Class 11 Mathematics Trigonometric Equations MCQs Set 07

SINGLE ANSWER QUESTIONS

Question. For any real \( \theta \), the maximum value of \( \cos^2(\cos \theta) + \sin^2(\sin \theta) \) is
(a) 1
(b) \( 1+\sin^2 1 \)
(c) \( 1+\cos^2 1 \)
(d) does not exist
Answer: (b) \( 1+\sin^2 1 \)

Question. The value of \( \cot \frac{7\pi}{16} + 2\cot \frac{3\pi}{8} + \cot \frac{15\pi}{16} \) is
(a) 4
(b) 2
(c) -2
(d) -4
Answer: (d) -4

Question. If \( 0 < \alpha < \frac{\pi}{6} \) then \( \alpha(\operatorname{cosec} \alpha) \) is
(a) less than \( \frac{\pi}{6} \)
(b) greater than \( \frac{\pi}{6} \)
(c) less than \( \frac{\pi}{3} \)
(d) greater than \( \frac{\pi}{3} \)
Answer: (c) less than \( \frac{\pi}{3} \)

Question. Let \( P = \{\theta : \sin \theta - \cos \theta = \sqrt{2}\cos \theta\} \) and \( Q = \{\theta : \sin \theta + \cos \theta = \sqrt{2}\sin \theta\} \) be two sets. Then
(a) P \( \subset \) Q and P \( \neq \emptyset \)
(b) Q \( \not\subset \) P
(c) P \( \not\subset \) Q
(d) P = Q
Answer: (d) P = Q

Question. The maximum value of \( (\cos \alpha_1)(\cos \alpha_2) \dots (\cos \alpha_n) \) under the restrictions \( 0 \leq \alpha_1, \alpha_2, \dots, \alpha_n \leq \frac{\pi}{2} \) and \( \cot \alpha_1 \cot \alpha_2 \dots \cot \alpha_n = 1 \) is
(a) \( \frac{1}{2^{n/2}} \)
(b) \( \frac{1}{2^n} \)
(c) \( \frac{1}{2n} \)
(d) 1
Answer: (a) \( \frac{1}{2^{n/2}} \)

Question. If \( \tan \theta - \cot \theta = a \) and \( \cos \theta - \sin \theta = b \) then \( (a^2+4)(b^2-1)^2 \)
(a) 1
(b) 3
(c) 4
(d) 5
Answer: (c) 4

Question. The maximum value of the expression \( \left|\sqrt{(\sin^2 x + 2a^2)} - \sqrt{(2a^2 - 1 - \cos^2 x)}\right| \) where a and x are real numbers is
(a) \( \sqrt{3} \)
(b) \( \sqrt{2} \)
(c) 1
(d) \( \sqrt{5} \)
Answer: (b) \( \sqrt{2} \)

MULTIPLE ANSWER QUESTIONS

Question. If the equation \( \sin x(\sin x + \cos x) = k \) has real solutions then k may lie in the interval
(a) \( \left[0, \frac{\sqrt{2}+1}{2}\right] \)
(b) \( [2-\sqrt{3}, 2+\sqrt{3}] \)
(c) \( [0, 2-\sqrt{3}] \)
(d) \( \left[\frac{1-\sqrt{2}}{2}, \frac{1+\sqrt{2}}{2}\right] \)
Answer: (a) \( \left[0, \frac{\sqrt{2}+1}{2}\right] \), (c) \( [0, 2-\sqrt{3}] \), (d) \( \left[\frac{1-\sqrt{2}}{2}, \frac{1+\sqrt{2}}{2}\right] \)

Question. If \( \cos x + \cos y + \cos z = 0 = \sin x + \sin y + \sin z \) then the possible value of \( \cos\left(\frac{x-y}{2}\right) = \)
(a) \( \frac{1}{2} \)
(b) \( -\frac{1}{2} \)
(c) 1
(d) -1
Answer: (a) \( \frac{1}{2} \), (b) \( -\frac{1}{2} \)

Question. If \( x = \sin(\alpha-\beta)\sin(\gamma-\delta) \), \( y = \sin(\beta-\gamma)\sin(\alpha-\delta) \) and \( z = \sin(\gamma-\alpha)\sin(\beta-\delta) \) then
(a) \( x+y+z = 0 \)
(b) \( x+y-z = 0 \)
(c) \( y+z-x = 0 \)
(d) \( x^3+y^3+z^3 = 3xyz \)
Answer: (a) \( x+y+z = 0 \), (d) \( x^3+y^3+z^3 = 3xyz \)

Question. If \( \frac{\tan 3A}{\tan A} = k \) (\( k \neq 1 \)) then
(a) \( \frac{\cos A}{\cos 3A} = \frac{k^2-1}{2k} \)
(b) \( \frac{\sin 3A}{\sin A} = \frac{2k}{k-1} \)
(c) \( k < \frac{1}{3} \)
(d) \( k > 3 \)
Answer: (b) \( \frac{\sin 3A}{\sin A} = \frac{2k}{k-1} \), (c) \( k < \frac{1}{3} \), (d) \( k > 3 \)

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

Question. For \( 0 < \phi < \pi/2 \), if \( x = \sum_{n=0}^{\infty} \cos^{2n} \phi \), \( y = \sum_{n=0}^{\infty} \sin^{2n} \phi \) and \( z = \sum_{n=0}^{\infty} \cos^{2n} \phi \sin^{2n} \phi \) then xyz =
(a) xy + z
(b) xz + y
(c) x + y + z
(d) yz + x
Answer: (a) xy + z, (c) x + y + z

Question. Let \( f(x) = a_1\cos(\alpha_1+x) + a_2\cos(\alpha_2+x) + \dots + a_n\cos(\alpha_n+x) \). If \( f(x) \) vanishes for \( x = 0 \) and \( x = x_1 \) (where \( x_1 \neq k\pi, k \in \mathbb{Z} \)) then
(a) \( a_1\cos\alpha_1 + a_2\cos\alpha_2 + \dots + a_n\cos\alpha_n = 0 \)
(b) \( a_1\sin\alpha_1 + a_2\sin\alpha_2 + \dots + a_n\sin\alpha_n = 0 \)
(c) \( f(x) = 0 \) has only two solutions \( 0, x_1 \)
(d) \( f(x) \) is identically zero \( \forall x \)
Answer: (a) \( a_1\cos\alpha_1 + a_2\cos\alpha_2 + \dots + a_n\cos\alpha_n = 0 \), (b) \( a_1\sin\alpha_1 + a_2\sin\alpha_2 + \dots + a_n\sin\alpha_n = 0 \), (d) \( f(x) \) is identically zero \( \forall x \)

Question. For \( \alpha = \pi/7 \) which of the following hold(s) good?
(a) \( \tan \alpha \tan 2\alpha \tan 3\alpha = \tan 3\alpha - \tan 2\alpha - \tan \alpha \)
(b) \( \operatorname{cosec} \alpha = \operatorname{cosec} 2\alpha + \operatorname{cosec} 4\alpha \)
(c) \( \cos \alpha - \cos 2\alpha + \cos 3\alpha = 1/2 \)
(d) \( 8\cos \alpha \cos 2\alpha \cos 4\alpha = 1 \)
Answer: (a) \( \tan \alpha \tan 2\alpha \tan 3\alpha = \tan 3\alpha - \tan 2\alpha - \tan \alpha \), (b) \( \operatorname{cosec} \alpha = \operatorname{cosec} 2\alpha + \operatorname{cosec} 4\alpha \), (c) \( \cos \alpha - \cos 2\alpha + \cos 3\alpha = 1/2 \)

Question. Which of the following is/are correct?
(a) \( (\tan x)^{\ln(\sin x)} > (\cot x)^{\ln(\sin x)} \forall x \in \left(0, \frac{\pi}{4}\right) \)
(b) \( 4^{\ln \operatorname{cosec} x} < 5^{\ln \operatorname{cosec} x} \forall x \in \left(0, \frac{\pi}{2}\right) \)
(c) \( \left(\frac{1}{2}\right)^{\ln(\cos x)} < \left(\frac{1}{3}\right)^{\ln(\cos x)} \forall x \in \left(0, \frac{\pi}{2}\right) \)
(d) \( 2^{\ln(\tan x)} > 2^{\ln(\sin x)} \forall x \in \left(0, \frac{\pi}{2}\right) \)
Answer: (a) \( (\tan x)^{\ln(\sin x)} > (\cot x)^{\ln(\sin x)} \forall x \in \left(0, \frac{\pi}{4}\right) \), (b) \( 4^{\ln \operatorname{cosec} x} < 5^{\ln \operatorname{cosec} x} \forall x \in \left(0, \frac{\pi}{2}\right) \), (c) \( \left(\frac{1}{2}\right)^{\ln(\cos x)} < \left(\frac{1}{3}\right)^{\ln(\cos x)} \forall x \in \left(0, \frac{\pi}{2}\right) \), (d) \( 2^{\ln(\tan x)} > 2^{\ln(\sin x)} \forall x \in \left(0, \frac{\pi}{2}\right) \)

COMPREHENSION TYPE

Given, \( \cos 2^m \theta \cos 2^{m+1} \theta \dots \cos 2^n \theta = \frac{\sin 2^{n+1}\theta}{2^{n-m+1}\sin 2^m\theta} \)
where \( 2^m\theta \neq k\pi, n,m,k \in I \) solve the following

Question. \( \sin \frac{9\pi}{14} \sin \frac{11\pi}{14} \sin \frac{13\pi}{14} \) is equal to
(a) 1/64
(b) -1/64
(c) 1/8
(d) -1/8
Answer: (c) 1/8

Question. \( \cos 2^3\frac{\pi}{10} \cos 2^4\frac{\pi}{10} \cos 2^5\frac{\pi}{10} \dots \cos 2^{10}\frac{\pi}{10} \) is equal to
(a) 1/128
(b) 1/256
(c) \( \frac{1}{512}\sin \frac{\pi}{10} \)
(d) \( \frac{\sqrt{5}-1}{512}\sin \frac{3\pi}{10} \)
Answer: (b) 1/256

Question. \( \cos \frac{\pi}{11} \cos \frac{2\pi}{11} \cos \frac{3\pi}{11} \dots \cos \frac{10\pi}{11} \) is equal to
(a) -1/32
(b) 1/512
(c) 1/1024
(d) -1/1024
Answer: (d) -1/1024

In a \( \Delta ABC \), if \( \cos A \cos B \cos C = \frac{\sqrt{3}-1}{8} \)
and \( \sin A \sin B \sin C = \frac{3+\sqrt{3}}{8} \) then
On the basis of above information, answer the following questions:

Question. The value of tan A + tan B + tan C is
(a) \( \frac{3+\sqrt{3}}{\sqrt{3}-1} \)
(b) \( \frac{\sqrt{3}+4}{\sqrt{3}-1} \)
(c) \( \frac{6-\sqrt{3}}{\sqrt{3}-1} \)
(d) \( \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-1} \)
Answer: (a) \( \frac{3+\sqrt{3}}{\sqrt{3}-1} \)

Question. The value of tan A tan B + tan B tan C + tan C tan A is
(a) \( 5-4\sqrt{3} \)
(b) \( 5+4\sqrt{3} \)
(c) \( 6+\sqrt{3} \)
(d) \( 6-\sqrt{3} \)
Answer: (b) \( 5+4\sqrt{3} \)

Question. The angles of \( \Delta ABC \) are
(a) \( 45^\circ, 30^\circ, 105^\circ \)
(b) \( 45^\circ, 60^\circ, 75^\circ \)
(c) \( 45^\circ, 45^\circ, 90^\circ \)
(d) \( 45^\circ, 60^\circ, 70^\circ \)
Answer: (b) \( 45^\circ, 60^\circ, 75^\circ \)

If \( 7\theta = (2n+1)\pi \), when n = 0, 1, 2, 3, 4, 5, 6 then on the basis of above information, answer the following questions

Question. The equation whose roots are \( \cos \pi/7, \cos 3\pi/7, \cos 5\pi/7 \) is
(a) \( 8x^3 + 4x^2 + 4x + 1 = 0 \)
(b) \( 8x^3 - 4x^2 - 4x + 1 = 0 \)
(c) \( 8x^3 - 4x^2 - 4x - 1 = 0 \)
(d) \( 8x^3 + 4x^2 + 4x - 1 = 0 \)
Answer: (b) \( 8x^3 - 4x^2 - 4x + 1 = 0 \)

Question. The value of \( \sec \pi/7 + \sec 3\pi/7 + \sec 5\pi/7 \) is
(a) 4
(b) -4
(c) 3
(d) -3
Answer: (a) 4

Consider the function defined by
\( f(x) = \frac{\sin x}{\sqrt{1+\tan^2 x}} + \frac{\cos x}{\sqrt{1+\cot^2 x}} \)
for permissible values. Then

Question. Which of the following is true ?
(a) \( f(x) = \sin 2x \) for all x
(b) \( f(x) = \sin x|\cos x| + \cos x|\sin x| \) for all x
(c) \( f(x) = 0 \) for \( \frac{\pi}{2} < x < \pi \)
(d) \( f(x) = \sin x \) for all x
Answer: (c) \( f(x) = 0 \) for \( \frac{\pi}{2} < x < \pi \)

Question. \( f(x) = 0 \) (n is an integer) for
(a) \( 2n\pi < x < 2n\pi + \frac{\pi}{2} \)
(b) \( (2n+1)\frac{\pi}{2} < x < (2n+2)\frac{\pi}{2} \)
(c) \( (2n+1)\pi < x < (2n+2)\pi \)
(d) \( 0 < x < 2n\pi \)
Answer: (b) \( (2n+1)\frac{\pi}{2} < x < (2n+2)\frac{\pi}{2} \)

Question. \( f(x) < 0 \) for all x in (n is any integer)
(a) \( 2n\pi < x < 2n\pi + \frac{\pi}{2} \)
(b) \( 2n\pi < x < (2n+1)\pi \)
(c) \( (2n+1)\pi < x < (4n+3)\frac{\pi}{2} \)
(d) \( 0 < x < 2n\pi \)
Answer: (c) \( (2n+1)\pi < x < (4n+3)\frac{\pi}{2} \)

MATRIX MATCHING QUESTIONS

Question. Match the following
COLUMN-I
A) When \( \theta \) is fixed constant then the maximum value of \( \{\cos(2A+\theta) + \cos(2B+\theta)\} \)
B) The maximum value of \( \{\cos 2A + \cos 2B\} \) where (A+B) is constant and \( A, B \in (0, \pi/2) \), is
C) The minimum value of \( \{\sec 2A + \sec 2B\} \), where (A+B) is constant and \( A, B \in (0, \pi/4) \), is
D) The minimum value of \( \sqrt{\{\tan \theta + \cot \theta - 2\cos 2(A+B)\}} \), where A, B are constant and \( \theta \in (0, \pi/2) \) is
COLUMN-II
p) \( 2\sin(A+B) \)
q) 2 sec (A+B)
r) 2 cos (A+B)
s) 2 cos (A-B)
Answer: A -> s; B -> r; C -> q; D -> p

Question. Match the following:
COLUMN-I
A) If maximum and minimum values of \( \frac{7 + 6\tan \theta - \tan^2 \theta}{(1 + \tan^2 \theta)} \) for all real values of \( \theta \neq (2n+1)\frac{\pi}{2} \) are \( \lambda \) and \( \mu \) respectively then
B) If maximum and minimum values of \( 5\cos \theta + 3\cos\left(\theta + \frac{\pi}{3}\right) + 3 \) for all real values of \( \theta \) are \( \lambda \) and \( \mu \) respectively then
C) If maximum and minimum values of \( 1 + \sin\left(\frac{\pi}{4} + \theta\right) + 2\cos\left(\frac{\pi}{4} - \theta\right) \) for all real values of \( \theta \) are \( \lambda \) and \( \mu \) respectively then
COLUMN-II
p) \( \lambda + \mu = 2 \)
q) \( \lambda - \mu = 6 \)
r) \( \lambda + \mu = 6 \)
s) \( \lambda - \mu = 10 \)
t) \( \lambda - \mu = 14 \)
Answer: A -> r, s; B -> r, t; C -> p, q

Question. Match the following:
COLUMN-I
A) In an acute angled triangle ABC, the least values of \( \sum \sec A \) and \( \sum \tan^2 A \) are \( \lambda \) and \( \mu \) respectively then
B) In a triangle ABC, the least values of \( \sum \operatorname{cosec}\left(\frac{A}{2}\right) \) and \( \sum \sec^2\left(\frac{A}{2}\right) \) are \( \lambda \) and \( \mu \) respectively then
C) In a triangle ABC, the least values of \( \prod \operatorname{cosec}\left(\frac{A}{2}\right) \) and \( \sum \operatorname{cosec}^2 A \) are \( \lambda \) and \( \mu \) respectively then
COLUMN-II
p) \( \lambda - \mu = 2 \)
q) \( \mu - \lambda = 3 \)
r) \( \lambda - \mu = 4 \)
s) \( 3\lambda - 2\mu = 0 \)
t) \( 2\lambda - 3\mu = 0 \)
Answer: A -> q, s; B -> p, t; C -> r

ASSERTION-REASONING QUESTIONS

This section contains 2 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. STATEMENT – 1: The minimum value of the expression \( \sin \alpha + \sin \beta + \sin \gamma \) where \( \alpha, \beta, \gamma \) are real numbers such that \( \alpha + \beta + \gamma = \pi \), is negative because
STATEMENT – 2: \( \alpha, \beta, \gamma \) are angles of a triangle
(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: (c) Statement–1 is True, Statement–2 is False

Question. STATEMENT–1: The equation \( \sin^2 x + \cos^2 y = 2\sec^2 z \) is only solvable if \( \sin x = 1, \cos y = 1 \) and \( \sec z = 1 \) where x, y, z \( \in \mathbb{R} \) because
STATEMENT – 2: Maximum value of sin x and cos y is 1 and minimum value of sec z is 1.
(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: (c) Statement–1 is True, Statement–2 is False

Question. STATEMENT–1 : The maximum and minimum values of the function \( f(x) = \frac{1}{3\sin x + 4\cos x - 2} \) does not exist
STATEMENT – 2: The given fuction is an unbounded function.
(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