CBSE Class 12 Mathematics Inverse Trigonometric Functions VBQs Set 04

Question. If \( \cos \left( \sin^{-1} \frac{2}{5} + \cos^{-1} x \right) = 0 \), then \( x \) is equal to : 
(a) \( \frac{1}{5} \)
(b) \( \frac{2}{5} \)
(c) \( 0 \)
(d) \( 1 \)
Answer: (b) \( \frac{2}{5} \)

Question. Which of the following corresponds to the principal value branch of \( \tan^{-1} x \)?
(a) \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \)
(b) \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)
(c) \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) - \{0\} \)
(d) \( (0, \pi) \)
Answer: (a) \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \)

Question. The principal value branch of \( \sec^{-1} x \) is:
(a) \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] - \{0\} \)
(b) \( [0, \pi] - \{ \frac{\pi}{2} \} \)
(c) \( (0, \pi) \)
(d) \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \)
Answer: (b) \( [0, \pi] - \{ \frac{\pi}{2} \} \)

Question. The principal value of the expression \( \cos^{-1} [\cos (-680^\circ)] \) is :
(a) \( \frac{2\pi}{9} \)
(b) \( -\frac{2\pi}{9} \)
(c) \( \frac{34\pi}{9} \)
(d) \( \frac{\pi}{9} \)
Answer: (a) \( \frac{2\pi}{9} \)

Question. The value of \( \cot (\sin^{-1} x) \) is:
(a) \( \frac{\sqrt{1+x^2}}{x} \)
(b) \( \frac{x}{\sqrt{1+x^2}} \)
(c) \( \frac{1}{x} \)
(d) \( \frac{\sqrt{1-x^2}}{x} \)
Answer: (d) \( \frac{\sqrt{1-x^2}}{x} \)

Question. The domain of \( \sin^{-1} 2x \) is:
(a) \( [0, 1] \)
(b) \( [-1, 1] \)
(c) \( [-\frac{1}{2}, \frac{1}{2}] \)
(d) \( [-2, 2] \)
Answer: (c) \( [-\frac{1}{2}, \frac{1}{2}] \)

Question. The principal value of \( \sin^{-1} \left( -\frac{\sqrt{3}}{2} \right) \) is:
(a) \( -\frac{2\pi}{3} \)
(b) \( -\frac{\pi}{3} \)
(c) \( \frac{4\pi}{3} \)
(d) \( \frac{5\pi}{3} \)
Answer: (b) \( -\frac{\pi}{3} \)

Question. The domain of \( y = \cos^{-1} (x^2 - 4) \) is:
(a) \( [3, 5] \)
(b) \( [0, \pi] \)
(c) \( [-\sqrt{5}, -\sqrt{3}] \cap [\sqrt{3}, \sqrt{5}] \)
(d) \( [-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}] \)
Answer: (d) \( [-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}] \)

Question. The domain of the function defined by \( f(x) = \sin^{-1} x + \cos x \) is:
(a) \( [-1, 1] \)
(b) \( [-1, \pi + 1] \)
(c) \( (-\infty, \infty) \)
(d) \( \phi \)
Answer: (a) \( [-1, 1] \)

Question. The value of \( \sin [2 \sin^{-1} (0.6)] \) is:
(a) \( 0.48 \)
(b) \( 0.96 \)
(c) \( 1.2 \)
(d) \( \sin 1.2 \)
Answer: (b) \( 0.96 \)

Question. The value of \( \tan \left\{ \cos^{-1} \frac{1}{5\sqrt{2}} - \sin^{-1} \frac{4}{\sqrt{17}} \right\} \) is :
(a) \( \frac{\sqrt{29}}{3} \)
(b) \( \frac{29}{3} \)
(c) \( \frac{\sqrt{3}}{29} \)
(d) \( \frac{3}{29} \)
Answer: (d) \( \frac{3}{29} \)

Question. If \( \cot^{-1} \left( -\frac{1}{5} \right) = \theta \), the value of \( \sin \theta \) is :
(a) \( \frac{\sqrt{26}}{5} \)
(b) \( -\frac{5}{\sqrt{26}} \)
(c) \( \frac{\sqrt{5}}{\sqrt{26}} \)
(d) \( \frac{5}{\sqrt{26}} \)
Answer: (d) \( \frac{5}{\sqrt{26}} \)

Question. If \( \alpha \le 2 \sin^{-1} x + \cos^{-1} x \le \beta \), then:
(a) \( \alpha = -\frac{\pi}{2}, \beta = \frac{\pi}{2} \)
(b) \( \alpha = 0, \beta = \pi \)
(c) \( \alpha = -\frac{\pi}{2}, \beta = \frac{3\pi}{2} \)
(d) \( \alpha = 0, \beta = 2\pi \)
Answer: (b) \( \alpha = 0, \beta = \pi \)

Question. The value of \( \tan^2 (\sec^{-1} 2) + \cot^2 (\text{cosec}^{-1} 3) \) is:
(a) \( 5 \)
(b) \( 11 \)
(c) \( 13 \)
(d) \( 15 \)
Answer: (b) \( 11 \)

Question. The value of \( \tan \left[ \frac{1}{2} \cos^{-1} \left( \frac{\sqrt{5}}{3} \right) \right] \) is : 
(a) \( \frac{3 + \sqrt{5}}{2} \)
(b) \( \frac{3 - \sqrt{5}}{2} \)
(c) \( \frac{-3 + \sqrt{5}}{2} \)
(d) \( \frac{-3 - \sqrt{5}}{2} \)
Answer: (b) \( \frac{3 - \sqrt{5}}{2} \)

Question. The principal value of \( \tan^{-1} \left( \frac{1}{\sqrt{3}} \right) \) is:
(a) \( \frac{\pi}{2} \)
(b) \( \frac{\pi}{6} \)
(c) \( \frac{\pi}{3} \)
(d) \( \pi \)
Answer: (b) \( \frac{\pi}{6} \)

Question. The principal value of \( \sec^{-1} \left( -\frac{2}{\sqrt{3}} \right) \) is :
(a) \( \frac{5\pi}{6} \)
(b) \( \frac{2\pi}{3} \)
(c) \( \frac{\pi}{3} \)
(d) None of the options
Answer: (a) \( \frac{5\pi}{6} \)

Question. The inverse of cosine function is defined in the intervals :
(a) \( [-\pi, 0] \)
(b) \( [-\frac{\pi}{2}, 0] \)
(c) \( [0, \frac{\pi}{2}] \)
(d) \( [\frac{\pi}{2}, \pi] \)
Answer: (a) \( [-\pi, 0] \)

Question. If \( \sin^{-1} x = y \), then :
(a) \( 0 \le y \le x \)
(b) \( -\frac{\pi}{2} \le y \le \frac{\pi}{2} \)
(c) \( 0 < y < \pi \)
(d) \( -\frac{\pi}{2} < y < \frac{\pi}{2} \)
Answer: (b) \( -\frac{\pi}{2} \le y \le \frac{\pi}{2} \)

Question. \( \sin \left( \frac{\pi}{3} - \sin^{-1} \left( -\frac{1}{2} \right) \right) \) is equal to :
(a) \( 1/2 \)
(b) \( 1/3 \)
(c) \( 1/4 \)
(d) \( 1 \)
Answer: (d) \( 1 \)

Question. The value of \( \tan^{-1} \left( -\frac{1}{\sqrt{3}} \right) + \cot^{-1} \left( \frac{1}{\sqrt{3}} \right) + \tan^{-1} \left( \sin \left( -\frac{\pi}{2} \right) \right) \) is :
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{12} \)
(c) \( -\frac{\pi}{12} \)
(d) \( -\frac{\pi}{2} \)
Answer: (c) \( -\frac{\pi}{12} \)

Question. The value of \( \tan^{-1} \left[ 2 \sin \left( 2 \cos^{-1} \frac{\sqrt{3}}{2} \right) \right] \) is :
(a) \( \frac{\pi}{3} \)
(b) \( \frac{2\pi}{3} \)
(c) \( -\frac{\pi}{3} \)
(d) \( \frac{\pi}{6} \)
Answer: (a) \( \frac{\pi}{3} \)

Question. The value of \( \tan^{-1} \left( \tan \frac{5\pi}{6} \right) + \cos^{-1} \left( \cos \frac{13\pi}{6} \right) \) is :
(a) \( 0 \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{6} \)
(d) \( \frac{2\pi}{3} \)
Answer: (a) \( 0 \)

Question. The domain of the function \( \cos^{-1} (2x - 1) \) is :
(a) \( [0, 1] \)
(b) \( [-1, 1] \)
(c) \( (-1, 1) \)
(d) \( [0, \pi] \)
Answer: (a) \( [0, 1] \)

Question. The domain of the function defined by \( f(x) = \sin^{-1} \sqrt{x - 1} \) is : 
(a) \( [1, 2] \)
(b) \( [-1, 1] \)
(c) \( [0, 1] \)
(d) None of the options
Answer: (a) \( [1, 2] \)

Question. The value of \( \cos^{-1} \left( \cos \frac{3\pi}{2} \right) \) is equal to : 
(a) \( \frac{\pi}{2} \)
(b) \( \frac{3\pi}{2} \)
(c) \( \frac{5\pi}{2} \)
(d) \( \frac{7\pi}{2} \)
Answer: (a) \( \frac{\pi}{2} \)

Question. Solve \( \sin (\tan^{-1} x), |x| < 1 \) is equal to :
(a) \( \frac{x}{\sqrt{1-x^2}} \)
(b) \( \frac{1}{\sqrt{1-x^2}} \)
(c) \( \frac{1}{\sqrt{1+x^2}} \)
(d) \( \frac{x}{\sqrt{1+x^2}} \)
Answer: (d) \( \frac{x}{\sqrt{1+x^2}} \)

Assertion and Reason Questions

Choose the correct option :
(a) Both (A) and (R) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.

Question. Assertion (A) : \( \sin^{-1} (\sin 3) = 3 \)
Reason (R) : For principal values \( \sin^{-1} (\sin x) = x \)
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (d) A is false but R is true.

Question. Assertion (A) : The solution of system of equations \( \cos^{-1} x + (\sin^{-1} y)^2 = \frac{p\pi^2}{4} \) and \( (\cos^{-1} x)(\sin^{-1} y)^2 = \frac{\pi^4}{16} \) is \( x = \cos \frac{\pi^2}{4} \) and \( y = \pm 1 \), \( \forall p \in I \).
Reason (R) : \( AM \ge GM \)
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (a) Both (A) and (R) are true and R is the correct explanation A.

Question. Assertion (A) : If \( \sum_{i=1}^{2n} \sin^{-1} x_i = n\pi, n \in N \). Then, \( \sum_{i=1}^{2n} x_i = \sum_{i=1}^{2n} x_i^2 = \sum_{i=1}^{2n} x_i^3 = n \)
Reason (R) : \( -\frac{\pi}{2} \le \sin^{-1} x \le \frac{\pi}{2}, \forall x \in [-1, 1] \)
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (a) Both (A) and (R) are true and R is the correct explanation A.

Question. Assertion : The equation \( 2(\sin^{-1} x)^2 - 5(\sin^{-1} x) + 2 = 0 \).
Reason : \( \sin^{-1} (\sin x) = x \) if \( x \in [-1.57, 1.57] \).
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (d) A is false but R is true.