Class 11 Mathematics Trigonometric Equations MCQs Set 01

Question. If \( 4 \cos^2 \theta = 3 \) then \( \theta = \) -----
(a) \( \frac{5\pi}{6}, \frac{\pi}{6} \)
(b) \( \frac{3\pi}{4}, \frac{\pi}{4} \)
(c) \( \frac{2\pi}{3}, \frac{\pi}{3} \)
(d) \( \pm \frac{\pi}{2} \)
Answer: (a) \( \frac{5\pi}{6}, \frac{\pi}{6} \)

Question. If \( \tan \theta + \sec \theta = \sqrt{3} \), then the principal value of \( \left( \theta + \frac{\pi}{6} \right) \) is
(a) \( \frac{\pi}{3} \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{2\pi}{3} \)
(d) \( \frac{\pi}{2} \)
Answer: (a) \( \frac{\pi}{3} \)

Question. If \( \tan^2 \theta = \sqrt{3} + (\sqrt{3}-1)\tan \theta \) and \( \theta \) lies in \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) then \( \theta = \)
(a) \( \left\{ -\frac{\pi}{3}, \frac{\pi}{4} \right\} \)
(b) \( \left\{ \frac{\pi}{3}, -\frac{\pi}{4} \right\} \)
(c) \( \left\{ \frac{\pi}{6}, -\frac{\pi}{3} \right\} \)
(d) \( \left\{ -\frac{\pi}{12}, \frac{\pi}{12} \right\} \)
Answer: (b) \( \left\{ \frac{\pi}{3}, -\frac{\pi}{4} \right\} \)

Question. If \( \sin \left( \frac{\pi \cot \theta}{4} \right) = \cos \left( \frac{\pi \tan \theta}{4} \right) \) and \( \theta \) is in the first quadrant then \( \theta = \) -----
(a) \( \frac{\pi}{3} \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{\pi}{4} \)
(d) \( \frac{\pi}{6} \)
Answer: (c) \( \frac{\pi}{4} \)

Question. If \( 3\tan^4 \alpha - 10\tan^2 \alpha + 3 = 0 \) then principal values of '\( \alpha \)' are
(a) \( \pm 45^\circ, \pm 36^\circ \)
(b) \( \pm 30^\circ, \pm 60^\circ \)
(c) \( \pm 75^\circ, \pm 36^\circ \)
(d) \( \pm 60^\circ, \pm 15^\circ \)
Answer: (b) \( \pm 30^\circ, \pm 60^\circ \)

Question. If \( \sin(x + 28^\circ) = \cos(3x - 78^\circ) \) then \( x = \)
(a) \( 37^\circ, 8^\circ \)
(b) \( 39^\circ \)
(c) \( 35^\circ, 8^\circ \)
(d) \( 47^\circ \)
Answer: (c) \( 35^\circ, 8^\circ \)

Question. If \( \sin 2\theta - \cos 3\theta = 0 \) and \( \theta \) is acute then \( \theta = \)
(a) \( 36^\circ \)
(b) \( 54^\circ \)
(c) \( 18^\circ \)
(d) \( 30^\circ \)
Answer: (c) \( 18^\circ \)

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

Question. The most general value of \( \theta \) satisfying the equations \( \sin \theta = \frac{1}{\sqrt{2}} \), \( \cos \theta = -\frac{1}{\sqrt{2}} \) is
(a) \( 2n\pi + \frac{\pi}{4}, \forall n \in Z \)
(b) \( 2n\pi + \frac{3\pi}{4}, \forall n \in Z \)
(c) \( n\pi + \frac{\pi}{6}, \forall n \in Z \)
(d) \( n\pi + \frac{\pi}{3}, \forall n \in Z \)
Answer: (b) \( 2n\pi + \frac{3\pi}{4}, \forall n \in Z \)

Question. The general solution of \( \frac{\tan 2x - \tan x}{1 + \tan x \cdot \tan 2x} = 1 \) is
(a) \( n\pi + \frac{\pi}{4}, \forall n \in Z \)
(b) \( n\pi \pm \frac{\pi}{4}, \forall n \in Z \)
(c) \( \phi \)
(d) \( n\pi + \frac{\pi}{6}, \forall n \in Z \)
Answer: (c) \( \phi \)

Question. If \( 11\sin^2 x + 7\cos^2 x = 8 \) then \( x = \) -----
(a) \( n\pi \pm \frac{\pi}{6}, \forall n \in Z \)
(b) \( n\pi \pm \frac{\pi}{4}, \forall n \in Z \)
(c) \( n\pi \pm \frac{\pi}{3}, \forall n \in Z \)
(d) \( n\pi \pm \frac{\pi}{2}, \forall n \in Z \)
Answer: (a) \( n\pi \pm \frac{\pi}{6}, \forall n \in Z \)

Question. The general solution of \( \sin 2x = 4\cos x \) is
(a) \( (2n + 1)\frac{\pi}{2}, \forall n \in Z \)
(b) \( n\pi, \forall n \in Z \)
(c) Empty Set
(d) \( 2n\pi, \forall n \in Z \)
Answer: (a) \( (2n + 1)\frac{\pi}{2}, \forall n \in Z \)

Question. \( \tan \left( \frac{\pi}{4} + \frac{A}{2} \right) + \tan \left( \frac{\pi}{4} - \frac{A}{2} \right) = \frac{4}{\sqrt{3}} \) ; then the general solution of A =
(a) \( 2n\pi \pm \frac{\pi}{6}, \forall n \in Z \)
(b) \( n\pi + \frac{\pi}{4}, \forall n \in Z \)
(c) \( 2n\pi \pm \frac{\pi}{4}, \forall n \in Z \)
(d) \( n\pi, \forall n \in Z \)
Answer: (a) \( 2n\pi \pm \frac{\pi}{6}, \forall n \in Z \)

Question. If \( \tan x + 2\tan 2x + 4\tan 4x + 8\cot 8x = \sqrt{3} \) then the general solution of x =
(a) \( n\pi + \frac{\pi}{3}, \forall n \in Z \)
(b) \( n\pi + \frac{\pi}{6}, \forall n \in Z \)
(c) \( n\pi + \frac{\pi}{4}, \forall n \in Z \)
(d) \( n\pi, \forall n \in Z \)
Answer: (b) \( n\pi + \frac{\pi}{6}, \forall n \in Z \)

Question. The general value of \( \theta \) satisfies the equation \( \tan \theta \tan(120^\circ + \theta) \tan(120^\circ - \theta) = \frac{1}{\sqrt{3}} \) is
(a) \( (6n + 1)\frac{\pi}{18}, \forall n \in Z \)
(b) \( (3n + 1)\frac{\pi}{3}, \forall n \in Z \)
(c) \( (6n + 1)\frac{\pi}{6}, \forall n \in Z \)
(d) \( (3n + 1)\frac{\pi}{6}, \forall n \in Z \)
Answer: (a) \( (6n + 1)\frac{\pi}{18}, \forall n \in Z \)

Question. If \( \tan m\theta = \cot n\theta \) then the G.S of \( \theta = \)
(a) \( \frac{(k + 1)\pi}{2(m + n)}, \forall k \in Z \)
(b) \( \frac{(2k + 1)\pi}{2(m + n)}, \forall k \in Z \)
(c) \( \frac{(2k + 1)\pi}{m + n}, \forall k \in Z \)
(d) \( \frac{(2k + 1)\pi}{m - n}, \forall k \in Z \)
Answer: (b) \( \frac{(2k + 1)\pi}{2(m + n)}, \forall k \in Z \)

Question. If \( \cos x + \cos y = 1 \) and \( \cos x \cos y = 1/4 \) then the G.S are
(a) \( x = 2n\pi \pm \frac{\pi}{4}, n \in Z, y = 2m\pi \pm \frac{\pi}{4}, m \in Z \)
(b) \( x = 2n\pi \pm \frac{\pi}{3}, n \in Z, y = 2m\pi \pm \frac{\pi}{3}, m \in Z \)
(c) \( x = 2n\pi \pm \frac{\pi}{6}, n \in Z, y = 2m\pi \pm \frac{\pi}{6}, m \in Z \)
(d) \( x = 2n\pi \pm \frac{\pi}{5}, n \in Z, y = 2m\pi \pm \frac{\pi}{5}, m \in Z \)
Answer: (b) \( x = 2n\pi \pm \frac{\pi}{3}, n \in Z, y = 2m\pi \pm \frac{\pi}{3}, m \in Z \)

Question. If \( \sin x \cdot \sin(60^\circ + x) \cdot \sin(60^\circ - x) = \frac{1}{8} \), then x =
(a) \( n\pi + (-1)^n \frac{\pi}{6} \)
(b) \( \frac{n\pi}{3} + (-1)^n \frac{\pi}{18} \)
(c) \( n\pi + (-1)^n \frac{\pi}{3} \)
(d) \( \frac{n\pi}{3} + (-1)^n \frac{\pi}{9} \)
Answer: (b) \( \frac{n\pi}{3} + (-1)^n \frac{\pi}{18} \)

Question. The value of \( \theta \) satisfying \( \sin 7\theta = \sin 4\theta - \sin \theta \) in \( 0 < \theta < \frac{\pi}{2} \) are
(a) \( \frac{\pi}{9}, \frac{\pi}{4} \)
(b) \( \frac{\pi}{3}, \frac{\pi}{9} \)
(c) \( \frac{\pi}{6}, \frac{\pi}{4} \)
(d) \( 0, 1 \)
Answer: (a) \( \frac{\pi}{9}, \frac{\pi}{4} \)

Question. If \( 1 + \sin^2 \theta = 3\sin \theta \cos \theta \) then the solution set in \( \left( 0, \frac{\pi}{2} \right) \) is
(a) \( \left\{ \frac{\pi}{4}, \cos^{-1}\frac{1}{3} \right\} \)
(b) \( \left\{ \frac{\pi}{4}, \tan^{-1}\frac{1}{2} \right\} \)
(c) \( \left\{ \frac{\pi}{3}, \tan^{-1}\frac{1}{3} \right\} \)
(d) \( \left\{ \frac{\pi}{6}, \sin^{-1}\frac{1}{3} \right\} \)
Answer: (b) \( \left\{ \frac{\pi}{4}, \tan^{-1}\frac{1}{2} \right\} \)

Question. The equation \( \sqrt{3}\sin x + \cos x = 4 \) has
(a) Only one solution
(b) Two solutions
(c) Infinitely many solutions
(d) No solution
Answer: (d) No solution

Question. If \( y + \cos \theta = \sin \theta \) has a real solution then
(a) \( -\sqrt{2} \le y \le \sqrt{2} \)
(b) \( y > \sqrt{2} \)
(c) \( y \le -\sqrt{2} \)
(d) \( y \le 1 \)
Answer: (a) \( -\sqrt{2} \le y \le \sqrt{2} \)

Question. If \( 3\tan\theta = \cot\theta \) then \( \theta = \) - - - - -
(a) \( \pm 30^\circ \)
(b) \( \pm 60^\circ \)
(c) \( \pm 45^\circ \)
(d) \( \pm 15^\circ \)
Answer: (a) \( \pm 30^\circ \)

Question. The principal value of \( \left(\theta + \frac{\pi}{4}\right) \) where \( \sin\theta + \cos\theta = 1 \) is
(a) \( 0^\circ \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{4} \)
(d) \( \frac{\pi}{2} \)
Answer: (c) \( \frac{\pi}{4} \)

Question. If \( \cos 2\theta = 2\sin^2\theta \) then \( \theta = \)
(a) \( \pm 30^\circ \)
(b) \( \pm 60^\circ \)
(c) \( \pm 45^\circ \)
(d) \( \pm 90^\circ \)
Answer: (a) \( \pm 30^\circ \)

Question. If \( \tan(\pi\cos\theta) = \cot(\pi\sin\theta) \) then the value(s) of \( \cos\left(\theta - \frac{\pi}{4}\right) \) is (are)
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{\sqrt{2}} \)
(c) \( \pm \frac{1}{2\sqrt{2}} \)
(d) \( \frac{1}{3\sqrt{2}} \)
Answer: (c) \( \pm \frac{1}{2\sqrt{2}} \)

Question. If A and B are acute angles such that \( \sin A = \sin^2 B \) and \( 2\cos^2 A = 3\cos^2 B \) then A=
(a) \( \frac{\pi}{4} \)
(b) \( \frac{\pi}{6} \)
(c) \( \frac{\pi}{3} \)
(d) \( \frac{\pi}{2} \)
Answer: (b) \( \frac{\pi}{6} \)

Question. If \( \sqrt{3} \cos\theta - \sin\theta = 1 \) then \( \theta = \)
(a) \( \pi \)
(b) \( \frac{\pi}{2} \)
(c) \( 1 \)
(d) \( \frac{\pi}{6} \)
Answer: (d) \( \frac{\pi}{6} \)

Question. If \( \cot\theta - \tan\theta = 2 \) then principal value of \( \theta \)
(a) \( \frac{\pi}{4} \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{\pi}{8} \)
(d) \( \frac{3\pi}{4} \)
Answer: (c) \( \frac{\pi}{8} \)

Question. If \( \sin A = \sin B \), \( \cos A = \cos B \) then A =
(a) \( n\pi + B, \forall n \in Z \)
(b) \( 2n\pi + B, \forall n \in Z \)
(c) \( 2n\pi - B, \forall n \in Z \)
(d) \( 2n\pi \pm B, \forall n \in Z \)
Answer: (b) \( 2n\pi + B, \forall n \in Z \)

Question. The general solution of \( \frac{\tan 5x - \tan 4x}{1 + \tan 5x \tan 4x} = 1 \) is
(a) \( n\pi + \frac{\pi}{4}, \forall n \in Z \)
(b) \( n\pi \pm \frac{\pi}{4}, \forall n \in Z \)
(c) \( \phi \)
(d) \( n\pi + \frac{\pi}{6}, \forall n \in Z \)
Answer: (a) \( n\pi + \frac{\pi}{4}, \forall n \in Z \)

Question. Solution of \( \cot^2 \theta + \left(\sqrt{3} + \frac{1}{\sqrt{3}}\right)\cot\theta + 1 = 0 \) is
(a) \( n\pi - \frac{\pi}{6}, n\pi - \frac{\pi}{3}, \forall n \in Z \)
(b) \( n\pi + \frac{\pi}{6}, n\pi + \frac{\pi}{3}, \forall n \in Z \)
(c) \( n\pi + \frac{\pi}{12}, \forall n \in Z \)
(d) \( n\pi + \frac{\pi}{4}, \forall n \in Z \)
Answer: (a) \( n\pi - \frac{\pi}{6}, n\pi - \frac{\pi}{3}, \forall n \in Z \)

Question. \( 3\sin x + 4\cos x - 6 = 0 \) then the general solution of x=
(a) \( n\pi + (-1)^n \frac{\pi}{6}, \forall n \in Z \)
(b) \( n\pi + (-1)^n \frac{\pi}{4}, \forall n \in Z \)
(c) \( n\pi + (-1)^n \frac{\pi}{3}, \forall n \in Z \)
(d) Empty Set
Answer: (d) Empty Set

Question. \( \tan^2\left(\frac{\pi}{4} + \frac{x}{2}\right) - \tan^2\left(\frac{\pi}{4} - \frac{x}{2}\right) = 2 \) then the general solution of x =
(a) \( n\pi \pm \frac{\pi}{4}, \forall n \in Z \)
(b) \( n\pi + \frac{\pi}{4}, \forall n \in Z \)
(c) \( n\pi - \frac{\pi}{4}, \forall n \in Z \)
(d) \( n\pi, \forall n \in Z \)
Answer: (b) \( n\pi + \frac{\pi}{4}, \forall n \in Z \)

Question. If \( \tan A + \tan 2A + \sqrt{3} \tan A \tan 2A = \sqrt{3} \) then the general solution of \( \frac{A}{2} = \)
(a) \( \frac{n\pi}{3} + \frac{\pi}{9}, \forall n \in Z \)
(b) \( n\pi + \frac{\pi}{9}, \forall n \in Z \)
(c) \( \frac{n\pi}{6} + \frac{\pi}{18}, \forall n \in Z \)
(d) \( \frac{n\pi}{3} + \frac{\pi}{4}, \forall n \in Z \)
Answer: (c) \( \frac{n\pi}{6} + \frac{\pi}{18}, \forall n \in Z \)

Question. Solution of \( \tan x + \tan(120^\circ + x) + \tan(x - 120^\circ) = 0 \) is
(a) \( \frac{n\pi}{3}, \forall n \in Z \)
(b) \( \frac{n\pi}{6}, \forall n \in Z \)
(c) \( \frac{n\pi}{4}, \forall n \in Z \)
(d) \( \frac{n\pi}{2}, \forall n \in Z \)
Answer: (a) \( \frac{n\pi}{3}, \forall n \in Z \)

Question. If \( \tan p\theta = \cot q\theta \) then solutions of \( \theta \) are in -
(a) A.P
(b) G.P
(c) H.P
(d) A.G.P
Answer: (a) A.P

Question. If \( \sin x + \cos x = \sqrt{2} \cos \alpha \)
\( \implies \) x = ----
(a) \( 2n\pi - \frac{\pi}{4} \pm \alpha \)
(b) \( n\pi - \frac{\pi}{4} + \alpha \)
(c) \( 2n\pi + \frac{\pi}{4} \pm \alpha \)
(d) \( n\pi - \frac{\pi}{4} \pm \alpha \)
Answer: (c) \( 2n\pi + \frac{\pi}{4} \pm \alpha \)

Question. The solution of \( (\sin x - 2)^2 + (\cos x - 3/2)^2 = 0 \) is
(a) x = \( \pi/2 \)
(b) x = \( \pi/3 \)
(c) x = \( \pi/4 \)
(d) No solution
Answer: (d) No solution

Question. The values of \( \theta \) satisfying \( \sin 5\theta = \sin 3\theta - \sin \theta \) and \( 0 < \theta < \frac{\pi}{2} \) are
(a) \( \frac{\pi}{6}, \frac{\pi}{3} \)
(b) \( \frac{\pi}{6}, \frac{\pi}{4} \)
(c) \( \frac{\pi}{4}, \frac{\pi}{3} \)
(d) \( \frac{\pi}{4}, \frac{\pi}{2} \)
Answer: (a) \( \frac{\pi}{6}, \frac{\pi}{3} \)