Class 11 Mathematics Trigonometric Functions MCQs Set 05

Question. If \( \cos p\theta = \cos q\theta, p \neq q \), then
(a) \( \theta = 2n\pi \)
(b) \( \theta = \frac{2n\pi}{p \pm q} \)
(c) \( \theta = \frac{n\pi}{p + q} \)
(d) None of these
Answer: B

Question. General solution of \( \tan 5\theta = \cot 2\theta \) is
(a) \( \theta = \frac{n\pi}{7} + \frac{\pi}{14} \)
(b) \( \theta = \frac{n\pi}{7} + \frac{\pi}{5} \)
(c) \( \theta = \frac{n\pi}{7} + \frac{\pi}{2} \)
(d) \( \theta = \frac{n\pi}{7} + \frac{\pi}{3}, n \in Z \)
Answer: A

Question. If \( \tan 2\theta \tan \theta = 1 \), then \( \theta = \)
(a) \( \frac{\pi}{3} \)
(b) \( (6n \pm 1)\frac{\pi}{6} \)
(c) \( (4n \pm 1)\frac{\pi}{6} \)
(d) None of these
Answer: B

Question. If \( \tan \theta + \cot \theta = 2 \), then \( \theta = \)
(a) \( n\pi \)
(b) \( n\pi + \frac{\pi}{4} \)
(c) \( n\pi \pm \frac{\pi}{4} \)
(d) \( n\pi \pm \frac{\pi}{3} \)
Answer: B

Question. If \( \cot \theta + \tan \theta = 2 \csc \theta \), then general value of \( \theta \) is
(a) \( n\pi \pm \frac{\pi}{3} \)
(b) \( n\pi \pm \frac{\pi}{6} \)
(c) \( 2n\pi \pm \frac{\pi}{3} \)
(d) \( 2n\pi \pm \frac{\pi}{6} \)
Answer: C

Question. If \( 1 + \cot \theta = \csc \theta \), then the general value of \( \theta \) is
(a) \( n\pi + \frac{\pi}{2} \)
(b) \( 2n\pi - \frac{\pi}{2} \)
(c) \( 2n\pi + \frac{\pi}{2} \)
(d) None of these
Answer: A

Question. The value of \( \cos y \cos(\frac{\pi}{2} - x) - \cos(\frac{\pi}{2} - y) \cos x + \sin y \cos(\frac{\pi}{2} - x) + \cos x \sin(\frac{\pi}{2} - y) \) is zero if
(a) x = 0
(b) y = 0
(c) x = y
(d) \( x = n\pi - \frac{\pi}{4} + y (n \in I) \)
Answer: D

Question. If \( \sin^2 \theta - 2 \cos \theta + \frac{1}{4} = 0 \), then the general value of \( \theta \) is
(a) \( n\pi \pm \frac{\pi}{3} \)
(b) \( 2n\pi \pm \frac{\pi}{3} \)
(c) \( 2n\pi \pm \frac{\pi}{6} \)
(d) \( n\pi \pm \frac{\pi}{6} \)
Answer: B

Question. The general value of \( \theta \) satisfying \( \sin^2 \theta + \sin \theta = 2 \) is
(a) \( n\pi + (-1)^n \frac{\pi}{6} \)
(b) \( 2n\pi + \frac{\pi}{4} \)
(c) \( n\pi + (-1)^n \frac{\pi}{2} \)
(d) \( n\pi + (-1)^n \frac{\pi}{3} \)
Answer: C

Question. If \( \sec 4\theta - \sec 2\theta = 2 \), then the general value of \( \theta \) is
(a) \( (2n + 1)\frac{\pi}{4} \)
(b) \( (2n + 1)\frac{\pi}{10} \)
(c) \( n\pi + \frac{\pi}{2} \) or \( \frac{n\pi}{5} + \frac{\pi}{10} \)
(d) None of these
Answer: C

Question. If \( \sin(\frac{\pi}{4} \cot \theta) = \cos(\frac{\pi}{4} \tan \theta) \), then \( \theta = \)
(a) \( n\pi + \frac{\pi}{4} \)
(b) \( 2n\pi \pm \frac{\pi}{4} \)
(c) \( n\pi - \frac{\pi}{4} \)
(d) \( 2n\pi \pm \frac{\pi}{6} \)
Answer: A

Question. The general solution of \( \sin x - 3 \sin 2x + \sin 3x = \cos x - 3 \cos 2x + \cos 3x \) is
(a) \( n\pi + \frac{\pi}{8} \)
(b) \( \frac{n\pi}{2} + \frac{\pi}{8} \)
(c) \( (-1)^n \frac{n\pi}{2} + \frac{\pi}{8} \)
(d) \( 2n\pi + \cos^{-1} \frac{3}{2} \)
Answer: B

Question. The solution of the equation \( \sec \theta - \csc \theta = \frac{4}{3} \) is
(a) \( \frac{1}{2} [n\pi + (-1)^n \sin^{-1}(\frac{3}{4})] \)
(b) \( n\pi + (-1)^n \sin^{-1}(\frac{3}{4}) \)
(c) \( \frac{n\pi}{2} + (-1)^n \sin^{-1}(\frac{3}{4}) \)
(d) None of these
Answer: A

Question. If \( 4 \sin^4 x + \cos^4 x = 1 \), then \( x \) equal to
(a) \( n\pi \)
(b) \( n\pi \pm \sin^{-1} \sqrt{\frac{2}{5}} \)
(c) \( n\pi + \frac{\pi}{6} \)
(d) None of these
Answer: A

Question. If \( 2 \sin^2 (\sin x \cdot \cos 2x) + \sin 2x (1 + 2 \sin x) - 2 \cos x = 0 \), then \( x = \)
(a) \( (2n \pm 1) \pi \)
(b) \( (2n \pm 1) \frac{\pi}{6} \)
(c) \( (n \pm 1) \frac{\pi}{4} \)
(d) \( (n \pm 1) \frac{\pi}{6} \)
Answer: B

Question. Expression \( 2^{\sin \theta} + 2^{-\cos \theta} \) is minimum when \( \theta = \) ............ and its minimum value is ....................
(a) \( 2n\pi + \frac{\pi}{4}, n \in I ; 2 \)
(b) \( 2n\pi + \frac{7\pi}{4}, n \in I ; 2^{1 - \frac{1}{\sqrt{2}}} \)
(c) \( n\pi \pm \frac{\pi}{4}, n \in I ; 2^{1 - \frac{1}{\sqrt{2}}} \)
(d) None of these
Answer: B

Question. The general solution of the equation \( 2^{\cos 2x} + 1 = 3 \cdot 2^{-\sin^2 x} \) is
(a) \( n\pi \)
(b) \( n\pi + \pi \)
(c) \( n\pi - \pi \)
(d) None of these
Answer: A

Question. If \( \theta = \tan^{-1}(2 \tan^2 \theta) - \frac{1}{2} \sin^{-1} \left( \frac{3 \sin 2\theta}{5 + 4 \cos 2\theta} \right) \), then the general value of \( \theta \)
(a) \( n\pi \)
(b) \( n\pi + \frac{\pi}{4} \)
(c) \( n\pi + \tan^{-1}(-2) \)
(d) All of these
Answer: D

Question. If the expression \( \frac{\sin \frac{x}{2} + \cos \frac{x}{2} - i \tan x}{1 + 2i \sin \frac{x}{2}} \) is real, then \( x \) is equal to
(a) \( 2n\pi + \tan^{-1} K, K \in R, n \in Z \)
(b) \( 2n\pi + 2 \tan^{-1} K, \text{where } K \in (0,1), n \in Z \)
(c) \( 2n\pi + 2 \tan^{-1} K, \text{where } K \in (1,2), n \in Z \)
(d) \( 2n\pi + 2 \tan^{-1} K, K \in (2,3), n \in Z \)
Answer: B

Question. If \( \frac{1}{6} \sin x, \cos x, \tan x \) are in G.P., then \( x \) is equal to
(a) \( n\pi \pm \frac{\pi}{3}, n \in Z \)
(b) \( 2n\pi \pm \frac{\pi}{3}, n \in Z \)
(c) \( n\pi + (-1)^n \frac{\pi}{3}, n \in Z \)
(d) None of these
Answer: B

Question. If \( 32 \tan^8 \theta = 2 \cos^2 \alpha - 3 \cos \alpha \) and \( 3 \cos 2\theta = 1 \), then the general value of \( \alpha \) is
(a) \( 2n\pi \pm \frac{\pi}{3} \)
(b) \( 2n\pi \pm \cos^{-1} 2 \)
(c) \( 2n\pi \pm \frac{2\pi}{3} \)
(d) None of these
Answer: C

Question. If \( \max_{\theta \in R} \{ 5 \sin \theta + 3 \sin (\theta - \alpha) \} = 7 \), then the set of possible values of \( \alpha \) is
(a) \( \{ x \mid x = 2n\pi \pm \frac{\pi}{3}, n \in Z \} \)
(b) \( \{ x \mid x = 2n\pi \pm \frac{2\pi}{3}, n \in Z \} \)
(c) \( [\frac{\pi}{3}, \frac{2\pi}{3}] \)
(d) None of these
Answer: B

Question. If \( | \cos x |^{\sin^2 x - \frac{3}{2} \sin x + \frac{1}{2}} = 1 \), then possible values of \( x \) are
(a) \( n\pi \) or \( n\pi + (-1)^n \frac{\pi}{6}, n \in I \)
(b) \( n\pi \) or \( 2n\pi + \frac{\pi}{2} \) or \( n\pi + (-1)^n \frac{\pi}{6}, n \in I \)
(c) \( n\pi + (-1)^n \frac{\pi}{6}, n \in I \)
(d) None of these
Answer: B

Question. The general solution of \( \cos^{50} x - \sin^{50} x = 1 \) is
(a) \( n\pi \)
(b) \( 2n\pi \)
(c) \( n\pi + \frac{\pi}{2} \)
(d) \( 2n\pi + \frac{\pi}{2} \)
Answer: A

Question. Let [x] = the greatest integer less than or equal to x and let \( f(x) = \sin x + \cos x \). Then the most general solution of \( f(x) = [f(\frac{\pi}{10})] \) are
(a) \( 2n\pi + \frac{\pi}{2}, n \in Z \)
(b) \( n\pi, n \in Z \)
(c) \( 2n\pi, n \in Z \)
(d) None of these
Answer: C

Question. The most general values of \( x \) for which \( \sin x + \cos x = \min_{a \in R} \{ 1, a^2 - 4a + 6 \} \) are given by
(a) \( 2n\pi \)
(b) \( 2n\pi + \frac{\pi}{2} \)
(c) \( n\pi + (-1)^n \frac{\pi}{4} - \frac{\pi}{4} \)
(d) None of these
Answer: C

Number of Solutions and Solutions in the Interval of a Trigonometric Equation

Basic Level

Question. The number of values of \( \theta \) in \( [0, 2\pi] \) satisfying the equation \( 2 \sin^2 \theta = 4 + 3 \cos \theta \) are
(a) 0
(b) 1
(c) 2
(d) 3
Answer: A

Question. The equation \( 3 \cos x + 4 \sin x = 6 \) has....solution
(a) Finite
(b) Infinite
(c) One
(d) No
Answer: D

Question. The solution of the equation \( \cos^2 \theta + \sin \theta + 1 = 0 \), lies in the interval
(a) \( (-\frac{\pi}{4}, \frac{\pi}{4}) \)
(b) \( (\frac{\pi}{4}, \frac{3\pi}{4}) \)
(c) \( (\frac{3\pi}{4}, \frac{5\pi}{4}) \)
(d) \( (\frac{5\pi}{4}, \frac{7\pi}{4}) \)
Answer: D

Question. The solution set of the system of equations \( x + y = \frac{2\pi}{3}, \cos x + \cos y = \frac{3}{2} \), where x and y are real in
(a) A finite non-empty set
(b) Null set
(c) Infinite
(d) None of these
Answer: B

Question. The equation \( 2 \cos^2 (\frac{x}{2}) \sin^2 x = x^2 + \frac{1}{x^2}, 0 \leq x \leq \frac{\pi}{2} \) has
(a) No solution
(b) One real solution
(c) More than one real solution
(d) None of these
Answer: A

Question. The smallest positive root of the equation \( \tan x - x = 0 \) lies on
(a) \( (0, \frac{\pi}{2}) \)
(b) \( (\frac{\pi}{2}, \pi) \)
(c) \( (\pi, \frac{3\pi}{2}) \)
(d) \( (\frac{3\pi}{2}, 2\pi) \)
Answer: C