NCERT Solutions Class 12 Mathematics Part II . 674 Pages (668-1342). Free Flip-Book by Study Innovations

Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 13:
Find the value of if sin−1 x = y, then

(A) (B)

(C) (D)
Answer
It is given that sin−1 x = y.

We know that the range of the principal value branch of sin−1 is

Therefore, .

Question 14: is equal to
Find the value of

(A) π (B) (C) (D)
Answer

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Question 1: Exercise 2.2

Prove
Answer

To prove:
Let x = sinθ. Then,
We have,

R.H.S. =

= 3θ

= L.H.S.

Question 2:

Prove
Answer

To prove:
Let x = cosθ. Then, cos−1 x =θ.
We have,

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 3:

Prove
Answer

To prove:

Question 4:

Prove
Answer

To prove:

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 5:
Write the function in the simplest form:
Answer

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 6:
Write the function in the simplest form:
Answer
Put x = cosec θ ⇒ θ = cosec−1 x

Question 7:
Write the function in the simplest form:

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Answer

Question 8:
Write the function in the simplest form:
Answer

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 9:
Write the function in the simplest form:

Answer

Question 10:
Write the function in the simplest form:
Answer

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 11:

Find the value of
Answer

Let . Then,

Question 12:

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Find the value of
Answer

Question 13:

Find the value of
Answer
Let x = tan θ. Then, θ = tan−1 x.

Let y = tan Φ. Then, Φ = tan−1 y.

Question 14:

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths
, then find the value of x.
If
Answer

On squaring both sides, we get:
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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Hence, the value of x is
Question 15:

If , then find the value of x.
Answer

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Hence, the value of x is
Question 16:
Find the values of
Answer

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

We know that sin−1 (sin x) = x if , which is the principal value branch of
sin−1x.

Here,

Now, can be written as:

Question 17:

Find the values of
Answer

We know that tan−1 (tan x) = x if , which is the principal value branch of
tan−1x.

Here,

Now, can be written as:

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 18:

Find the values of
Answer

Let . Then,

Question 19:

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Find the values of is equal to

(A) (B) (C) (D)
Answer

We know that cos−1 (cos x) = x if , which is the principal value branch of cos
−1x.

Here,

Now, can be written as:

The correct answer is B.
Question 20:

Find the values of is equal to

(A) (B) (C) (D) 1
Answer

Let . Then,

We know that the range of the principal value branch of .
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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

∴
The correct answer is D.

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Question 1: Miscellaneous Solutions

Find the value of , which is the principal value branch of cos
Answer
We know that cos−1 (cos x) = x if
−1x.

Here,

Now, can be written as:

Question 2:

Find the value of
Answer

We know that tan−1 (tan x) = x if , which is the principal value branch of
tan −1x.

Here,

Now, can be written as:

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 3:
Prove
Answer

Now, we have:

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 4:

Prove
Answer

Now, we have:

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 5:

Prove
Answer

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Now, we will prove that:
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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 6:

Prove
Answer

Now, we have:
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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 7:
Prove
Answer

Using (1) and (2), we have
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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 8:

Prove
Answer

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 9:
Prove
Answer

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths
Question 10:

Prove
Answer

Question 11: [Hint: putx = cos 2θ]

Prove
Answer

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 12:
Prove
Answer

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 13:

Solve
Answer

Question 14:

Solve
Answer

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Question 15: is equal to
Solve

(A) (B) (C) (D)
Answer

Let tan−1 x = y. Then,

The correct answer is D.
Question 16:

Solve , then x is equal to

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

(A) (B) (C) 0 (D)
Answer

Therefore, from equation (1), we have
Put x = sin y. Then, we have:

But, when , it can be observed that:

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

is not the solution of the given equation.
Thus, x = 0.
Hence, the correct answer is C.

Question 17:

Solve is equal to

(A) (B). (C) (D)
Answer

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Class XII Chapter 2 – Inverse Trigonometric Functions Maths

Hence, the correct answer is C.

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Class XII Chapter 5 – Continuity and Differentiability Maths
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Question 1: Exercise 5.1
is continuous at
Prove that the function
Answer

Therefore, f is continuous at x = 0 .

Therefore, f is continuous at x = −3

Therefore, f is continuous at x = 5
Question 2:
Examine the continuity of the function
Answer

Thus, f is continuous at x = 3

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Class XII Chapter 5 – Continuity and Differentiability Maths

Question 3:
Examine the following functions for continuity.
(a) (b)

(c) (d)
Answer
(a) The given function is
It is evident that f is defined at every real number k and its value at k is k − 5.

It is also observed that,

Hence, f is continuous at every real number and therefore, it is a continuous function.
(b) The given function is
For any real number k ≠ 5, we obtain

Hence, f is continuous at every point in the domain of f and therefore, it is a continuous
function.
(c) The given function is
For any real number c ≠ −5, we obtain

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Class XII Chapter 5 – Continuity and Differentiability Maths

Hence, f is continuous at every point in the domain of f and therefore, it is a continuous
function.
(d) The given function is
This function f is defined at all points of the real line.
Let c be a point on a real line. Then, c < 5 or c = 5 or c > 5
Case I: c < 5
Then, f (c) = 5 − c

Therefore, f is continuous at all real numbers less than 5.
Case II : c = 5
Then,

Therefore, f is continuous at x = 5
Case III: c > 5

Therefore, f is continuous at all real numbers greater than 5.
Hence, f is continuous at every real number and therefore, it is a continuous function.

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Class XII Chapter 5 – Continuity and Differentiability Maths

Question 4:

Prove that the function is continuous at x = n, where n is a positive integer.

Answer

The given function is f (x) = xn

It is evident that f is defined at all positive integers, n, and its value at n is nn.

Therefore, f is continuous at n, where n is a positive integer.

Question 5:
Is the function f defined by

continuous at x = 0? At x = 1? At x = 2?
Answer

The given function f is
At x = 0,
It is evident that f is defined at 0 and its value at 0 is 0.

Therefore, f is continuous at x = 0
At x = 1,
f is defined at 1 and its value at 1 is 1.
The left hand limit of f at x = 1 is,

The right hand limit of f at x = 1 is,

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Class XII Chapter 5 – Continuity and Differentiability Maths

Therefore, f is not continuous at x = 1
At x = 2,
f is defined at 2 and its value at 2 is 5.

Therefore, f is continuous at x = 2

Question 6:
Find all points of discontinuity of f, where f is defined by

Answer

It is evident that the given function f is defined at all the points of the real line.
Let c be a point on the real line. Then, three cases arise.
(i) c < 2
(ii) c > 2
(iii) c = 2
Case (i) c < 2

Therefore, f is continuous at all points x, such that x < 2
Case (ii) c > 2

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Class XII Chapter 5 – Continuity and Differentiability Maths

Therefore, f is continuous at all points x, such that x > 2
Case (iii) c = 2
Then, the left hand limit of f at x = 2 is,

The right hand limit of f at x = 2 is,

It is observed that the left and right hand limit of f at x = 2 do not coincide.
Therefore, f is not continuous at x = 2
Hence, x = 2 is the only point of discontinuity of f.
Question 7:
Find all points of discontinuity of f, where f is defined by

Answer

The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:

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Class XII Chapter 5 – Continuity and Differentiability Maths

Therefore, f is continuous at all points x, such that x < −3
Case II:

Therefore, f is continuous at x = −3
Case III:

Therefore, f is continuous in (−3, 3).
Case IV:
If c = 3, then the left hand limit of f at x = 3 is,

The right hand limit of f at x = 3 is,

It is observed that the left and right hand limit of f at x = 3 do not coincide.
Therefore, f is not continuous at x = 3
Case V:

Therefore, f is continuous at all points x, such that x > 3
Hence, x = 3 is the only point of discontinuity of f.
Question 8:
Find all points of discontinuity of f, where f is defined by

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Class XII Chapter 5 – Continuity and Differentiability Maths

Answer

It is known that,
Therefore, the given function can be rewritten as

The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x < 0
Case II:
If c = 0, then the left hand limit of f at x = 0 is,

The right hand limit of f at x = 0 is,

It is observed that the left and right hand limit of f at x = 0 do not coincide.
Therefore, f is not continuous at x = 0
Case III:

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Class XII Chapter 5 – Continuity and Differentiability Maths

Therefore, f is continuous at all points x, such that x > 0
Hence, x = 0 is the only point of discontinuity of f.
Question 9:
Find all points of discontinuity of f, where f is defined by

Answer

It is known that,
Therefore, the given function can be rewritten as

Let c be any real number. Then,

Also,
Therefore, the given function is a continuous function.
Hence, the given function has no point of discontinuity.

Question 10:
Find all points of discontinuity of f, where f is defined by

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Class XII Chapter 5 – Continuity and Differentiability Maths

Answer

The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x, such that x < 1
Case II:

The left hand limit of f at x = 1 is,

The right hand limit of f at x = 1 is,

Therefore, f is continuous at x = 1
Case III:

Therefore, f is continuous at all points x, such that x > 1
Hence, the given function f has no point of discontinuity.

Question 11:
Find all points of discontinuity of f, where f is defined by

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Class XII Chapter 5 – Continuity and Differentiability Maths

Answer

The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x, such that x < 2
Case II:

Therefore, f is continuous at x = 2
Case III:

Therefore, f is continuous at all points x, such that x > 2
Thus, the given function f is continuous at every point on the real line.
Hence, f has no point of discontinuity.
Question 12:
Find all points of discontinuity of f, where f is defined by

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Class XII Chapter 5 – Continuity and Differentiability Maths

Answer

The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x, such that x < 1
Case II:
If c = 1, then the left hand limit of f at x = 1 is,

The right hand limit of f at x = 1 is,

It is observed that the left and right hand limit of f at x = 1 do not coincide.
Therefore, f is not continuous at x = 1
Case III:

Therefore, f is continuous at all points x, such that x > 1
Thus, from the above observation, it can be concluded that x = 1 is the only point of
discontinuity of f.

Question 13:
Is the function defined by

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Class XII Chapter 5 – Continuity and Differentiability Maths

a continuous function?
Answer

The given function is

The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x, such that x < 1
Case II:

The left hand limit of f at x = 1 is,

The right hand limit of f at x = 1 is,

It is observed that the left and right hand limit of f at x = 1 do not coincide.
Therefore, f is not continuous at x = 1
Case III:

Therefore, f is continuous at all points x, such that x > 1
Thus, from the above observation, it can be concluded that x = 1 is the only point of
discontinuity of f.

Question 14:

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Class XII Chapter 5 – Continuity and Differentiability Maths

Discuss the continuity of the function f, where f is defined by

Answer
The given function is
The given function is defined at all points of the interval [0, 10].
Let c be a point in the interval [0, 10].
Case I:

Therefore, f is continuous in the interval [0, 1).
Case II:

The left hand limit of f at x = 1 is,

The right hand limit of f at x = 1 is,

It is observed that the left and right hand limits of f at x = 1 do not coincide.
Therefore, f is not continuous at x = 1
Case III:

Therefore, f is continuous at all points of the interval (1, 3).
Case IV:

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Class XII Chapter 5 – Continuity and Differentiability Maths

The left hand limit of f at x = 3 is,

The right hand limit of f at x = 3 is,

It is observed that the left and right hand limits of f at x = 3 do not coincide.
Therefore, f is not continuous at x = 3
Case V:

Therefore, f is continuous at all points of the interval (3, 10].
Hence, f is not continuous at x = 1 and x = 3

Question 15:
Discuss the continuity of the function f, where f is defined by

Answer

The given function is

The given function is defined at all points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x, such that x < 0
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Class XII Chapter 5 – Continuity and Differentiability Maths
Case II:

The left hand limit of f at x = 0 is,

The right hand limit of f at x = 0 is,

Therefore, f is continuous at x = 0
Case III:

Therefore, f is continuous at all points of the interval (0, 1).
Case IV:

The left hand limit of f at x = 1 is,

The right hand limit of f at x = 1 is,

It is observed that the left and right hand limits of f at x = 1 do not coincide.
Therefore, f is not continuous at x = 1
Case V:

Therefore, f is continuous at all points x, such that x > 1
Hence, f is not continuous only at x = 1

Question 16:
Discuss the continuity of the function f, where f is defined by

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Class XII Chapter 5 – Continuity and Differentiability Maths

Answer

The given function is defined at all points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x, such that x < −1
Case II:
The left hand limit of f at x = −1 is,
The right hand limit of f at x = −1 is,

Therefore, f is continuous at x = −1
Case III:

Therefore, f is continuous at all points of the interval (−1, 1).
Case IV:

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