Class 11 Mathematics Trigonometric Equations MCQs Set 02

Question. If \(\cos 2\theta \cdot \cos 3\theta \cdot \cos \theta = \frac{1}{4}\) for \(0 < \theta < \pi\), then \(\theta =\)
(a) \(\frac{\pi}{7}, \frac{5\pi}{7}, \pi\)
(b) \(\frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{\pi}{2}\)
(c) \(\frac{\pi}{8}, \frac{3\pi}{8}, \frac{5\pi}{8}, \frac{7\pi}{8}, \frac{\pi}{3}, \frac{2\pi}{3}\)
(d) \(\frac{\pi}{9}, \frac{4\pi}{9}, \frac{5\pi}{9}\)
Answer: (c) \(\frac{\pi}{8}, \frac{3\pi}{8}, \frac{5\pi}{8}, \frac{7\pi}{8}, \frac{\pi}{3}, \frac{2\pi}{3}\)

Question. If \(\alpha\) and \(\beta\) are two different solutions lying between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) of the equation \(2\tan \theta + \sec \theta = 2\) then \(\tan \alpha + \tan \beta\) is
(a) 0
(b) 1
(c) 4/3
(d) 8/3
Answer: (d) 8/3

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

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) \(n\pi \pm \frac{\pi}{3}\)
(b) \(2n\pi \pm \frac{2\pi}{3}\)
(c) \(n\pi \pm \pi\)
(d) \(2n\pi \pm \frac{\pi}{2}\)
Answer: (b) \(2n\pi \pm \frac{2\pi}{3}\)

Question. If \(\cos 20^\circ = k\) and \(\cos x = 2k^2 - 1\), then the possible values of \(x\) between \(0^\circ\) and \(360^\circ\) are
(a) \(140^\circ\)
(b) \(40^\circ\) and \(140^\circ\)
(c) \(40^\circ\) and \(320^\circ\)
(d) \(50^\circ\) and \(130^\circ\)
Answer: (c) \(40^\circ\) and \(320^\circ\)

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

Question. In a right angled triangle, the hypotenuse is \(2\sqrt{2}\) times the length of the perpendicular drawn from the opposite vertex to the hypotenuse. Then the other two angles are
(a) \(\frac{\pi}{3}, \frac{\pi}{6}\)
(b) \(\frac{\pi}{4}, \frac{\pi}{4}\)
(c) \(\frac{3\pi}{8}, \frac{\pi}{8}\)
(d) \(\frac{5\pi}{12}, \frac{\pi}{12}\)
Answer: (c) \(\frac{3\pi}{8}, \frac{\pi}{8}\)

Question. Number of solutions of the equation \(\tan x + \sec x = 2\cos x\) in the interval \([0, 2\pi]\) is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (c) 2

Question. The number of integral values of \(k\) for which the equation \(7\cos x + 5\sin x = 2k + 1\) has a solution is
(a) 4
(b) 8
(c) 10
(d) 12
Answer: (b) 8

Question. The number of distinct real roots of \(\begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} = 0\) in the interval \(-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}\) is
(a) 0
(b) 2
(c) 1
(d) 3
Answer: (c) 1

Question. If the solutions for \(\theta\) of \(\cos p\theta + \cos q\theta = 0\), \(p > q > 0\) are in A.P then numerically smallest common difference of A.P is
(a) \(\frac{\pi}{p+q}\)
(b) \(\frac{2\pi}{p+q}\)
(c) \(\frac{\pi}{2(p+q)}\)
(d) \(\frac{1}{p+q}\)
Answer: (b) \(\frac{2\pi}{p+q}\)

Question. The solution set of \(\tan(4k+2)x - \tan(4k+1)x - \tan(4k+2)x\tan(4k+1)x = 1\), \(k \in Z\) is
(a) \(\phi\)
(b) \(\frac{\pi}{4}\)
(c) \(\left\{ n\pi + \frac{\pi}{4} : n \in Z \right\}\)
(d) \(\left\{ 2n\pi + \frac{\pi}{4} : n \in Z \right\}\)
Answer: (a) \(\phi\)

Question. The general solution of the equation \(\sin^2 \theta \sec \theta + \sqrt{3}\tan \theta = 0\) is
(a) \(\theta = n\pi + (-1)^n \frac{\pi}{3}, \forall n \in Z\)
(b) \(\theta = n\pi, \forall n \in Z\)
(c) \(\theta = n\pi + (-1)^n \frac{\pi}{4}, \forall n \in Z\)
(d) \(\theta = \frac{n\pi}{6}, \forall n \in Z\)
Answer: (b) \(\theta = n\pi, \forall n \in Z\)

Question. \(\cos^3 \alpha + \cos^3(120^\circ + \alpha) + \cos^3(120^\circ - \alpha) = \frac{3\sqrt{3}}{4}\) then the general solution of \(\alpha\) is
(a) \(\phi\)
(b) \(2n\pi \pm \frac{\pi}{3}, \forall n \in Z\)
(c) \((2n+1)\frac{\pi}{2}, \forall n \in Z\)
(d) \(n\pi, \forall n \in Z\)
Answer: (a) \(\phi\)

Question. The general solution of \(\sin^{100} x - \cos^{100} x = 1\) is
(a) \(2n\pi + \frac{\pi}{3}, n \in I\)
(b) \(n\pi + \frac{\pi}{4}, n \in I\)
(c) \(n\pi + \frac{\pi}{2}, n \in I\)
(d) \(2n\pi - \frac{\pi}{3}, n \in I\)
Answer: (c) \(n\pi + \frac{\pi}{2}, n \in I\)

Question. The equation \(4\sin^2 x + 4\sin x + a^2 - 3 = 0\) has a solution if
(a) \(-2 \leq a \leq 2\)
(b) \(-1 \leq a \leq 1\)
(c) \(-3 \leq a \leq 3\)
(d) \(-4 \leq a \leq 4\)
Answer: (a) \(-2 \leq a \leq 2\)

Question. If the complex numbers \(\sin x + i\cos 2x\) and \(\cos x - i\sin 2x\) are conjugate to each other, then the set of values of \(x =\)
(a) \(n\pi\)
(b) \((2n+1)\frac{\pi}{2}\)
(c) \(\{0\}\)
(d) \(\phi\)
Answer: (d) \(\phi\)

Question. Given that \(\tan A\), \(\tan B\) are the roots of the equation \(x^2 - bx + c = 0\), the value of \(\sin^2(A + B)\) is
(a) \(\frac{b}{(b+c)^2}\)
(b) \(\frac{b^2}{b^2+c^2}\)
(c) \(\frac{b^2}{c^2+(1-b^2)}\)
(d) \(\frac{b^2}{b^2+(1-c)^2}\)
Answer: (d) \(\frac{b^2}{b^2+(1-c)^2}\)

Question. The smallest positive \(x\) satisfying \(\log_{\cos x} \sin x + \log_{\sin x} \cos x = 2\) is
(a) \(\frac{\pi}{2}\)
(b) \(\frac{\pi}{3}\)
(c) \(\frac{\pi}{4}\)
(d) \(\frac{\pi}{6}\)
Answer: (c) \(\frac{\pi}{4}\)

Question. The number of solutions of the equation \(2^{\cos x} = |\sin x|\) in \([-2\pi, 2\pi]\)
(a) 1
(b) 4
(c) 3
(d) 8
Answer: (d) 8

Question. If the equation \(k\cos x - 3\sin x = k + 1\) has a solution for '\(x\)' then
(a) \(K \leq 4\)
(b) \(K \geq 4\)
(c) \(1 \leq k \leq 6\)
(d) \(0 \leq k \leq 8\)
Answer: (a) \(K \leq 4\)

Question. If \( \cos 6\theta + \cos 4\theta + \cos 2\theta + 1 = 0 \) for \( 0 \le \theta \le \pi \), then \( \theta = \)
(a) \( \frac{\pi}{7}, \frac{5\pi}{7}, \pi \)
(b) \( \frac{\pi}{2}, \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{6} \)
(c) \( \frac{\pi}{8}, \frac{3\pi}{8}, \frac{5\pi}{8}, \frac{7\pi}{8}, \frac{\pi}{3}, \frac{2\pi}{3} \)
(d) \( \frac{\pi}{9}, \frac{4\pi}{9}, \frac{5\pi}{9} \)
Answer: (b) \( \frac{\pi}{2}, \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{6} \)

Question. If \( \alpha, \beta, \gamma, \delta \) are the four solutions of the equation \( \tan\left(\theta + \frac{\pi}{4}\right) = 3\tan 3\theta \). No two of which have equal tangents, then the value of \( \tan\alpha + \tan\beta + \tan\gamma + \tan\delta = \)
(a) 1
(b) 0
(c) -1
(d) 4
Answer: (b) 0

Question. If \( 0 < x, y < \frac{\pi}{2} \) then the system of equations \( \sin x \cdot \sin y = 3/4 \) and \( \tan x \cdot \tan y = 3 \) has a solution at
(a) \( x = \frac{\pi}{6}, y = \frac{\pi}{6} \)
(b) \( x = \frac{\pi}{3}, y = \frac{\pi}{3} \)
(c) \( x = \frac{\pi}{12}, y = \frac{\pi}{12} \)
(d) \( x = \frac{\pi}{4}, y = \frac{\pi}{4} \)
Answer: (b) \( x = \frac{\pi}{3}, y = \frac{\pi}{3} \)

Question. The solution set of \( \sin\left(x+\frac{\pi}{4}\right) = \sin 2x \)
(a) \( \frac{n\pi - \left(\frac{\pi}{4}\right)}{1 + (-1)^n 2} \)
(b) \( \frac{n\pi + \left(\frac{\pi}{4}\right)}{1 + (-1)^n 2} \)
(c) \( \frac{n\pi + \left(\frac{\pi}{4}\right)}{1 - (-1)^n 2} \)
(d) \( \frac{n\pi - \left(\frac{\pi}{4}\right)}{1 - (-1)^n 2} \)
Answer: (d) \( \frac{n\pi - \left(\frac{\pi}{4}\right)}{1 - (-1)^n 2} \)

Question. In a \( \Delta ABC \), the angle A is greater than angle B. If the values of angles A and B satisfy the equation \( 3\sin x - 4\sin^3 x - k = 0, 0 < k < 1 \), then the measure of angle C =
(a) \( \frac{\pi}{3} \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{2\pi}{3} \)
(d) \( \frac{5\pi}{6} \)
Answer: (c) \( \frac{2\pi}{3} \)

Question. The equation \( 8\cos x \cdot \cos 2x \cdot \cos 4x = \frac{\sin 6x}{\sin x} \) has a solution if
(a) \( \sin x = 0 \)
(b) \( \cos 7x = 0 \)
(c) \( \sin 7x = 0 \)
(d) \( \sin 8x = 0 \)
Answer: (b) \( \cos 7x = 0 \)

Question. If \( x \in (-\pi, \pi) \) such that \( y = 1 + |\cos x| + |\cos^2 x| + |\cos^3 x| + ........ \) and \( 8^y = 64 \), then no.of values of \( x \) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (d) 4

Question. If a is any real number, the number of roots of \( \cot x - \tan x = a \) in the first quadrant is
(a) 2
(b) 0
(c) 1
(d) 4
Answer: (c) 1

Question. The number of roots of the equation \( x^3 + x^2 + 2x + \sin x = 0 \) in \( (-2\pi, 2\pi) \)
(a) 4
(b) 3
(c) 2
(d) 1
Answer: (d) 1

Question. If \( \begin{vmatrix} \cos(A+B) & -\sin(A+B) & \cos 2B \\ \sin A & \cos A & \sin B \\ -\cos A & \sin A & \cos B \end{vmatrix} = 0 \) then B =
(a) \( (2n+1)\frac{\pi}{2} \)
(b) \( n\pi \)
(c) \( (2n+1)\pi \)
(d) \( 2n\pi \)
Answer: (a) \( (2n+1)\frac{\pi}{2} \)

Question. If \( \frac{1}{6} \sin x \), \( \cos x \), \( \tan x \) are in G.P. then \( x = \)
(a) \( n\pi \pm \frac{\pi}{3}, n \in Z \)
(b) \( 2n\pi \pm \frac{\pi}{3}, n \in Z \)
(c) \( n\pi \pm (-1)^n \frac{\pi}{3}, n \in Z \)
(d) \( n\pi \pm \frac{\pi}{6}, n \in Z \)
Answer: (b) \( 2n\pi \pm \frac{\pi}{3}, n \in Z \)

Question. The sum of the solutions of the equation \( \tan x \cdot \tan 4x = 1 \) for \( 0 < x < \pi \) is
(a) \( 10\pi \)
(b) \( \frac{3\pi}{2} \)
(c) \( \frac{5\pi}{2} \)
(d) \( 2\pi \)
Answer: (d) \( 2\pi \)

Question. If \( \tan^2 2\theta = \cot^2 \alpha \) then the general solution is
(a) \( \theta = \left\{\frac{1}{4}\right\}\left[n\pi \pm \left(\frac{\pi}{2} - \alpha\right)\right], \forall n \in Z \)
(b) \( \theta = \left\{\frac{1}{2}\right\}\left[n\pi \pm \left(\frac{\pi}{4} - \alpha\right)\right], \forall n \in Z \)
(c) \( \theta = \left\{\frac{1}{2}\right\}\left[n\pi \pm \left(\frac{\pi}{2} - \alpha\right)\right], \forall n \in Z \)
(d) \( \theta = \left\{\frac{1}{4}\right\}\left[n\pi \pm \left(\frac{\pi}{2} + \alpha\right)\right], \forall n \in Z \)
Answer: (c) \( \theta = \left\{\frac{1}{2}\right\}\left[n\pi \pm \left(\frac{\pi}{2} - \alpha\right)\right], \forall n \in Z \)

Question. If \( \sin x + \cos x = 1 + \sin x \cdot \cos x \) then \( x = \) ---
(a) \( n\pi + \frac{\pi}{3} \)
(b) \( n\pi + (-1)^n \frac{\pi}{6} \)
(c) \( n\pi + (-1)^n \frac{\pi}{2} \cup 2n\pi \)
(d) \( n\pi \)
Answer: (c) \( n\pi + (-1)^n \frac{\pi}{2} \cup 2n\pi \)

Question. The inequation \( 3^{\sin^2 \theta} + 3^{\cos^2 \theta} \ge 2\sqrt{3} \) is true
(a) for all real values of \( \theta \)
(b) for some real values of \( \theta \)
(c) for imaginary values of \( \theta \)
(d) for no value of \( \theta \)
Answer: (a) for all real values of \( \theta \)

Question. The set of values of a for which the equation \( \sin^4 x + \cos^4 x = a \) has a solution is
(a) \( \left[ -\frac{1}{2}, \frac{1}{2} \right] \)
(b) \( \left[ \frac{1}{4}, 1 \right] \)
(c) \( \left[ \frac{1}{2}, 1 \right] \)
(d) \( \left[ \frac{1}{4}, \frac{1}{2} \right] \)
Answer: (c) \( \left[ \frac{1}{2}, 1 \right] \)

Question. If \( \sin 2x \cdot \cos 2x \cdot \cos 4x = \gamma \) has a solution then \( \gamma \) lies in the interval
(a) \( \left[ -\frac{1}{2}, \frac{1}{2} \right] \)
(b) \( \left[ -\frac{1}{4}, \frac{1}{4} \right] \)
(c) \( \left[ -\frac{1}{3}, \frac{1}{3} \right] \)
(d) \( [-1, 1] \)
Answer: (b) \( \left[ -\frac{1}{4}, \frac{1}{4} \right] \)

Question. If \( \alpha \) and \( \beta \) satisfying the equation \( \sin \alpha + \sin \beta = \sqrt{3}(\cos \alpha - \cos \beta) \), then
(a) \( \sin 3\alpha + \sin 3\beta = 1 \)
(b) \( \sin 3\alpha + \sin 3\beta = 0 \)
(c) \( \sin 3\alpha - \sin 3\beta = 0 \)
(d) \( \sin 3\alpha - \sin 3\beta = 1 \)
Answer: (b) \( \sin 3\alpha + \sin 3\beta = 0 \)

Question. If \( \frac{3 + 2i\sin \theta}{1 - 2i\sin \theta} \) is a real number and \( 0 < \theta < 2\pi \), then \( \theta = \)
(a) \( \pi \)
(b) \( \pi / 2 \)
(c) \( \pi / 3 \)
(d) \( \pi / 6 \)
Answer: (a) \( \pi \)