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### Instruction: help me answer this question Find the discriminant of the quadratic equation x^2 + x + 1/β2 = 0., think step by step
### Assistant: , The answer is The discriminant (D) of the quadratic equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac. For the given equation, a = 1, b = 1, and c = 1/β2. Substituting these values into the formula, we get D = (1)^2 - 4*1*1/β2 = 2 - 2/β2 = 2*(β2 - 1).
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### Instruction: help me answer this question Find the slope of the line that makes an angle of 30Β° with the positive direction of the y-axis measured anticlockwise., think step by step
### Assistant: , The answer is The slope of the given line is tan 120Β° = tan (180Β° β 60Β°) = β tan 60Β° = ββ3
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### Instruction: help me answer this question The shaded area in the graph is the solution area for the given inequalities x + 2y β€ 10, x + y β₯ 1, x - y β€ 0, x β₯ 0, y β₯ 0, think step by step
### Assistant: , The answer is The solution area for the given inequalities is the shaded area in the graph, which satisfies all the given inequalities simultaneously. This area is to the left of the line graph of x + 2y = 10, above the line graph of x + y = 1, below the line graph of x - y = 0, and in the first quadrant. The required points are (0, 5) and (10, 0).
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### Instruction: help me answer this question (v) {x: x is a triangle in a plane}β¦{x: x is a rectangle in the plane}, think step by step
### Assistant: Not every triangle in a plane is a rectangle in the same plane. , The answer is {x: x is a triangle in a plane} β {x: x is a rectangle in the plane}
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### Instruction: help me answer this question Given 3x + 4y β€ 60, what are the required points?, think step by step
### Assistant: These points are obtained by putting values of x = 0 and y = 0 in the equation one by one. , The answer is The required points are (0, 15) and (20, 0).
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### Instruction: help me answer this question The shaded area in the graph is the required solution area which satisfies all the given inequalities at the same time.
, think step by step
### Assistant: , The answer is N/A
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### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.
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### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.
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### Instruction: help me answer this question Given, β 3x + 2y β₯ β 6, think step by step
### Assistant: , The answer is The solution region of the given inequality is above the line β 3x + 2y β₯ β 6. (That is, the origin is included in the region)
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### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.
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### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step
### Assistant: , The answer is The domain of the function is `x β₯ 1`.
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### Instruction: help me answer this question How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?, think step by step
### Assistant: , The answer is 9 Γ 9 Γ 8 = 648
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### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.
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### Instruction: help me answer this question If a = 1, b = 31, n = m + 2, then find the value of m., think step by step
### Assistant: We are given that a = 1, b = 31, n = m + 2. We have to find the value of m. From the given data, we can form the following equation: 31 = 1 + (m + 2 β 1) (d) Simplifying the above equation, we get: 30 = (m + 1) d Further simplification leads us to: d = 30/ (m + 1) ...(1) As there is no additional information provided to find the value of d, we cannot proceed further. However, we have been asked to find the value of m, which can be found by rearranging the above equation as follows: m = 30/d - 1 Since we are not provided with a specific value for d, we can find the value of m for any given value of d by substituting it in the above equation. Therefore, the value of m cannot be determined uniquely and depends on the value of d. , The answer is The value of m is 14.
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### Instruction: help me answer this question Expand (99)5 using the binomial theorem., think step by step
### Assistant: The binomial theorem is applied with 99 expressed as 100 - 1. , The answer is 9509900499
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### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.
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### Instruction: help me answer this question a^2 + 8a β 2a β 16 = 0, think step by step
### Assistant: , The answer is Adding -2a to both sides of the equation a^2 + 6a β 16 = 0, we get a^2 + 8a β 2a β 16 = 0.
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### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers.
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### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step
### Assistant: , The answer is The domain of the function is `x β₯ 1`.
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### Instruction: help me answer this question (v) 1β A is incorrect, think step by step
### Assistant: , The answer is An element of a set can never be a subset of itself.
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### Instruction: help me answer this question (iii) {1, 2, 3} β {1, 3, 5}, think step by step
### Assistant: Here 2 is an element of {1, 2, 3} but not an element of {1, 3, 5}. , The answer is False
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### Instruction: help me answer this question Is event A a simple event?, think step by step
### Assistant: , The answer is Yes, A = (HHH) is a simple event.
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### Instruction: help me answer this question Find the equation of the tangent to the curve `y = x^2 - 3x + 2` at `x = 2`., think step by step
### Assistant: , The answer is The equation of the tangent is `y = 4x - 6`.
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### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step
### Assistant: , The answer is The domain of the function is `x β₯ 1`.
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### Instruction: help me answer this question For all the values of x, the value of y would be the same in the given inequality.
, think step by step
### Assistant: , The answer is N/A
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### Instruction: help me answer this question Which of the following is the LCM of 12 and 18?, think step by step
### Assistant: , The answer is The LCM of 12 and 18 is 36.
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### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers.
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### Instruction: help me answer this question Calculate the standard deviation for the group A data., think step by step
### Assistant: , The answer is The standard deviation for group A is 15.09.
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### Instruction: help me answer this question x2 = -16y, think step by step
### Assistant: Since the coefficient of y is negative, the parabola opens downwards. On comparing the given equation with x2 = -4ay, we get -4a = -16, which gives a = 4. Thus, the coordinates of the focus are (0, a) = (0, -4), and the equation of directrix is y = a, which gives y = 4. The length of the latus rectum is 4a = 16. , The answer is The coordinates of the focus are (0, -4), the axis of the parabola is the y-axis, the equation of the directrix is y = 4, and the length of the latus rectum is 16.
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### Instruction: help me answer this question Find the equation of the line passing through point (x1, y1) and having slope m = -A/B is, think step by step
### Assistant: This is the formula for finding the equation of a line when the slope and a point through which the line passes is given. , The answer is The equation of the line passing through point (x1, y1) and having slope m = -A/B is y β y1 = -A/B (x β x1) or B (y β y1) = -A (x β x1)
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### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step
### Assistant: , The answer is The domain of the function is `x β₯ 1`.
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### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers.
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### Instruction: help me answer this question If the ratio of the fifth term from the beginning to the fifth term from the end of the expansion of (a + b)βΏ is β6:1, find n., think step by step
### Assistant: The given text provides a detailed explanation for this calculation. , The answer is The value of n is 10.
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### Instruction: help me answer this question What is the Cartesian product of sets A = {1, 2} and B = {3, 4}?, think step by step
### Assistant: The Cartesian product of sets A and B is the set of all ordered pairs (a, b), where a is an element of set A and b is an element of set B. , The answer is A Γ B = {(1, 3), (1, 4), (2, 3), (2, 4)}
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### Instruction: help me answer this question Vertex (0, 0); focus (β2, 0), think step by step
### Assistant: , The answer is The equation of the parabola is y2 = -8x
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### Instruction: help me answer this question Prove that, think step by step
### Assistant: The solution to this problem is not provided in the text, but the reader is directed to access other solutions related to the binomial theorem for further assistance. , The answer is The solution to this problem is not provided in the text. It is suggested to access other exercise solutions of Class 11 Maths Chapter 8 β Binomial Theorem, specifically Exercise 8.2 Solutions 12 Questions and Miscellaneous Exercise on Chapter 8 Solutions 10 Questions.
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### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step
### Assistant: , The answer is The domain of the function is `x β₯ 1`.
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### Instruction: help me answer this question Write the equation of the ellipse., think step by step
### Assistant: , The answer is xΒ²/52 + yΒ²/13 = 1
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### Instruction: help me answer this question Find the equation of the line if point R (h, k) divides a line segment between the axes in the ratio 1:2., think step by step
### Assistant: , The answer is The equation of the line is 2x/3h + y/3k = 1.
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### Instruction: help me answer this question If f(x) = x^2, find f(2), f(-1), and f(0)., think step by step
### Assistant: , The answer is f(2) = 2^2 = 4, f(-1) = (-1)^2 = 1, and f(0) = 0^2 = 0.
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### Instruction: help me answer this question Determine the domain and range of the relation R defined by R = {(x, x + 5): x β {0, 1, 2, 3, 4, 5}}. , think step by step
### Assistant: , The answer is Domain of R = {0, 1, 2, 3, 4, 5} and Range of R = {5, 6, 7, 8, 9, 10}
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### Instruction: help me answer this question (x + 1)/3 β€ (x - 1)/2, think step by step
### Assistant: , The answer is x β (-β, -4] βͺ [1, β)
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### Instruction: help me answer this question Prove that (cos x β cos y)^2 + (sin x β sin y)^2 = 2 - 2cos(x β y), think step by step
### Assistant: This involves the use of trigonometric formulas and simplification. , The answer is LHS = (cos^2 x + cos^2 y - 2cos x cos y) + (sin^2 x + sin^2 y - 2sin x sin y) = 2 - 2cos(x β y) = RHS
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### Instruction: help me answer this question Find the variance for Y., think step by step
### Assistant: , The answer is The variance for Y is 4.
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### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step
### Assistant: , The answer is The domain of the function is `x β₯ 1`.
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### Instruction: help me answer this question Evaluate the given limit:, think step by step
### Assistant: Given limit, substitute x = 0, we get (a*0 + b) / (c*0 + 1) = b , The answer is b
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### Instruction: help me answer this question Question: 2 (x β 1) < x + 5, 3 (x + 2) > 2 β x, think step by step
### Assistant: The given inequalities are 2 (x β 1) < x + 5 and 3 (x + 2) > 2 β x. After solving these inequalities, we get the solution as (-1, 7). , The answer is The solutions of the given inequalities are all real numbers x such that -1 < x < 7.
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### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers.
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### Instruction: help me answer this question What is the intersection of set B and set D?, think step by step
### Assistant: , The answer is (HHT, HTH)
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### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step
### Assistant: , The answer is The domain of the function is `x β₯ 1`.
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### Instruction: help me answer this question Vertices (0, Β± 5), foci (0, Β± 8)., think step by step
### Assistant: Given vertices (0, Β± 5) and foci (0, Β± 8) are on the y-axis, so the equation of the hyperbola is of the form y2/a2 β x2/b2 = 1. Here, a = 5 and c = 8. We know that a2 + b2 = c2, so b2 = 39. , The answer is The equation of the hyperbola is y2/25 β x2/39 = 1.
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### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.
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### Instruction: help me answer this question Foci (Β± 4, 0), the latus rectum is of length 12., think step by step
### Assistant: Given foci (Β± 4, 0) are on the x-axis, and the latus rectum is of length 12, so the equation of the hyperbola is of the form x2/a2 β y2/b2 = 1. Here, a = 2 and c = 4. We know that a2 + b2 = c2, so b2 = 3. , The answer is The equation of the hyperbola is x2/4 β y2/3 = 1.
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### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step
### Assistant: , The answer is The domain of the function is `x β₯ 1`.
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### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers.
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### Instruction: help me answer this question Find the equation of the parabola with equation x^2 = -16y., think step by step
### Assistant: On comparing the given equation with x^2 = -4ay, we get a = 4. The coordinates of the focus are (0,-a) = (0,-4). The given equation involves x^2, so the axis of the parabola is the y-axis. Therefore, the equation of directrix is y = a, which is y = 4. The length of latus rectum is 4a = 4(4) = 16. , The answer is x^2 = -16y.
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### Instruction: help me answer this question If A = Ξ¦ we get n (A) = 0, think step by step
### Assistant: Therefore, P (A) has one element. , The answer is n [P(A)] = 1
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### Instruction: help me answer this question Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour., think step by step
### Assistant: None. , The answer is 6C3 * 5C3 * 5C3
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### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.
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### Instruction: help me answer this question 6) Where, A = 92.5, h = 5, think step by step
### Assistant: This is not a question but a part of the instructions for calculating the mean, variance, and standard deviation. , The answer is Not applicable.
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### Instruction: help me answer this question What are the possible values of C?, think step by step
### Assistant: , The answer is The possible values of C are 3 and 6.
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### Instruction: help me answer this question Length of minor axis is 16 and foci (0, Β±6). What is the equation of the ellipse?, think step by step
### Assistant: The foci are on the y-axis, so the major axis is along the y-axis. The equation of the ellipse will be of the form x2/b2 + y2/a2 = 1. Since the length of the minor axis is 16, b = 8. Also, c = 6. Using the formula a2 = b2 + c2, we get a = 10. Therefore, the equation of the ellipse is x2/64 + y2/100 = 1. , The answer is The equation of the ellipse is x2/64 + y2/100 = 1.
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### Instruction: help me answer this question What is (iv) [β23, 5) in interval notation?, think step by step
### Assistant: [β23, 5) represents all real numbers greater than or equal to -23 and less than 5. , The answer is {x: x β R, β23 β€ x < 5}
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### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step
### Assistant: , The answer is The domain of the function is `x β₯ 1`.
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### Instruction: help me answer this question Prove that sin2 6x β sin2 4x = sin 2x sin 10x, think step by step
### Assistant: Using sum and difference formulas, sin 6x can be written as sin (2x + 4x) and sin 4x can be written as sin (2x + 2x). Then applying the formula for sin (a + b) and sin (a - b) and using some trigonometric identities, the result can be obtained. , The answer is sin2 6x β sin2 4x = sin 2x sin 10x
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### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step
### Assistant: , The answer is The domain of the function is `x β₯ 1`.
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### Instruction: help me answer this question Which of the conditions for a valid probability distribution do not hold for the assignment of throwing a die?, think step by step
### Assistant: , The answer is The second condition, which states that the sum of probabilities should be greater than or equal to 1, does not hold for the assignment of throwing a die.
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### Instruction: help me answer this question Prove that the line through the point (x1, y1) and parallel to the line Ax + By + C = 0 is A (x β x1) + B (y β y1) = 0., think step by step
### Assistant: The slope of the given line is found, then the formula of the line passing through a point and having slope m is used, which gives the required equation. , The answer is The equation is A(x β x1) + B(y β y1) = 0
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### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step
### Assistant: , The answer is The domain of the function is `x β₯ 1`.
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### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step
### Assistant: , The answer is The domain of the function is `x β₯ 1`.
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### Instruction: help me answer this question Expand (102)5, think step by step
### Assistant: Solution:
Given, (102)5
102 can be expressed as the sum or difference of two numbers, and then the binomial theorem can be applied.
The given question can be written as 102 = 100 + 2
(102)5 = (100 + 2)5
= 5C0 (100)5 + 5C1 (100)4 (2) + 5C2 (100)3 (2)2 + 5C3 (100)2 (2)3 + 5C4 (100) (2)4 + 5C5 (2)5
= (100)5 + 5 (100)4 (2) + 10 (100)3 (2)2 + 5 (100) (2)3 + 5 (100) (2)4 + (2)5 , The answer is 11040808032
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### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step
### Assistant: , The answer is The domain of the function is `x β₯ 1`.
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### Instruction: help me answer this question Find the distance between the following pairs of points:
(i) (2, 3, 5) and (4, 3, 1), think step by step
### Assistant: The distance is calculated using the formula Distance PQ = β[(x2 β x1)2 + (y2 β y1)2 + (z2 β z1)2]. The values of x1, y1, z1, x2, y2, z2 are substituted in the formula for the first pair of points, and the square roots of the sums are calculated. , The answer is 2β5 units
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### Instruction: help me answer this question Find the coordinates of the foci, vertices, length of major axis, length of minor axis, eccentricity, and length of latus rectum for the equation 4x^2 + 9y^2 = 36., think step by step
### Assistant: , The answer is {'foci': ['(β5, 0)', '(-β5, 0)'], 'vertices': ['(3, 0)', '(-3, 0)'], 'major axis length': 6, 'minor axis length': 4, 'eccentricity': 'β5/3', 'latus rectum length': '8/3'}
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### Instruction: help me answer this question Exercise 1.4 Solutions, think step by step
### Assistant: None , The answer is Not applicable, as this is a section header, not a question.
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### Instruction: help me answer this question Express each of the complex numbers given in Exercises 1 to 10 in the form a + ib., think step by step
### Assistant: The complex numbers (5i) and (-3/5i) can be expressed in the form a + ib by multiplying the factors and using the property i2 = -1. , The answer is
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### Instruction: help me answer this question The dimensions of the circle are Diameter = 40 cm, find the Radius., think step by step
### Assistant: , The answer is Radius = Diameter / 2 = 40 / 2 = 20 cm
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### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.
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### Instruction: help me answer this question Show that the equation of the line is x/a + y/b = 2, if P (a, b) is the mid-point of a line segment between axes., think step by step
### Assistant: , The answer is Proved.
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### Instruction: help me answer this question Find the sum to n terms of the series 12 + (12 + 22) + (12 + 22 + 32) + β¦, think step by step
### Assistant: , The answer is The sum to n terms of the series can be expressed as (1/3)n(n+1)(n+5).
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### Instruction: help me answer this question Expand (1 - 3x)β· using the binomial theorem., think step by step
### Assistant: The given text provides a detailed explanation for this calculation. , The answer is The expanded expression of (1 - 3x)β· is 1 - 21x + 210xΒ² - 1260xΒ³ + 4725xβ΄ - 11340xβ΅ + 17010xβΆ - 15120xβ· + 6237xβΈ.
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### Instruction: help me answer this question `f(x) = sqrt(x - 1)`, think step by step
### Assistant: , The answer is The domain of the function is `x β₯ 1`.
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### Instruction: help me answer this question Is number 3 odd?, think step by step
### Assistant: An odd number is a number that cannot be divided evenly by 2. Number 3 cannot be divided evenly by 2, therefore, 3 is an odd number. , The answer is Yes, number 3 is odd.
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### Instruction: help me answer this question If a, b, c and d are in G.P., show that (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2, think step by step
### Assistant: As a, b, c, and d are in G.P., we have b^2 = ac and c^2 = bd. We first express the given expression as a quadratic in ac, find its roots using the quadratic formula, and then show that these roots are indeed b^2 and d^2, thus verifying the identity. , The answer is (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2
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### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.
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### Instruction: help me answer this question Write the first five terms of the sequence whose nth term is an = (3n β 1)/2, think step by step
### Assistant: , The answer is 1, 5/2, 4, 11/2, 7
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### Instruction: help me answer this question (iii) {{3, 4}} β A is correct, think step by step
### Assistant: , The answer is {3, 4} β {{3, 4}} and {3, 4} β A.
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### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.
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### Instruction: help me answer this question Find the sum to n terms of the series 3 Γ 8 + 6 Γ 11 + 9 Γ 14 + β¦, think step by step
### Assistant: , The answer is The sum of n terms of the series can be expressed as the sum of the first n terms of the sequence an, where an = 3n + 3.
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### Instruction: help me answer this question How many words, with or without meaning, can be made from the letters of the word MONDAY, assuming that no letter is repeated, if
(i) 4 letters are used at a time
(ii) All letters are used at a time
(iii) all letters are used, but the first letter is a vowel, think step by step
### Assistant: The formulas used are permutations and combinations. , The answer is { (i) The number of 4-letter words that can be formed is 15, (ii) The number of words that can be formed by using all the letters is 720, (iii) The number of words that can be formed by using all the letters, with the first letter being a vowel, is 120 }
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### Instruction: help me answer this question `f(x) = 1/(x^2 + 1)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers.
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### Instruction: help me answer this question Is event D a compound event?, think step by step
### Assistant: , The answer is Yes, D = (HHH, HHT, HTH, HTT) is a compound event.
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### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.
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### Instruction: help me answer this question (12.1 Exercises Solutions), think step by step
### Assistant: The data contains a link to access the solutions for Exercise 12.1, but does not include the solutions directly. , The answer is Not provided in the data.
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### Instruction: help me answer this question a8 if an = 2^n, think step by step
### Assistant: , The answer is a8 = 256
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### Instruction: help me answer this question `f(x) = (x + 3)/(x - 4)`, think step by step
### Assistant: , The answer is The domain of the function is all real numbers except `x = 4`.
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### Instruction: help me answer this question Question : 3x + 2y β€ 12, x β₯ 1, y β₯ 2, think step by step
### Assistant: The required area to be included would be on the left of the graph x β₯1 and below the graph y β₯2. , The answer is The solution for 3x + 2y β€ 12 is the area below the line 3x + 2y = 12. The solution for x β₯ 1 is the area on the right side of the line x = 1 and for y β₯ 2 is the area below the line y = 2.
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### Instruction: help me answer this question Given 2x β y >1, find the region of the solution., think step by step
### Assistant: Putting values of x = 0 and y = 0 in the equation, we get points (0,-1) and (0.5, 0). Checking for the origin, putting (0, 0), we get 0 >1, which is false. Hence, the origin does not lie in the solution region. , The answer is The region of the solution for 2x β y >1 is the area on the right of the line graph.
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### Instruction: help me answer this question Calculate the mean deviation about median age for the age distribution of 100 persons given below:, think step by step
### Assistant: The solution for this question involves calculating mean deviation about median age for the given age distribution. , The answer is None
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### Instruction: help me answer this question 2x + 19 β€ 6x +47, think step by step
### Assistant: , The answer is x β₯ -7
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