CBSE Class 12 Mathematics Inverse Trigonometric Functions MCQs Set 07

Multiple Choice Questions

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 value of \( \sin^{-1} \left( \cos \frac{\pi}{9} \right) \) is
(a) \( \frac{\pi}{9} \)
(b) \( \frac{5\pi}{9} \)
(c) \( \frac{-5\pi}{9} \)
(d) \( \frac{7\pi}{18} \)
Answer: (d) \( \frac{7\pi}{18} \)

Question. The value of \( \tan^{-1}(1) + \cos^{-1}\left(-\frac{1}{2}\right) + \sin^{-1}\left(-\frac{1}{2}\right) \) corresponding to principal branches is
(a) \( \frac{3\pi}{4} \)
(b) \( \frac{\pi}{4} \)
(c) \( -\frac{\pi}{4} \)
(d) \( \frac{3\pi}{4} \)
Answer: (a) \( \frac{3\pi}{4} \)

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) \( \left[ -\frac{1}{2}, \frac{1}{2} \right] \)
(d) \( [-2, 2] \)
Answer: (c) \( \left[ -\frac{1}{2}, \frac{1}{2} \right] \)

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. Let \( \theta = \sin^{-1}(\sin(-600^\circ)) \), then value of \( \theta \) is
(a) \( \frac{\pi}{3} \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{2\pi}{3} \)
(d) \( \frac{-2\pi}{3} \)
Answer: (a) \( \frac{\pi}{3} \)

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. If \( \sin^{-1} x + \sin^{-1} y = \frac{\pi}{2} \), then value of \( \cos^{-1} x + \cos^{-1} y \) is
(a) \( \frac{\pi}{2} \)
(b) \( \pi \)
(c) \( 0 \)
(d) \( \frac{2\pi}{3} \)
Answer: (a) \( \frac{\pi}{2} \)

Question. The value of \( \tan \left( \cos^{-1} \frac{3}{5} + \tan^{-1} \frac{1}{4} \right) \) is
(a) \( \frac{19}{8} \)
(b) \( \frac{8}{19} \)
(c) \( \frac{19}{12} \)
(d) \( \frac{3}{4} \)
Answer: (a) \( \frac{19}{8} \)

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 \( \tan^2(\sec^{-1} 2) + \cot^2(\operatorname{cosec}^{-1} 3) \) is
(a) 5
(b) 11
(c) 13
(d) 15
Answer: (b) 11

Question. \( \sin(\tan^{-1} x), |x| < 1 \) 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) \( \frac{x}{\sqrt{1 + x^2}} \)

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

Question. If \( \tan^{-1} \frac{1}{2} + \tan^{-1} \frac{2}{11} = \tan^{-1} a \), then \( a \) is equal to
(a) \( \frac{1}{4} \)
(b) \( \frac{1}{2} \)
(c) \( \frac{3}{4} \)
(d) 1
Answer: (c) \( \frac{3}{4} \)

Question. If \( 3\tan^{-1} x + \cot^{-1} x = \pi \), then \( x \) equals
(a) 0
(b) 1
(c) \( -1 \)
(d) \( \frac{1}{2} \)
Answer: (b) 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. If \( \sin^{-1} \left( \frac{2a}{1 + a^2} \right) + \cos^{-1} \left( \frac{1 - a^2}{1 + a^2} \right) = \tan^{-1} \left( \frac{2x}{1 - x^2} \right) \), where \( a, x \in (0, 1) \), then the value of \( x \) is
(a) 0
(b) \( \frac{a}{2} \)
(c) \( a \)
(d) \( \frac{2a}{1 - a^2} \)
Answer: (d) \( \frac{2a}{1 - a^2} \)

Question. The value of \( \cot \left[ \cos^{-1} \left( \frac{7}{25} \right) \right] \) is
(a) \( \frac{25}{24} \)
(b) \( \frac{25}{7} \)
(c) \( \frac{24}{25} \)
(d) \( \frac{7}{24} \)
Answer: (d) \( \frac{7}{24} \)

Question. If \( \cos^{-1} \alpha + \cos^{-1} \beta + \cos^{-1} \gamma = 3\pi \), then \( \alpha(\beta + \gamma) + \beta(\gamma + \alpha) + \gamma(\alpha + \beta) \) equals
(a) 0
(b) 1
(c) 6
(d) 12
Answer: (c) 6