Class 11 Mathematics Trigonometric Equations MCQs Set 08

SINGLE ANSWER QUESTIONS

Question. If \( \cos(A+B+C) = \cos A \cos B \cos C \), then
\( \frac{8\sin(B+C)\sin(C+A)\sin(A+B)}{\sin 2A \sin 2B \sin 2C} = \)
(a) \( 1 \)
(b) \( -1 \)
(c) \( \frac{1}{2} \)
(d) \( -\frac{1}{2} \)
Answer: (b) \( -1 \)

Question. \( \sin^4 \frac{\pi}{16} + \sin^4 \frac{3\pi}{16} + \sin^4 \frac{5\pi}{16} + \sin^4 \frac{7\pi}{16} = \)
(a) \( 1 \)
(b) \( 1/2 \)
(c) \( 3/2 \)
(d) \( 2 \)
Answer: (c) \( 3/2 \)

Question. \( a^2 + b^2 + 2ab \cos \theta = 1 \), \( c^2 + d^2 + 2cd \cos \theta = 1 \) and \( ac + bd + (ad + bc)\cos \theta = 0 \) then \( a^2 + c^2 = \)
(a) \( \sec^2 \theta \)
(b) \( \operatorname{cosec}^2 \theta \)
(c) \( \cot^2 \theta \)
(d) \( \tan^2 \theta \)
Answer: (b) \( \operatorname{cosec}^2 \theta \)

Question. If \( a \sin x + b \cos (x+\theta) + b \cos (x-\theta) = d \) then the minimum value of \( |\cos \theta| \) is equal to (where \( x \) is variable)
(a) \( \frac{1}{2|b|} \sqrt{d^2 - a^2} \)
(b) \( \frac{1}{2|a|} \sqrt{d^2 - a^2} \)
(c) \( \frac{1}{2|d|} \sqrt{d^2 - a^2} \)
(d) \( \frac{1}{|d|} \sqrt{d^2 - a^2} \)
Answer: (a) \( \frac{1}{2|b|} \sqrt{d^2 - a^2} \)

Question. \( \cos^6 \frac{\pi}{16} + \cos^6 \frac{3\pi}{16} + \cos^6 \frac{5\pi}{16} + \cos^6 \frac{7\pi}{16} = \)
(a) \( 0 \)
(b) \( 1/4 \)
(c) \( 3/4 \)
(d) \( 5/4 \)
Answer: (d) \( 5/4 \)

Question. If the equation \( \cot^4 x - 2 \operatorname{cosec}^2 x + a^2 = 0 \) has at least one solution, then the sum of all possible integral values of 'a' is equal to
(a) \( 4 \)
(b) \( 3 \)
(c) \( 2 \)
(d) \( 0 \)
Answer: (d) \( 0 \)

Question. The value of \( \sum_{n=1}^{\infty} \frac{\tan \left(\frac{\theta}{2^n}\right)}{2^{n-1} \cos \frac{\theta}{2^{n-1}}} \) is
(a) \( \frac{2}{\sin 2\theta} - \frac{1}{\theta} \)
(b) \( \frac{2}{\sin 2\theta} + \frac{1}{\theta} \)
(c) \( \frac{1}{\sin 2\theta} - \frac{1}{\theta} \)
(d) \( \frac{1}{\sin \theta} - \frac{1}{\theta} \)
Answer: (a) \( \frac{2}{\sin 2\theta} - \frac{1}{\theta} \)

Question. The sum of the series,
\( \sin \theta \sec 3\theta + \sin 3\theta \sec 3^2 \theta + \sin 3^2 \theta \sec 3^3 \theta + \dots \) upto \( n \) terms, is
(a) \( \frac{1}{2} \left[ \tan 3^n \theta - \tan 3^{n-1} \theta \right] \)
(b) \( \left[ \tan 3^n \theta - \tan \theta \right] \)
(c) \( \frac{1}{2} \left[ \tan 3^n \theta - \tan \theta \right] \)
(d) \( \frac{1}{2} \left( \tan 3^n \theta - 1 \right) \)
Answer: (c) \( \frac{1}{2} \left[ \tan 3^n \theta - \tan \theta \right] \)

Question. If \( \frac{\sec^8 \theta}{a} + \frac{\tan^8 \theta}{b} = \frac{1}{a+b} \) then
(a) \( ab \leq 0 \)
(b) \( ab \geq 0 \)
(c) \( a+b=0 \)
(d) \( ab = 0 \)
Answer: (a) \( ab \leq 0 \)

Question. If the mapping \( f(x) = ax + b, a < 0 \) maps [-1,1] onto [0,2] then for all values of \( \theta, A = \cos^2 \theta + \sin^4 \theta \) is such that
(a) \( f\left(\frac{1}{4}\right) \leq A \leq f(0) \)
(b) \( f(0) \leq A \leq f(-1) \)
(c) \( f\left(\frac{1}{3}\right) \leq A \leq f(0) \)
(d) \( f(1) < A \leq f(-1) \)
Answer: (a) \( f\left(\frac{1}{4}\right) \leq A \leq f(0) \)

Question. If \( x \sin a + y \sin 2a + z \sin 3a = \sin 4a \)
\( x \sin b + y \sin 2b + z \sin 3b = \sin 4b \),
\( x \sin c + y \sin 2c + z \sin 3c = \sin 4c \).
Then the roots of the equation
\( t^3 - \left(\frac{z}{2}\right)t^2 - \left(\frac{y+z}{4}\right)t + \left(\frac{z-x}{8}\right) = 0, a,b,c \neq n\pi \), are
(a) \( \sin a, \sin b, \sin c \)
(b) \( \cos a, \cos b, \cos c \)
(c) \( \sin 2a, \sin 2b, \sin 2c \)
(d) \( \cos 2a, \cos 2b, \cos 2c \)
Answer: (b) \( \cos a, \cos b, \cos c \)

Question. If \( \alpha, \beta, \gamma \) are acute angles and
\( \cos \theta = \sin \beta / \sin \alpha, \cos \phi = \sin \gamma / \sin \alpha \) and
\( \cos (\theta - \phi) = \sin \beta \sin \gamma \) then the value of
\( \tan^2 \alpha - \tan^2 \beta - \tan^2 \gamma \) is equal to
(a) \( -1 \)
(b) \( 0 \)
(c) \( 1 \)
(d) \( 2 \)
Answer: (b) \( 0 \)

Question. The value of \( \sum_{k=1}^{100} \sin(kx) \cos(101 - k)x \) is equal to
(a) \( \frac{101}{2} \sin(101x) \)
(b) \( 99\sin(101x) \)
(c) \( 50\sin(101x) \)
(d) \( 100\sin(101x) \)
Answer: (c) \( 50\sin(101x) \)

MULTIPLE ANSWERS QUESTIONS

Question. \( x = \sqrt{a^2 \cos^2 \alpha + b^2 \sin^2 \alpha} + \sqrt{a^2 \sin^2 \alpha + b^2 \cos^2 \alpha} \),
then \( x^2 = a^2 + b^2 + 2\sqrt{p(a^2+b^2) - p^2} \),
where \( p \) is equal to
(a) \( a^2 \cos^2 \alpha + b^2 \sin^2 \alpha \)
(b) \( a^2 \sin^2 \alpha + b^2 \cos^2 \alpha \)
(c) \( \frac{1}{2} \left[ a^2 + b^2 + (a^2 - b^2)\cos 2\alpha \right] \)
(d) \( \frac{1}{2} \left[ a^2 + b^2 - (a^2 - b^2)\cos 2\alpha \right] \)
Answer: (a), (b), (c), (d)

Question. If \( (x-a) \cos \theta + y \sin \theta = (x-a) \cos \phi + y \sin \phi = a \)
and \( \tan(\theta/2) - \tan(\phi/2) = 2b \) then
(a) \( y^2 = 2ax - (1-b^2)x^2 \)
(b) \( \tan \frac{\theta}{2} = \frac{1}{x} (y + bx) \)
(c) \( y^2 = 2bx - (1-a^2)x^2 \)
(d) \( \tan \frac{\phi}{2} = \frac{1}{x} (y - bx) \)
Answer: (a), (b), (d)

Question. If \( \cos x - \sin \alpha \cot \beta \sin x = \cos \alpha \) then the value of \( \tan(x/2) \) is
(a) \( -\tan(\alpha/2)\cot(\beta/2) \)
(b) \( \tan(\alpha/2)\tan(\beta/2) \)
(c) \( -\cot(\alpha/2)\tan(\beta/2) \)
(d) \( \cot(\alpha/2)\cot(\beta/2) \)
Answer: (a), (b)

Question. If \( \sin x + \cos x + \tan x + \cot x + \sec x + \operatorname{cosec} x = 7 \) and \( \sin 2x = a - b \sqrt{7} \) then
(a) \( a = 8 \)
(b) \( b = 22 \)
(c) \( a = 22 \)
(d) \( b = 8 \)
Answer: (c), (d)

Question. The expression \( (a \tan g + b \cot g)(a \cot g + b \tan g) - 4ab \cot^2 2g \) is
(a) independent of a, b
(b) independent of g
(c) dependent on g
(d) dependent on a, b
Answer: (b), (d)

Question. The equation \( x^3 - \frac{3}{4}x = - \frac{\sqrt{3}}{8} \) is satisfied by
(a) \( x = \cos\left(\frac{5\pi}{18}\right) \)
(b) \( x = \cos\left(\frac{7\pi}{18}\right) \)
(c) \( x = \cos\left(\frac{23\pi}{18}\right) \)
(d) \( x = -\sin\left(\frac{7\pi}{9}\right) \)
Answer: (a), (b), (c), (d)

COMPREHENSION QUESTIONS

Passage:
\( \alpha, \beta, \gamma, \delta \) are angles in I, II, III and IV quadrant respectively and no one of them is an integral multiple of \( \pi / 2 \). They form an increasing arithmetic progression.

Question. Which of the following holds
(a) \( \cos(\alpha + \delta) > 0 \)
(b) \( \cos(\alpha + \delta) = 0 \)
(c) \( \cos(\alpha + \delta) < 0 \)
(d) \( \cos(\alpha + \delta) > 0 \) or \( \cos(\alpha + \delta) < 0 \)
Answer: (a) \( \cos(\alpha + \delta) > 0 \)

Question. Which of the following does not hold
(a) \( \sin(\beta + \gamma) = \sin(\alpha + \delta) \)
(b) \( \sin(\beta - \gamma) = \sin(\alpha - \delta) \)
(c) \( \tan 2(\alpha - \beta) = \tan(\beta - \delta) \)
(d) \( \cos(\alpha + \gamma) = \cos 2\beta \)
Answer: (a), (c), (d)

Question. If \( \alpha + \beta + \gamma + \delta = \theta \) and \( \alpha = 70^\circ \),
(a) \( 400^\circ < \theta < 580^\circ \)
(b) \( 470^\circ < \theta < 650^\circ \)
(c) \( 680^\circ < \theta < 860^\circ \)
(d) \( 540^\circ < \theta < 900^\circ \)
Answer: (c) \( 680^\circ < \theta < 860^\circ \)

Passage:
A line OA of length \( r \) starts from its initial position OX and traces an angle AOB=\( \alpha \) in the anticlockwise direction. It then traces back in the clockwise direction an angle BOC=\( 3\theta \) (where \( \alpha > 3\theta \)). L is the foot of the perpendicular from C on OA.
\( \frac{\sin^3 \theta}{CL} = \frac{\cos^3 \theta}{OL} = 1 \)

Question. \( \frac{1 - r\cos\alpha}{r\sin\alpha} \) is equal to
(a) \( \tan 2\theta \)
(b) \( \cot 2\theta \)
(c) \( \sin 2\theta \)
(d) \( \cos 2\theta \)
Answer: (a) \( \tan 2\theta \)

Question. \( \frac{2r\sin\alpha}{1 + 2r\cos\alpha} \) is equal to
(a) \( \tan^2 \theta \)
(b) \( \cot^2 \theta \)
(c) \( \cot 2\theta \)
(d) \( \tan 2\theta \)
Answer: (d) \( \tan 2\theta \)

Question. \( \frac{2r^2 - 1}{r} \) is equal to
(a) \( \sin \alpha \)
(b) \( \cos \alpha \)
(c) \( \sin \theta \)
(d) \( \cos \theta \)
Answer: (b) \( \cos \alpha \)

MATRIX MATCHING QUESTIONS

Question. Let \( f_n(\theta) = \frac{\cos \frac{\theta}{2} + \cos 2\theta + \cos \frac{7\theta}{2} + \dots + \cos(3n-2)\frac{\theta}{2}}{\sin \frac{\theta}{2} + \sin 2\theta + \sin \frac{7\theta}{2} + \dots + \sin(3n-2)\frac{\theta}{2}} \).
Then match the entries of column-I with their corresponding values given in column-II.
                       Column-I                                        Column-II
(A) \( f_3\left(\frac{3\pi}{16}\right) \)                 (P) \( 2 - \sqrt{3} \)
(B) \( f_5\left(\frac{\pi}{28}\right) \)                   (Q) \( \sqrt{3+2\sqrt{2}} \)
(C) \( f_7\left(\frac{\pi}{60}\right) \)                   (R) \( \sqrt{2} - 1 \)
                                                                         (S) \( \sqrt{7+4\sqrt{3}} \)
Answer: (A) R; (B) Q; (C) S

ASSERTION-REASONING QUESTIONS

This section contains two questions. Each question contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct.
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(b) Statement-1 is True, Statement-2 is True; Statement-2 NOT a correct explanation for Statement-1.
(c) Statement-1 is True, Statement-2 is False
(d) Statement -1 is False, Statement-2 is True.

Question. Let \( \alpha, \beta \) and \( \gamma \) satisfy \( 0 < \alpha < \beta < \gamma < 2\pi \) and \( \cos(x+\alpha) + \cos(x+\beta) + \cos(x+\gamma) = 0 \forall x \in R \)
Statement 1: \( \gamma - \alpha = \frac{2\pi}{3} \)
Statement 2: \( \cos \alpha + \cos \beta + \cos \gamma = 0 \) and \( \sin \alpha + \sin \beta + \sin \gamma = 0 \)
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(b) Statement-1 is True, Statement-2 is True; Statement-2 NOT a correct explanation for Statement-1
(c) Statement-1 is True, Statement-2 is False
(d) Statement -1 is False, Statement-2 is True
Answer: (d) Statement -1 is False, Statement-2 is True

Question. Statement 1: In any triangle ABC
\( \ln\left(\cot \frac{A}{2} + \cot \frac{B}{2} + \cot \frac{C}{2}\right) = \ln \cot \frac{A}{2} + \ln \cot \frac{B}{2} + \ln \cot \frac{C}{2} \)
Statement 2: \( \ln(1+\sqrt{3}+(2+\sqrt{3})) = \ln 1 + \ln \sqrt{3} + \ln(2+\sqrt{3}) \)
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(b) Statement-1 is True, Statement-2 is True; Statement-2 NOT a correct explanation for Statement-1
(c) Statement-1 is True, Statement-2 is False
(d) Statement -1 is False, Statement-2 is True
Answer: (b) Statement-1 is True, Statement-2 is True; Statement-2 NOT a correct explanation for Statement-1

Question. Let \( 0 < a, b, g < \frac{\pi}{2} \)
Statement-1: If \( \tan^3 a \), \( \tan^3 b \), \( \tan^3 g \) are the roots of the cubic equation \( x^3 - 6x^2 + kx - 8 = 0 \), then \( \tan a = \tan b = \tan g \). because
Statement-2: If \( a^3 + b^3 + c^3 = 3abc \) and a, b, c are positive numbers then \( a = b = c \).
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(b) Statement-1 is True, Statement-2 is True; Statement-2 NOT a correct explanation for Statement-1
(c) Statement-1 is True, Statement-2 is False
(d) Statement -1 is False, Statement-2 is True
Answer: (a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1