Class 11 Mathematics Trigonometric Functions MCQs Set 06

Question. If \( a \) is any real number, then the number of roots of \( \cot x - \tan x = a \) in the first quadrant are
(a) 2
(b) 0
(c) 1
(d) None of these
Answer: C

Question. The number of solutions of the equation \( \sin(\frac{\pi x}{2\sqrt{3}}) = x^2 - 2\sqrt{3}x + 4 \)
(a) Form an empty set
(b) 1
(c) 2
(d) > 2
Answer: B

Question. The number of all possible triplets \( (a_1, a_2, a_3) \) such that \( a_1 + a_2 \cos 2x + a_3 \sin^2 x = 0 \) for all \( x \) is
(a) Zero
(b) 1
(c) 2
(d) Infinite
Answer: D

Question. The equation \( (\cos p - 1)x^2 - (\cos p)x + \sin p = 0 \), where x is a variable, has real roots. Then the interval of p may be any one of the followings
(a) \( (0, 2\pi) \)
(b) \( (-\pi, 0) \)
(c) \( (-\frac{\pi}{2}, \frac{\pi}{2}) \)
(d) \( (0, \pi) \)
Answer: D

Question. Let n be an odd integer if \( \sin n\theta = \sum_{r=0}^{n} b_r \sin^r \theta \), for every value of \( \theta \), then
(a) \( b_0 = 1, b_1 = 3 \)
(b) \( b_0 = 0, b_1 = 4 \)
(c) \( b_0 = 0, b_1 = n \)
(d) \( b_0 = 0, b_1 = n^2 - n + 3 \)
Answer: C

Solution of Simultaneously Solving Equation

Basic Level

Question. The most general value of \( \theta \) which will satisfy both the equations \( \sin \theta = -\frac{1}{2} \) and \( \tan \theta = \frac{1}{\sqrt{3}} \) is
(a) \( n\pi + (-1)^n \frac{\pi}{6} \)
(b) \( n\pi + \frac{\pi}{6} \)
(c) \( 2n\pi \pm \frac{\pi}{6} \)
(d) None of these
Answer: D

Question. The most general value of \( \theta \) satisfying the equations \( \sin \theta = \sin \alpha \) and \( \cos \theta = \cos \alpha \) is
(a) \( 2n\pi + \alpha \)
(b) \( 2n\pi - \alpha \)
(c) \( n\pi + \alpha \)
(d) \( n\pi - \alpha \)
Answer: A

Question. If \( \sin A = \sin B, \cos A = \cos B \), then the value of A in terms of B is
(a) \( n\pi + B \)
(b) \( n\pi + (-1)^n B \)
(c) \( 2n\pi + B \)
(d) \( 2n\pi - B \)
Answer: C

Question. If \( \cos \theta = -\frac{1}{\sqrt{2}} \) and \( \tan \theta = 1 \), then the general value of \( \theta \) is
(a) \( 2n\pi + \frac{\pi}{4} \)
(b) \( (2n + 1)\pi + \frac{\pi}{4} \)
(c) \( n\pi + \frac{\pi}{4} \)
(d) \( n\pi \pm \frac{\pi}{4} \)
Answer: B

Question. If \( \sin 2x + \sin 4x = 2 \sin 3x \), then \( x = \)
(a) \( \frac{n\pi}{3} \)
(b) \( n\pi + \frac{\pi}{3} \)
(c) \( 2n\pi \pm \frac{\pi}{3} \)
(d) None of these
Answer: A

Question. If \( \tan \theta = \sqrt{3} \) and \( \csc \theta = -\frac{2}{\sqrt{3}} \), then the most general value of \( \theta \) satisfying both the equations is
(a) \( 2n\pi + \frac{\pi}{3} \)
(b) \( 2n\pi + \frac{2\pi}{3} \)
(c) \( 2n\pi + \frac{4\pi}{3} \)
(d) None of these
Answer: C

Question. If \( \sin \theta = \frac{1}{2} \) and \( \cos \theta = \frac{\sqrt{3}}{2} \), then the general value of \( \theta \) which satisfies both the equations is
(a) \( 2n\pi + \frac{\pi}{3} \)
(b) \( 2n\pi + \frac{\pi}{4} \)
(c) \( 2n\pi + \frac{\pi}{6} \)
(d) None of these
Answer: C

Question. General solution of \( (1 + 2 \sin \theta)^2 + (\sqrt{3} \tan \theta - 1)^2 = 0 \) is
(a) \( n\pi + \frac{\pi}{6} \)
(b) \( 2n\pi + \frac{11\pi}{6} \)
(c) \( 2n\pi + \frac{7\pi}{6} \)
(d) None of these
Answer: B

Question. If \( 3 \sin^2 \theta + 2 \sin^2 \phi = 1 \) and \( 3 \sin 2\theta = 2 \sin 2\phi, 0 < \theta < \frac{\pi}{2} \) and \( 0 < \phi < \frac{\pi}{2} \), then the value of \( \theta + 2\phi \) is
(a) \( \frac{\pi}{2} \)
(b) \( \frac{\pi}{4} \)
(c) 0
(d) None of these
Answer: A

Principal Value and Particular Values of a Trigonometrical Equations

Basic Level

Question. The smallest value of \( \theta \) satisfying \( \sqrt{3}(\cot \theta + \tan \theta) = 4 \) is
(a) \( \frac{2\pi}{3} \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{6} \)
(d) \( \frac{\pi}{12} \)
Answer: C

Question. If \( \sin \theta = \sqrt{3} \cos \theta, -\pi < \theta < 0 \), then \( \theta = \)
(a) \( -\frac{5\pi}{6} \)
(b) \( -\frac{4\pi}{6} \)
(c) \( \frac{4\pi}{6} \)
(d) \( \frac{5\pi}{6} \)
Answer: B

Question. If \( \tan \theta = -\frac{1}{\sqrt{3}}, \sin \theta = \frac{1}{2} \) and \( \cos \theta = -\frac{\sqrt{3}}{2} \), then the principal value of \( \theta \) will be
(a) \( \frac{\pi}{6} \)
(b) \( \frac{5\pi}{6} \)
(c) \( \frac{7\pi}{6} \)
(d) \( -\frac{\pi}{6} \)
Answer: B

Question. The smallest positive angle satisfying the equation \( \sin^2 \theta - 2 \cos \theta + \frac{1}{4} = 0 \) is
(a) \( \frac{\pi}{2} \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{4} \)
(d) \( \frac{\pi}{6} \)
Answer: B

Question. If \( \sin 5x + \sin 3x + \sin x = 0 \), then the value of x other than 0 lying between \( 0 \leq x \leq \frac{\pi}{2} \) is
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{12} \)
(c) \( \frac{\pi}{3} \)
(d) \( \frac{\pi}{4} \)
Answer: C

Question. The value of x satisfying the equation \( \sin x + \frac{1}{\sin x} = \frac{7}{2\sqrt{3}} \) is given by
(a) \( 10^\circ \)
(b) \( 30^\circ \)
(c) \( 45^\circ \)
(d) \( 60^\circ \)
Answer: D

Question. If \( \sin 2\theta = \cos 3\theta \) and \( \theta \) is an acute angle, then \( \sin \theta \) is equal to
(a) \( \frac{\sqrt{5} - 1}{4} \)
(b) \( \frac{-\sqrt{5} - 1}{4} \)
(c) 0
(d) None of these
Answer: A

Question. The values of \( \theta \) satisfying \( \sin 7\theta = \sin 4\theta - \sin \theta \) and \( 0 < \theta < \frac{\pi}{2} \) are
(a) \( \frac{\pi}{9}, \frac{\pi}{4} \)
(b) \( \frac{\pi}{3}, \frac{\pi}{9} \)
(c) \( \frac{\pi}{6}, \frac{\pi}{9} \)
(d) \( \frac{\pi}{3}, \frac{\pi}{4} \)
Answer: C

Question. If \( (2 \cos x - 1)(3 + 2 \cos x) = 0, 0 \leq x \leq 2\pi \), then x =
(a) \( \frac{\pi}{3} \)
(b) \( \frac{\pi}{3}, \frac{5\pi}{3} \)
(c) \( \frac{\pi}{3}, \frac{5\pi}{3}, \cos^{-1}(-\frac{3}{2}) \)
(d) \( \frac{5\pi}{3} \)
Answer: B

Question. If \( 2 \cos^2 x + 3 \sin x - 3 = 0, 0 \leq x \leq 180^\circ \), then \( x = \)
(a) \( 30^\circ, 90^\circ, 150^\circ \)
(b) \( 60^\circ, 120^\circ, 180^\circ \)
(c) \( 0^\circ, 30^\circ, 150^\circ \)
(d) \( 45^\circ, 90^\circ, 135^\circ \)
Answer: A

Question. If \( r \sin \theta = 3, r = 4(1 + \sin \theta), 0 \leq \theta \leq 2\pi \), then \( \theta = \)
(a) \( \frac{\pi}{6}, \frac{\pi}{3} \)
(b) \( \frac{\pi}{6}, \frac{5\pi}{6} \)
(c) \( \frac{\pi}{3}, \frac{\pi}{4} \)
(d) \( \frac{\pi}{2}, \pi \)
Answer: B

Question. If \( \tan(\theta + x) \tan(\theta - x) = 1 \) for all x, then value of \( \theta \) must be
(a) \( 0^\circ \)
(b) \( 30^\circ \)
(c) \( 45^\circ \)
(d) \( 60^\circ \)
Answer: C

Question. If \( 2 \sin^2 \theta = 3 \cos \theta, \text{where } 0 \leq \theta \leq 2\pi \), then \( \theta = \)
(a) \( \frac{\pi}{6}, \frac{7\pi}{6} \)
(b) \( \frac{\pi}{3}, \frac{5\pi}{3} \)
(c) \( \frac{\pi}{3}, \frac{7\pi}{3} \)
(d) None of these
Answer: B

Question. \( \cot \theta = \sin 2\theta (\theta \neq n\pi, n \text{ is integer}), \text{if } \theta = \)
(a) \( 45^\circ \text{ and } 60^\circ \)
(b) \( 45^\circ \text{ and } 90^\circ \)
(c) \( 45^\circ \text{ only} \)
(d) \( 90^\circ \text{ only} \)
Answer: A

Question. If \( \cos \theta = -\frac{1}{2} \) and \( 0 < \theta < 360^\circ \), then the values of \( \theta \) are
(a) \( 120^\circ \text{ and } 300^\circ \)
(b) \( 60^\circ \text{ and } 120^\circ \)
(c) \( 120^\circ \text{ and } 240^\circ \)
(d) \( 60^\circ \text{ and } 240^\circ \)
Answer: C

Question. If \( \tan(\pi \cos \theta) = \cot(\pi \sin \theta) \), then \( \sin(\theta + \frac{\pi}{4}) \) equals
(a) \( \frac{1}{\sqrt{2}} \)
(b) \( \frac{1}{2} \)
(c) \( \frac{1}{2\sqrt{2}} \)
(d) \( \frac{\sqrt{3}}{2} \)
Answer: C

Question. If \( 2 \sec(2\alpha) = \tan \beta + \cot \beta \), then one of the values of \( (\alpha + \beta) \) is
(a) \( \pi \)
(b) \( n\pi - \frac{\pi}{4} \)
(c) \( \frac{\pi}{4} \)
(d) None of these
Answer: C

Advance Level

Question. If \( 3 \sin 2\theta = 2 \sin 3\theta \text{ and } 0 < \theta < \pi \), then value of \( \sin \theta \) is
(a) \( \frac{\sqrt{2}}{3} \)
(b) \( \frac{\sqrt{3}}{\sqrt{5}} \)
(c) \( \frac{\sqrt{15}}{4} \)
(d) \( \frac{\sqrt{2}}{\sqrt{5}} \)
Answer: C

Question. \( 2 \sin^2 x + \sin^2 2x = 2, -\pi < x < \pi, \text{then } x = \)
(a) \( \pm \frac{\pi}{6} \)
(b) \( \pm \frac{\pi}{4} \)
(c) \( \frac{3\pi}{2} \)
(d) None of these
Answer: B

Question. If \( 5 \cos 2\theta + 2 \cos^2 \frac{\theta}{2} + 1 = 0, -\pi < \theta < \pi \), then \( \theta = \)
(a) \( \cos^{-1}(\frac{3}{5}) \)
(b) \( \cos^{-1}(\frac{1}{5}) \)
(c) \( \cos^{-1}(-\frac{3}{5}) \)
(d) None of these
Answer: C