2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2

Students can Download Maths Chapter 2 Inverse Trigonometric Functions Ex 2.2 Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2

2nd PUC Maths Inverse Trigonometric Functions NCERT Text Book Questions and Answers Ex 2.2

Question 1.
\(3 \sin ^{-1} x=\sin ^{-1}\left(3 x-4 x^{3}\right), \quad x \in\left[-\frac{1}{2}, \frac{1}{2}\right]\)
Answer:
Let sin-1 x be θ ⇒ x = sin θ
In RHS putting the value of x
sin-1 (3x – 4x3) ⇒ sin-1 (3 sinθ – 4 sin3θ)
= sin-1 (sin 3θ)
[∵ 3sinθ – 4sin3 θ = sin3θ]
= 3θ
putting the value of θ
3θ = 3 sin-1 x = LHS
∴ 3sin-1 x = sin-1 (3x – 4x3)
Hence proved.

KSEEB Solutions

Question 2.
\(3 \cos ^{-1} x=\cos ^{-1}\left(4 x^{3}-3 x\right), x \in\left[\frac{1}{2}, 1\right]\)
Answer:
Let cos-1 x be θ ⇒ x = cos θ
In RHS, putting the value of x
cos-1 (4x3 – 3x) ⇒ cos1 (4cos3 θ – 3 cosθ)
⇒ cos-1 (cos 3θ)
[∵ 4cos3 θ – 3cosθ = cos 3θ]
= 30
putting the value of 0 = 30
⇒ 3 cos-1 x = LHS
LHS = RHS
Hence proved

Question 3.
\(\tan ^{-1} \frac{2}{11}+\tan ^{-1} \frac{7}{24}=\tan ^{-1} \frac{1}{2}\)
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 1

Question 4.
\(2 \tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{7}=\tan ^{-1} \frac{31}{17}\)
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 2
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 3

KSEEB Solutions

Write the following function in the simplest form:

Question 5.
\(\tan ^{-1} \frac{\sqrt{1+x^{2}}-1}{x}, x \neq 0\)
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 4

Question 6.
\(\tan ^{-1} \frac{1}{\sqrt{x^{2}-1}},|x|>1\)
Answer:
Let cosec-1 x = 0
⇒ x = cosec 0
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 5

Question 7.
\(\tan ^{-1}(\sqrt{\frac{1-\cos x}{1+\cos x}}), 0<x<\pi\)
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 6

Question 8.
\(\tan ^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right), \frac{-\pi}{4}<x<\frac{3 \pi}{4}\)
Answer:
Divide each term of numerator and denominator inside the brackets by cos x
\(\Rightarrow \tan ^{-1}\left[\frac{(\cos x-\sin x) / \cos x}{(\cos x+\sin x) / \cos x}\right] \Rightarrow \tan ^{-1}\left[\frac{1-\tan x}{1+\tan x}\right]\)
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 7

Question 9.
\(\tan ^{ -1 } \frac { { x } }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } ,|{ x }|<{ a }\)
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 8

KSEEB Solutions

Question 10.
\(\tan ^{-1}\left(\frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}\right), a>0 ; \frac{-a}{\sqrt{3}} \leq x \leq \frac{a}{\sqrt{3}}\)
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 9

Find the values of each of the following

Question 11.
\(\tan ^{-1}\left[2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 10

Question 12.
cot (tan-1 a + cot-1 a)
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 11

Question 13.
\(\begin{aligned}&\tan \frac{1}{2}\left[\sin ^{-1} \frac{2 x}{1+x^{2}}+\cos ^{-1} \frac{1-y^{2}}{1+y^{2}}\right]\\&|x|<1, y>0 \text { and } x y<1\end{aligned}\)
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 12
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 13

KSEEB Solutions

Question 14.
In \(\sin \left(\sin ^{-1} \frac{1}{5}+\cos ^{-1} x\right)=1\)
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 14

Question 15.
\(\text { If } \tan ^{-1} \frac{x-1}{x-2}+\tan ^{-1} \frac{x+1}{x+2}=\frac{\pi}{4}\)Then find the value of x
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 15
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 16
Find the values of each of the expressions in Exercises 16 to 18.

Question 16.
\(\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)\)
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 17

Question 17.
\( \tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)\)
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 18

KSEEB Solutions

Question 18.
\(\tan \left(\sin ^{-1} \frac{3}{5}+\cot ^{-1}\left(\frac{3}{2}\right)\right)\)
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 19
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 20

Choose the correct Answer:

Question 19.
\(\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right) \text { is equal to }\)
(A) \(\frac{7 \pi}{6}\)
(B) \(\frac{5 \pi}{6}\)
(C) \(\frac{\pi}{3}\)
(D) \(\frac{\pi}{6}\)
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 21

Question 20.
\(\sin \left(\frac{\pi}{3}-\sin ^{-1}\left(-\frac{1}{2}\right)\right)\)is equal to
(A) π
(B) \(-\frac{\pi}{2}\)
(C) 0
(D) \(2 \sqrt{3}\)
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 22
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 23

Question 21.
\(\tan ^{-1} \sqrt{3}-\cot ^{-1}(-\sqrt{3})\) is equal to
(A) π
(B) \(-\frac{\pi}{2}\)
(C) 0
(D) \(2 \sqrt{3}\)
Answer:
2nd PUC Maths Question Bank Chapter 2 Inverse Trigonometric Functions Ex 2.2 24

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