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1 | 118-121 | The contrapositive of a conditional can be formed by interchanging the conclusion
and the hypothesis and negating both Example 6 Prove that the function f : R →
→
→
→
→ R defined by f (x) = 2x + 5 is one-one Solution A function is one-one if f (x1) = f (x2) ⇒ x1 = x2 Using this we have to show that “2x1+ 5 = 2x2 + 5” ⇒ “x1 = x2” |
1 | 119-122 | Example 6 Prove that the function f : R →
→
→
→
→ R defined by f (x) = 2x + 5 is one-one Solution A function is one-one if f (x1) = f (x2) ⇒ x1 = x2 Using this we have to show that “2x1+ 5 = 2x2 + 5” ⇒ “x1 = x2” This is of the form
p ⇒ q, where, p is 2x1+ 5 = 2x2 + 5 and q : x1 = x2 |
1 | 120-123 | Solution A function is one-one if f (x1) = f (x2) ⇒ x1 = x2 Using this we have to show that “2x1+ 5 = 2x2 + 5” ⇒ “x1 = x2” This is of the form
p ⇒ q, where, p is 2x1+ 5 = 2x2 + 5 and q : x1 = x2 We have proved this in Example 2
of “direct method” |
1 | 121-124 | Using this we have to show that “2x1+ 5 = 2x2 + 5” ⇒ “x1 = x2” This is of the form
p ⇒ q, where, p is 2x1+ 5 = 2x2 + 5 and q : x1 = x2 We have proved this in Example 2
of “direct method” We can also prove the same by using contrapositive of the statement |
1 | 122-125 | This is of the form
p ⇒ q, where, p is 2x1+ 5 = 2x2 + 5 and q : x1 = x2 We have proved this in Example 2
of “direct method” We can also prove the same by using contrapositive of the statement Now
contrapositive of this statement is ~ q ⇒ ~ p, i |
1 | 123-126 | We have proved this in Example 2
of “direct method” We can also prove the same by using contrapositive of the statement Now
contrapositive of this statement is ~ q ⇒ ~ p, i e |
1 | 124-127 | We can also prove the same by using contrapositive of the statement Now
contrapositive of this statement is ~ q ⇒ ~ p, i e , contrapositive of “ if f (x1) = f (x2),
then x1 = x2” is “if x1 ≠x2, then f (x1) ≠ f (x2)” |
1 | 125-128 | Now
contrapositive of this statement is ~ q ⇒ ~ p, i e , contrapositive of “ if f (x1) = f (x2),
then x1 = x2” is “if x1 ≠x2, then f (x1) ≠ f (x2)” Now
x1 ≠ x2
⇒
2x1 ≠ 2x2
⇒
2x1+ 5 ≠ 2x2 + 5
⇒
f (x1) ≠ f (x2) |
1 | 126-129 | e , contrapositive of “ if f (x1) = f (x2),
then x1 = x2” is “if x1 ≠x2, then f (x1) ≠ f (x2)” Now
x1 ≠ x2
⇒
2x1 ≠ 2x2
⇒
2x1+ 5 ≠ 2x2 + 5
⇒
f (x1) ≠ f (x2) Since “~ q ⇒ ~ p”, is equivalent to “p ⇒ q” the proof is complete |
1 | 127-130 | , contrapositive of “ if f (x1) = f (x2),
then x1 = x2” is “if x1 ≠x2, then f (x1) ≠ f (x2)” Now
x1 ≠ x2
⇒
2x1 ≠ 2x2
⇒
2x1+ 5 ≠ 2x2 + 5
⇒
f (x1) ≠ f (x2) Since “~ q ⇒ ~ p”, is equivalent to “p ⇒ q” the proof is complete Example 7 Show that “if a matrix A is invertible, then A is non singular” |
1 | 128-131 | Now
x1 ≠ x2
⇒
2x1 ≠ 2x2
⇒
2x1+ 5 ≠ 2x2 + 5
⇒
f (x1) ≠ f (x2) Since “~ q ⇒ ~ p”, is equivalent to “p ⇒ q” the proof is complete Example 7 Show that “if a matrix A is invertible, then A is non singular” Solution Writing the above statement in symbolic form, we have
p ⇒ q, where, p is “matrix A is invertible” and q is “A is non singular”
Instead of proving the given statement, we prove its contrapositive statement, i |
1 | 129-132 | Since “~ q ⇒ ~ p”, is equivalent to “p ⇒ q” the proof is complete Example 7 Show that “if a matrix A is invertible, then A is non singular” Solution Writing the above statement in symbolic form, we have
p ⇒ q, where, p is “matrix A is invertible” and q is “A is non singular”
Instead of proving the given statement, we prove its contrapositive statement, i e |
1 | 130-133 | Example 7 Show that “if a matrix A is invertible, then A is non singular” Solution Writing the above statement in symbolic form, we have
p ⇒ q, where, p is “matrix A is invertible” and q is “A is non singular”
Instead of proving the given statement, we prove its contrapositive statement, i e ,
if A is not a non singular matrix, then the matrix A is not invertible |
1 | 131-134 | Solution Writing the above statement in symbolic form, we have
p ⇒ q, where, p is “matrix A is invertible” and q is “A is non singular”
Instead of proving the given statement, we prove its contrapositive statement, i e ,
if A is not a non singular matrix, then the matrix A is not invertible Rationalised 2023-24
MATHEMATICS
194
If A is not a non singular matrix, then it means the matrix A is singular, i |
1 | 132-135 | e ,
if A is not a non singular matrix, then the matrix A is not invertible Rationalised 2023-24
MATHEMATICS
194
If A is not a non singular matrix, then it means the matrix A is singular, i e |
1 | 133-136 | ,
if A is not a non singular matrix, then the matrix A is not invertible Rationalised 2023-24
MATHEMATICS
194
If A is not a non singular matrix, then it means the matrix A is singular, i e ,
|A| = 0
Then
A–1 =
A
adj|A|
does not exist as |A| = 0
Hence, A is not invertible |
1 | 134-137 | Rationalised 2023-24
MATHEMATICS
194
If A is not a non singular matrix, then it means the matrix A is singular, i e ,
|A| = 0
Then
A–1 =
A
adj|A|
does not exist as |A| = 0
Hence, A is not invertible Thus, we have proved that if A is not a non singular matrix, then A is not invertible |
1 | 135-138 | e ,
|A| = 0
Then
A–1 =
A
adj|A|
does not exist as |A| = 0
Hence, A is not invertible Thus, we have proved that if A is not a non singular matrix, then A is not invertible i |
1 | 136-139 | ,
|A| = 0
Then
A–1 =
A
adj|A|
does not exist as |A| = 0
Hence, A is not invertible Thus, we have proved that if A is not a non singular matrix, then A is not invertible i e |
1 | 137-140 | Thus, we have proved that if A is not a non singular matrix, then A is not invertible i e , ~ q ⇒ ~ p |
1 | 138-141 | i e , ~ q ⇒ ~ p Hence, if a matrix A is invertible, then A is non singular |
1 | 139-142 | e , ~ q ⇒ ~ p Hence, if a matrix A is invertible, then A is non singular (iii)
Proof by a counter example
In the history of Mathematics, there are occasions when all attempts to find a
valid proof of a statement fail and the uncertainty of the truth value of the statement
remains unresolved |
1 | 140-143 | , ~ q ⇒ ~ p Hence, if a matrix A is invertible, then A is non singular (iii)
Proof by a counter example
In the history of Mathematics, there are occasions when all attempts to find a
valid proof of a statement fail and the uncertainty of the truth value of the statement
remains unresolved In such a situation, it is beneficial, if we find an example to falsify the statement |
1 | 141-144 | Hence, if a matrix A is invertible, then A is non singular (iii)
Proof by a counter example
In the history of Mathematics, there are occasions when all attempts to find a
valid proof of a statement fail and the uncertainty of the truth value of the statement
remains unresolved In such a situation, it is beneficial, if we find an example to falsify the statement The example to disprove the statement is called a counter example |
1 | 142-145 | (iii)
Proof by a counter example
In the history of Mathematics, there are occasions when all attempts to find a
valid proof of a statement fail and the uncertainty of the truth value of the statement
remains unresolved In such a situation, it is beneficial, if we find an example to falsify the statement The example to disprove the statement is called a counter example Since the disproof
of a proposition p ⇒ q is merely a proof of the proposition ~ (p ⇒ q) |
1 | 143-146 | In such a situation, it is beneficial, if we find an example to falsify the statement The example to disprove the statement is called a counter example Since the disproof
of a proposition p ⇒ q is merely a proof of the proposition ~ (p ⇒ q) Hence, this is
also a method of proof |
1 | 144-147 | The example to disprove the statement is called a counter example Since the disproof
of a proposition p ⇒ q is merely a proof of the proposition ~ (p ⇒ q) Hence, this is
also a method of proof Example 8 For each n,
22
1
n + is a prime (n ∈ N) |
1 | 145-148 | Since the disproof
of a proposition p ⇒ q is merely a proof of the proposition ~ (p ⇒ q) Hence, this is
also a method of proof Example 8 For each n,
22
1
n + is a prime (n ∈ N) This was once thought to be true on the basis that
212
1
+ = 22 + 1 = 5 is a prime |
1 | 146-149 | Hence, this is
also a method of proof Example 8 For each n,
22
1
n + is a prime (n ∈ N) This was once thought to be true on the basis that
212
1
+ = 22 + 1 = 5 is a prime 222
1
+ = 24 + 1 = 17 is a prime |
1 | 147-150 | Example 8 For each n,
22
1
n + is a prime (n ∈ N) This was once thought to be true on the basis that
212
1
+ = 22 + 1 = 5 is a prime 222
1
+ = 24 + 1 = 17 is a prime 223
1
+ = 28 + 1 = 257 is a prime |
1 | 148-151 | This was once thought to be true on the basis that
212
1
+ = 22 + 1 = 5 is a prime 222
1
+ = 24 + 1 = 17 is a prime 223
1
+ = 28 + 1 = 257 is a prime However, at first sight the generalisation looks to be correct |
1 | 149-152 | 222
1
+ = 24 + 1 = 17 is a prime 223
1
+ = 28 + 1 = 257 is a prime However, at first sight the generalisation looks to be correct But, eventually it was
shown that
225
1
+ = 232 + 1 = 4294967297
which is not a prime since 4294967297 = 641 × 6700417 (a product of two numbers) |
1 | 150-153 | 223
1
+ = 28 + 1 = 257 is a prime However, at first sight the generalisation looks to be correct But, eventually it was
shown that
225
1
+ = 232 + 1 = 4294967297
which is not a prime since 4294967297 = 641 × 6700417 (a product of two numbers) So the generalisation “For each n,
22
1
n + is a prime (n ∈ N)” is false |
1 | 151-154 | However, at first sight the generalisation looks to be correct But, eventually it was
shown that
225
1
+ = 232 + 1 = 4294967297
which is not a prime since 4294967297 = 641 × 6700417 (a product of two numbers) So the generalisation “For each n,
22
1
n + is a prime (n ∈ N)” is false Just this one example
225
1
+ is sufficient to disprove the generalisation |
1 | 152-155 | But, eventually it was
shown that
225
1
+ = 232 + 1 = 4294967297
which is not a prime since 4294967297 = 641 × 6700417 (a product of two numbers) So the generalisation “For each n,
22
1
n + is a prime (n ∈ N)” is false Just this one example
225
1
+ is sufficient to disprove the generalisation This is the
counter example |
1 | 153-156 | So the generalisation “For each n,
22
1
n + is a prime (n ∈ N)” is false Just this one example
225
1
+ is sufficient to disprove the generalisation This is the
counter example Thus, we have proved that the generalisation “For each n,
22
1
n + is a prime
(n ∈ N)” is not true in general |
1 | 154-157 | Just this one example
225
1
+ is sufficient to disprove the generalisation This is the
counter example Thus, we have proved that the generalisation “For each n,
22
1
n + is a prime
(n ∈ N)” is not true in general Rationalised 2023-24
PROOFS IN MATHEMATICS
195
Example 9 Every continuous function is differentiable |
1 | 155-158 | This is the
counter example Thus, we have proved that the generalisation “For each n,
22
1
n + is a prime
(n ∈ N)” is not true in general Rationalised 2023-24
PROOFS IN MATHEMATICS
195
Example 9 Every continuous function is differentiable Proof We consider some functions given by
(i)
f (x) = x2
(ii)
g(x) = ex
(iii)
h(x) = sin x
These functions are continuous for all values of x |
1 | 156-159 | Thus, we have proved that the generalisation “For each n,
22
1
n + is a prime
(n ∈ N)” is not true in general Rationalised 2023-24
PROOFS IN MATHEMATICS
195
Example 9 Every continuous function is differentiable Proof We consider some functions given by
(i)
f (x) = x2
(ii)
g(x) = ex
(iii)
h(x) = sin x
These functions are continuous for all values of x If we check for their
differentiability, we find that they are all differentiable for all the values of x |
1 | 157-160 | Rationalised 2023-24
PROOFS IN MATHEMATICS
195
Example 9 Every continuous function is differentiable Proof We consider some functions given by
(i)
f (x) = x2
(ii)
g(x) = ex
(iii)
h(x) = sin x
These functions are continuous for all values of x If we check for their
differentiability, we find that they are all differentiable for all the values of x This
makes us to believe that the generalisation “Every continuous function is differentiable”
may be true |
1 | 158-161 | Proof We consider some functions given by
(i)
f (x) = x2
(ii)
g(x) = ex
(iii)
h(x) = sin x
These functions are continuous for all values of x If we check for their
differentiability, we find that they are all differentiable for all the values of x This
makes us to believe that the generalisation “Every continuous function is differentiable”
may be true But if we check the differentiability of the function given by “φ(x) = | x|”
which is continuous, we find that it is not differentiable at x = 0 |
1 | 159-162 | If we check for their
differentiability, we find that they are all differentiable for all the values of x This
makes us to believe that the generalisation “Every continuous function is differentiable”
may be true But if we check the differentiability of the function given by “φ(x) = | x|”
which is continuous, we find that it is not differentiable at x = 0 This means that the
statement “Every continuous function is differentiable” is false, in general |
1 | 160-163 | This
makes us to believe that the generalisation “Every continuous function is differentiable”
may be true But if we check the differentiability of the function given by “φ(x) = | x|”
which is continuous, we find that it is not differentiable at x = 0 This means that the
statement “Every continuous function is differentiable” is false, in general Just this
one function “φ(x) = | x|” is sufficient to disprove the statement |
1 | 161-164 | But if we check the differentiability of the function given by “φ(x) = | x|”
which is continuous, we find that it is not differentiable at x = 0 This means that the
statement “Every continuous function is differentiable” is false, in general Just this
one function “φ(x) = | x|” is sufficient to disprove the statement Hence, “φ(x) = | x|”
is called a counter example to disprove “Every continuous function is differentiable” |
1 | 162-165 | This means that the
statement “Every continuous function is differentiable” is false, in general Just this
one function “φ(x) = | x|” is sufficient to disprove the statement Hence, “φ(x) = | x|”
is called a counter example to disprove “Every continuous function is differentiable” —v
v
v
v
v—
Rationalised 2023-24
196
MATHEMATICS
A |
1 | 163-166 | Just this
one function “φ(x) = | x|” is sufficient to disprove the statement Hence, “φ(x) = | x|”
is called a counter example to disprove “Every continuous function is differentiable” —v
v
v
v
v—
Rationalised 2023-24
196
MATHEMATICS
A 2 |
1 | 164-167 | Hence, “φ(x) = | x|”
is called a counter example to disprove “Every continuous function is differentiable” —v
v
v
v
v—
Rationalised 2023-24
196
MATHEMATICS
A 2 1 Introduction
In class XI, we have learnt about mathematical modelling as an attempt to study some
part (or form) of some real-life problems in mathematical terms, i |
1 | 165-168 | —v
v
v
v
v—
Rationalised 2023-24
196
MATHEMATICS
A 2 1 Introduction
In class XI, we have learnt about mathematical modelling as an attempt to study some
part (or form) of some real-life problems in mathematical terms, i e |
1 | 166-169 | 2 1 Introduction
In class XI, we have learnt about mathematical modelling as an attempt to study some
part (or form) of some real-life problems in mathematical terms, i e , the conversion of
a physical situation into mathematics using some suitable conditions |
1 | 167-170 | 1 Introduction
In class XI, we have learnt about mathematical modelling as an attempt to study some
part (or form) of some real-life problems in mathematical terms, i e , the conversion of
a physical situation into mathematics using some suitable conditions Roughly speaking
mathematical modelling is an activity in which we make models to describe the behaviour
of various phenomenal activities of our interest in many ways using words, drawings or
sketches, computer programs, mathematical formulae etc |
1 | 168-171 | e , the conversion of
a physical situation into mathematics using some suitable conditions Roughly speaking
mathematical modelling is an activity in which we make models to describe the behaviour
of various phenomenal activities of our interest in many ways using words, drawings or
sketches, computer programs, mathematical formulae etc In earlier classes, we have observed that solutions to many problems, involving
applications of various mathematical concepts, involve mathematical modelling in one
way or the other |
1 | 169-172 | , the conversion of
a physical situation into mathematics using some suitable conditions Roughly speaking
mathematical modelling is an activity in which we make models to describe the behaviour
of various phenomenal activities of our interest in many ways using words, drawings or
sketches, computer programs, mathematical formulae etc In earlier classes, we have observed that solutions to many problems, involving
applications of various mathematical concepts, involve mathematical modelling in one
way or the other Therefore, it is important to study mathematical modelling as a separate
topic |
1 | 170-173 | Roughly speaking
mathematical modelling is an activity in which we make models to describe the behaviour
of various phenomenal activities of our interest in many ways using words, drawings or
sketches, computer programs, mathematical formulae etc In earlier classes, we have observed that solutions to many problems, involving
applications of various mathematical concepts, involve mathematical modelling in one
way or the other Therefore, it is important to study mathematical modelling as a separate
topic In this chapter, we shall further study mathematical modelling of some real-life
problems using techniques/results from matrix, calculus and linear programming |
1 | 171-174 | In earlier classes, we have observed that solutions to many problems, involving
applications of various mathematical concepts, involve mathematical modelling in one
way or the other Therefore, it is important to study mathematical modelling as a separate
topic In this chapter, we shall further study mathematical modelling of some real-life
problems using techniques/results from matrix, calculus and linear programming A |
1 | 172-175 | Therefore, it is important to study mathematical modelling as a separate
topic In this chapter, we shall further study mathematical modelling of some real-life
problems using techniques/results from matrix, calculus and linear programming A 2 |
1 | 173-176 | In this chapter, we shall further study mathematical modelling of some real-life
problems using techniques/results from matrix, calculus and linear programming A 2 2 Why Mathematical Modelling |
1 | 174-177 | A 2 2 Why Mathematical Modelling Students are aware of the solution of word problems in arithmetic, algebra, trigonometry
and linear programming etc |
1 | 175-178 | 2 2 Why Mathematical Modelling Students are aware of the solution of word problems in arithmetic, algebra, trigonometry
and linear programming etc Sometimes we solve the problems without going into the
physical insight of the situational problems |
1 | 176-179 | 2 Why Mathematical Modelling Students are aware of the solution of word problems in arithmetic, algebra, trigonometry
and linear programming etc Sometimes we solve the problems without going into the
physical insight of the situational problems Situational problems need physical insight
that is introduction of physical laws and some symbols to compare the mathematical
results obtained with practical values |
1 | 177-180 | Students are aware of the solution of word problems in arithmetic, algebra, trigonometry
and linear programming etc Sometimes we solve the problems without going into the
physical insight of the situational problems Situational problems need physical insight
that is introduction of physical laws and some symbols to compare the mathematical
results obtained with practical values To solve many problems faced by us, we need a
technique and this is what is known as mathematical modelling |
1 | 178-181 | Sometimes we solve the problems without going into the
physical insight of the situational problems Situational problems need physical insight
that is introduction of physical laws and some symbols to compare the mathematical
results obtained with practical values To solve many problems faced by us, we need a
technique and this is what is known as mathematical modelling Let us consider the
following problems:
(i)
To find the width of a river (particularly, when it is difficult to cross the river) |
1 | 179-182 | Situational problems need physical insight
that is introduction of physical laws and some symbols to compare the mathematical
results obtained with practical values To solve many problems faced by us, we need a
technique and this is what is known as mathematical modelling Let us consider the
following problems:
(i)
To find the width of a river (particularly, when it is difficult to cross the river) (ii)
To find the optimal angle in case of shot-put (by considering the variables
such as : the height of the thrower, resistance of the media, acceleration due to
gravity etc |
1 | 180-183 | To solve many problems faced by us, we need a
technique and this is what is known as mathematical modelling Let us consider the
following problems:
(i)
To find the width of a river (particularly, when it is difficult to cross the river) (ii)
To find the optimal angle in case of shot-put (by considering the variables
such as : the height of the thrower, resistance of the media, acceleration due to
gravity etc ) |
1 | 181-184 | Let us consider the
following problems:
(i)
To find the width of a river (particularly, when it is difficult to cross the river) (ii)
To find the optimal angle in case of shot-put (by considering the variables
such as : the height of the thrower, resistance of the media, acceleration due to
gravity etc ) (iii)
To find the height of a tower (particularly, when it is not possible to reach the top
of the tower) |
1 | 182-185 | (ii)
To find the optimal angle in case of shot-put (by considering the variables
such as : the height of the thrower, resistance of the media, acceleration due to
gravity etc ) (iii)
To find the height of a tower (particularly, when it is not possible to reach the top
of the tower) (iv)
To find the temperature at the surface of the Sun |
1 | 183-186 | ) (iii)
To find the height of a tower (particularly, when it is not possible to reach the top
of the tower) (iv)
To find the temperature at the surface of the Sun Appendix 2
MATHEMATICAL MODELLING
Rationalised 2023-24
MATHEMATICAL MODELLING 197
(v)
Why heart patients are not allowed to use lift |
1 | 184-187 | (iii)
To find the height of a tower (particularly, when it is not possible to reach the top
of the tower) (iv)
To find the temperature at the surface of the Sun Appendix 2
MATHEMATICAL MODELLING
Rationalised 2023-24
MATHEMATICAL MODELLING 197
(v)
Why heart patients are not allowed to use lift (without knowing the physiology
of a human being) |
1 | 185-188 | (iv)
To find the temperature at the surface of the Sun Appendix 2
MATHEMATICAL MODELLING
Rationalised 2023-24
MATHEMATICAL MODELLING 197
(v)
Why heart patients are not allowed to use lift (without knowing the physiology
of a human being) (vi)
To find the mass of the Earth |
1 | 186-189 | Appendix 2
MATHEMATICAL MODELLING
Rationalised 2023-24
MATHEMATICAL MODELLING 197
(v)
Why heart patients are not allowed to use lift (without knowing the physiology
of a human being) (vi)
To find the mass of the Earth (vii)
Estimate the yield of pulses in India from the standing crops (a person is not
allowed to cut all of it) |
1 | 187-190 | (without knowing the physiology
of a human being) (vi)
To find the mass of the Earth (vii)
Estimate the yield of pulses in India from the standing crops (a person is not
allowed to cut all of it) (viii) Find the volume of blood inside the body of a person (a person is not allowed to
bleed completely) |
1 | 188-191 | (vi)
To find the mass of the Earth (vii)
Estimate the yield of pulses in India from the standing crops (a person is not
allowed to cut all of it) (viii) Find the volume of blood inside the body of a person (a person is not allowed to
bleed completely) (ix)
Estimate the population of India in the year 2020 (a person is not allowed to wait
till then) |
1 | 189-192 | (vii)
Estimate the yield of pulses in India from the standing crops (a person is not
allowed to cut all of it) (viii) Find the volume of blood inside the body of a person (a person is not allowed to
bleed completely) (ix)
Estimate the population of India in the year 2020 (a person is not allowed to wait
till then) All of these problems can be solved and infact have been solved with the help of
Mathematics using mathematical modelling |
1 | 190-193 | (viii) Find the volume of blood inside the body of a person (a person is not allowed to
bleed completely) (ix)
Estimate the population of India in the year 2020 (a person is not allowed to wait
till then) All of these problems can be solved and infact have been solved with the help of
Mathematics using mathematical modelling In fact, you might have studied the methods
for solving some of them in the present textbook itself |
1 | 191-194 | (ix)
Estimate the population of India in the year 2020 (a person is not allowed to wait
till then) All of these problems can be solved and infact have been solved with the help of
Mathematics using mathematical modelling In fact, you might have studied the methods
for solving some of them in the present textbook itself However, it will be instructive if
you first try to solve them yourself and that too without the help of Mathematics, if
possible, you will then appreciate the power of Mathematics and the need for
mathematical modelling |
1 | 192-195 | All of these problems can be solved and infact have been solved with the help of
Mathematics using mathematical modelling In fact, you might have studied the methods
for solving some of them in the present textbook itself However, it will be instructive if
you first try to solve them yourself and that too without the help of Mathematics, if
possible, you will then appreciate the power of Mathematics and the need for
mathematical modelling A |
1 | 193-196 | In fact, you might have studied the methods
for solving some of them in the present textbook itself However, it will be instructive if
you first try to solve them yourself and that too without the help of Mathematics, if
possible, you will then appreciate the power of Mathematics and the need for
mathematical modelling A 2 |
1 | 194-197 | However, it will be instructive if
you first try to solve them yourself and that too without the help of Mathematics, if
possible, you will then appreciate the power of Mathematics and the need for
mathematical modelling A 2 3 Principles of Mathematical Modelling
Mathematical modelling is a principled activity and so it has some principles behind it |
1 | 195-198 | A 2 3 Principles of Mathematical Modelling
Mathematical modelling is a principled activity and so it has some principles behind it These principles are almost philosophical in nature |
1 | 196-199 | 2 3 Principles of Mathematical Modelling
Mathematical modelling is a principled activity and so it has some principles behind it These principles are almost philosophical in nature Some of the basic principles of
mathematical modelling are listed below in terms of instructions:
(i)
Identify the need for the model |
1 | 197-200 | 3 Principles of Mathematical Modelling
Mathematical modelling is a principled activity and so it has some principles behind it These principles are almost philosophical in nature Some of the basic principles of
mathematical modelling are listed below in terms of instructions:
(i)
Identify the need for the model (for what we are looking for)
(ii)
List the parameters/variables which are required for the model |
1 | 198-201 | These principles are almost philosophical in nature Some of the basic principles of
mathematical modelling are listed below in terms of instructions:
(i)
Identify the need for the model (for what we are looking for)
(ii)
List the parameters/variables which are required for the model (iii)
Identify the available relevent data |
1 | 199-202 | Some of the basic principles of
mathematical modelling are listed below in terms of instructions:
(i)
Identify the need for the model (for what we are looking for)
(ii)
List the parameters/variables which are required for the model (iii)
Identify the available relevent data (what is given |
1 | 200-203 | (for what we are looking for)
(ii)
List the parameters/variables which are required for the model (iii)
Identify the available relevent data (what is given )
(iv)
Identify the circumstances that can be applied (assumptions)
(v)
Identify the governing physical principles |
1 | 201-204 | (iii)
Identify the available relevent data (what is given )
(iv)
Identify the circumstances that can be applied (assumptions)
(v)
Identify the governing physical principles (vi)
Identify
(a) the equations that will be used |
1 | 202-205 | (what is given )
(iv)
Identify the circumstances that can be applied (assumptions)
(v)
Identify the governing physical principles (vi)
Identify
(a) the equations that will be used (b) the calculations that will be made |
1 | 203-206 | )
(iv)
Identify the circumstances that can be applied (assumptions)
(v)
Identify the governing physical principles (vi)
Identify
(a) the equations that will be used (b) the calculations that will be made (c) the solution which will follow |
1 | 204-207 | (vi)
Identify
(a) the equations that will be used (b) the calculations that will be made (c) the solution which will follow (vii)
Identify tests that can check the
(a) consistency of the model |
1 | 205-208 | (b) the calculations that will be made (c) the solution which will follow (vii)
Identify tests that can check the
(a) consistency of the model (b) utility of the model |
1 | 206-209 | (c) the solution which will follow (vii)
Identify tests that can check the
(a) consistency of the model (b) utility of the model (viii) Identify the parameter values that can improve the model |
1 | 207-210 | (vii)
Identify tests that can check the
(a) consistency of the model (b) utility of the model (viii) Identify the parameter values that can improve the model Rationalised 2023-24
198
MATHEMATICS
The above principles of mathematical modelling lead to the following: steps for
mathematical modelling |
1 | 208-211 | (b) utility of the model (viii) Identify the parameter values that can improve the model Rationalised 2023-24
198
MATHEMATICS
The above principles of mathematical modelling lead to the following: steps for
mathematical modelling Step 1: Identify the physical situation |
1 | 209-212 | (viii) Identify the parameter values that can improve the model Rationalised 2023-24
198
MATHEMATICS
The above principles of mathematical modelling lead to the following: steps for
mathematical modelling Step 1: Identify the physical situation Step 2: Convert the physical situation into a mathematical model by introducing
parameters / variables and using various known physical laws and symbols |
1 | 210-213 | Rationalised 2023-24
198
MATHEMATICS
The above principles of mathematical modelling lead to the following: steps for
mathematical modelling Step 1: Identify the physical situation Step 2: Convert the physical situation into a mathematical model by introducing
parameters / variables and using various known physical laws and symbols Step 3: Find the solution of the mathematical problem |
1 | 211-214 | Step 1: Identify the physical situation Step 2: Convert the physical situation into a mathematical model by introducing
parameters / variables and using various known physical laws and symbols Step 3: Find the solution of the mathematical problem Step 4: Interpret the result in terms of the original problem and compare the result
with observations or experiments |
1 | 212-215 | Step 2: Convert the physical situation into a mathematical model by introducing
parameters / variables and using various known physical laws and symbols Step 3: Find the solution of the mathematical problem Step 4: Interpret the result in terms of the original problem and compare the result
with observations or experiments Step 5: If the result is in good agreement, then accept the model |
1 | 213-216 | Step 3: Find the solution of the mathematical problem Step 4: Interpret the result in terms of the original problem and compare the result
with observations or experiments Step 5: If the result is in good agreement, then accept the model Otherwise modify
the hypotheses / assumptions according to the physical situation and go to
Step 2 |
1 | 214-217 | Step 4: Interpret the result in terms of the original problem and compare the result
with observations or experiments Step 5: If the result is in good agreement, then accept the model Otherwise modify
the hypotheses / assumptions according to the physical situation and go to
Step 2 The above steps can also be viewed through the following diagram:
Fig A |
1 | 215-218 | Step 5: If the result is in good agreement, then accept the model Otherwise modify
the hypotheses / assumptions according to the physical situation and go to
Step 2 The above steps can also be viewed through the following diagram:
Fig A 2 |
1 | 216-219 | Otherwise modify
the hypotheses / assumptions according to the physical situation and go to
Step 2 The above steps can also be viewed through the following diagram:
Fig A 2 1
Example 1 Find the height of a given tower using mathematical modelling |
1 | 217-220 | The above steps can also be viewed through the following diagram:
Fig A 2 1
Example 1 Find the height of a given tower using mathematical modelling Solution Step 1 Given physical situation is “to find the height of a given tower” |
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