Class 11 Mathematics Trigonometric Functions MCQs Set 11

Question. The acute angle of a rhombus whose side is a mean proportional between its diagonals is
(a) \( 15^{\circ} \)
(b) \( 20^{\circ} \)
(c) \( 30^{\circ} \)
(d) \( 80^{\circ} \)
Answer: C

Question. The minimum value of \( 3 \cos x + 4 \sin x + 8 \) is
(a) 5
(b) 9
(c) 2
(d) 3
Answer: D

Question. The maximum value of \( \sin(\theta + \pi/6) + \cos(\theta + \pi/6) \) is attained at
(a) \( \frac{\pi}{12} \)
(b) \( \frac{\pi}{6} \)
(c) \( \frac{\pi}{3} \)
(d) \( \frac{\pi}{2} \)
Answer: A

Question. Minimum value of \( 5 \sin^2 \theta + 4 \cos^2 \theta \) is -
(a) 1
(b) 2
(c) 3
(d) 4
Answer: D

Question. If \( x + \frac{1}{x} = 2 \cos \theta \), then \( x^3 + \frac{1}{x^3} = \)
(a) \( \cos 3\theta \)
(b) \( 2 \cos 3\theta \)
(c) \( \frac{1}{2} \cos 3\theta \)
(d) \( \frac{1}{3} \cos 3\theta \)
Answer: B

Question. If \( (\sec A - \tan A) (\sec B - \tan B) (\sec C - \tan C) = (\sec A + \tan A) (\sec B + \tan B) (\sec C + \tan C) \), then every side is equal to -
(a) \( \pm 1 \)
(b) 1
(c) \( - 1 \)
(d) None of these
Answer: A

Question. Let \( A_0 A_1 A_2 A_3 A_4 A_5 \) be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments \( A_0 A_1, A_0 A_2 \) and \( A_0 A_4 \) is
(a) 3/4
(b) \( 3\sqrt{3} \)
(c) 3
(d) \( \frac{3\sqrt{3}}{2} \)
Answer: C

Question. If \( A + B + C = \pi \), then \( \frac{\tan A + \tan B + \tan C}{\tan A \cdot \tan B \cdot \tan C} = \)
(a) 0
(b) 2
(c) 1
(d) –1
Answer: C

Question. If \( A + B + C = \pi \) and \( \cos A = \cos B \cos C \), then \( \tan B \tan C \) is equal to
(a) 1/2
(b) 2
(c) 1
(d) –1/2
Answer: B

Question. If \( A + B + C = \frac{3\pi}{2} \), then \( \cos 2A + \cos 2B + \cos 2C + 4 \sin A \sin B \sin C = \)
(a) 0
(b) 1
(c) 2
(d) 3
Answer: B

Question. Let \( A, B \) and \( C \) are the angles of a plain triangle and \( \tan \frac{A}{2} = \frac{1}{3}, \tan \frac{B}{2} = \frac{2}{3} \). Then \( \tan \frac{C}{2} \) is equal to
(a) 7/9
(b) 2/9
(c) 1/3
(d) 2/3
Answer: A

Question. If \( f(\theta) = 5 \cos \theta + 3 \cos \left(\theta + \frac{\pi}{3}\right) + 3 \), then range of \( f(\theta) \) is-
(a) [–5, 11]
(b) [–3, 9]
(c) [–2, 10]
(d) [–4, 10]
Answer: D

Question. If \( A + B + C = \pi \), then \( \sum \tan \frac{A}{2} \tan \frac{B}{2} \) is equal to -
(a) 0
(b) –1
(c) 1/2
(d) 1
Answer: D

Question. Maximum value of \( \sin \theta + \cos \beta \) is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: C

Question. The maximum value of the expression \( \frac{12}{9 + 3 \cos x + 4 \sin x} \) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: C

Question. If \( \tan A + \tan B + \tan C = \tan A \cdot \tan B \cdot \tan C \) then
(a) \( A, B, C \) must be angles of a triangle
(b) the sum of any two of \( A, B, C \) is equal to the third
(c) \( A + B + C \) must be an integral multiple of \( \pi \)
(d) None of these
Answer: C

Question. Find the minimum value of the expression \( \frac{1}{5 + 4 \cos x} \)
(a) 1/6
(b) 1/7
(c) 1/8
(d) 1/9
Answer: D

Question. If \( A + B + C = 180^{\circ} \), then the value of \( (\cot B + \cot C) (\cot C + \cot A) (\cot A + \cot B) \) will be
(a) \( \sec A \sec B \sec C \)
(b) \( \csc A \csc B \csc C \)
(c) \( \tan A \tan B \tan C \)
(d) 1
Answer: B

Question. The greatest value of \( \sin^4 \theta + \cos^4 \theta \) is
(a) 1/2
(b) 1
(c) 2
(d) 3
Answer: B

Question. If \( \sin \theta_1 + \sin \theta_2 + \sin \theta_3 = 3 \), then \( \cos \theta_1 + \cos \theta_2 + \cos \theta_3 \) is equal to
(a) 3
(b) 2
(c) 1
(d) 0
Answer: D

Question. If \( \csc \theta - \cot \theta = q \), then the value of \( \csc \theta \) is
(a) \( q + \frac{1}{q} \)
(b) \( q - \frac{1}{q} \)
(c) \( \frac{1}{2} \left( q + \frac{1}{q} \right) \)
(d) None of these
Answer: C

Question. If \( \sin x + \cos x = \frac{\sqrt{7}}{2} \) where \( x \in [0, \pi/4] \), then \( \tan \frac{x}{2} \) is equal to
(a) \( \frac{3 - \sqrt{7}}{3} \)
(b) \( \frac{\sqrt{7} - 2}{3} \)
(c) \( \frac{4 - \sqrt{7}}{4} \)
(d) none of these
Answer: B

Question. If \( \sin x + \csc x = 2 \), then \( \sin^n x + \csc^n x \) is equal to
(a) 2
(b) \( 2^n \)
(c) \( 2^{n-1} \)
(d) \( 2^{n-2} \)
Answer: A

Question. If \( f(x) = 3 \left[ \sin^4 \left(\frac{3\pi}{2} - x\right) + \sin^4 (3\pi + x) \right] - 2 \left[ \sin^6 \left(\frac{\pi}{2} + x\right) + \sin^6 (5\pi - x) \right] \) then, for all permissible values of \(x, f(x)\) is
(a) – 1
(b) 0
(c) 1
(d) not a constant function
Answer: C

Question. The difference \( (\sin^8 75^{\circ} - \cos^8 75^{\circ}) \) is equal to
(a) 1
(b) \( \frac{3\sqrt{3}}{8} \)
(c) \( \frac{3\sqrt{3}}{16} \)
(d) \( \frac{7\sqrt{3}}{16} \)
Answer: C

Question. If \( \tan \theta + \sin \theta = m \) and \( \tan \theta - \sin \theta = n \), then
(a) \( m^2 - n^2 = 4mn \)
(b) \( m^2 + n^2 = 4mn \)
(c) \( m^2 - n^2 = m^2 + n^2 \)
(d) \( m^2 - n^2 = 4\sqrt{mn} \)
Answer: D

Question. If \( \tan \theta = - \frac{4}{3} \), then \( \sin \theta \) is
(a) \( - \frac{4}{5} \) but not \( \frac{4}{5} \)
(b) \( - \frac{4}{5} \) or \( \frac{4}{5} \)
(c) \( \frac{4}{5} \) but not \( - \frac{4}{5} \)
(d) None of these
Answer: B

Question. If \( \alpha + \beta + \gamma = 2\pi \), then
(a) \( \tan \frac{\alpha}{2} + \tan \frac{\beta}{2} + \tan \frac{\gamma}{2} = \tan \frac{\alpha}{2} \tan \frac{\beta}{2} \tan \frac{\gamma}{2} \)
(b) \( \tan \frac{\alpha}{2} \tan \frac{\beta}{2} + \tan \frac{\beta}{2} \tan \frac{\gamma}{2} + \tan \frac{\gamma}{2} \tan \frac{\alpha}{2} = 1 \)
(c) \( \tan \frac{\alpha}{2} + \tan \frac{\beta}{2} + \tan \frac{\gamma}{2} + \tan \frac{\alpha}{2} \tan \frac{\beta}{2} \tan \frac{\gamma}{2} = 1 \)
(d) \( \tan \frac{\alpha}{2} \tan \frac{\beta}{2} + \tan \frac{\beta}{2} \tan \frac{\gamma}{2} + \tan \frac{\gamma}{2} \tan \frac{\alpha}{2} = - \tan \frac{\alpha}{2} \tan \frac{\beta}{2} \tan \frac{\gamma}{2} \)
Answer: A

Question. Given \( A = \sin^2 \theta + \cos^4 \theta \), then for all real \( \theta \),
(a) \( 1 \le A \le 2 \)
(b) \( 3/4 \le A \le 1 \)
(c) \( 13/16 \le A \le 1 \)
(d) \( 3/4 \le A \le 13/16 \)
Answer: B

Question. The value of \( \left(1 + \cos \frac{\pi}{8}\right) \left(1 + \cos \frac{3\pi}{8}\right) \left(1 + \cos \frac{5\pi}{8}\right) \left(1 + \cos \frac{7\pi}{8}\right) \)
(a) 1/4
(b) 3/4
(c) 1/8
(d) 3/8
Answer: C