Class 11 Mathematics Trigonometric Ratios MCQs Set 03

Question. If \( \theta \) is not in 4th quadrant and \( \tan \theta = -4/3 \) then \( 5 \sin \theta + 10 \cos \theta + 9 \sec \theta + 16 \csc \theta + 4 \cot \theta = \)
(a) -1
(b) 2/5
(c) 4/5
(d) 0
Answer: (d) 0

Question. \( f(x) = x^3 - 3x + 5 \) then \( f\left(\sin \frac{3\pi}{2}\right) + f\left(\cos \frac{3\pi}{2}\right) = \)
(a) 10
(b) 12
(c) 14
(d) 16
Answer: (b) 12

Question. If \( \tan \theta, 2 \tan \theta + 2, 3 \tan \theta + 3 \) are in G.P then the value of \( \frac{7 - 5 \cot \theta}{9 - 4 \sqrt{\sec^2 \theta - 1}} \) is
(a) \( \frac{12}{5} \)
(b) \( \frac{-33}{28} \)
(c) \( \frac{33}{100} \)
(d) \( \frac{12}{13} \)
Answer: (b) \( \frac{-33}{28} \)

Question. In a triangle ABC, \( \text{C} = 90^{\circ} \), then the equation whose roots are \( \tan \text{A}, \tan \text{B} \) is
(a) \( ab x^2 + c^2 x + ab = 0 \)
(b) \( ab x^2 + c^2 x - ab = 0 \)
(c) \( ab x^2 - c^2 x - ab = 0 \)
(d) \( ab x^2 - c^2 x + ab = 0 \)
Answer: (d) \( ab x^2 - c^2 x + ab = 0 \)

Question. Which of the following is correct?
(a) \( \sin 1^{\circ} > \sin 1 \)
(b) \( \sin 1^{\circ} < \sin 1 \)
(c) \( \sin 1^{\circ} = \sin 1 \)
(d) \( \sin 1^{\circ} = \frac{\pi}{180} \sin 1 \)
Answer: (b) \( \sin 1^{\circ} < \sin 1 \)

Question. If \( a = \cos 3 \) and \( b = \sin 8 \) then
(a) \( a > 0, b > 0 \)
(b) \( ab < 0 \)
(c) \( a > b \)
(d) \( ab > 0 \)
Answer: (b) \( ab < 0 \)

Question. \( \csc \text{A} = 4p + \frac{1}{16p} \)
\( \implies \csc \text{A} + \cot \text{A} = \)
(a) \( 8p \)
(b) \( \frac{1}{8p} \)
(c) \( -8p \) (or) \( \frac{1}{8p} \)
(d) \( 8p \) (or) \( -\frac{1}{8p} \)
Answer: (d) \( 8p \) (or) \( -\frac{1}{8p} \)

Question. If \( \sqrt{\frac{1 - \sin \text{A}}{1 + \sin \text{A}}} = \sec \text{A} - \tan \text{A} \) then A lies in the quadrants
(a) I, II
(b) II, III
(c) I, IV
(d) I, III
Answer: (c) I, IV

Question. If \( \sin \theta + \cos \theta = m \) and \( \sec \theta + \csc \theta = n \), then \( n(m + 1)(m - 1) = \)
(a) m
(b) n
(c) 2m
(d) 2n
Answer: (c) 2m

Question. If \( 1 + \sin x + \sin^2 x + \dots \text{to } \infty = 4 + 2\sqrt{3}, 0 < x < \pi \) and \( x \neq \frac{\pi}{2} \) then \( x = \)
(a) \( 30^{\circ}, 60^{\circ} \)
(b) \( 60^{\circ}, 120^{\circ} \)
(c) \( 90^{\circ}, 120^{\circ} \)
(d) \( 30^{\circ}, 45^{\circ} \)
Answer: (b) \( 60^{\circ}, 120^{\circ} \)

Question. \( \cos^2 5^{\circ} + \cos^2 10^{\circ} + \cos^2 15^{\circ} + \dots + \cos^2 360^{\circ} = \)
(a) 18
(b) 27
(c) 36
(d) 45
Answer: (c) 36

Question. If \( x = a \cos^2 \theta \sin \theta \) and \( y = a \sin^2 \theta \cos \theta \), then \( \frac{(x^2 + y^2)^3}{x^2 y^2} = \)
(a) a
(b) \( a^3 \)
(c) \( a^2 \)
(d) \( a^5 \)
Answer: (c) \( a^2 \)

Question. \( a \sec \theta + b \tan \theta = 1, a \sec \theta - b \tan \theta = 5 \)
\( \implies a^2(b^2 + 4) = \)
(a) \( 3b^2 \)
(b) \( 9b^2 \)
(c) \( b^2 \)
(d) \( 4b^2 \)
Answer: (b) \( 9b^2 \)

Question. \( \tan \text{A} = a \tan \text{B}, \sin \text{A} = b \sin \text{B} \)
\( \implies \frac{b^2 - 1}{a^2 - 1} = \)
(a) \( \sin^2 \text{A} \)
(b) \( \sin^3 \text{A} \)
(c) \( \cos^2 \text{A} \)
(d) \( \cos^3 \text{A} \)
Answer: (c) \( \cos^2 \text{A} \)

Question. If \( \frac{\cos^2 \theta}{a} = \frac{\sin^2 \theta}{b} \) then \( \frac{\cos^4 \theta}{a} + \frac{\sin^4 \theta}{b} = \)
(a) \( \frac{1}{a + b} \)
(b) \( \frac{1}{(a + b)^2} \)
(c) \( \frac{1}{a^2} + \frac{1}{b^2} \)
(d) \( a + b \)
Answer: (a) \( \frac{1}{a + b} \)

Question. If \( a = x \cos^2 \text{A} + y \sin^2 \text{A} \) then \( (x - a)(y - a) + (x - y)^2 \sin^2 \text{A} \cos^2 \text{A} = \)
(a) 0
(b) 1
(c) \( xy + a^2 \)
(d) \( xy - a^2 \)
Answer: (a) 0

Question. If \( a \cos^3 \alpha + 3a \cos \alpha \sin^2 \alpha = m \) and \( a \sin^3 \alpha + 3a \cos^2 \alpha \sin \alpha = n \) then \( (m + n)^{2/3} + (m - n)^{2/3} = \)
(a) \( 2a^2 \)
(b) \( 2a^{1/3} \)
(c) \( 2a^{2/3} \)
(d) \( 2a^3 \)
Answer: (c) \( 2a^{2/3} \)

Question. \( \cos^4 \alpha - \sin^4 \alpha = a \) then \( \frac{1 - a}{1 + a} = \)
(a) \( \tan^2 \alpha \)
(b) \( \cot^2 \alpha \)
(c) \( -\tan^2 \alpha \)
(d) \( -\cot^2 \alpha \)
Answer: (a) \( \tan^2 \alpha \)

Question. \( a = \frac{1 + \sin x}{1 - \cos x + \sin x} \)
\( \implies \frac{1 + \cos x + \sin x}{2 \sin x} = \)
(a) a
(b) \( \frac{1}{a} \)
(c) \( a^2 \)
(d) \( \frac{1}{a^2} \)
Answer: (a) a

Question. Eliminate \( \theta \) from \( x = 1 + \tan \theta, y = 2 + \cot \theta \)
(a) \( xy + 1 = x + y \)
(b) \( xy + 2 = 2x + y \)
(c) \( xy + 1 = 2x + y \)
(d) \( xy + 1 = 2y + x \)
Answer: (c) \( xy + 1 = 2x + y \)

Question. \( f(x) = \sin^2 x + \csc^2 x \)
\( \implies \)
(a) \( f(x) < 1 \)
(b) \( f(x) = 1 \)
(c) \( 1 < f(x) < 2 \)
(d) \( f(x) \ge 2 \)
Answer: (d) \( f(x) \ge 2 \)

Question. Which of the following is not possible ?
(a) \( \sin \theta = \frac{5}{7} \)
(b) \( \cos \theta = \frac{1 + a^2}{1 - a^2}, |a| \ne 1 \)
(c) \( \tan \theta = 100 \)
(d) \( \sec \theta = \frac{5}{2} \)
Answer: (b) \( \cos \theta = \frac{1 + a^2}{1 - a^2}, |a| \ne 1 \)

Question. If \( \theta = \frac{11\pi}{6} \), then \( \cos \theta + \sin \theta = \)
(a) \( \frac{\sqrt{3} + 1}{\sqrt{2}} \)
(b) \( \frac{\sqrt{3} - 1}{\sqrt{2}} \)
(c) \( \frac{\sqrt{3} - 1}{2} \)
(d) \( \frac{\sqrt{3} + 1}{2} \)
Answer: (c) \( \frac{\sqrt{3} - 1}{2} \)

Question. The value of \( \sin \left(n\pi + (-1)^n \frac{\pi}{4}\right), n \in I \), is
(a) 0
(b) \( \frac{1}{\sqrt{2}} \)
(c) \( \frac{-1}{\sqrt{2}} \)
(d) \( \frac{\sqrt{3}}{2} \)
Answer: (b) \( \frac{1}{\sqrt{2}} \)

Question. If \( e^{\left(\sin^2 x + \sin^4 x + \sin^6 x + \dots \infty\right) \log 2} = 8 \) and \( 0 < x < \frac{\pi}{2} \) then \( \frac{\cos x}{\cos x + \sin x} = \)
(a) \( \frac{\sqrt{3} + 1}{2} \)
(b) \( \frac{\sqrt{3} - 1}{2} \)
(c) \( \frac{2}{\sqrt{3} + 1} \)
(d) \( \frac{2}{\sqrt{3} - 1} \)
Answer: (b) \( \frac{\sqrt{3} - 1}{2} \)

Question. If \( \cos x = \tan y, \cot y = \tan z \) and \( \cot z = \tan x \), then \( \sin x = \)
(a) \( \frac{\sqrt{5} + 1}{4} \)
(b) \( \frac{\sqrt{5} - 1}{4} \)
(c) \( \frac{\sqrt{5} + 1}{2} \)
(d) \( \frac{\sqrt{5} - 1}{2} \)
Answer: (d) \( \frac{\sqrt{5} - 1}{2} \)

Question. If \( \sin \theta + \cos \theta = p \) and \( \sin^3 \theta + \cos^3 \theta = q \), then \( p(p^2 - 3) = \)
(a) q
(b) 2q
(c) -q
(d) -2q
Answer: (d) -2q

Question. If \( 0 \le x \le \pi, 81^{\sin^2 x} + 81^{\cos^2 x} = 30 \) then \( x = \)
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{15} \)
(d) \( \frac{\pi}{8} \)
Answer: (a) \( \frac{\pi}{6} \)

Question. \( \csc^2 \alpha = \frac{4xy}{(x + y)^2} \quad [(x, y) \ne (0, 0)] \) is
(a) Possible for x = y
(b) Impossible for x = y
(c) Possible for x = - y
(d) Not possible
Answer: (a) Possible for x = y

Question. If \( a \sin \theta + b \cos \theta = c \) then \( \frac{a - b \tan \theta}{b + a \tan \theta} = \)
(a) \( \frac{\pm \sqrt{b^2 - c^2 + a^2}}{c} \)
(b) \( \frac{\pm \sqrt{b^2 + c^2 - a^2}}{c} \)
(c) \( \frac{\pm \sqrt{b^2 - c^2 - a^2}}{c} \)
(d) \( \frac{\pm \sqrt{a^2 + b^2 + c^2}}{b^2 - c^2} \)
Answer: (a) \( \frac{\pm \sqrt{b^2 - c^2 + a^2}}{c} \)

Question. If \( \cos \theta = \frac{3}{5} \) and \( \theta \) is not in the first quadrant, then \( \frac{5 \tan(\pi + \theta) + 4 \cos(\pi + \theta)}{5 \sec(2\pi - \theta) - 4 \cot(2\pi + \theta)} = \)
(a) \( \frac{4}{5} \)
(b) \( -\frac{4}{5} \)
(c) \( \frac{5}{4} \)
(d) \( -\frac{5}{4} \)
Answer: (b) \( -\frac{4}{5} \)

Question. \( f(x) = x^3 - 2x^2 + 3x - 5 \)
\( \implies f\left[\sin \left(\frac{5\pi}{2}\right)\right] + f\left[\sin \left(\frac{3\pi}{2}\right)\right] = \)
(a) 10
(b) -10
(c) 14
(d) -14
Answer: (d) -14

Question. \(\cos \text{A}, \sin \text{A}, \cot \text{A}\) are in GP then \( \tan^6 \text{A} - \tan^2 \text{A} = \)
(a) -1
(b) 0
(c) 1
(d) 2
Answer: (c) 1

Question. If ABCD is a cyclic quadrilateral such that \( 12 \tan A - 5 = 0 \) and \( 5 \cos B + 3 = 0 \), then \( \cos C \tan D = \)
(a) \( \frac{-16}{13} \)
(b) \( \frac{16}{13} \)
(c) \( \frac{-13}{16} \)
(d) \( \frac{23}{16} \)
Answer: (a) \( \frac{-16}{13} \)

Question. \( x = \cos 1^{\circ}, y = \cos 1 \)
\( \implies \)
(a) x = y
(b) x > y
(c) x < y
(d) 2x = y
Answer: (b) x > y

Question. \( a = \sec 2^{\circ}, b = \sec 2 \)
\( \implies \)
(a) a = b
(b) a < b
(c) b < a
(d) 2a = b
Answer: (c) b < a

Question. \( \tan \theta = P - \frac{1}{4p} \)
\( \implies \sec \theta - \tan \theta = \)
(a) \( 2p \) (or) \( \frac{1}{2p} \)
(b) \( \frac{1}{2p} \) (or) \( -2p \)
(c) \( -\frac{1}{2p} \) (or) \( 2p \)
(d) \( -\frac{1}{2p} \) (or) \( -2p \)
Answer: (b) \( \frac{1}{2p} \) (or) \( -2p \)

Question. If \( \sqrt{\frac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A \), then A lies in the quadrants
(a) I, II
(b) II, III
(c) I, III
(d) I, IV
Answer: (d) I, IV

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

Question. \( 1 + \cos x + \cos^2 x + \dots \text{to } \infty = 4 + 2\sqrt{3} \) then x =
(a) \( 30^{\circ} \)
(b) \( 60^{\circ} \)
(c) \( 45^{\circ} \)
(d) \( 90^{\circ} \)
Answer: (a) \( 30^{\circ} \)