NCERT Solutions For Class 12 Maths Chapter 2 – Inverse Trigonometric Functions

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NCERT Solutions Of Class 12 Maths Chapter 2  – Inverse Trigonometric Functions

Name OF The Section Topic Name
2 Inverse Trigonometric Functions
2.1 Introduction
2.2 Basic Concepts
2.3 Properties of Inverse Trigonometric Functions

 Class 12 Maths Chapter 2 NCERT Solutions – 2.1 Introduction

Question 1:
Find the principal value of NCERT Solutions Class 12 Chapter 2 Ex 2.1 Q 1
Answer:
Let sin-1 NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 1(a) Then sin y = NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 1(b)

We know that the range of the principal value branch of sin−1 is
NCERT Solutions Class 12 Maths Chapter2 Ex 2.1 Q 1(c) and sin NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 1(d)
Therefore, the principal value of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 1(e)


Question 2:
Find the principal value of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 2
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 2(a)

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We know that the range of the principal value branch of cos−1 is
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 2(b)
Therefore, the principal value of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 2(c)


Question 3:

Find the principal value of cosec−1 (2)
Answer:
Let cosec−1 (2) = y. Then, NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 3
We know that the range of the principal value branch of cosec−1 is
NCERT Solutions Class 12 Maths Chapter 2 Ex2.1 Q 3(a)
Therefore, the principal value of

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Question 4:
Find the principal value of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 4
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 4(a)
We know that the range of the principal value branch of tan−1 is
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 4(c)
Therefore, the principal value of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 4(b)


Question 5:
Find the principal value of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 5
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 5(a)

We know that the range of the principal value branch of cos−1 is
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 5(b)
Therefore, the principal value of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 5(c)


Question 6:

Find the principal value of tan−1 (−1)
Answer:
Let tan−1 (−1) = y. Then, NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 6

We know that the range of the principal value branch of tan−1 is
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 6(a)
Therefore, the principal value of NCERT Solutions Class 12 Maths Chapter 2 eX 2.1 Q 6(b)


Question 7:
Find the principal value of NCERT Solutions Class 12 Maths Chapter 2 ex 2.1 Q 7
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 7(a)

We know that the range of the principal value branch of sec−1 is
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 7(b)
Therefore, the principal value of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 7(c)


Question 8:
Find the principal value of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 8
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 8(a)

We know that the range of the principal value branch of cot−1 is (0,π) and
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 8(b)
Therefore, the principal value of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 8(c)


Question 9:
Find the principal value of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 9
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 9(a)
We know that the range of the principal value branch of cos−1 is [0,π] and
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 9(b)
Therefore, the principal value of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 9(c)


Question 10:
Find the principal value of NCERT Solutions Class 12 Maths Chapter 2 ex 2.1 Q 10
Answer:
NCERT Solutions Class 12 Maths Chapter 2 ex 2.1 Q 10(a)
We know that the range of the principal value branch of cosec−1 is
NCERT Solutions Class 12 Maths Chapter 2 ex 2.1 Q 10(b)
Therefore, the principal value of NCERT Solutions Class 12 Maths Chapter 2 EX 2.1 Q 10(c)


Question 11:
Find the value of NCERT Solutions Class 12 Maths Chapter 2 eX 2.1 Q 10(d)
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 11(a)



Question 12:
Find the value of NCERT Solutions Class 12 Maths Chapter 2 ex 2.1 Q 12
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 12(a)



Question 13:

Find the value of if sin−1 y, then
(A) NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 13 (B) NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 13(a)
(C) NCERT Solutions Class 12 Maths Chapter 2 ex 2.1 Q 13(b) (D) NCERT Solutions Class 12 Maths Chapter 2 ex 2.1 Q 13(c)
Answer:

It is given that sin−1 y.

We know that the range of the principal value branch of sin−1 is
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 13(d)
Therefore, NCERT Solutions Class 12 Maths Chapter 2 ex 2.1 Q 13(e)


Question 14:
Find the value of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 14 is equal to
(A) π (B) NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 16(a) (C) NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 14(b) (D) NCERT Solutions Class 12 Maths Chapter 2 Ex 2.1 Q 14(c)
Answer:
NCERT Solutions Class 12 Maths Chapter 2 ex 2.1 Q 14(d)

 

NCERT Solutions For Class 12 Maths Chapter 2 Inverse Trigonometric Functions – Exercise 2.2

 

Question 1:
Prove NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 1
Answer:
To prove,
Let x = sinθ. Then, NCERT Solutions Class 12 Maths Chapter 2 ex 2.2 Q 1(a)

We have,
R.H.S. = NCERT Solutions Class 12 Maths Chapter 2 ex 2.2 Q 1(b)
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 1(c)
= 3θ

NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 1(d)
= L.H.S.


Question 2:
Prove NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 2
Answer:
To prove:

Let x = cosθ. Then, cos−1 x =θ.

We have,
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 2(a)


Question 3:
Prove NCERT Solutions Class 12 Maths Cahpter 2 Ex 2.2 Q 3
Answer:
To prove:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 3(a)


Question 4:
Prove NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 4
Answer:
To prove:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 4(a)


Question 5:

Write the function in the simplest form:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 5
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 5(a)



Question 6:

Write the function in the simplest form: NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 6
Answer:

Put x = cosec θ ⇒ θ = cosec−1 x
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 6(a)



Question 7:

Write the function in the simplest form:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 7
Answer:
NCERT Solutions Class 12 Maths Chapter 2 EX 2.2 Q 7(a)



Question 8:

Write the function in the simplest form:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 8
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 8(a)
Dividing numerator and denominator by cos x
NCERT Solutions Class 12 Maths chapter 2 Ex 2.2 Q 8(b)



Question 9:

Write the function in the simplest form:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 9
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 9(a)



Question 10:

Write the function in the simplest form:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 10
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 10(a)



Question 11:
Find the value of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 11
Answer:
Let NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 11(a) Then, NCERT Solutions Class 12 Maths CHApter 2 Ex 2.2 Q 11(b)
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 11(c)


Question 12:
Find the value of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 12
Answer:
NCERT Solutions Class 12 Maths Cahpter 2 Ex 2.2 Q 12(a)



Question 13:
Find the value of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 13
Answer:

Let x = tan θ. Then, θ = tan−1 x.
NCERT Solutions Class 12 Maths Chapter 2 ex 2.2 Q 13(a)

Let y = tan Φ. Then, Φ = tan−1 y.
NCERT Solutions Class 12 Chapter 2 Ex 2.2 Q 13(b)

Let y = tan Φ. Then, Φ = tan−1 y.
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 13(c)


Question 14:
If NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 14, then find the value of x
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 14
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 14(a)

On squaring both sides, we get:
NCERT Solutions class 12 Maths Chaapter 2 Ex 2.2 Q 14(b)
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 14(c)Hence, the value of x is


Question 15:
If NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 15 then find the value of x.
Answer:
NCERT Solutions Class 12 Maths Chapter 2 ex 2.2 Q 15(a)
Hence, the value of x is


Question 16:
Find the values of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 16
Answer:
We know that sin−1 (sin x) = x if NCERT Solutions Class 12 Maths Chapter 2 ex 2.2 Q 16(a) which is the principal value branch of sin−1x.
Here, NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 16(b)
Now, can be written as:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 16(c)
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 16(d)


Question 17:
Find the values of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 17
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 17
We know that tan−1 (tan x) = x if NCERT Solutions class 12 Maths Chapter 2 Ex 2.2 Q 17(a) which is the principal value branch of tan−1x.
Here NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 17(b)
Now, NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 17 can be written as:
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 17(c)


Question 18:
Find the values of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 18
Answer:
Let NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 18(a) Then, NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 18(b)
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 18(c)


Question 19:
Find the values of NCERT Solutions Class 12 Maths Chapter 2Ex 2.2Q 19 is equal to
(A) NCERT Solutions Class 12 Maths Cahpter 2 Ex 2.2 Q 19(a) (B) NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 19(b) (C) NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 19(c) (D) NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 19(d)
Answer:
We know that cos−1 (cos x) = x if NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 19(e) which is the principal value branch of cos −1x.
Here, NCERT Solutions Class 12 Maths Chapter 2 EX 2.2 Q 19(f)
Now, NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 19(g) can be written as:
cos-1cos7π6 = cos-1cosπ+π6cos-1cos7π6 = cos-1- cosπ6             as, cosπ+θ = – cos θcos-1cos7π6  = cos-1- cosπ-5π6cos-1cos7π6 = cos-1– cos 5π6   as, cosπ-θ = – cos θ
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 19(h)

The correct answer is B.


Question 20:
Find the values of NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 20 is equal to
(A) NCERT Solutions Class 12 Maths Chapter 2 ex 2.2 Q 20(a) (B) NCERT Solutions Class 12 Maths Chapter 2 EX 2.2 Q 20(b) (c) NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 20(c) (D) 1
Answer:
Let NCERT Solutions Class 12 Maths Chapter 2 eX 2.2 Q 20(d) Then, NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 20(e)
We know that the range of the principal value branch of
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 20(f)
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 20(g)
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 20(h)

The correct answer is D.


Question 21:
Find the values of NCERT Solutions Class 12 Maths Chapter 2 ex 2.2 Q 21 is equal to
(A) π (B) NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 21(a) (C)0 (D) NCERT Solutions Class 12 Maths Chapter 2 ex 2.2 Q 21(b)
Answer:
Let, NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 21(c)Then, NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 21(e)
We know that the range of the principal value branch of
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 21(f)
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 21(g)
Let NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 21(h)
NCERT Solutions Class 12 Maths Chapter 2 ex 2.2 Q 21(i)
The range of the principal value branch of NCERT Solutions Class 12 Maths chapter 2 Ex 2.2 Q 21(j)
NCERT Solutions Class 12 Maths Chapter 2 Ex 2.2 Q 21(k)
The correct answer is B.

NCERT Solutions Class 12 Maths Chapter 2 Inverse Trigonometric Functions – Miscellaneous Solutions

Question 1:
Find the value of NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 1
Answer:
We know that cos−1 (cos x) = x if NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 1(a), which is the principal value branch of cos −1x.
Here, NCERT Solutions Class 12 Chapter 2 Miscellaneous Solutions Q 1(c)
Now, NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 1(c)can be written as:
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 1(d)


Question 2:
Find the value of NCERT Solutions Class 12 Maths Miscellaneous Solutions Q 2
Answer:
We know that tan−1 (tan x) = x if NCERT Solutions Class 12 Chapter 2 Miscellaneous Solutions Q 2(b), which is the principal value branch of tan −1x.
Here, NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 2(c)
Now,
NCERT Solutions Class 12 Maths Miscellaneous Solutions Q 2can be written as:
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 2(d)
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 2(e)


Question 3:
Prove NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 3
Answer:
NCERT Solutions Class 12 Chapter 2 Miscellaneous Solutions Q 3

Now, we have:
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 3(b)


Question 4:
Prove NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 4
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 4(a)

Now, we have:
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 4(b)


Question 5:
Prove NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 5
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 5(a)

Now, we will prove that:
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 5(b)


Question 6:
Prove NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 6
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 6(a)

Now, we have:
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 6(b)


Question 7:
Prove NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 7
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 7(a)

Using (1) and (2), we have
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 7(b)


Question 8:
Prove NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 8
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Q 8(b)



Question 9:
Prove NCERT Solutions Class 12 Maths Chapter 2 Q 9
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Q 9(a)



Question 10:
Prove NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 10
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 10(a)



Question 11:
Prove NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 11[Hint: put x = cos 2θ] Answer:
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 11(a)
Question 12:
Prove NCERT Solutins Class 12 Chapter 2 Miscellaneous Solutions Q 12
Answer:
NCERT Solutions Class 12 Maths Miscellaneous Solutions Q 12(a)



Question 13:
Solve NCERT Solutions Class 12 Maths Miscellaneous Solutions Q 13
Answer:
NCERT Solutions Class 12 Maths Miscellaneous Solutions Q 13(a)



Question 14:
Solve NCERT Solutions Class 12 Maths Miscellaneous Solutions Q 14
Answer:
NCERT Solutions Class 12 Maths Miscellaneous Solutions Q 14(a)



Question 15:
Solve NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 15 is equal to
(A) NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 15(a)(B) NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 15(b)(C) NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 15(c)(D) NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 15(d)
Answer:
Let tan−1x = y. Then, NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 15(e)
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 15(f)


The correct answer is D.
Question 16:
Solve NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 16, then x is equal to
(A) NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 16(a)(B) NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 16(b)(C) 0 (D) NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 16(c)
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 16(d)

Therefore, from equation (1), we have
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 16(e)

Put x = sin y. Then, we have:
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 16(f)
But, when

it can be observed that:
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 16(h)
is not the solution of the given equation.

Thus, x = 0.

Hence, the correct answer is C.


Question 17:
Solve NCERT Solutiuons Class 12 Maths Chapter 2 Miscellaneious Solutions Q 17is equal to
(A) NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 17(a)(B) NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 17(b)(C) NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 17(c)(D) NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 17(d)
Answer:
NCERT Solutions Class 12 Maths Chapter 2 Miscellaneous Solutions Q 17(e)
Hence, the correct answer is C.