π/2 for your values of theta.
Surely for the area within the ellipse between P and the point where the ellipse touches the x-axis i.e: (x,y) (5,0).. you need to integrate between theta=0 and theta=π/4 .. right? Because when y=0, theta=0
so first if you integrate under the ellipse between @=0 and @=π/4
= INT [y.(dx/[email protected])[email protected]]
= INT [([email protected])([email protected])[email protected]]
= INT [-20.([email protected])^[email protected]]
= -10. INT [1 - [email protected]@]
= -10.[@ - 1/2 (sin [email protected])]
= -10.{[0 - 1/2 (sin 0)] - [π/4 - 1/2 (sin(π/2)]}
= (5π/2 - 5) units^2
________________
and then find the area of the triangle PQR, where Q ((5/2)root2 , 0)..
= [(5root2) - (5/2)root2]. root2
= 10 - 5 = 5 units^2
_______
= 5 - (5π/2 - 5) = (10 - 5π/2) units^2
anyone agree?! (ppppplllllllease?!)
2. I seem to be getting a different answer to everyone else....I integrated the curve between x=5√2 and x=5 (i.e. from the point P to where the curve crosses the x-axis). So in terms of 'θ' the integration is from pi/4 to 0
So, in the end my answer was: 0.5 (20 - 5pi)
3. (Original post by Sahir)
for part d, narb..
You integrated between π/4 and π/2 for your values of theta.
Surely for the area within the ellipse between P and the point where the ellipse touches the x-axis i.e: (x,y) (5,0).. you need to integrate between theta=0 and theta=π/4 .. right? Because when y=0, theta=0
so first if you integrate under the ellipse between @=0 and @=π/4
= INT [y.(dx/[email protected])[email protected]]
= INT [([email protected])([email protected])[email protected]]
= INT [-20.([email protected])^[email protected]]
= -10. INT [1 - [email protected]@]
= -10.[@ - 1/2 (sin [email protected])]
= -10.{[0 - 1/2 (sin 0)] - [π/4 - 1/2 (sin(π/2)]}
= (5π/2 - 5) units^2
________________
and then find the area of the triangle PQR, where Q ((5/2)root2 , 0)..
= [(5root2) - (5/2)root2]. root2
= 10 - 5 = 5 units^2
_______
= 5 - (5π/2 - 5) = (10 - 5π/2) units^2
anyone agree?! (ppppplllllllease?!)
4. (Original post by mockel)
yay, good stuff that was well tough!
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# The equation of the ellipse whose focus is (1,0) and the directrix is x+ y+1=0 and eccentricity is equal to 1/√2 is
A
(x1)2+y2=(x+y+1)2
B
2(x1)2+Y2=(x+y+1)2
C
4{(x1)2+y2}=(x+y+1)2
D
none of these
Video Solution
Text Solution
Verified by Experts
## Let S(1,0) be the focus and ZZ' be the directrix let P(x,y) be any point on the ellipse and PM be perpendicular from P on directrix . Then by definition , we have . SP=ePM, where e=1/√2 ⇒SP2=e2PM2 ⇒(x−1)2+(y−0)2=(1√2)2{x+y+1√1+1}2 ⇒4{(x−1)2+y2}=(x+y+1)2
|
Updated on:21/07/2023
### Knowledge Check
• Question 1 - Select One
## The equation of the ellipse whose focus is (1,-1), directrix x−y−3=0 and eccentricity equals 12 is :
A7x2+2xy+7y210x+10y+7=0
B7x2+2xy+7y2+7=0
C7x2+2xy+7y2+10x10y7=0
D7x2+2xy+7y2+10x10y7=0
• Question 2 - Select One
## If question of the ellipse whose focus is (1,−1), then directrix the line x−y−3=0 and eccentricity 12 is
A7x2+2xy+7y210x+10y+7=0
B7x2+2xy+7y2+7=0
C7x2+2xy+7y2+10x10y7=0
Dnone of these
• Question 3 - Select One
## The equation of an ellipse whose focus is (−1,1), directrix is x−y+3=0 and eccentricity is 12, is given by
A7x2+2xy+7y2+10x10y+7=0
B7x22xy+7y210x+19y+7=0
C7x22xy+7y210x10y7=0
D7x22xy+7y2+10x10y7=0
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# INFOSYS by chintuu1990
VIEWS: 736 PAGES: 18
• pg 1
``` Aptitude Questions
1. There is a merry-go-round race going on.One person says,"1/3 of those in front of me and 3/4 of those behind me, give the total number of children in the race". Then the number of children took part in the race? (repeated from previous papers) Ans : 13 [ Assume there are x participants in the race.In a round race,no: of participants in front of a person wil be x-1 an that behind him wil b x-1. i.e, 1/3(x-1) + 3/4(x-1) = x ; solving x = 13 ] 2. In an Island the natives lie and visitors speak truth. A man wants to know whether a salesman beside him in a bar is a native or visitor. He asked him to ask a woman beside him whether she is a native or visitor. He replied "she says she is a visitor". Then he knew that the salesman is a native or visitor. salesman is in which category , native or visitor? Ans : Native [ Draw table and see ] 3.A man fixed an appointment to meet the manager, Manager asked him to come two days after the day before the day after tomorrow. Today is Friday. When will the manager expect him? (repeated from previous papers) Ans: Monday [Don't confuse it with Tuesday.the correct answer is Monday]
5.A man said he spent 1/6 of his as a child, 1/12 as salesman in a liquor shop, 1/7 and 5 years as a politician and a good husband respectively. At that time Jim was born. Jim was elected as Alderman four years back.when he was half of his age. What is his age? (repeated from previous papers) Ans: 84 years
[Assume that he lived x years. X/6 + x/12 + x/7 + 5 + 4 + x/2 = x. Solving x= 84, Same as Question in Shakundala Devi book] 6.Jack,Doug and Ann, 3 children had a running race while returning from school.Mom asked who won the race. Then Jack replied" I wont tell u.I wil give u a clue,When Ann takes 28 steps Doug takes 24 steps, meantime I take 21 steps. Jack explained that his 6 steps equals Droug's 7 steps and Ann's 8 steps. Who won the race? (repeated from previous papers) Ans: Doug [ Ann steps = 8,16,24,28 --- finished by 3 & half full steps Doug steps=7,14,21,24 --- finished before 3 & half full steps Jack steps= 6,12,18,21 --- finished by 3 & half full steps So Doug won the race ] 7. Every day a cyclist meets a car at the station.The road is straight and both are travelling in the same direction. The cyclist travels with a speed of 12 mph.One day the cyclist comes late by 20 min. and meets the car 5miles before the Station. What is the speed of the car? Ans: 60 mph [Very similar to Shakuntala Devi puzzles to puzzle you problem no: 38 ] 9.A lady goes for shopping. She bought some shoestrings. 4 times the number of shoestrings, she bought pins and 8 times, handkerchiefs. She paid each item with their count as each piece's cost. She totally spent Rs. 3.24.How many handkerchiefs did she buy? (repeated from previous papers) 10. Complete the series : a) b) " Ans : 3,6,13,26,33,66,____(repeated from previous papers) 364,361,19,16,4,1,___( " " ) a) 63 b) 1
11. Lucia is a wonderful grandmother. Her age is between 50 and 70.Each of her sons have as many sons as they have brothers. Their combined number gives Lucia?s age. What is the age? Ans: 64
12.There are two towers A and B. Their heights are 200ft and 150ft respectively and the foot of the towers are 250ft apart. Two birds on top of each tower fly down with the same speed and meet at the same instant on the ground to pick a grain. What is the distance between the foot of tower A and the grain? Ans:90ft 13 Grass It takes whole of How many Ans: 120 13. Four tourists A,B,C,D and four languages English, German, French and Italian. They are not able to converse among themselves in one language. Though A does not know English he can act as an interpreter between B and C. No one spoke both French and German. A knows German and was able to converse with D who doesn?t know a word in German. Only one language was spoken by more than two persons. Each spoke two languages. Find who spoke what. Ans : ABcDGerman,Italian French,Italian German,English Italian,English in lawn grows equally thick and in a uniform rate. 40 days for 40 cows and 60 days for 30 cows to eat the the grass. days does it take for 20 cows to do the same?
14. There is a five digit number. It has two prime digits (1 is not a prime number). Third digit is the highest. Second digit is the lowest. First digit is one less than the third digit. The fifth digit is half of the fourth. The sum of 4th and 5th is less than the first. Find the number. Ans ? 71842 15.6. Four persons A, B, C and D are playing cards. Each person has one card, laid down on the table below him, which has two different colours on either side. No card has the same color on both sides. The colours visible on the table are Red, Green, Red and Blue respectively. They see the color on the reverse side and give the following comment.
A: B: C: D:
Yellow or Green Neither Blue nor Green Blue or Yellow Blue or Yellow
Given that out of the 4 people 2 always lie find out the colours on the cards each person. Ans: ABCDYellow Yellow Green Red
16. A 1 k.m. long wire is held by n poles. If one pole is removed, the length of the gap becomes 12/3m. What is the number of poles initially? Ans:6km 17. Find the digits X,Y,Z X X X X Y Y Y Y + Z Z Z Z -------------Y X X X Z ---------------X Y Z 9 1 8
Ans:
18. A man starts walking at 3 pm . ha walks at a speed of 4 km/hr on level ground and at a speed of 3 km/hr on uphill , 6 km/hr downhill and then 4 km/hr on level ground to reach home at 9 pm. What is the distance covered on one way? Ans: 12 km 19. A grandma has many sons; each son has as many sons as his brothers. What is her age if it?s the product of the no: of her sons and grandsons plus no: of her sons?(age b/w 70 and 100). Ans: 81 20. An electric wire runs for 1 km b/w some no: of poles. If one pole is removed the distance b/w each pole increases by 1 2/6 (mixed fraction). How many poles were there initially? 21. There is a church tower 150 feet tall and another catholic tower at a distance of 350 feet from it which is 200 feet tall. There is one each bird sitting on top of both the towers. They fly at a constant speed
and time to reach a grain in b/w the towers at the same time. At what distance from the church is the grain? Ans: 90 22. A person wants to meet a lawyer and as that lawyer is busy he asks him to come three days after the before day of the day after tomorrow? on which day the lawyer asks the person to come? ans: thursday 23. A person is 80 years old in 490 and only 70 years old in 500 in which year is he born? ans: 470 24.A person says that their speed while going to a city was 10mph however while returning as there is no much traffic they came with a speed of 15mph. what is their average speed? ans: 12mph 25. There is a peculiar island where a man always tells truth and a women never says two 2 consecutive truth or false statements that is if she says truth statement then she says false statement next and vice versa. A boy and girl also goes in the same way. one day i asked a child " what r u a boy or a girl" however the child replied in their language that i dint understand but the parents knew my language and one parent replied that " kibi is a boy" the other one said that "no kibi is a girl, kibi lied". a: is kibi a boy or a girl b: who ansered first mother or father? ans: kibi is a girl and mother answered first. 26. The boy goes to school reaches railway station at his 1/3 of his journey& mill at 1/4 of his journey the time taken him to walk between railway station & mill is 5 mins. Also he reaches railway station at 7.35amwhen he started from house& when he reaches school? Ans: 7:15to8.15 27. if a person is sitting in a exam having 30 questions (objective type) the examiner use the formula to calculate the score is S=30+4cw here c is number of correct answer and w is number of wrong answer , the examiner find the score is more than 80, tell how may questions are correct ? if the score is little less
but still more than 80 then u wont be able to answer. ans :- 16 28. if a person having 1000 rs and he want to distribute this to his five children in the manner that ecah son having 20 rs more than the younger one , what will be the share of youngest child ans- 160 29.raju having some coins want to distribute to his 5 son , 5 daughter and driver in a manner that , he gave fist coin to driver and 1/5 of remaining to first son he again gave one to driver and 1/5 to 2nd son and so on.... at last he equally distributed all the coins to 5 daughters. how many coins raju initially have??? ans:-881 30.if ravi binded his book and the binder cut the pages of the book , ravi decided to mark the pages by himself own , what he found that number of three appears 61 times find of number of pages answer ans - 300 31. a painter went in a exhibition to purchases some pictures where T,U,V,W,X,Y,Z pictures were remaining , he want to buy only five in the condition on that if T is there then X should not be there, if U is there than y should be there if if v is there then X should be there which is the combination the painter can have (a) T,U,V,W,Y (b)T,Z,U,W,X (c)T,X,U,V,W (d)T,U,Y,W,Z ans (d) 32.There are 100 men in town. Out of which 85% were married, 70% have a phone, 75% own a car, 80% own a house. What is the maximum number of people who are married, own a phone, own a car and own a house ? ( 3 marks) Sol: 15%
33. There are 10 Red, 10 Blue, 10 Green, 10 Yellow, 10 White balls in a bag. If you are blindfolded and asked to pick up the balls from the bag, what is the minimum number of balls required to get a pair of atleast one colour ? ( 2 Marks) Sol :6 balls. 34. Triplet who usually wear same kind and size of shoes, namely, Annie, Danny, Fanny. Once one of them broke a glass in kitchen and their shoe prints were there on floor of kitchen. When their mother asked who broke Annie said, ?I didn?t do it?; Fanny said ?Danny did it?; Danny said ?Fanny is lieing?; here two of them are lieing, one is speaking truth. Can you find out who broke it ? (3 Marks) Sol : Annie 35. 4 players were playing a card game. Cards had different colours on both sides. Neither of cards had same colour on both sides. Colours were 2 Red, 2 Blue, 2 Green, 2 Yellow. Cards were lying in front of each player. Now, each player knew the colour on other side of his card. They are required to tell their colour. Statement given by each of them was : Annie : Blue or Green Bobby : Neither Blue nor Green Cindy : Blue or Yellow Danny : Blue or Yellow colours of cards that are visible to all were Red, Blue, Green, Blue in order of their names. Exactly two of them are telling truth and exactly two of them are lieing. Can you tell the colour on other face of card for each player ? (6 Marks) Sol : Annie Bobby Cindy Danny : : : : Yellow (Lieing) Yellow (Telling truth) Blue (Telling truth) Green (Lieing)
36. In a game i won 12 games, each game if i loose i will give u one chocolate, You have 8 chocolates how many games played. Ans : 32 38. 75 persons Major in physics, 83 major in chemistry, 10 not at major in these subjects u want to find number of students majoring in both subjects Ans 68. 39. if A wins in a race against B by 10 mts in a 100 Meter
race. If B is behind of A by 10 mts. Then they start running race, who will won? Ans A 40. A+B+C+D=D+E+F+G=G+H+I=17 given A=4.Find value of G and H? Ans : G = 5 E=1 41. One guy has Rs. 100/- in hand. He has to buy 100 balls. One football costs Rs. 15/, One Cricket ball costs Re. 1/- and one table tennis ball costs Rs. 0.25 He spend the whole Rs. 100/- to buy the balls. How many of each balls he bought? ans :F=3,T=56,C=41 42. The distance between Station Atena and Station Barcena is 90 miles. A train starts from Atena towards Barcena. A bird starts at the same time from Barcena straight towards the moving train. On reaching the train, it instantaneously turns back and returns to Barcena. The bird makes these journeys from Barcena to the train and back to Barcena continuously till the train reaches Barcena. The bird finally returns to Barcena and rests. Calculate the total distance in miles the bird travels in the following two cases: (a) The bird flies at 90 miles per hour and the speed of the train is 60 miles per hour. (b) the bird flies at 60 miles per hour and the speed of the train is 90 miles per hour Ans: time of train=1hr.so dist of bird=60*1=60miles 43. A tennis championship is played on a knock-out basis, i.e., a player is out of the tournament when he loses a match. (a) How many players participate in the tournament if 15 matches are totally played? (b) How many matches are played in the tournament if 50 players totally participate? Ans: (a)16 (b)49
44.When I add 4 times my age 4 years from now to 5 times my age 5 years from now, I get 10 times my current age. How old will I be 3 years from now? Ans:Age=41 years. 45.A rich merchant had collected many gold coins. He did not
want anybody to know about them. One day, his wife asked, "How many gold coins do we have?" After pausing a moment, he replied, "Well! If I divide the coins into two unequal numbers, then 37 times the difference between the two numbers equals the difference between the squares of the two numbers." The wife looked puzzled. Can you help the merchant's wife by finding out how many gold R Ans:37 46. A set of football matches is to be organized in a "roundrobin" fashion, i.e., every participating team plays a match against every other team once and only once. If 21 matches are totally played, how many teams participated? Ans :7 47. Glenn and Jason each have a collection of cricket balls. Glenn said that if Jason would give him 2 of his balls they would have an equal number; but, if Glenn would give Jason 2 of his balls, Jason would have 2 times as many balls as Glenn. How many balls does Jason have? Ans: 14 48. Suppose 8 monkeys take 8 minutes to eat 8 bananas. a) How many minutes would it take 3 monkeys to eat 3 bananas? (b) How many monkeys would it take to eat 48 bananas in 48 minutes Ans: a)48 B)6 49. It was vacation time, and so I decided to visit my cousin's home. What a grand time we had! In the mornings, we both would go for a jog. The evenings were spent on the tennis court. Tiring as these activities were, we could manage only one per day, i.e., either we went for a jog or played tennis each day. There were days when we felt lazy and stayed home all day long. Now, there were 12 mornings when we did nothing, 18 evenings when we stayed at home, and a total of 14 days when we jogged or played tennis. For how many days did I stay at my cousin's place? Ans : 22 days 50 A 31" x 31" square metal plate needs to be fixed by a carpenter on to a wooden board. The carpenter
uses nails all along the edges of the square such that there are 32 nails on each side of the square. Each nail is at the same distance from the neighboring nails. How many nails does the carpenter use? Ans :124
Top
51. A man starts his walking at 3PM from point A, he walks at the rate of 4km/hr in plains and 3km/hr in hills to reach the point B. During his return journey he walks at the rate of 6km/hr in hills and 4km/hr in plains and reaches the point A at 9PM. What is the distance between A and B? Ans: 12km 52.2. A boy asks his father, " what is the age of grand father?". Father replied " He is x years old in x^2 years", and also said, "we are talking about 20th century". what is the year of birth of grand father? Ans: 1892 53. A boy travels in a scooter after covering 2/3rd of the distance the wheel got punctured he covered the remaining distance by walk. Walking time is twice that of the time the boy?s riding time. How many times the riding speed as that of the walking speed? Ans: 4 times. 54. In a Knockout tournament 51 teams are participated, every team thrown out of the tournament if they lost twice. How many matches to be held to choose the winner? Ans: 101 matches 55. A man sold 2 pens. Initial cost of each pen was Rs. 12. If he sell it together one at 25% profit and another 20% loss. Find the amount of loss or gain, if he sells them seperately. Ans: 60 Paise gain 56. Find the 3 digit no. whose last digit is the squareroot of the first digit and second digit is the sum of the other two digits. Ans: 462
57. Meera was playing with her brother using 55 blocks.She gets bored playing and starts arranging the blocks such that the no. of blocks in each row is one less than that in the lower row. Find how many were there in the bottom most row? Ans: 10 58. Two people are playing with a pair of dies. Instead of numbers, the dies have different colors on theirsides. The first person wins if the same color appears on both the dies and the second person wins if the colors are different. The odds of their winning are equal. If the first dice has 5 red sides and 1 blue side, find the color(s) on the second one. Ans: 3 Red, 3 Blue 59. A person travels in a car with uniform speed. He observes the milestone,which has 2 digits. After one hour he observes another milestone with same digits reversed. After another hour he observes another milestone with same 2 digits separated by 0. Find the speed of the car? Ans : 45 60. Three persons A, B &C went for a robbery in different directions and they theft one horse, one mule and one camel. They were caught by the police and when interrogated gave the following statements A: B has stolen the horse B: I didn't rob anything. C: both A & B are false and B has stolen the mule. The person who has stolen the horse always tell the truth and The person who has stolen the camel always tell the lie. Find who has stolen which animal? Ans: A- camel B- mule C- horse 61. One quarter of the time till now from midnight and half of the time
remaining from now up to midnight adds to the present time. What is the present time? Ans: 9:36AM 62. After world war II three departments did as follows First department gave some tanks to 2nd &3rd departments equal to the number they are having. Then 2nd department gave some tanks to 1st & 3rd departments equal to the number they are having. Then 3rd department gave some tanks to 2nd &1st departments equal to the number they are having. Then each department has 24 tanks. Find the initial number of tanks of each department? Ans ; A-39 B-21 C-12 63. A, B, C, D&E are having their birthdays on consecutive days of the week not ecessarily in the same order. A 's birthday comes before G's as many days as B's birthday comes after E's. D is older than E by 2 days. This time G's birthday came on wednesday. Then find the day of each of their birthdays? Ans: Birthday of D on SUNDAY Birthday of B on MONDAY Birthday of E on TUESDAY Birthday of G on WEDNESDAY Birthday of A on THURSDAY 64. A girl 'A' told to her friend about the size and color of a snake she has seen in the beach. It is one of the colors brown/black/green and one of the sizes 35/45/55. If it were not green or if it were not of length 35 it is 55. If it were not black or if it were not of length 45 it is 55. If it were not black or if it were not of length 35 it is 55.
a) What is the color of the snake? b) What is the length of the snake? Ans: a) brown b) 55 65. There are 2 pesons each having same amount of marbles in the beginning. after that 1 person gain 20 more from second person n he eventually lose two third of it during the play n the second person now have 4 times marble of what 1st person is having now. find out how much marble did each had in the beginning. ANSWER - 100 each 66. A lady was out for shopping. she spent half of her money in buying A and gave 1 doller to bagger. futher she spent half of her remaining money and gave 2 doller to charity. futher she spent half of remaining money n gave 3 dollor to some childrans. now she has left with 1 doller. how much she had in the beginning? Ans \$42 67. There are certain diamonds in a shop. 1 thief stole half of diamonds and 2 more. 2 thief stole half of remaining and 2 more 3. same as above 4 same as above. 5 came nothing was left for that. how many dimands was there??? Ans 60 diamonds
68. There are three frens A B C. 1. Either A or B is oldest 2. Either C is oldest or A is youngest. Who is Youngest and who is Oldest? Ans A is youngest n B is oldest. 69. Father says my son is five times older than my daughter. my wife is 5 times older that my son. I am twice old from my wife and altogether (sum of our ages) is equal to my mother 's age and she is celebrating her 81 birthday. so what is my son's age? Ans - 5 years. 70.. In Mulund, the shoe store is closed every Monday, the boutique is closed every Tuesday, the grocery store is closed every Thursday and the bank is open only on Monday, Wednesday and Friday. Everything is closed on Sunday. One day A, B, C and D went shopping together, each with a different place to go. They made the following statements: A D and I wanted to go earlier in the week but there wasn?t day when we could both take care of our errands. B I did not want to come today but tomorrow I will not be able to do what I want to do. C I could have gone yesterday or the day before just as well as today. D Either yesterday or tomorrow would have suited me. Which place did each person visit ? Ans : A-BOUTIQUE B-BANK C-GROCERY D-SHOE 71. Fodder, pepsi and cereale often eat dinner out. each orders either coffee or tea after dinner. if fodder orders coffee, then pepsi orders the drink that
cereale orders if pepsi orders coffee, then fodder orders the drink that cereale doesnot oder if cereale orders tea, then fodder orders the drink that pepsi orders which person/persons always orders the same drink after dinner ? Ans:Fodder 72. At a recent birthday party there were four mothers and their children. Aged 1,2,3 and 4. from the clues below can you work out whose child is whose and their relevant ages ? It was jane?s child?s birthday party. Brian is not the oldest child. Sarah had Anne just over a year ago. Laura?s Child will be next birthday. Daniel is older than Charlie is. Teresa?s child is the oldest. Charlie is older than Laura?s child. Ans: Jane ? Charlie -3 Laura ? Brian ? 2 Teresa ? Daniel ? 4 Sarah ? Anne - 1 73. We are given 100 pieces of a puzzle. If fixing two components together is counted as 1 move ( a component can be one piece or an already fixed set of pieces), how many moves do we need to fix the entire puzzle. Ans: 99 74. Two guys work at some speed...After some time one guy realises he has done only half of the other guy completed which is equal to half of what is left !!! #\$%#\$ So how much faster than the other is this guy supposed to do to finish with the first. Ans: one and half times or 3/2 75. There is a square cabbage patch.He told his sister that i have a larger patch than last year and hence more cabbages thios year.Then how many cabbages i have this year.? Ans:106*106=11236 76. There are three guesses on the color of a mule 1 says:itz not black
2 says:itz brown or grey 3 says: itz brown Atlest one of them is wrong and one of them is true.....Then whatz the color of mule? Ans: Grey 77. Jim,Bud and sam were rounded up by the police yesterday. because one of them was suspected of having robbed the local bank. The three suspects made the following statements under intensive questioning. Jim: I'm innocent Bud: I'm innocent Sam: Bud is the guilty one. If only one of the statements turned out to be true, who robbed the bank? Ans:BUD. 78. There are two containers on a table. A and B . A is half full of wine, while B, which is twice A's size, is onequarter full of wine . Both containers are filled with water and the contents are poured into a third container C. What portion of container C's mixture is wine ? Ans:33.33% 79. A man was on his way to a marriage in a car with a constant speed. After 2 hours one of the tier is punctured and it took 10 minutes to replace it.
After that they traveled with a speed of 30 miles/hr and reached the marriage 30 minutes late to the scheduled time. The driver told that they would be late by 15 minutes only if the 10 minutes was not waste. Find the distance between the two towns? Ans: 120 miles 80. A bargainhunter bought some plates for \$ 1.30 from a sale on saturday,where price 2cents was marked off at each article .On monday she went to return them at regular prices,and bought some cups and saucers from that much amount of money only.the normal price of plate were equal to the price of 'one cup and one saucer'. In total she bought 16 items more than previous. saucers were only of 3 cents hence she brought 10 saucers more than the cups, How many cups and saucers she bought and at what price? Ans: 8,18 Price: 12,3. 81. Mr. T has a wrong weighing pan.One arm is lengthier than other.1 kilogram on left balances 8 melons on right.1 kilogram on right balances 2 melons on left.If all melons are equal in weight,what is the weight of a single melon? Ans:200 gms 82. A card boarb of 34 * 14 has to be attached to a wooden box and a total of 35 pins are to be used on the each side of the cardbox.Find the total number of pins used . Ans: 210
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Latest update
# Calculate Out of Plane Deflection Curve
2018-03-21 17:24:28
First of all, this is not a homework question.
I simply have a hard time finding relevant sources or help. Therefore, I am asking this question here. Hopefully, someone can help me.
I would like to calculate the analytic solution of the deflection curve of the L-Shaped cantilever beam as shown in the following image:
It is difficult, because the force acts out of the plane, and will therefore induce an out of plane bending of the cantilever. The cantilever will exhibit torsinal and bending stress.
For the L2 part the deflection is \$\$PL^3/3EI \$\$ considering it a simple cantilever beam with L= L2-W1
The L1 with the L = L1+ W2 has two deflections. first acting as a cantilever under load P. Second twisting under the torque P(L2-W1).
This is simplifying the stresses in the small corner rectangular W1.W2.
From Roark Formulas hanbook,
twist per inch length = T/(KG)
\$\$K = ab^3*[16/3-3.36*(b/a)*(1-((b/a)^4)/12)]\$\$
a = long side, W ; b = short side, T.
T =Torque in
• For the L2 part the deflection is \$\$PL^3/3EI \$\$ considering it a simple cantilever beam with L= L2-W1
The L1 with the L = L1+ W2 has two deflections. first acting as a cantilever under load P. Second twisting under the torque P(L2-W1).
This is simplifying the stresses in the small corner rectangular W1.W2.
From Roark Formulas hanbook,
twist per inch length = T/(KG)
\$\$K = ab^3*[16/3-3.36*(b/a)*(1-((b/a)^4)/12)]\$\$
a = long side, W ; b = short side, T.
T =Torque in-lb
L= Length in inches
G = Modulus of rigidity
This is twist per inch length, you multiply by L.
You now add the deflection of the L1 under cantilever action to the axis of L1.
You can modify theese results by considerin the small rectangula W1.W2 strains too.
This is kind of a rough first estimate. Just to give you preliminary numbers.
2018-03-21 18:06:41
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calculus
posted by .
Let A be the bounded region enclosed by
the graphs of
f(x) = x , g(x) = x3 .
Find the volume of the solid obtained by rotating the region A about the line x + 2 = 0.
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0
# What is the diagonal length of a rectangle that is 80 inches long and 60 inches wide?
Updated: 4/28/2022
Wiki User
11y ago
Because the diagonal would be the hypotenuse of a right triangle, its length is the square root of the sum of 802 square inches and 602 square inches, which is exactly 100 inches.
A quicker and easier way to determine the length of the diagonal is to recognize right away that you are working with a 3-4-5 right triangle, one of the classic forms used a lot in education, especially appealing because the length of every side is a whole number. The one in this problem is scaled by a factor of 20, but if you recognize the 3 to 4 ratio between the sides or the 3 to 5 or 4 to 5 ratios between a side and the hypotenuse, your work becomes a lot easier.
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You use the pythagorous theorm to calculate the hypotenuse of the triangle, which is the same line as the diagonal. 7(7)+ 10(10)= diagonal x diagonal 149= diagonal x diagonal Diagonal= square root of 149: this approximates to 12.207in Visit quickanswerz.com for more math help/tutoring! Consider a rectangle with dimensions 7 inches by 10 inches. Let ABCD be the rectangle. We need to find the length of the diagonal. We know that the diagonals of a rectangle are same in length. So, it is enough to find the length of the diagonal BD. From the rectangle ABCD, it is clear that the triangle BCD is a right angled triangle. So, we can find the length of the diagonal using the Pythagorean Theorem. BD2 = BC2 + DC2 BD2 = 102 + 72 BD2 = 100 + 49 BD2 = 149 BD = √149 BD = 12.207 So, the length of the diagonal is 12.21 inches. Source: www.icoachmath.com
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# P and C
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P and C [#permalink]
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26 Oct 2011, 17:45
In how many ways can one choose 6 cards from a normal deck of cards so as to
have all suits present?
a. (13^4) x 48 x 47
b. (13^4) x 27 x 47
c. 48C6
d. 13^4
e. (13^4) x 48C6
I have one basic question. For the problem above..
13c1 * 13c1 * 13c1 * 13c1 ... we need to have 2 cards.. when we choose 2 cards is the order in which we choose the cards is important?? like taking 2 and 5 is different from 5 and 2 ??? why is it 48 * 47 instead of 48c2 ??
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Re: P and C [#permalink]
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30 Oct 2011, 16:13
It says how many ways, so I think that orders do matter in this case. Otherwise, the answer should be totally revised, and the order of the first four cards should be considered.
Re: P and C [#permalink] 30 Oct 2011, 16:13
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Why register? ...To Enhance Your Experience
1. ## Hose weight
I am doing a little bit of research on the weight of charged hose lines and need some help figuring.
How much does a charged 50 foot section of 1-3/4", 2-1/2", 3" weigh?
What's the formula for figuring?
2. Using the nominial diameter as the inside diameter (e.g. assume the flow x-section of 1 3/4" hose is 1 3/4"), and the unit wieght of water at 62.4 lbs/cubic foot.
Calulate the flow area of the hose: (PI)x(diameter - in feet)^2.. result in square feet
Multiply the area by length (50 feet) gives the the volume of water in cubic feet
Multiply by the unit wieght of water gives pounds per 50' length
Results:
1 3/4" hose = 52 lbs per 50' length
2 1/2" hose = 106 lbs per 50' length
3" hose = 153 lbs per 50' length
4" hose = 272 lbs per 50' length
5" hose = 425 lbs per 50' length
3. One minor, but significant adjustment to the equation given by MichaelsDad:
Area = [(PI)x(diameter - in feet)^2]/4 (you must divide by 4)
Also, I have used the following equation which incorporates all of the factors used by MichaelsDad (BTW, Nice job).
Weight of water = 0.34 X d^2 X Hose Length
where "d" is in inches and Hose Length is in feet.
Of course, you must add the weight of the hose which is manufacturer specific.
And, if you REALLY REALLY want to be correct, add the weight of water absorbed by the cotton jacket of the hose.
4. Originally Posted by FireH2O
And, if you REALLY REALLY want to be correct, add the weight of water absorbed by the cotton jacket of the hose.
Don't forget to add in the weight of the dirt and mud too!
5. You forgot the couplings!
6. Or you could just put the hose on a scale!!!!
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# Fisher’s Exact Test
Fisher’s Exact Test is a statistical analysis tool used to determine the significance of the relationship between two categorical variables. It is commonly used when the sample size is small, and traditional statistical tests such as the chi-squared test may not be suitable. This test calculates the probability of obtaining a particular distribution of data, assuming that there is no association between the two variables. It is often used in medical research and other fields where small sample sizes are common. Fisher’s Exact Test provides a more accurate and reliable measure of significance compared to other tests in these situations.
## What is Fisher’s Exact Test?
Fisher’s Exact Test is a statistical test used to determine if the proportions of categories in two group variables significantly differ from each other. To use this test, you should have two group variables with two or more options and you should have fewer than 10 values per cell. See more below.
Fisher’s Exact Test is also called the Fisher Irwin Test, Fisher’s Exact Test of Independence, Fisher’s Test.
## Assumptions for Fisher’s Exact Test
Every statistical method has assumptions. Assumptions mean that your data must satisfy certain properties in order for statistical method results to be accurate.
The assumptions for Fisher’s Exact Test include:
1. Random Sample
2. Independence
3. Mutually exclusive groups
Let’s dive into what that means.
#### Random Sample
The data points for each group in your analysis must have come from a simple random sample. This is important because if your groups were not randomly determined then your analysis will be incorrect. In statistical terms this is called bias, or a tendency to have incorrect results because of bad data.
#### Independence
Each of your observations (data points) should be independent. This means that each value of your variables doesn’t “depend” on any of the others. For example, this assumption is usually violated when there are multiple data points over time from the same unit of observation (e.g. subject/customer/store), because the data points from the same unit of observation are likely to be related or affect one another.
#### Mutually Exclusive Groups
The two groups of your categorical variable should be mutually exclusive. For example, if your categorical variable is hungry (yes/no), then your groups are mutually exclusive, because one person cannot belong to both groups at once.
## When to use Fisher’s Exact Test?
You should use Fisher’s Exact Test in the following scenario:
1. You want to test the difference between two variables
2. Your variable of interest is proportional or categorical
3. You have only two options
4. You have independent samples
5. You have less than 10 in a cell
Let’s clarify these to help you know when to use Fisher’s Exact Test.
#### Difference
You are looking for a statistical test to look at how a variable differs between two groups. Other types of analyses include testing for a relationship between two variables or predicting one variable using another variable (prediction).
#### Proportional or Categorical
For this test, your variable of interest must be proportional or categorical. A categorical variable is a variable that contains categories without a natural order. Examples of categorical variables are eye color, city of residence, type of dog, etc. Proportional variables are derived from categorical variables, for instance: the number of people that converted on two different versions of your website (10% vs 15%), percentages, the number of people who voted vs people who did not vote, the proportion of plants that died vs survived an experimental treatment, etc.
If you want to compare two continuous variables, you may want to use an Independent Samples T-Test.
#### Two Options
Your categorical variables should have only two possible options. Some examples of variables like this are made a purchase (yes/no), color (if just black/white), recovered from disease (yes/no).
#### Independent Samples
Independent samples means that your two groups are not related in any way. For example, repeated measurements from the same group over time are often not independent samples, because each observation from the same person is likely related to other samples from that person.
If you have repeated measures from a single sample, you should consider using the McNemar Test.
#### Less than 10 in a Cell
The rule-of-thumb we recommend is to use this test when you have around 10 or fewer observations in each cell. “Cell” in this case refers simply to the count of values in each group. For example, if I have a list of survey responses with 5 “yes” and 1 “no”, there are 5 and 1 value(s) per cell, respectively.
If you have more than 10 in a cell, we recommend using the Two-Proportion Z-Test. And if you have more than 10 in every cell and more than 1000 total observations, we recommend using the G-Test.
## Fisher’s Exact Test Example
Group: Treatment (A/B)
Variable: Recovered from disease (yes/no)
In this example, we are interested in investigating whether our two treatment groups differ significantly in rate of recovery from disease. The null hypothesis is that there is no difference between recovery rates between the two groups.
Because our variable is binary with only two possible values (yes/no), and our data meet all other assumptions, we know that Fisher’s Exact Test is appropriate to use here.
The analysis will result in a probability or p-value. The p-value represents the chance of seeing our results if there was actually no difference in recovery rate between the two treatment types. A p-value less than or equal to 0.05 means that our result is statistically significant and we can trust that the difference is not due to chance alone.
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## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)
$x\leq 12,500$ dollars
Let the cost of production be $x$. The cost of the production of the software cannot exceed 12,500 dollars can be translated as: $x$ is less than or equal to 12,500. $x\leq 12,500$ dollars
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# Thread: use a Maclaurin to find a Maclaurin
1. ## use a Maclaurin to find a Maclaurin
Use a known Maclaurin polynomial to find a Maclaurin polynomial of degree 8 for f(x) = xsin(2x)
I am not sure what "Use a known Maclaurin polynomial" means. I know that sin(2x) = 2sin(x)cos(x). I know how to find a nth degree Maclaurin polynomial, keep diriving the function n times, plug in 0 for x, and divide by the n! multiple the result to the corrisponding x^n.
2. i tried finding the second dirivative and that took a while. finding the 8th will take a lot of work. Swapping xsin(2x) with 2xsin(x)cos(x) doesnt make it any easier. So can i break it part like g = x, h = sin (2x), f = g * h. find the 8th deg maclaurin for g and h and multiply them together?
3. The polynomial for $\sin x$ is $x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040}$
Plug in $2x$ for $x$ to get $2x-\frac{(2x)^3}{6}+\frac{(2x)^5}{120}-\frac{(2x)^7}{5040} = 2x-\frac{4x^3}{3}+\frac{4x^5}{15}-\frac{8x^7}{315}$
Multiply through by $x$ to get your final answer: $2x^2-\frac{4x^4}{3}+\frac{4x^6}{15}-\frac{8x^8}{315}$
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Search a number
4453 = 6173
BaseRepresentation
bin1000101100101
320002221
41011211
5120303
632341
715661
oct10545
96087
104453
113389
1226b1
132047
1418a1
1514bd
hex1165
4453 has 4 divisors (see below), whose sum is σ = 4588. Its totient is φ = 4320.
The previous prime is 4451. The next prime is 4457. The reversal of 4453 is 3544.
Adding to 4453 its reverse (3544), we get a palindrome (7997).
Subtracting from 4453 its reverse (3544), we obtain a palindrome (909).
It can be divided in two parts, 44 and 53, that multiplied together give a palindrome (2332).
It is a semiprime because it is the product of two primes, and also a brilliant number, because the two primes have the same length.
It can be written as a sum of positive squares in 2 ways, for example, as 1089 + 3364 = 33^2 + 58^2 .
It is a cyclic number.
It is not a de Polignac number, because 4453 - 21 = 4451 is a prime.
It is an Ulam number.
It is a Duffinian number.
It is a plaindrome in base 11 and base 15.
It is a congruent number.
It is not an unprimeable number, because it can be changed into a prime (4451) by changing a digit.
It is a polite number, since it can be written in 3 ways as a sum of consecutive naturals, for example, 25 + ... + 97.
It is an arithmetic number, because the mean of its divisors is an integer number (1147).
24453 is an apocalyptic number.
It is an amenable number.
4453 is a deficient number, since it is larger than the sum of its proper divisors (135).
4453 is an equidigital number, since it uses as much as digits as its factorization.
4453 is an evil number, because the sum of its binary digits is even.
The sum of its prime factors is 134.
The product of its digits is 240, while the sum is 16.
The square root of 4453 is about 66.7308024828. The cubic root of 4453 is about 16.4519569738.
The spelling of 4453 in words is "four thousand, four hundred fifty-three".
Divisors: 1 61 73 4453
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Explain the difference between experimental probability and theoretical probability using an example. What is the difference between theoretical and experimental probability? Next, we complete a quick experiment. Theoretical And Experimental Probability - Displaying top 8 worksheets found for this concept. At all times, probability is given as a number between 0 and 1, where 1 and 0 imply that the event will definitely occur and the event will not occur respectively. Probabilities are calculated using the simple formula: Probability = Number of desired outcomes ÷ Number of possible outcomes. The theoretical probability of getting a 6 is $\frac{1}{6}$. Experimental probability. Practice: Simple probability. I take out a coin, I ask students to remind me about the theoretical probability of flipping a coin on heads (1/2).
Then the probability of getting head is 3/10. Percentage into Ratio Step I: Obtain the percentage. Probability is the measure of expectation that a specific event will occur or a statement will be true. What is the probability it will land on tails?”) and then ask, “is this an example of theoretical or experimental probability?”. Compare theoretical and experimental probability. Intro to theoretical probability. Simple probability: yellow marble. Intuitive sense of probabilities ... Email. Experimental probability is also useful when a theoretical probability is too difficult to compute, or when events are not equally likely. Experimental Probability Vs Theoretical Probability. around the world. I display these examples (i.e. https://www.onlinemathlearning.com/theoretical-experimental-probability.html Conduct the experiment to get the experimental probability. Let’s go back to the die tossing example. Theoretical probability is what is expected to happen. A good example of this is weather. roll a die or conduct a survey). Example: You asked your 3 friends Shakshi, Shreya and Ravi to toss a fair coin 15 times each in a row and the outcome of this experiment is given as below: This means that in 12 throws we would have expected to get 6 twice. Simple probability: non-blue marble. Answer to 1. Practice: Experimental probability. Please update your bookmarks accordingly. Theoretical vs Experimental Probability . This is the currently selected item. Experimental probability is the result of an experiment. Math Module 2 Notes Lesson one – Odds and Probability Review 1. Experimental probability. “we flip a coin. We have moved all content for this concept to for better organization. 2. You can compare that to the theoretical probability. If after 12 throws you get one 6, then the experimental probability is $\frac{1}{12}$. Basic probability. Experimental Probability Example.
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David Neubelt
Member
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1. Questions on Baked GI Spherical Harmonics
For the figure on page (18) it was generated with SH9 baked into light map texels. However, you're approach with a grid will work as well if it's dense enough.The occlusion will come naturally and there are no additional techniques you need to apply because at every position in shadow the cubemap will receive less light. This works for both diffuse and ambient. I suspect the reason you aren't seeing any occlusion is because you are adding an ambient term artificially flattening out the scene. If there is a bright light on the opposite side of the probe then yes. Ringing can occur when you have a bright light on one side which causes that side of the probe to generate really high numbers. On the opposite side of the probe facing away from the light it will need a really high negative number to counter act the high number. The negative numbers are the cause of the ringing
2. The Order 1886: Spherical Gaussian Lightmaps
1) To uniformly distribute the basis vectors - you can choose any distribution you want but that is just the one we went with. 2) Yes, the free parameters you want are the sharpness and amplitude. There really is no rotation its just use the lights direction vector. -= Dave
3. View matrix - explanation of its elements
To expand, If you have a transformation that is a rotation and a translation, e.g. the 3d camera's position is t and it's rotation is R then it's world transform is, x' = Rx + t, where R is a 3x3 matrix, x is a 3x1 matrix(column vector) and t is a 3x1 matrix(column vector). The output is x', your transformed vector. To get a view matrix you want to bring all vertices in the frame of reference of the camera so the camera sits at the origin looking down one of the Cartesian axis. You simply need to invert the above equation and solve for x. x' = Rx + t x' - t = Rx R-1(x' - t) = (R-1)Rx RT(x' - t) = x // because a rotation matrix is orthogonal then its transpose equals its inverse x = RTx' - RTt So to invert a 4x4 camera transformation matrix, efficiently, you will want to transpose the upper 3x3 block of the matrix and for the translation part of the matrix you will want to negate and transform your translation vector by the transposed rotation matrix. -= Dave
4. Spherical Worlds using Spherical Coordinates
Geodesic grids work with minimal distortion. They work by starting with an icosahedron and subdividing. http://kiwi.atmos.colostate.edu/BUGS/geodesic/text.html
6. Gaussian filter confusion
Hmm, I'd imagine the RTR is a typo because typically you don't want to scale and you want to be normalized to 1 when you integrate over the domain -\inf to \inf. That's what the term out in the front is for. I don't have the book on me at the moment to see in context what they are talking about. -= Dave
They aren't implausible assumptions otherwise we wouldn't make them to begin with! Hopefully,no infinities creep in to our answer otherwise we we would get flux of infinite values and that's not very useful. Our end goal is to obtain plausible results that are close to the ground truth. The only reason we use simplifying assumptions and simpler models is that we want simpler calculations. We know that with our simplified models we get results that are very close to the ground truth. An example is to look at the general form of the radiation transfer equation (it describes how much flux is transfered between two surface elements). The general form requires double integrals over both surfaces which is computationally expensive. Sometimes its good enough to say that we can approximate the result as two point sources and ask the question how much energy is transfered between two point sources. These approximations will be valid and come to the right solution, for our purposes, if the two points are far enough away and the surfaces are small enough. Since the point-to-point radiation transfer equation gave us the same answer, within our acceptable tolerance, and with no integrals we are happy to use it. Additionally, with some mathematical foot work you can show that the point-to-point transfer equation is derived directly from the defitions of irradiance, intensity and solid angle so its mathematically sound with its feet firmly planted in the ground of physical plausability. In the same vein, it's ok to use a pinhole model and if you do it correctly and you make some assumptions about your scene, light and camera then the result should be very similiar to if you had an aperature, integrated over sensor area, over time and all wavelengths. For example, you could write a simple raytracer that did a montecarlo integration at every pixel with a very small aperature for one spherical area light very far away from the scene and it would come very close to a rasterizer that used the point light source and a pinhole camera. Hope that makes sense. -= Dave
Sorry for the late reply I've been busy. 1 The above should be a little more clear. We want to develop a BRDF that for the intensity of the light coming in one direction, \L_i, should be equal to the light going out in the direction of reflection. Yah, sorry you want to setup the dirac delta function so that the angle away from the normal axis is equivalent and the other angle is equivalent but rotated 180 degrees. Right, as you point out, you will want to integrate over the visible hemisphere(aperture), sensor area, and time (equivalent to shutter speed) which beyond just controlling exposure will give you phenomenon such as motion blur and depth of field. If instead of integrating RGB you additionally integrate over the wavelengths then you can get other phenomenon such as chromatic aberration. -= Dave
The fundamental radiation transfer equation deals with integrating out all of the differential terms to get flux arriving at a sensor (integrating over area's of the surfaces). When you have an image plane it will have sensors with finite area that will detect, even in the case of a pinhole camera, radiation coming from a finite surface area in a scene. In your diagrams you are simply point sampling a contiguous signal of that flux.
The easiest way is to work backwards. You know you want to reflect out what is coming in so you construct a BRDF to match those conditions. In this case, you want a BRDF the reflects all energy in the mirror direction and to be valid it must maintain energy conservation and reciprocity. The first conditions requires the BRDF to zero everywhere except where, 1 and the second condition to conserve energy will require that, 2 thus, f must equal to 3 and the third condition is easily verified. P.S. Sorry, I tried to use latex but the forum blocked me from using latex images. I tried using the equation tags but the latex wouldn't parse so I just posted the link instead
What are you guys doing for indirect specular? There are details in the course notes but specular probes.
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invest_3ed.pdf
# R calculate an empirical p value for the ratio you
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(r) Calculate an empirical p-value for the ratio you calculated in (m). What conclusion would you draw based on this p-value? Probability Result: For large samples, this sampling distribution is well modeled by an F distribution with parameters number of groups ± 1 and overall sample size ± number of groups , the degrees of freedom of the numerator and denominator respectively.
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Chance/Rossman, 2015 ISCAM III Investigation 5.4 342 Terminology Detour We will compare I group means, where each group has n i observations. The overall sample size will be denoted by N = 6 n i . H 0 : There is no treatment effect or H 0 : P 1 = … = P I H a : There is a treatment effect or H a : at least one P i differs from the rest The between-group variability will be measured by looking at the sum of the squared deviations of the group means to the overall mean, . Each group mean is weighted by the sample size of that group. We will refer to this quantity as the “sum of squares for treatment,” SST. SST = ³ ´ ¦ ± I i i i x x n 1 2 We will then “average” these valu es by considering how many groups were involved. This quantity will be referred to as the “mean square for treatments.” MST = ) 1 ( ± I SST Note, if we fix the overall mean, once we know I ± 1 of the group means, the value of the i th mean is determined. So the degrees of freedom of this quantity is I ± 1. The within-group variability will be measured by the pooled variance. In general, each term will be weighted by the sample size of that group. We will again divide by an indication of the o verall sample size across the groups. We will refer to this quantity as the “mean squares for error,” MSE. MSE = I N s n I i i i ± ± ¦ 1 2 ) 1 ( which has N ± I degrees of freedom. The test statistic is then the ratio of these “mean square” quantities: MSE MST F When the null hypothesis is true, this test statistic should be close to 1. So larger values of F provide evidence against the null hypothesis. The corresponding p-value comes from a probability distribution called the F distribution with I ± 1 and N ± I degrees of freedom. We will use this F distribution to approximate both the sampling distribution of this test statistic in repeated samples from the same population (H 0 : P 1 = = P , ´ and the randomization distribution for a randomized experiment (H 0 : no treatment effect) as long as the technical conditions (see below) are met. Because we are focusing on the variance of group means, this procedure is termed Analysis of Variance (ANOVA). With one explanatory variable, this is called one-way ANOVA. x
Chance/Rossman, 2015 ISCAM III Investigation 5.4 343 Technology Detour ANOVA In R x If you have the data in a response vector and an explanatory vector, you can use summary(aov(response~explanatory)) This will output the Mean Square values in the Mean Sq column, first for the treatment group (the disability row) and then for the error term (Residuals).
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Question Video: Finding the First Derivative of a Function Involving Negative Exponents Using the Power Rule | Nagwa Question Video: Finding the First Derivative of a Function Involving Negative Exponents Using the Power Rule | Nagwa
Question Video: Finding the First Derivative of a Function Involving Negative Exponents Using the Power Rule Mathematics • Second Year of Secondary School
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Find ππ¦/ππ₯, given that π¦ = β43/π₯βΈ.
02:19
Video Transcript
Find ππ¦ ππ₯, given that π¦ equals negative 43 over π₯ to the power of eight.
So to find ππ¦ ππ₯, what weβre gonna need to do is actually differentiate our function. So the first thing Iβm gonna do is Iβm actually going to rewrite our function. And in order to do that, what Iβm gonna use is an exponent rule. And the exponent rule is that one over π to the power of π is equal to π to the power of negative π. So therefore, we can say that π¦ is equal to negative 43π₯ to the power of negative eight. And Iβve done this so that we can actually remove the fraction and it makes it easier to differentiate.
Okay, now, weβre actually gonna move on to the differentiation. And if we want to differentiate, we think about our function in the form ππ₯ to the power of π. So if we have a function in this form, then we can say that the derivative which Iβve denoted here is π ππ₯ of π π₯ is gonna be equal to ππ π₯ to the power of π minus one. So what Iβve done here is Iβve actually multiplied our coefficient by our exponents β so π and π. And then, weβve reduced the exponent by one. So weβve got π minus one.
Okay, great, we know what to do. Itβs to use this to actually differentiate our function. So then, we can say that the derivative of our function or ππ¦ ππ₯ is equal to negative 43 multiplied by negative eight cause thatβs our coefficient multiplied by our exponent and then π₯ to the power of negative eight minus one. Right, so now, we can actually simplify this and this gives us 344π₯ to the power of negative nine. And we got 344 because we got negative 43 multiplied by negative eight. And a negative multiplied by a negative gives us a positive. And weβve got π₯ to the power of negative nine because negative eight minus one gives us negative nine.
So fantastic, weβve actually reached an answer here. Weβve differentiated our term. The final thing weβre gonna do is actually rewrite it in our original form β so include a fraction again. And in order to do this, what Iβll use is the exponent rule that I used earlier just in the opposite way. And that rule was the one over π to the power of π equals π to the power of negative π. So we can say that given that π¦ equals negative 43 over π₯ to the power of eight ππ¦ ππ₯ is equal to 344 over π₯ to the power of nine.
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# Replace two consecutive equal values with one greater
Difficulty Level Easy
ArrayViews 713
## Problem Statement
Suppose you have an integer array. The problem “Replace two consecutive equal values with one greater” asks to replace all those pair values say ‘a’ which comes consecutively with a number “a+1” 1 greater than them (two consecutive numbers), such that even after the modification or repetition there will be no new consecutive pair remaining.
## Example
`arr[]={5, 2, 1, 1, 2, 2}`
`5 4`
Explanation
As 1 is a consecutive number it is replaced by a value greater than that means 2 and arr will become {5,2,2,2,2}
Now 2 is a consecutive number so will be replaced by a number greater than it i.e., 3 and arr will become {5,3,2,2}
Again 2 is a consecutive number so will be replaced by 3 and arr will become {5,3,3}
Now 3 is the consecutive number so will be replaced by 4 and arr will become {5,4}
`arr[]={2,5,5,6,2,7,7}`
`2 7 2 8`
Explanation
As 5 is a consecutive number it is replaced with a value 1 greater than that means 6. So arr will look like arr[]={2,6,6,2,7,7}
6 come in place of that but the next number is also 6, so it is also replaced with the value 1 greater than 6, means 7 and arr will become {2,7,2,7,7}
As 7 also occurs consecutively at last, so it is replaced with the 8 and arr will become {2,7,2,8}
## Algorithm
```1. Set the position’s value to 0.
2. Traverse the array from o to n(n is the length of the array).
1. Copy the value of arr[i] to arr[position] and increase the value of the position by 1.
2. While the position is greater than 1, its previous two values are equal or not.
1. Decrease the value of a position by 1,
2. and increase the value of arr[position -1] by 1.
3. Print the array from index 0 to position.```
Explanation
We have given an array of integers. We have asked to replace all those values which come consecutively with a number 1 greater than the number itself. If 4 come in arrays consecutively, then it will be replaced with the value 5. That is 1 greater than the number 4. Now with one traversal, we can only make a modification. Suppose, there are 3 numbers present 4, 4, 5. Then we will be converting 4, 4 to 5 and then also 5 is a consecutive number. Because its next number is same as the number itself. So we will be doing this using a nested loop.
Traverse the array from 0 to n. Open a loop, so it will become a nested loop. With the outer loop, we will be handling the traversals. And with the inner loop, we are going to update the values or replace the values according to the given condition. In the outer loop, we are going to copy the values in the same array up to two values.
Only after two traversals in the outer loop, it will go in the inner loop. In a while loop, we are going to check if the indices value is greater than 1. Because we are going to compare if the previous two values are equal. That’s why we leave that condition that two values must be copied into the array positioned values. Then simply decrease the values of position and update the array element with values 1 greater than the number itself. We will just keep on with this loop and this method. It will continuously replace all those values with the consecutive ones.
Now print the array from 0 to index position which was last updated, it will give the desired array.
## Code
### C++ code to Replace two consecutive equal values with one greater
```#include<iostream>
using namespace std;
void replaceValues(int arr[], int n)
{
int position = 0;
for (int i = 0; i < n; i++)
{
arr[position++] = arr[i];
while (position > 1 && arr[position - 2] == arr[position - 1])
{
position--;
arr[position - 1]++;
}
}
for (int i = 0; i < position; i++)
cout << arr[i] << " ";
}
int main()
{
int arr[] = { 2,5,5,6,2,7,7};
int n = sizeof(arr) / sizeof(int);
replaceValues(arr, n);
return 0;
}
```
`2 7 2 8`
### Java code to Replace two consecutive equal values with one greater
```class replaceConsecutiveValues
{
public static void replaceValues(int arr[], int n)
{
int position = 0;
for (int i = 0; i < n; i++)
{
arr[position++] = arr[i];
while (position > 1 && arr[position - 2] == arr[position - 1])
{
position--;
arr[position - 1]++;
}
}
for (int i = 0; i < position; i++)
System.out.print( arr[i] + " ");
}
public static void main(String args[])
{
int arr[] = {2,5,5,6,2,7,7};
int n = arr.length;
replaceValues (arr, n);
}
}
```
`2 7 2 8`
## Complexity Analysis
### Time Complexity
O(n2where“n” is the number of elements in the array. Because we have used two nested loops which made the algorithm to run in polynomial time.
### Space Complexity
O(1), that is independent of the number of elements in the array. The algorithm itself takes constant space but the program as a whole takes O(N) space(for input).
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# Find by integration the area of the region bounded by the curve y=2x−x2 and the x-axis.
A
23 sq. units
B
43 sq. units
C
53 sq. units
D
83 sq. units
Video Solution
Text Solution
Verified by Experts
## Given equation of curve is y=2x−x2 ⇒x2−2x=−y ⇒x2−2x+1=−y+1 ⇒(x−1)2=−(y−1) This is the equation of parabola having vertex (1, 1) and open downward. The parabola intersect the X-axis, put y = 0, we get 0=2x−x2⇒x(2−x)=0 ⇒x=0,2 ∴ Area of bounded region between the curve and X-axis =∫20ydx =∫20(2x−x2)dx=[2x22−x33]20 =[4−83−0−0]=43 sq units.
|
Updated on:21/07/2023
### Knowledge Check
• Question 1 - Select One
## The area of the region bounded by the curve y=2x−x2 and x - axis is
A23 sq , units
B43 sq,units
C53 sq , units
D83 sq. units
• Question 2 - Select One
## The area of the region bounded by the curve y=2x−x2and X- axis is
A23 sq.units
B43 sq.units
C53 sq.units
D83 sq.units
• Question 3 - Select One
## Area of the region bounded by the curve y=x2−5x+4 and the X-axis is
A32
B52
C72
D92
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# Birthday problem extension to unequal probabilities and multiple collisions
Let $$p_1, ... ,p_k$$ denote the probabilities of drawing bin $$1, .. ,k$$, where $$\sum_{i = 1}^{k} p_i= 1$$. My question is if we draw $$n$$ times, how can I show that the probability that all bins are drawn less than $$C$$ times is maximized by $$p_1 = p_2 = ... = p_k = 1/k$$ ?
I've tried using conditional probability to solve inductively such as assuming bin $$i$$ is drawn $$ times and trying to reason about the remaining $$k-1$$, but no luck so far. Computing explicitly seems to be really difficult.
Here is a proof using generating functions. Let $$c\geq 2$$ and $$k\geq 2$$ be fixed.
Let $$X=(X_1(n),\ldots,X_k(n))$$ be the $$k$$-tuple of occupancy numbers at time $$n$$, i.e. $$X_i(n)$$ = number of bins of type $$i$$ drawn at time $$n$$.
Clearly $$X$$ has the multinomial distribution with parameters $$n$$ and $$p=(p_1,\ldots,p_k)$$. Therefore
$$\mathbb{P}( \max X_i(n)\leq c-1)=n!\,[t^n] \prod _{i=1}^n q_c(p_it)$$
where $$q_c(t):=\sum_{j=0}^{c-1}\frac{t^j}{j!}$$ is the $$c$$th partial sum of $$\exp(t)$$.
The following lemma is the key.
Lemma
When $$p_i,p_j$$ are each replaced by their arithmetic mean $$a:=(p_i+p_j)/2$$
$$\mathbb{P}(\max X_i(n) \leq c-1 )$$ will not decrease.
If $$c\leq n \leq k(c-1)$$ and all other $$p_\ell$$ are positive $$\mathbb{P}(\max X_i(n) \leq c-1 )$$ will strictly increase if $$p_i\not = p_j$$.
Proof Consider first the case of two factors. Let $$x,y\in\mathbb{R}_+,x. We show that $$[t^m] q_c(xt)q_c(yt)\leq [t^m] \left(q_c(((x+y)/2)t\right)^2\;\;.$$It is easy to see that equality holds for $$m\leq c-1$$ and $$m>2c-2$$. Let $$c\leq m \leq 2c-2$$. We have $$m!\,[t^m] q_{c}(xt)q_c(yt)=(x+y)^m-\sum_{i=0}^{m-c} {m \choose i}\left(x^i y^{m-i}+y^i x^{m-i}\right)$$ For fixed sum $$s=x+y$$ the function $$x \mapsto f(x):=\sum_{i=0}^{m-c} {m \choose i}\left(x^i y^{m-i}+y^i x^{m-i}\right)$$ has the derivative $$f^\prime(x)=(m-c+1){m \choose c} \left(x^{m-c+1}(s-x)^c-(s-x)^{m-c+1}x^c\right)$$. Thus $$x\mapsto f(x)$$ is strictly decreasing (resp. increasing) on $$[0,s/2]$$ (resp. $$[s/2,s]$$), attaining its minimum at $$x=s/2$$. The rest is easy. End proof.
For any distribution $$p=(p_1,\ldots,p_k)$$ we have that $$\mathbb{P}( \max X_i(n)\leq c-1)=1$$ for $$n\leq c-1$$, and $$\mathbb{P}( \max X_i(n)\leq c-1)=0$$ for $$n > k(c-1)$$, so that only the cases $$c\leq n \leq k(c-1)$$ are of interest.
Corollary
For $$c\leq n \leq k(c-1)$$ the uniform distribution $$p_1=\ldots=p_k=\tfrac{1}{k}$$ uniquely maximises $$\mathbb{P}(\max X_i(n) \leq c-1 )$$.
Proof: Since we are maximising a continuous function (multivariate polynomial) on a compact set (simplex) the maximum is attained. Let $$p$$ be a maximising distribution. No $$p_i$$ can be $$0$$ (else replacing the last $$0$$ strictly increases the probability). But then $$p$$ can only be maximising if $$p$$ is the uniform distribution. End proof.
Remark Lemma 2 shows that the probability $$\mathbb{P}(\max X_i(n) \leq c-1 )$$ is a Schur-concave function on the simplex $$p_i\geq 0, \sum_{i=1}^k p_i=1$$. Possibly the result can be found under this heading in the literature (but a brief Google-search didn't reveal anything).
Experts may have a better answer.
This is I believe very hard if not impossible to do exactly, but if you use poissonization, i.e., model $$X_i$$ as independent poisson arrivals into bin $$i$$ with rate $$np_i$$ then the new process $$(X_1,\ldots, X_k)$$ equals the original bin arrival process $$(Y_1,\ldots,Y_k)$$ in expectation.
There is also a property that the probability of a monotone event can only be at most a factor of $$2$$ different in the bin arrival model and this model (I saw this in Mitzenmacher and Upfal's Probability and Computing textbook). Your event
$$A=\{ \omega: \textrm{Max } X_i \leq C\}$$ is monotone. Maybe you can show strong concentration with upper and lower bounds on $$A$$ in the poisson model.
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# Lesson 15
Multiplying Rational Numbers
## 15.1: Which One Doesn’t Belong: Expressions (5 minutes)
### Warm-up
This warm-up prompts students to compare four expressions. It encourages students to explain their reasoning, hold mathematical conversations, and gives you the opportunity to hear how they use terminology and talk about the expressions in comparison to one another. To allow all students to access the activity, each expression has one obvious reason it does not belong.
### Launch
Arrange students in groups of 2–4. Display the expressions for all to see. Ask students to indicate when they have noticed one expression that does not belong and can explain why. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, tell each student to share their reasoning why a particular expression does not belong and together find at least one reason each expression doesn’t belong.
### Student Facing
Which expression doesn’t belong?
$$7.9x$$
$$7.9\boldcdot (\text- 10)$$
$$7.9 + x$$
$$\text-79$$
### Activity Synthesis
Ask each group to share one reason why a particular expression does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given make sense.
## 15.2: Rational Numbers Multiplication Grid (10 minutes)
### Optional activity
In this optional activity, students revisit the representation of a multiplication chart, which may be familiar from previous grades; however, in this activity, the multiplication chart is extended to include negative numbers. Students identify and continue patterns (MP8) to complete the chart and see that it fits the patterns in the chart for the product of two negative numbers to be a positive number.
The blackline master has a multiplication chart that also includes the factors 1.5, -1.5, 3.2, and -3.2, so that students can see how the patterns extend to rational numbers that are not integers. Encourage students to complete the rows and columns for the integers first and then come back to 1.5, -1.5, 3.2, and -3.2 later. Directions are included on the blackline master for a way that students can fold their papers to hide the non-integers while they fill in the integers. If you want students to do this, it would be good to demonstrate and walk them through the process of folding their paper.
### Launch
Arrange students in groups of 3. If desired, distribute 1 copy of the blackline master to every student and instruct students to ignore the chart printed in their books or devices. (Also if desired, instruct students to fold their papers according to the directions on the top and right sides of the chart, so that the decimal rows and columns are temporarily hidden.) Give students 30 seconds of quiet think time. Have them share what patterns they notice about the numbers that are already filled in. Give the groups 5 minutes of work time followed by whole-class discussion.
If students have access to the digital materials, students can use the applet to complete the chart. The applet helps students focus on fewer of the numbers and patterns at a time, similar to the purpose of folding the blackline master. Also, the applet gives students immediate feedback on whether their answers are correct which helps them test their theories about ramifications of multiplying by a negative number.
Representation: Internalize Comprehension. Activate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
### Student Facing
Look at the patterns along the rows and columns and continue those patterns to complete the table. When you have filled in all the boxes you can see, click on the "More Boxes" button.
What does this tell you about multiplication by a negative?
### Launch
Arrange students in groups of 3. If desired, distribute 1 copy of the blackline master to every student and instruct students to ignore the chart printed in their books or devices. (Also if desired, instruct students to fold their papers according to the directions on the top and right sides of the chart, so that the decimal rows and columns are temporarily hidden.) Give students 30 seconds of quiet think time. Have them share what patterns they notice about the numbers that are already filled in. Give the groups 5 minutes of work time followed by whole-class discussion.
If students have access to the digital materials, students can use the applet to complete the chart. The applet helps students focus on fewer of the numbers and patterns at a time, similar to the purpose of folding the blackline master. Also, the applet gives students immediate feedback on whether their answers are correct which helps them test their theories about ramifications of multiplying by a negative number.
Representation: Internalize Comprehension. Activate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
### Student Facing
1. Complete the shaded boxes in the multiplication square.
2. Look at the patterns along the rows and columns. Continue those patterns into the unshaded boxes.
3. Complete the whole table.
4. What does this tell you about multiplication with negative numbers?
### Anticipated Misconceptions
Some students may need a reminder of how a mutliplication chart works: the factors are listed at the end of the rows and columns, and their products go in the boxes.
### Activity Synthesis
The most important takeaway is that it makes sense for the product of two negative numbers to be a positive number, whether or not the numbers are integers. This fits in with the patterns in the extended multiplication chart. Those patterns depend on the distributive property. For example, the reason the numbers in the top row go up by 5s is that $$5(n+1) = 5n + 5$$. So when students extend the pattern to negative numbers, they are extending the distributive property.
Display a complete chart for all to see, and ask students to explain the ways in which the chart shows that the product of a negative and a negative is a positive. The general argument involves assuming that a pattern observed in a row or column will continue on the other side of 0.
## 15.3: Card Sort: Matching Expressions (10 minutes)
### Optional activity
This activity reminds students of the links between positive fractions and multiplication and prepares them to think about division as multiplication by the reciprocal; this will be important for dividing negative numbers. Students will use earlier work from grade 6 and their work in previous lessons in this unit to extend what they know about division of positive rationals to all rational numbers (MP7).
Ask students as they are working if there is an easy way to tell if two expressions are not equivalent, making note of students who reason about how many negative numbers are multiplied, and what the outcome will be. For example, they may have first gone through and marked whether each product would be positive or negative before doing any arithmetic.
Teacher Notes for IM 6–8 Accelerated
This activity is not optional for this course. It is an important opportunity for students to practice multiplying rational numbers, including unit fractions. This helps prepare students for dividing rational numbers in the next lesson.
### Launch
Ask students to recall the rules they have previously used about multiplication of signed numbers.
This is the first encounter with an expression where three integers are multiplied, so students might need to see an example of evaluating an expression like this one step at a time. Display an expression like this for all to see, and ask students how they might go about evaluating it: $$\displaystyle (\text-2)\boldcdot (\text-3)\boldcdot (\text-4)$$ The key insight is that you can consider only one product, and replace a pair of numbers with the product. In this example, you can replace $$(\text-2)\boldcdot (\text-3)$$ with 6. Then, you are just looking at $$(6)\boldcdot (\text-4)$$, which we already know how to evaluate.
Arrange students in groups of 2. Distribute sets of cards.
Representation: Internalize Comprehension. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity by beginning with fewer cards. For example, give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches.
Supports accessibility for: Conceptual processing; Organization
Conversing: MLR8 Discussion Supports. Arrange students in groups of 2. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “ ____ and ____ are equal because . . .”, and “I noticed ___ , so I matched . . .” Encourage students to challenge each other when they disagree. This will help students clarify their reasoning about multiplication of signed numbers.
Design Principle(s): Support sense-making; Maximize meta-awareness
### Student Facing
Your teacher will give you cards with multiplication expressions on them. Match the expressions that are equal to each other. There will be 3 cards in each group.
### Activity Synthesis
Ask the previously identified students to share their rationale for identifying those that do not match.
Consider highlighting the link between multiplying by a fraction and dividing by a whole number. If desired, ask students to predict the values of some division expressions with signed numbers. For example, students could use the expression $$\text-64 \boldcdot \frac{1}{8}$$ to predict the value of $$\text-64 \div 8$$. However, it is not necessary for students to learn rules for dividing signed numbers at this point. That will be the focus of future lessons.
## 15.4: Row Game: Multiplying Rational Numbers (10 minutes)
### Optional activity
This optional activity gives students an opportunity to practice multiplying signed numbers. The solutions to the problems in each row are the same, so students can check their work with a partner.
### Launch
Arrange students in groups of 2. Make sure students know how to play a row game. Give students 5–6 minutes of partner work time followed by whole-class discussion.
Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts. For example, after students have completed the first 2-3 rows of the table, check-in with either select groups of students or the whole class. Invite students to share how they have applied generalizations about multiplying signed numbers from the previous activity so far.
Supports accessibility for: Conceptual processing; Organization; Memory
### Student Facing
Evaluate the expressions in one of the columns. Your partner will work on the other column. Check in with your partner after you finish each row. Your answers in each row should be the same. If your answers aren’t the same, work together to find the error.
column A column B
$$790\div 10$$ $$(7.9)\boldcdot 10$$
$$\text- \frac67 \boldcdot 7$$ $$(0.1) \boldcdot \text- 60$$
$$(2.1) \boldcdot \text- 2$$ $$(\text-8.4) \boldcdot\frac12$$
$$(2.5) \boldcdot (\text-3.25)$$ $$\text{-} \frac52 \boldcdot \frac{13}{4}$$
$$\text-10 \boldcdot (3.2) \boldcdot (\text-7.3)$$ $$5\boldcdot (\text-1.6) \boldcdot (\text-29.2)$$
### Student Facing
#### Are you ready for more?
A sequence of rational numbers is made by starting with 1, and from then on, each term is one more than the reciprocal of the previous term. Evaluate the first few expressions in the sequence. Can you find any patterns? Find the 10th term in this sequence.
$$\displaystyle 1\qquad\quad 1+\frac{1}{1}\qquad\quad 1+\frac{1}{1+1}\qquad\quad 1+\frac{1}{1+\frac{1}{1+1}} \qquad \quad 1+\frac{1}{1+\frac{1}{1+\frac{1}{1+1}}}\qquad\quad \dots$$
### Activity Synthesis
Ask students, "Were there any rows that you and your partner did not get the same answer?" Invite students to share how they came to an agreement on the final answer for the problems in those rows.
Consider asking some of the following questions:
• "Did you and your partner use the same strategy for each row?"
• "What was the same and different about both of your strategies?"
• "Did you learn a new strategy from your partner?"
• "Did you try a new strategy while working on these questions?"
## Lesson Synthesis
### Lesson Synthesis
Display a number line with the numbers -1, 0, and 1 labeled. Ask students to give examples of multiplications problems with a product that is:
• greater than 1 (Sample responses: $$5 \boldcdot 3$$ or $$\text-5 \boldcdot \text-3$$)
• less than -1 (Sample responses: $$5 \boldcdot \text-3$$ or $$\text-5 \boldcdot \text-3 \boldcdot \text-1$$)
• between 0 and 1 (Sample responses: $$\frac15 \boldcdot \frac13$$ or $$\text-\frac15 \boldcdot \text-\frac13$$)
• between -1 and 0 (Sample responses: $$\frac15 \boldcdot \text-\frac13$$ or $$\text-\frac15 \boldcdot \text-\frac13 \boldcdot \text-1$$)
## Student Lesson Summary
### Student Facing
We can think of $$3\boldcdot 5$$ as $$5 + 5 + 5$$, which has a value of 15.
We can think of $$3\boldcdot (\text-5)$$ as $$\text-5 + \text-5 + \text-5$$, which has a value of -15.
We know we can multiply positive numbers in any order: $$3\boldcdot 5=5\boldcdot 3$$
If we can multiply signed numbers in any order, then $$(\text-5)\boldcdot 3$$ would also equal -15.
Now let’s think about multiplying two negatives.
We can find $$\text-5\boldcdot (3+\text-3)$$ in two ways:
• Applying the distributive property:
$$\text-5\boldcdot 3 + \text-5\boldcdot (\text-3)$$
• Adding the numbers in parentheses:
$$\text-5\boldcdot (0) = 0$$
This means that these expressions must be equal.
$$\text-5\boldcdot 3 + \text-5\boldcdot (\text-3) = 0$$
Multiplying the first two numbers gives
$$\text-15 + \text-5\boldcdot (\text-3) = 0$$
Which means that
$$\text-5\boldcdot (\text-3) = 15$$
There was nothing special about these particular numbers. This always works!
• A positive times a positive is always positive.
For example, $$\frac35 \boldcdot \frac78 = \frac{21}{40}$$.
• A negative times a negative is also positive.
For example, $$\text-\frac35 \boldcdot \text-\frac78 = \frac{21}{40}$$.
• A negative times a positive or a positive times a negative is always negative.
For example, $$\frac35 \boldcdot \text-\frac78 = \text-\frac35 \boldcdot \frac78 = \text-\frac{21}{40}$$.
• A negative times a negative times a negative is also negative.
For example, $$\text-3 \boldcdot \text-4 \boldcdot \text-5 = \text-60$$.
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# Homework Help: Expectation value for a superposition
1. Oct 28, 2007
### T-7
1. The problem statement, all variables and given/known data
$$u(x) = \sqrt{\frac{8}{5}}\left(\frac{3}{4}u_{1}(x)-\frac{1}{4}u_{3}(x)\right)$$
Determine the time-dependent expectation value of position of this wave function (the particle is in an infinite potential well between x = 0 and x = a).
3. The attempt at a solution
I make it a/2, ie. that the expectation value for this wave function is time-independent - we always expect it at the centre of the well. Does that seem reasonable for that superposition?
On integrating, I end up with three integrals, two of which amount to just the expectation values of the first and third eigenfunction times their respective probabilities (so I say [9/10]*[a/2] + [1/10]*[a/2], without bothering to unpack them) , and one integral which involves the product of u1 and u3 with x times a cosine component. This I evaluate as 0. Hence we have just a/2.
Does that seems stupid to anyone?
Cheers.
2. Oct 29, 2007
### Galileo
The wavefunction is supposed to represent the state of a physical system. Its time-dependence means the observable physical properties change with time. Wouldn't it be rather weird if you could change that by simply shifting your choice of axes???
The wavefunction is only time independent if it is in a stationary state. That is, an eigenstate of the Hamiltonian. (I guess that's what the u_i(x)'s stand for). Actually, a stationary state will pick up a time dependent phase-factor exp(-iwt), where the angular frequenty depends on the energy of the eigenstate.
When you have a linear combination of eigenstates, each with different energy, each term will pick up a different phase factor and it's not a stationary state anymore.
So what is the time dependence of u(x)?
3. Oct 29, 2007
### T-7
I expected time-dependence too for a superposition. But I still seem to be evaluating the relevant integral as zero:
<x> = (a/10).(a/2) + (1/10).(a/2) - (3/10).2cos$$\left(\frac{E_{3}-E_{1}}{\hbar}.t\right)\int^{+\infty}_{-\infty}u1.x.u3.x dx$$
4. Oct 29, 2007
### nrqed
Did you integrate from minus infinity to plus infinity or from 0 to a? You need to do it from 0 to a!
5. Oct 29, 2007
### T-7
Nope. From 0 to a. Tried it with Maple too.
The integral of sin(Pi*x/a) * x * (sin3*Pi*x/a) from 0 to a is zero.
int(sin(Pi*x/a)*x*sin(3*Pi*x/a),x=0..a);
ans: 0
If it was eigenfunction 1 with eigenfunction 4, say, it would be a different matter.
int(sin(Pi*x/a)*x*sin(4*Pi*x/a),x=0..a);
ans: (16a^2)/(-225*Pi^2)
Thinking about it, simply superimposing 1 and 3 does not create a net 'bulge' on either side of a/2 as it does with 1 and 2, say. Try drawing it.
The oscillating component seems to vanish because it is multiplying a zero integral.
Last edited: Oct 29, 2007
6. Oct 29, 2007
### Galileo
Well, it looks good then.
7. Oct 29, 2007
### nrqed
Oh, I did not notice it was u1 and u3! Of course it's zero (for some reason I thought it was u1 and u2). It's clear that it's zero since it's odd with respect to the center of the well, as you say. <x> is zero whenever the complete wave is a sum of two wavefunctions with index differing by an even number.
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## 308 Days Before November 21, 2023
Want to figure out the date that is exactly three hundred eight days before Nov 21, 2023 without counting?
Your starting date is November 21, 2023 so that means that 308 days earlier would be January 17, 2023.
You can check this by using the date difference calculator to measure the number of days before Jan 17, 2023 to Nov 21, 2023.
January 2023
• Sunday
• Monday
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January 17, 2023 is a Tuesday. It is the 17th day of the year, and in the 3rd week of the year (assuming each week starts on a Sunday), or the 1st quarter of the year. There are 31 days in this month. 2023 is not a leap year, so there are 365 days in this year. The short form for this date used in the United States is 01/17/2023, and almost everywhere else in the world it's 17/01/2023.
### What if you only counted weekdays?
In some cases, you might want to skip weekends and count only the weekdays. This could be useful if you know you have a deadline based on a certain number of business days. If you are trying to see what day falls on the exact date difference of 308 weekdays before Nov 21, 2023, you can count up each day skipping Saturdays and Sundays.
Start your calculation with Nov 21, 2023, which falls on a Tuesday. Counting forward, the next day would be a Wednesday.
To get exactly three hundred eight weekdays before Nov 21, 2023, you actually need to count 432 total days (including weekend days). That means that 308 weekdays before Nov 21, 2023 would be September 15, 2022.
If you're counting business days, don't forget to adjust this date for any holidays.
September 2022
• Sunday
• Monday
• Tuesday
• Wednesday
• Thursday
• Friday
• Saturday
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September 15, 2022 is a Thursday. It is the 258th day of the year, and in the 258th week of the year (assuming each week starts on a Sunday), or the 3rd quarter of the year. There are 30 days in this month. 2022 is not a leap year, so there are 365 days in this year. The short form for this date used in the United States is 09/15/2022, and almost everywhere else in the world it's 15/09/2022.
### Enter the number of days and the exact date
Type in the number of days and the exact date to calculate from. If you want to find a previous date, you can enter a negative number to figure out the number of days before the specified date.
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# Mystery Maths Week 5
The temperature two days ago was 10 degrees Celsius and it dropped by 6 degrees Celsius yesterday. Today it has dropped a further 6 degrees. What is the temperature today?
1. Manon Hollingworth
10-6=4
4-6=-2
The answer is negative 2 and it is in class 6 on the left of the door when you come in.
2. Henry
the answer is -2 and it’s underneath the + and the minus sign near the clock.
3. freddie
this is how i did it
10 digrees -6=4
-6= -2
4. freddie
it is in class 6 below the add and subtract desplay
5. Charlie driscoll
it is -2 and i worked it out by putting the 6s toghether it makes 12 -12 from 10 you get -2 it is under the multiplication sighns
6. livy
7. SHANNON + MORGAN
-2 UNDER THE CLOCK IN CLASS 6
8. minnie and georgia
-2 in class 6 under the clock
9. reuben
under the maths display in class 6 -2
10. Manon Hollingworth
10-6=4
4-6=-2
The number( -2) is in class 6 behind my chair, a bit to the right.
11. Sophie
The answer is -2. It is below the clock in class 6.
6+6=12
10-12=-2
12. Ella and Elle
10-6= 4 celsius
4-6=-2
It is in class6 and its under the clock and the sign which says sign.
13. logan
10-6-6=-2 🙂
14. isobel
it equals -2 in class 6 under the maths signs
15. logan
under subtraction words 🙂
16. Felix G
The answer is -2 and it’s below subtraction words in class 6
17. Elliot jones
the answer is minus 2 under the add and subtraction board on the wall.
The answer is -2 under a laminated piece of paper that says sign, in class 6.
19. beatrice
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Next:Exercise 9.2 Up:No Title Previous:Exercise 8.3
# Exercise 9.1
1.
To set ideas work on this simple data set.
``` X 12 15 18
Y 16 19 25 28```
(a)
Obtain all 12 differences (Y minus X).
(b)
Next obtain the point estimate, the median of the differences.
(c)
Subtract this estimate from the Y's and obtain the value of the Wilcoxon test statistic.
2.
For the last problem, use the following list of random numbers to obtain 2 resampled median of differences.
``` 2 9 2 2 7 2 2 3 0 8 8 1 9 8 8
2 3 3 4 0 9 2 1 0 7 9 3 6 6 2
3 7 6 8 8 7 0 5 0 3 4 3 5 7 7
3 4 5 0 1```
3.
Consider the batting averages of the switch hitters and the left-handed hitters from the baseball data set. Using class code Two-Sample Hypothesis and CI (Wilcoxon), obtain the estimate of the difference (Left minus switch) of batting averages and determine a 95% confidence interval for the difference. What does the interval mean in terms of the problem?
``` Switch .212 .218 .236 .242 .251 .251 .254 .261 .270 .282
Left .238 .271 .279 .283 .284 .290 .300 .303```
4.
Consider the following samples of Italian and Etruscan skull sizes. Use class code, Two-Sample Hypothesis and CI (Wilcoxon), to obtain the estimate of difference between a typical Etruscan skull and an Italian skull. Obtain a 95% confidence interval and interpret it in terms of the problem.
``` Ital. 134 132 126 134 131 130 130 125 132 126
Etru. 141 145 145 146 142 126 144 146 154 149 143 131```
5.
Let be the difference in weight between a typical pitcher and hitter, professional baseball players. Using class code,Two-Sample Hypothesis and CI (Wilcoxon) , estimate and determine a 95% confidence interval for it based on the following data. What does the interval mean in terms of the problem?
``` Hitters:
155 155 160 160 160 166 170 175 175 175 180
185 185 185 185 185 185 185 190 190 190 190
190 195 195 195 195 200 205 207 210 211 230
Pitchers:
160 175 180 185 185 185 190 190 195 195 195
200 200 200 200 205 205 210 210 218 219 220
222 225 225 232```
Next:Exercise 9.2 Up:No Title Previous:Exercise 8.3
2000-08-26
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# Self Dual functions in Digital Logic
A function is said to be Self dual if and only if its dual is equivalent to the given function, i.e., if a given function is f(X, Y, Z) = (XY + YZ + ZX) then its dual is, fd(X, Y, Z) = (X + Y).(Y + Z).(Z + X) (fd = dual of the given function) = (XY + YZ + ZX), it is equivalent to the given function. So function is self dual.
In a dual function:
1. AND operator of a given function is changed to OR operator and vice-versa.
2. A constant 1 (or true) of a given function is changed to a constant 0 (or false) and vice-versa.
A Switching function or Boolean function is said to be Self dual if :
1. The given function is neutral i.e., (number of min terms is equal to the number of max terms).For more about min term and max term (see Canonical and standard Form).
2. The function does not contain two mutually exclusive terms.
Note: Mutually exclusive term of XYZ is (X’Y’Z’) i.e, compliment of XYZ. So, two mutually exclusive terms are compliment of each other.
Example:
SL NO. X Y Z
0 0 0 0
1 0 0 1
2 0 1 0
3 0 1 1
4 1 0 0
5 1 0 1
6 1 1 0
7 1 1 1
In the above table, Mutually exclusive terms are:
`(0,7), (1,6), (2,5), (3,4) `
Explanation:
• Compliment of (000) i.e, 0 is (111) i.e, 7 so, (0, 7 are mutually exclusive to each other.)
• Compliment of (001) i.e, 1 is (110) i.e, 6 so, (1, 6 are mutually exclusive to each other.)
• Compliment of (010) i.e, 2 is (101) i.e, 5 so, (2, 5 are mutually exclusive to each other.)
• Compliment of (011) i.e, 3 is (100) i.e, 4 so, (3, 4 are mutually exclusive to each other.)
Now, lets check number of Self dual functions possible for a given function.
Let, a function has n variables then,
` Number of pairs possible = 2n/2 = 2(n-1)`
Therefore, number of Self dual functions possible with n variables
`= 22(n-1) `
There are 2 possibilities for each pair.
Example: What is total number of self dual of a function which has 3 variables X, Y and Z ?
```= 22(3-1)
= 222
= 24
= 16 ```
Note:
1. Every Self dual function is neutral but every neutral function is not Self dual.
2. Self duality is closed under compliment i.e, compliment of a Self dual function is also Self dual.
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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## A librarian has a set of ten books, including four different ##### This topic has 2 expert replies and 2 member replies ### Top Member ## A librarian has a set of ten books, including four different ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult A librarian has a set of ten books, including four different books about Abraham Lincoln. The librarian wants to put the ten books on a shelf with the four Lincoln books next to each other, somewhere on the shelf among the other six books. How many different arrangements of the ten books are possible? (A) (10!)/(4!) (B) (4!)(6!) (C) (4!)(7!) (D) (4!)(10!) (E) (4!)(6!)(10!) OA C Source: Magoosh ### GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1114 messages Followed by: 29 members Upvotes: 59 Quote: A librarian has a set of ten different books, including four books about Abraham Lincoln. The librarian wants to put the ten books on a shelf with the four Lincoln books next to each other, somewhere on the shelf among the other six books. How many different arrangements of the ten books are possible? (A) (10!)/(4!) (B) (4!)(6!) (C) (4!)(7!) (D) (4!)(10!) (E) (4!)(6!)(10!) Source: Magoosh $$?\,\,\,:\,\,\,\# \,\,\,{\rm{possibilities}}\,{\rm{,}}\,\,{\rm{Abe}}\,\,{\rm{books}}\;\,{\rm{together}}$$ $$10\,\,{\rm{different}}\,\,{\rm{books,}}\,\,{\rm{4}}\,\,{\rm{about}}\,{\rm{Abe}}\,\,\,\, \Rightarrow \,\,\,\,\left\{ \matrix{ \,1\,\,{\rm{multiple - block}}\,\,\,\left( {4\,\,{\rm{Abe}}\,\,{\rm{books}}} \right) \hfill \cr \,6\,\,{\rm{single}}\,{\rm{blocks}} \hfill \cr} \right.$$ $$\left. \matrix{ {P_7} = 7!\,\,\,{\rm{permutation}}\,\,{\rm{of}}\,\,{\rm{all}}\,\,{\rm{blocks}} \hfill \cr {{\rm{P}}_{\rm{4}}} = 4!\,\,\,{\rm{permutation}}\,\,{\rm{of}}\,\,{\rm{Abe}}\,\,{\rm{books}}\,\,\,\, \hfill \cr} \right\}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,?\,\,\, = \,\,{P_7} \cdot {P_4}\,\, = \,\,7!4!\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {\rm{C}} \right)$$ This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator ( Math for the GMAT) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br Master | Next Rank: 500 Posts Joined 15 Oct 2009 Posted: 321 messages Upvotes: 27 BTGmoderatorDC wrote: A librarian has a set of ten books, including four different books about Abraham Lincoln. The librarian wants to put the ten books on a shelf with the four Lincoln books next to each other, somewhere on the shelf among the other six books. How many different arrangements of the ten books are possible? (A) (10!)/(4!) (B) (4!)(6!) (C) (4!)(7!) (D) (4!)(10!) (E) (4!)(6!)(10!) OA C Source: Magoosh Consider the 4 Lincoln books as one book for the purposes of arranging on the shelf, since they have to be together. Along with the 6 other books, you can see that there are then 7 positions occupied. With the idea that order matters, there are then 7! ways to arrange the books on the shelf. Going back to the 4 Lincoln books, since the problem stated that they are "different", we are being told that order matters, so the number of ways to arrange the 4 LIncoln books is 4!. Total ways to arrange the books is therefore C, 7!x4! ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 12540 messages Followed by: 1245 members Upvotes: 5254 GMAT Score: 770 BTGmoderatorDC wrote: A librarian has a set of ten books, including four different books about Abraham Lincoln. The librarian wants to put the ten books on a shelf with the four Lincoln books next to each other, somewhere on the shelf among the other six books. How many different arrangements of the ten books are possible? (A) (10!)/(4!) (B) (4!)(6!) (C) (4!)(7!) (D) (4!)(10!) (E) (4!)(6!)(10!) Take the task of arranging the 10 books and break it into stages. Stage 1: Arrange the 4 books about Abe Lincoln in a row We can arrange n objects in n! ways. So, we can arrange the 4 books in 4! ways IMPORTANT: Now we'll "glue" the 4 Abe Lincoln books together to form 1 SUPER BOOK (this will ensure that the 4 Abe Lincoln books remain together) So, we now have 1 Abe Lincoln SUPER BOOK, along with 6 non-Abe Lincoln books (for a total of 7 "books") Stage 2: Arrange the 7 "books" We can arrange n objects in n! ways. So, we can arrange the 7 books in 7! ways By the Fundamental Counting Principle (FCP), we can complete the 2 stages (and thus arrange all of the books) in [color=blue](4!)(7!) ways Answer: C -------------------------- Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch our free video: http://www.gmatprepnow.com/module/gmat-counting/video/775 You can also watch a demonstration of the FCP in action: https://www.gmatprepnow.com/module/gmat-counting/video/776 Then you can try solving the following questions: EASY - http://www.beatthegmat.com/what-should-be-the-answer-t267256.html - http://www.beatthegmat.com/counting-problem-company-recruitment-t244302.html - http://www.beatthegmat.com/picking-a-5-digit-code-with-an-odd-middle-digit-t273110.html - http://www.beatthegmat.com/permutation-combination-simple-one-t257412.html - http://www.beatthegmat.com/simple-one-t270061.html MEDIUM - http://www.beatthegmat.com/combinatorics-solution-explanation-t273194.html - http://www.beatthegmat.com/arabian-horses-good-one-t150703.html - http://www.beatthegmat.com/sub-sets-probability-t273337.html - http://www.beatthegmat.com/combinatorics-problem-t273180.html - http://www.beatthegmat.com/digits-numbers-t270127.html - http://www.beatthegmat.com/doubt-on-separator-method-t271047.html - http://www.beatthegmat.com/combinatorics-problem-t267079.html DIFFICULT - http://www.beatthegmat.com/wonderful-p-c-ques-t271001.html - http://www.beatthegmat.com/permutation-and-combination-t273915.html - http://www.beatthegmat.com/permutation-t122873.html - http://www.beatthegmat.com/no-two-ladies-sit-together-t275661.html - http://www.beatthegmat.com/combinations-t123249.html Cheers, Brent _________________ Brent Hanneson – Creator of GMATPrepNow.com Use our video course along with Sign up for our free Question of the Day emails And check out all of our free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMAT’s FREE 60-Day Study Guide and reach your target score in 2 months! ### Top Member Legendary Member Joined 29 Oct 2017 Posted: 732 messages Followed by: 4 members Let's first "glue" the 4 Lincoln books together to create one SUPER BOOK (this will ensure that the 4 books remain together) We now have 7 books: 6 regular books and 1 super book We can arrange these 7 books in 7! ways. KEY: For each of the 7! arrangements, we can take the 4 Lincoln books (that comprise the SUPER BOOK) and arrange them in 4! ways. So, the TOTAL number of arrangements = (7!)(4!) Hence, the correct answer is C • 5-Day Free Trial 5-day free, full-access trial TTP Quant Available with Beat the GMAT members only code • 1 Hour Free BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • 5 Day FREE Trial Study Smarter, Not Harder Available with Beat the GMAT members only code • Free Trial & Practice Exam BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • Get 300+ Practice Questions 25 Video lessons and 6 Webinars for FREE Available with Beat the GMAT members only code • Free Veritas GMAT Class Experience Lesson 1 Live Free Available with Beat the GMAT members only code • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • Magoosh Study with Magoosh GMAT prep Available with Beat the GMAT members only code • FREE GMAT Exam Know how you'd score today for$0
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LINES, RAYS, AND PLANES
# LINES, RAYS, AND PLANES
Télécharger la présentation
## LINES, RAYS, AND PLANES
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##### Presentation Transcript
1. LINES, RAYS, AND PLANES
2. Lines, rays and planes are all around us. Let’s look at each one and see if we can find examples in our environment.
3. LINES What are lines? Lines are long, thin continuous marks.
4. There are 3 main types of lines. PARALLEL LINES PERPENDICULAR LINES INTERSECTING LINES
5. PARALLEL LINES What are parallel lines? Parallel lines are always the same distance apart and never touch.
6. Can you name parallel lines in the diagram below? A B E C D F AG and BH CE and DF H G
7. PERPENDICULAR LINES What are perpendicular lines? Perpendicular lines are lines that intersect (cross) and make 4 right angles.
8. Can you name perpendicular lines in the diagram below? A B C D E AF and BC AF and DE F
9. INTERSECTING LINES What are intersecting lines? Intersecting lines cross one another but never form right angles. Intersecting lines can resemble an ‘X’.
10. Can you name intersecting lines in the diagram below? B A C CD and AE CD and BF F E D
11. RAYS What are rays? Rays are lines that have a definite starting point but no end. Rays continue on endlessly in one direction.
12. Can you name any rays in the diagram below? A B C D E F FA and ED
13. PLANE What is a plane?(We are not talking about the one that flys in the sky). A plane is a flat surface. It is named by at least 3 points. A C This is plane ABC. B
14. Look around the classroom. Do you see any planes? Chalkboard Door Window Top of Desk Book Cover Floor Wall Paper Ceiling Dry Erase Boards
15. Can you find any parallel lines in the picture?
16. Are there any perpendicular lines in this picture?
17. Do you see any intersecting lines in these pictures?
18. What types of lines do can you find in these pictures?
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# How is excavation measured?
Contents
## What is the unit of measure in excavation?
Get solutionsGet solutions Get solutions done loading Looking for the textbook? The measurement of excavation should be done in cubic yard, cubic meters, cubic feet or in other words it should be measure in the form of volume.
## How is earthwork excavation measured?
The determination of earthwork quantities is based upon field cross- sections taken in a specified manner before and after excavation. Cross- sections are vertical profiles taken at right angles to the survey centerline. Every section is an area formed by the subgrade, the sideslopes, and the original ground surface.
## Is code for deep excavation?
4.13 No excavation or earthwork below the level of any foundation of building or structure shall be commenced or continued unless adequate steps are taken to prevent danger to any person employed, from collapse of the structure or fall of any part thereof.
## How much should I charge for excavation?
Excavation Costs
A typical residential excavation job runs between \$1,438 and \$5,295 with an average of \$3,274. Though most companies charge anywhere from \$40 to \$150 an hour, residential jobs receive project bids. Project bids reflect cubic yards of dirt moved, anywhere from \$50 to \$200 per cubic yard.
IT IS INTERESTING: How much do excavators cost to rent?
## How do you calculate slope of excavation?
Slope Angle Calculations
This simple equation will tell you the proper opening width: (depth x 2) x type slope ratio + width of original excavation = top width. As an example, let’s calculate the slope angle of a simple trench that is 6 feet deep by 2 feet wide, factoring in the type of soil. Type A: (6 feet x 2) x .
## What is the average depth of excavation cave in?
Subsequently, question is, what is the average depth of an excavation cave in? Some parts of a trench are more than 5 feet deep, while other parts are less than 5 feet deep. The average of those measurements is less than 5 feet.
## How is trench excavation measured?
Determine the volume of the trench by using the formula: Volume = Width x Length x Depth. As an example, a trench 12 feet long with an average width of 2.3 feet and an average depth of 5 feet has a volume of (12 x 2.3 x 5) cubic feet.
## What is the unit of earthwork?
1- Earthwork: It is measured in CUM or m3. This includes all types of excavation. Vegetation removal and other extra work is measured separately. 2- Backfilling: It is measured in CUM or m3.
## What is the unit of Distempering on wall?
Distempering Unit= Square Feet, or “sft”. iv. Painting Unit= Square Feet, or “sft”.
## What is earthwork excavation?
Besides digging the earth, the word excavation signifies the all the works related such as dressing of the sides, ramming of bottom, disposing of any soil not required out of site, and keeping the outer edge of excavation. …
IT IS INTERESTING: Question: What is a small excavator called?
## What is Prismoidal formula?
1) Prismoidal Formula:
This formula is based on the assumption that A1 and A2 are the areas at the ends and Am is the area of mid section parallel to ends, L=Length between the ends. From mensuration, volume of a prism having end faces is in parallel planes: V=L/6*(A1+A2+4Am) This is known as prismoidal formula.
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SOLUTION 7: We are given the equation $$x^3+2= \sin{x} \ \ \ \ \longrightarrow \ \ \ \ x^3+2-\sin{x}=0$$ Let function $$f(x)=x^3+2-\sin{x} \ \ \ \ and \ choose \ \ \ \ m=0$$ This function is continuous for all values of $x$ since it is the DIFFERENCE of continuous functions; it is well-known that $\sin{x}$ is continuous for all values of $x$ and $x^3+2$ is continuous for all values of $x$ since it is a polynomial. To establish an appropriate interval consider the graph of this function. (Please note that the graph of the function is not necessary for a valid proof, but the graph will help us understand how to use the Intermediate Value Theorem. On many subsequent problems, we will solve the problem without using the "luxury" of a graph.)
Note that $$f(0)= (0)^3+2-\sin{(0)} =2>0 \ \ \ \ and \ \ \ \ f(-\pi)= (-\pi)^3+2-\sin{(-\pi)}= -\pi^3+2-0=2-\pi^3 \approx -29<0$$
so that $$f(-\pi) \approx -29 < m <2=f(0)$$
i.e., $m=0$ is between $f(-\pi)$ and $f(0)$. Now choose the interval to be $\ [-\pi, 0]$.
The assumptions of the Intermediate Value Theorem have now been met, so we can conclude that there is some number $c$ in the interval $[-\pi, 0]$ which satisfies $$f(c)=m$$ i.e., $$c^3+2-\sin{c}=0$$ and the equation is solvable.
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