Class 11 Mathematics Trigonometric Ratios MCQs Set 04

Question. \( \sin^2 5^{\circ} + \sin^2 10^{\circ} + \dots + \sin^2 180^{\circ} = \)
(a) 18
(b) 27
(c) 1
(d) 0
Answer: (a) 18

Question. If \( x = a \cos^3 \theta \sin^2 \theta, y = a \sin^3 \theta \cos^2 \theta \) and \( \frac{(x^2 + y^2)^p}{(xy)^q}, (p, q \in N) \) is independent of \( \theta \) then
(a) \( p + q = 6 \)
(b) \( 4p = 5q \)
(c) \( 4q = 5p \)
(d) \( pq = 16 \)
Answer: (b) \( 4p = 5q \)

Question. \( a = \sec \theta - \tan \theta, b = \csc \theta + \cot \theta \)
\( \implies a = \)
(a) \( \frac{b + 1}{b - 1} \)
(b) \( \frac{1 + b}{1 - b} \)
(c) \( \frac{b - 1}{b + 1} \)
(d) \( \frac{1 - b}{1 + b} \)
Answer: (c) \( \frac{b - 1}{b + 1} \)

Question. \( \cos A = a \cos B, \sin A = b \sin B \)
\( \implies (b^2 - a^2) \sin^2 B = \)
(a) \( 1 + a^2 \)
(b) \( 2 + a^2 \)
(c) \( 1 - a^2 \)
(d) \( 2 - a^2 \)
Answer: (c) \( 1 - a^2 \)

Question. If \( \frac{\sin x}{a} = \frac{\cos x}{b} = \frac{\tan x}{c} = k \) then \( bc + \frac{1}{ck} + \frac{ak}{1 + bk} \) is
(a) \( k \left(a + \frac{1}{a}\right) \)
(b) \( \frac{1}{k} \left(a + \frac{1}{a}\right) \)
(c) \( \frac{1}{k^2} \)
(d) \( \frac{a}{k} \)
Answer: (b) \( \frac{1}{k} \left(a + \frac{1}{a}\right) \)

Question. If \( m \cos^2 A + n \sin^2 A = p \), then \( \cot^2 A = \)
(a) \( \frac{p + n}{m + p} \)
(b) \( \frac{p - n}{p - m} \)
(c) \( \frac{p - n}{m - p} \)
(d) \( \frac{n + 1}{m + p} \)
Answer: (c) \( \frac{p - n}{m - p} \)

Question. If \( a \sin^3 x + b \cos^3 x = \sin x \cos x \) and \( a \sin x = b \cos x \) then \( a^2 + b^2 = \)
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (b) 1

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

Question. If \( \frac{\sin \theta - \cos \theta + 1}{\sin \theta + \cos \theta - 1} = \frac{x}{\tan \theta - \sec \theta + 1} \) then \( x = \)
(a) 0
(b) 2
(c) \( \tan \theta - \sec \theta + 1 \)
(d) \( \tan \theta + \sec \theta - 1 \)
Answer: (d) \( \tan \theta + \sec \theta - 1 \)

Question. If \( \tan \theta + \sin \theta = m, \tan \theta - \sin \theta = n \), then \( (m^2 - n^2)^2 = \)
(a) \( 16 mn \)
(b) \( 4 mn \)
(c) \( 32 mn \)
(d) \( 8 mn \)
Answer: (a) \( 16 mn \)

Question. If \( \sec \theta + \cos \theta = 2 \) then \( \sin^2 \theta + \tan^2 \theta = \)
(a) 0
(b) 1
(c) 2
(d) 4
Answer: (a) 0

Question. Which of the following possible?
(a) \( \sin \theta = 2 \)
(b) \( \cos \theta = \frac{17}{5} \)
(c) \( \tan \theta = 2 \)
(d) \( \sec \theta = \frac{4}{5} \)
Answer: (c) \( \tan \theta = 2 \)

Question. If \( \theta = \frac{\pi}{21} \), then \( \frac{\sin 23\theta - \sin 7\theta}{\sin 2\theta + \sin 14\theta} = \)
(a) 0
(b) 1
(c) -1
(d) 2
Answer: (c) -1

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

Question. If \( e^{(1 + \sin^2 x + \sin^4 x + \dots \infty) \log 2} = 16 \), then \( \tan^2 x = \)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. If \( 2 \sin x + 5 \cos y + 7 \sin z = 14 \) then \( 7 \tan \frac{x}{2} + 4 \cos y - 6 \cos z = \)
(a) 4
(b) -3
(c) 11
(d) 5
Answer: (c) 11

Question. If \( \sin \theta \) and \( \cos \theta \) are the roots of \( px^2 + qx + r = 0 \) then \( q^2 - p^2 = \)
(a) 0
(b) \( -2pr \)
(c) \( 2qr \)
(d) \( 2rp \)
Answer: (d) \( 2rp \)

Question. If \( 0 \le x \le \pi, 4^{\sin^2 x} + 4^{\cos^2 x} = 5 \), then x =
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{4} \)
(d) \( \frac{\pi}{2} \)
Answer: (d) \( \frac{\pi}{2} \)

Question. If \( \alpha, \beta \) are complementary angles, \( \sin \alpha = \frac{3}{5} \), then \( \cos \alpha \cos \beta - \sin \alpha \sin \beta = \)
(a) 1
(b) 2
(c) 4/5
(d) 0
Answer: (d) 0
\( \sin \alpha = \frac{3}{5}, \cos \beta = \frac{3}{5}, \quad \cos \alpha = \frac{4}{5}, \sin \beta = \frac{4}{5} \)

Question. \( a^{2} \cos^{2} \frac{2\pi}{3} - 4a^{2} \tan^{2} \frac{3\pi}{4} + 2a^{2} \sin^{2} \frac{2\pi}{3} = \)
(a) \( a^{2} \)
(b) 0
(c) \( \frac{3a^{2}}{4} \)
(d) \( \frac{-9a^{2}}{4} \)
Answer: (d) \( \frac{-9a^{2}}{4} \)

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

Question. If \( a\sec \theta + b\tan \theta = c \) then \( (a\tan \theta + b\sec \theta)^{2} = \)
(a) \( a^{2}+b^{2}+c^{2} \)
(b) \( -a^{2}+b^{2}+c^{2} \)
(c) \( a^{2}-b^{2}+c^{2} \)
(d) \( a^{2}+b^{2}-c^{2} \)
Answer: (b) \( -a^{2}+b^{2}+c^{2} \)

Question. If \( \alpha, \beta \) are complementary angles and \( \sin \alpha = \frac{3}{5} \), then \( \sin \alpha \cos \beta - \cos \alpha \sin \beta = \)
(a) \( \frac{7}{25} \)
(b) \( -\frac{7}{25} \)
(c) \( \frac{25}{7} \)
(d) \( -\frac{25}{7} \)
Answer: (b) \( -\frac{7}{25} \)

Question. \( 4a^2 \sin^2\left(\frac{3\pi}{4}\right) - 3[a \tan 225^\circ]^2 + [2a \cos 315^\circ]^2 = \)
(a) 0
(b) a
(c) \( \sqrt{2}a \)
(d) \( a^2 \)
Answer: (d) \( a^2 \)

Question. If \( \theta \) lies in the first quadrant and \( 5 \tan \theta = 4 \), then \( \frac{5 \sin \theta - 3 \cos \theta}{\sin \theta + 2 \cos \theta} = \)
(a) 5/14
(b) 3/14
(c) 1/14
(d) 0
Answer: (a) 5/14

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

Question. \( \tan^2 \alpha = 1 - p^2 \), then \( \sec \alpha + \tan^3 \alpha \csc \alpha = \)
(a) \( (2 + p^2)^{\frac{3}{2}} \)
(b) \( (1 + p^2)^{\frac{3}{2}} \)
(c) \( (2 - p^2)^{\frac{3}{2}} \)
(d) \( (1 - p^2)^{\frac{3}{2}} \)
Answer: (c) \( (2 - p^2)^{\frac{3}{2}} \)

Question. \( \frac{\pi}{2} < \alpha < \pi \)
\( \implies \) \( \sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}} + \sqrt{\frac{1 + \cos \alpha}{1 - \cos \alpha}} = \)
(a) \( 2\sec\alpha \)
(b) \( -2\sec\alpha \)
(c) \( 2\csc\alpha \)
(d) \( -2\csc\alpha \)
Answer: (c) \( 2\csc\alpha \)

Question. If \( \sin \beta \) is the G.M. between \( \sin\alpha \) and \( \cos\alpha \) then \( (\cos \alpha - \sin \alpha)^2 - 2\cos^2 \beta = \)
(a) 0
(b) 1
(c) 2
(d) -1
Answer: (d) -1

Question. If \( \sin^2 \theta = \frac{x^2 + y^2 + 1}{2x} \) then x must be
(a) -3
(b) -2
(c) 1
(d) 3
Answer: (c) 1

Question. \( \sin x + \sin^2 x + \sin^3 x = 1 \)
\( \implies \) \( \cos^6 x - 4\cos^4 x + 8\cos^2 x = \)
(a) 4
(b) 2
(c) 1
(d) 0
Answer: (a) 4

Question. \( \cos \theta + \cos^2 \theta = 1 \),
\( a\sin^{12} \theta + b\sin^{10} \theta + c\sin^8 \theta + d\sin^6 \theta = 1 \)
\( \implies \) \( \frac{b + c}{a + d} = \)
(a) 2
(b) 3
(c) 4
(d) 6
Answer: (b) 3

Question. \( a \sin x = b \cos x = \frac{2c \tan x}{1 - \tan^2 x} \) and \( (a^2 - b^2)^2 = kc^2 (a^2 + b^2) \)
\( \implies \) \( k = \)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (d) 4

Question. The greatest among \( (\sin1 + \cos1) \), \( (\sqrt{\sin1} + \sqrt{\cos1}) \), \( (\sin1 - \cos1) \) and 1 is
(a) \( \sin1 + \cos1 \)
(b) \( \sqrt{\sin1} + \sqrt{\cos1} \)
(c) \( \sin1 - \cos1 \)
(d) 1
Answer: (b) \( \sqrt{\sin1} + \sqrt{\cos1} \)

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

Question. If \( \tan^2 \alpha \tan^2 \beta + \tan^2 \beta \tan^2 \gamma + \tan^2 \gamma \tan^2 \alpha + 2 \tan^2 \alpha \tan^2 \beta \tan^2 \gamma = 1 \) then the value of \( \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = \)
(a) 0
(b) -1
(c) 1
(d) 2
Answer: (c) 1

Question. Let \( \theta \in \left(0, \frac{\pi}{4}\right) \) and \( t_1 = (\tan \theta)^{\tan \theta} \), \( t_2 = (\tan \theta)^{\cot \theta} \), \( t_3 = (\cot \theta)^{\tan \theta} \), \( t_4 = (\cot \theta)^{\cot \theta} \) then
(a) \( t_1 > t_2 > t_3 > t_4 \)
(b) \( t_4 > t_3 > t_1 > t_2 \)
(c) \( t_3 > t_1 > t_2 > t_4 \)
(d) \( t_2 > t_3 > t_1 > t_4 \)
Answer: (b) \( t_4 > t_3 > t_1 > t_2 \)

Question. Two arcs of same length of two different circles subtended angles of 25° and 75° at their centres respectively. Then the ratio of the radii of the circles is
(a) 3 : 1
(b) 1 : 3
(c) 1 : 2
(d) 2 : 1
Answer: (a) 3 : 1

Question. If \( f_k(x) = \frac{1}{k} (\sin^k x + \cos^k x) \) where \( x \in \mathbb{R}, k \geq 1 \) then \( f_4(x) - f_6(x) = \) (MAINS-2014)
(a) \( \frac{1}{6} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{1}{9} \)
(d) \( \frac{1}{12} \)
Answer: (d) \( \frac{1}{12} \)

Question. The expression \( \frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A} \) can be written as (MAINS-2013)
(a) \( \sin A \cos A + 1 \)
(b) \( \sec A \csc A + 1 \)
(c) \( \tan A + \cot A \)
(d) \( \sec A + \csc A \)
Answer: (b) \( \sec A \csc A + 1 \)

Question. In a \( \Delta PQR \), if \( 3 \sin P + 4 \cos Q = 6 \) and \( 4 \sin Q + 3 \cos P = 1 \) then the acute angle 'R' is equal to [AIEEE-2012]
(a) \( \frac{\pi}{4} \)
(b) \( \frac{3\pi}{4} \)
(c) \( \frac{5\pi}{6} \)
(d) \( \frac{\pi}{6} \)
Answer: (d) \( \frac{\pi}{6} \)

Question. Assertion A: In a right angled triangle \( \sin^2 A + \sin^2 B + \sec^2 C = 2 \).
Reason R: If \( \alpha, \beta \) are complementary angles then \( \sin^2 \alpha + \sin^2 \beta = 1 \)
(a) A, R are true and R is the correct explanation of A
(b) A, R are true and R is not the correct explanation of A
(c) A is true, R is false
(d) A is false, R is true
Answer: (d) A is false, R is true

Question. Observe the following lists. Let \( \frac{\sin x}{a} = \frac{\cos x}{b} = \frac{\tan x}{c} = k \).
                                List - I                                ​​​​​​​                ​​​​​​​List-II
1) \( bc \)                ​​​​​​​                ​​​​​​​                ​​​​​​​        ​​​​​​​a) \( \frac{1}{b^2 k^4} \)
2) \( a^2 + b^2 \)     ​​​​​​​                ​​​​​​​  ​​​​​​​     ​​​​​​​         ​​​​​​​     ​​​​​​​  b) \( \frac{1}{ak} \)
3) \( \frac{1}{ck} + \frac{ak}{1 + bk} \)            c) \( \frac{a}{k} \)
4) \( a^2 + b^2 + c^2 \)                     ​​​​​​​             d) \( \frac{1}{k^2} \)
          ​​​​​​​              ​​​​​​​          ​​​​​​​              ​​​​​​​          ​​​​​​​            ​​​​​​​The correct match for List-I form List-II is
(a) 1 - c, 2 - d, 3 - b, 4 - a
(b) 1 - d, 2 - a, 3 - c, 4 - b
(c) 1 - a, 2 - b, 3 - d, 4 - c
(d) 1 - b, 2 - c, 3 - a, 4 - d
Answer: (a) 1 - c, 2 - d, 3 - b, 4 - a

Question. Satement I:The number of values of \( \theta \) satisfying \( \sec \theta + \cos \theta = 1 \) is 2.
Statement II: If \( 7 \sin^2 \alpha + 3 \cos^2 \alpha = 4 \), then \( \tan \alpha = \sqrt{3} \).
Which of the above statements is correct ?
(a) only I is true
(b) only II is true
(c) Both I and II are true
(d) Neither I nor II is true
Answer: (d) Neither I nor II is true

Question. If \( \alpha = \cos 10^\circ - \sin 10^\circ \), \( \beta = \cos 45^\circ - \sin 45^\circ \), \( \gamma = \cos 70^\circ - \sin 70^\circ \) then the descending order of \( \alpha, \beta, \gamma \) is
(a) \( \alpha, \beta, \gamma \)
(b) \( \gamma, \beta, \alpha \)
(c) \( \alpha, \gamma, \beta \)
(d) \( \beta, \alpha, \gamma \)
Answer: (a) \( \alpha, \beta, \gamma \)

Question. Match the following:
                ​​​​​​​                ​​​​​​​List - I                ​​​​​​​                ​​​​​​​                List - II
1) \( 3 \tan x + 27 \cot x \geq \) (\( x \in Q_1 \))                a) 24
2) \( 5 \sec^2 x + 125 \cos^2 x \geq \)                             ​​​​​​​b) 18
3) \( 16 \csc^2 x + 9 \sin^2 x \geq \)                                ​​​​​​​c) 50
(a) 1-a, 2-b, 3-c
(b) 1-c, 2-a, 3-b
(c) 1-b, 2-c, 3-a
(d) 1-c, 2-b, 3-a
Answer: (c) 1-b, 2-c, 3-a

Question. A = tan1, B = tan2, C = tan3, then the descending order of A, B, C is
(a) A, B, C
(b) C, B, A
(c) A, C, B
(d) B, C, A
Answer: (c) A, C, B