Class 11 Mathematics Trigonometric Functions MCQs Set 10

Question. \( \frac{\sin \theta + \sin 2\theta}{1 + \cos \theta + \cos 2\theta} = \)
(a) \( \frac{1}{2} \tan \theta \)
(b) \( \frac{1}{2} \cot \theta \)
(c) \( \tan \theta \)
(d) \( \cot \theta \)
Answer: C

Question. The sign of the product \( \sin 2 \sin 3 \sin 5 \) is -
(a) Negative
(b) Positive
(c) 0
(d) None of these
Answer: A

Question. \( \cos \theta \cdot \cos \left( \frac{\pi}{3} + \theta \right) \cdot \cos \left( \frac{\pi}{3} - \theta \right) \) is equal to
(a) \( \cos 2\theta \)
(b) \( \cos 3\theta \)
(c) \( \cos^3 \theta \)
(d) None of these
Answer: B

Question. If the perimeter of the sector of a circle is m times the radius of the circle, then the angle subtended by the sector at the center of the circle is
(a) \( m^c \)
(b) \( (m - 2)\pi^c \)
(c) \( (m - 1)^c \)
(d) \( (m - 2)^c \)
Answer: D

Question. Which of the following number(s) is rational-
(a) \( \sin 15^{\circ} \)
(b) \( \cos 15^{\circ} \)
(c) \( \sin 15^{\circ} \cos 15^{\circ} \)
(d) \( \sin 15^{\circ} \cos 75^{\circ} \)
Answer: C

Question. If a semi perimeter of a circle of radius r equals perimeter of a sector of the same circle subtending and angle \( \theta \) at the center then,
(a) \( \theta = (\pi + 2)^c \)
(b) area of the mentioned sector \( = (\pi - 2)r^2 \)
(c) length of the corresponding arc \( = (\pi - 2)r \)
(d) all of these
Answer: C

Question. Which of the following is not equal to \( \frac{\cot \theta + \csc \theta - 1}{\cot \theta - \csc \theta + 1} \)?
(a) \( \csc \theta + \cot \theta \)
(b) \( \frac{\csc \theta + 1}{\cot \theta} \)
(c) \( \frac{\sin \theta}{1 - \cos \theta} \)
(d) \( \frac{1 + \cos \theta}{\sin \theta} \)
Answer: B

Question. The value of \( \tan^2 30^{\circ} + 4 \sin^2 45^{\circ} + \frac{1}{3} \cos^2 30^{\circ} \) is
(a) \( 2 \frac{7}{12} \)
(b) \( 1 \frac{5}{12} \)
(c) \( - 2 \frac{5}{12} \)
(d) \( - 1 \frac{5}{12} \)
Answer: A

Question. Which of the following statements is not correct?
(a) \( \cos^4 \theta - \sin^4 \theta = \cos^2 \theta - \sin^2 \theta \)
(b) \( 1 + \tan^2 \theta = \sec^2 \theta \)
(c) \( \sin 40^{\circ} + \cos 50^{\circ} = 2 \sin 40 \)
(d) \( \sin^2 \theta + \cos^2 \theta = 2 \)
Answer: D

Question. The value of \( \frac{\tan^2 60^{\circ} - 2 \tan^2 45^{\circ} + \sec^2 30^{\circ}}{3 \sin^2 45^{\circ} \sin 90^{\circ} + \cos^2 60^{\circ} \cos 0^{\circ}} \)
(a) \( 49/12 \)
(b) \( 7/3 \)
(c) \( 14/9 \)
(d) \( 4/3 \)
Answer: D

Question. If \( \tan \theta = 4 \), then \( \left( \frac{\tan \theta}{\frac{\sin^3 \theta}{\cos \theta} + \sin \theta \cos \theta} \right) \) is equal to
(a) 0
(b) \( 2\sqrt{2} \)
(c) \( \sqrt{2} \)
(d) 1
Answer: D

Question. \( \sec^2 \theta + \csc^2 \theta \) is equal to
(a) \( \sec^2 \theta \cot^2 \theta \)
(b) \( \sec^2 \theta \tan^2 \theta \)
(c) \( \csc^2 \theta \cot^2 \theta \)
(d) \( \sec^2 \theta \csc^2 \theta \)
Answer: D

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

Question. If \( \sin \beta = \frac{12}{13} \), then the value of \( \frac{13 \sin \beta + 5 \sec \beta}{5 \tan \beta + 6 \csc \beta} \) is
(a) 0
(b) 1
(c) \( 50/37 \)
(d) \( 37/2 \)
Answer: C

Question. The value of the expression \( (\sin x + \csc x)^2 + (\cos x + \sec x)^2 - (\tan x + \cot x)^2 \), wherever defined, is equal to
(a) 0
(b) 5
(c) 7
(d) 9
Answer: C

Question. Each of the four statements given below are either True or False.
I. \( \sin 765^{\circ} = - \frac{1}{\sqrt{2}} \)
II. \( \csc(-1410^{\circ}) = 2 \)
III. \( \tan \frac{13\pi}{3} = \frac{1}{\sqrt{3}} \)
IV. \( \cot \left( - \frac{15\pi}{4} \right) = - 1 \)
Indicate the correct order of sequence, where 'T' stands for true and 'F' stands for false.
(a) F T F T
(b) F F T T
(c) T F F F
(d) F T F F
Answer: D

Question. The two legs of a right triangle are \( \sin \theta + \sin \left( \frac{3\pi}{2} - \theta \right) \) and \( \cos \theta - \cos \left( \frac{3\pi}{2} - \theta \right) \). The length of its hypotenuse is
(a) 1
(b) 2
(c) \( \sqrt{2} \)
(d) some function of \( \theta \)
Answer: C

Question. Suppose \( \sin \theta - \cos \theta = 1 \) then the value of \( \sin^3 \theta - \cos^3 \theta \) is (\( \theta \in R \))
(a) 1
(b) - 2
(c) - 1
(d) 0
Answer: A

Question. If \( \sin \theta + \csc \theta = 2 \), then value of \( \sin^3 \theta + \csc^3 \theta \) is
(a) 8
(b) -1
(c) 2
(d) 0
Answer: C

Question. The value of the expression \( \sqrt{3} \csc 20^{\circ} - \sec 20^{\circ} \) is equal to
(a) 2
(b) \( 2 \sin 20^{\circ} / \sin 40^{\circ} \)
(c) 4
(d) \( 4 \sin 20^{\circ} / \sin 40^{\circ} \)
Answer: C

Question. \( 3 (\sin x - \cos x)^4 + 6 (\sin x + \cos x)^2 + 4 (\sin^6 x + \cos^6 x) \) is equal to
(a) 11
(b) 12
(c) 13
(d) 14
Answer: C

Question. If \( K = \sin (\pi / 18) \sin (5\pi / 18) \sin (7\pi / 18) \), then the numerical value of \(K\) is __________.
(a) 1/8
(b) 1/6
(c) 1/3
(d) 3/7
Answer: A

Question. Let \( 0 < x < \frac{\pi}{4} \), then \( (\sec 2x - \tan 2x) \) equals
(a) \( \tan \left(x - \frac{\pi}{4}\right) \)
(b) \( \tan \left(\frac{\pi}{4} - x\right) \)
(c) \( \tan \left(x + \frac{\pi}{4}\right) \)
(d) \( \tan^2 \left(x + \frac{\pi}{4}\right) \)
Answer: B

Question. The expression \( \frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A} \) can be written as
(a) \( \sec A \csc A + 1 \)
(b) \( \tan A + \cot A \)
(c) \( \sec A + \csc A \)
(d) \( \sin A \cos A + 1 \)
Answer: A

Question. A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O on the ground is \( 45^{\circ} \). It flies off horizontally straight away from the point O. After one second, the elevation of the bird from O is reduced to \( 30^{\circ} \). Then the speed (in m/s) of the bird is
(a) \( 40(\sqrt{2} - 1) \)
(b) \( 40(\sqrt{3} - \sqrt{2}) \)
(c) \( 20\sqrt{2} \)
(d) \( 20(\sqrt{3} - 1) \)
Answer: D

Question. Let \( f_k(x) = \frac{1}{k} (\sin^k x + \cos^k x) \) where \( x \in \mathbb{R} \) and \( k \ge 1 \). Then \( f_4(x) - f_6(x) \) equals
(a) 1/6
(b) 1/3
(c) 1/4
(d) 1/12
Answer: D

Question. The value of \( \sin \frac{\pi}{14} \sin \frac{3\pi}{14} \sin \frac{5\pi}{14} \sin \frac{7\pi}{14} \sin \frac{9\pi}{14} \sin \frac{11\pi}{14} \sin \frac{13\pi}{14} \) is equal to _________
Answer: 1/64

Question. If \( A > 0, B > 0 \) and \( A + B = \pi / 3 \), then the maximum value of \( \tan A \tan B \) is ____________.
Answer: 1/3

Question. \( \sec^2 \theta = \frac{4xy}{(x + y)^2} \) is true if and only if
(a) \( x + y \neq 0 \)
(b) \( x = y, x \neq 0 \)
(c) \( x = y \)
(d) \( x \neq 0, y \neq 0 \)
Answer: B

Question. Let \( f(\theta) = \sin \theta (\sin \theta + \sin 3\theta) \). Then \( f(\theta) \) is
(a) \( \ge 0 \) only when \( \theta \ge 0 \)
(b) \( \le 0 \) for all real \( \theta \)
(c) \( \ge 0 \) for all real \( \theta \)
(d) \( \le 0 \) only when \( \theta \le 0 \)
Answer: C

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

Question. If \( \alpha + \beta = \pi/2 \) and \( \beta + \gamma = a \), then \( \tan \alpha \) equals
(a) \( 2(\tan \beta + \tan \gamma) \)
(b) \( \tan \beta + \tan \gamma \)
(c) \( \tan \beta + 2 \tan \gamma \)
(d) \( 2 \tan \beta + \tan \gamma \)
Answer: C

Question. The minimum value of the expression \( \sin \alpha + \sin \beta + \sin \gamma \), where \( \alpha, \beta, \gamma \) are real numbers satisfying \( \alpha + \beta + \gamma = \pi \) is
(a) Positive
(b) zero
(c) negative
(d) -3
Answer: B

Question. Which of the following number(s) is/are rational?
(a) \( \sin 15^{\circ} \)
(b) \( \cos 15^{\circ} \)
(c) \( \sin 15^{\circ} \cos 15^{\circ} \)
(d) \( \sin 15^{\circ} \cos 75^{\circ} \)
Answer: C

Question. For a positive integer \(n\), let \( f_n(\theta) = \left( \tan \frac{\theta}{2} \right) (1 + \sec \theta) (1 + \sec 2\theta) (1 + \sec 4\theta) \dots (1 + \sec 2^n \theta) \). Then
(a) \( f_2\left(\frac{\pi}{16}\right) = 1 \)
(b) \( f_3\left(\frac{\pi}{32}\right) = 1 \)
(c) \( f_4\left(\frac{\pi}{64}\right) = 1 \)
(d) \( f_5\left(\frac{\pi}{128}\right) = 1 \)
Answer: A, B, C, D

SUBJECTIVE

Question. If \( \tan a = \frac{m}{m + 1} \) and \( \tan \beta = \frac{1}{2m + 1} \), find the possible values of \( (\alpha + \beta) \).
Answer: \( \pi/4, 5\pi/4 \)

Question. If \( \cos(\alpha + \beta) = 4/5 \), \( \sin(\alpha - \beta) = \frac{5}{13} \), and \( \alpha, \beta \) lie between 0 and \( \pi / 4 \), find \( \tan 2\alpha \).
Answer: 56/33

Question. (a) The maximum value of the expression \( \frac{1}{\sin^2 \theta + 3 \sin \theta \cos \theta + 5 \cos^2 \theta} \) is
(b) Two parallel chords of a circle of radius 2 are at a distance \( \sqrt{3} + 1 \) apart. If the chords subtend at the center, angles of \( \frac{\pi}{k} \) and \( \frac{2\pi}{k} \) where \( k > 0 \), then the value of [k] is [Note : [k] denotes the largest integer less than or equal to k]
Answer: (a) 2 (b) 3

Question. The positive integer value of \( n > 3 \) satisfying the equation \( \frac{1}{\sin \left(\frac{\pi}{n}\right)} = \frac{1}{\sin \left(\frac{2\pi}{n}\right)} + \frac{1}{\sin \left(\frac{3\pi}{n}\right)} \) is
Answer: 7

Question. If \( \cos x + \cos y + \cos z = 0 = \sin x + \sin y + \sin z \), then possible value of \( \cos \frac{x - y}{2} \) is
Answer: \( \pm 1/2 \)