Class 11 Mathematics Trigonometric Ratios MCQs Set 02

Question. \( -11\frac{\pi}{3} \) radians =
(a) -390°
(b) -620°
(c) -610°
(d) -660°
Answer: (d) -660°
\( -11\frac{\pi}{3} \times \frac{180}{\pi} = -660^{\circ} \)

Question. \( \sin 4530^{\circ} = \)
(a) \( \frac{1}{2} \)
(b) \( -\frac{1}{2} \)
(c) \( \frac{\sqrt{3}}{2} \)
(d) \( -\frac{\sqrt{3}}{2} \)
Answer: (b) \( -\frac{1}{2} \)
\( \sin(4530^{\circ}) = \sin((12)360^{\circ} + 210^{\circ}) = \sin 210^{\circ} = -\sin 30^{\circ} = -\frac{1}{2} \)

Question. \( \frac{\sin 150^{\circ} - 5\cos 300^{\circ} + 7\tan 225^{\circ}}{\tan 135^{\circ} + 3\sin 210^{\circ}} = \)
(a) 10
(b) -5
(c) -2
(d) -3/2
Answer: (c) -2
Convert the corresponding values.

Question. \( \cos 23^{\circ} \text{cosec} 67^{\circ} - \sin 23^{\circ} \sec 67^{\circ} = \)
(a) 0
(b) 2
(c) 1
(d) -1
Answer: (a) 0
\( \cos 23^{\circ} \text{cosec}(90^{\circ}-23^{\circ}) - \sin 23^{\circ} \sec(90^{\circ}-23^{\circ}) = 1 - 1 = 0 \)

Question. \( \log \tan 18^{\circ} + \log \tan 36^{\circ} + \log \tan 54^{\circ} + \log \tan 72^{\circ} = \)
(a) \( \log 4 \)
(b) \( \log 3 \)
(c) \( \log 2 \)
(d) 0
Answer: (d) 0
\( \log(\tan 18^{\circ} \tan 36^{\circ} \tan 54^{\circ} \tan 72^{\circ}) = \log 1 = 0 \) [\( \because \text{If } A+B = 90^{\circ} \)
\( \implies \) \( \tan A \tan B = 1 \)]

Question. \( \cos 5^{\circ} + \cos 24^{\circ} + \cos 175^{\circ} + \cos 204^{\circ} + \cos 300^{\circ} = \)
(a) 1/2
(b) 1
(c) 3/2
(d) 2
Answer: (a) 1/2
\( \alpha + \beta = 180^{\circ} \)
\( \implies \) \( \cos \alpha + \cos \beta = 0 \)

Question. \( \text{cosec} A + \cot A = \frac{2}{3} \)
\( \implies \) \( \cos A = \)
(a) \( \frac{5}{13} \)
(b) \( \frac{13}{5} \)
(c) \( -\frac{5}{13} \)
(d) \( -\frac{13}{5} \)
Answer: (c) \( -\frac{5}{13} \)
Use \( (\text{cosec}^{2} \theta - \cot^{2} \theta) = 1 \)

Question. If \( \text{cosec} \theta - \cot \theta = 5 \) then \( \theta \) lies in the quadrant
(a) I
(b) II
(c) III
(d) IV
Answer: (b) II
Use \( (\text{cosec}^{2} A - \cot^{2} A) = 1 \). Find the quadrant in which \( A \) lies

Question. \( \tan 20^{\circ} + \tan 40^{\circ} + \tan 60^{\circ} + \dots + \tan 180^{\circ} = \)
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (a) 0
\( A+B = 180^{\circ} \)
\( \implies \) \( \tan A + \tan B = 0 \)

Question. \( (\sin \alpha + \text{cosec} \alpha)^{2} + (\sec \alpha + \cos \alpha)^{2} = k + \tan^{2} \alpha + \cot^{2} \alpha \)
\( \implies \) \( k = \)
(a) 9
(b) 7
(c) 5
(d) 3
Answer: (b) 7
Square and simplyfy

Question. \( \frac{1}{(1+\cot^{2} \alpha)^{2}} + \frac{\tan^{2} \alpha}{(1+\tan^{2} \alpha)^{2}} + \frac{1}{1+\tan^{2} \alpha} = \)
(a) -1
(b) 0
(c) 1
(d) 2
Answer: (c) 1
Put \( \alpha = 45^{\circ} \)

Question. \( \frac{\cos^{3} A + \sin^{3} A}{\cos A + \sin A} + \frac{\cos^{3} A - \sin^{3} A}{\cos A - \sin A} = k \) then \( k = \)
(a) 0
(b) 1
(c) 2
(d) -1
Answer: (c) 2
Put \( A = 0^{\circ} \)

Question. In \( \Delta ABC \), \( \tan(A - B - C) = \)
(a) \( \sin 2A \)
(b) 1
(c) \( \tan 2A \)
(d) 0
Answer: (c) \( \tan 2A \)
\( B+C = 180^{\circ}-A \)
\( \implies \) \( \tan(A+A-180^{\circ}) = \tan 2A \)

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

Question. \( \cos^{2}(125^{\circ} - x) - \cos^{2}(55^{\circ} + x) = \)
(a) -1
(b) 0
(c) 1
(d) 2
Answer: (b) 0
\( (125^{\circ}-x) + (55^{\circ}+x) = 180^{\circ} \)

Question. \( \sin^{2} \frac{\pi}{18} + \sin^{2} \frac{2\pi}{18} + \sin^{2} \frac{4\pi}{18} + \sin^{2} \frac{8\pi}{18} + \sin^{2} \frac{7\pi}{18} + \sin^{2} \frac{5\pi}{18} = \)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3
If \( (A+B) = \frac{\pi}{2} \) then \( \sin^{2} A + \sin^{2} B = 1 \)

Question. If A B C D is a quadrilateral then \( \tan \left( \frac{A+B}{4} \right) = \)
(a) \( \cos \left( \frac{C-D}{4} \right) \)
(b) \( \cot \left( \frac{C-D}{4} \right) \)
(c) \( \cos \left( \frac{C+D}{4} \right) \)
(d) \( \cot \left( \frac{C+D}{4} \right) \)
Answer: (d) \( \cot \left( \frac{C+D}{4} \right) \)
\( (A+B+C+D) = 360^{\circ} \)
\( \implies \) \( \frac{A+B}{4} = \frac{\pi}{2} - \frac{C+D}{4} \)

Question. \( \left[ \cos \left( \frac{\pi}{2} - x \right) + \cos(\pi - x) \right]^{2} + \left[ \sin \left( \frac{3\pi}{2} - x \right) + \sin(2\pi - x) \right]^{2} = \)
(a) 0
(b) 2
(c) 1
(d) 4
Answer: (b) 2
\( (\sin x - \cos x)^{2} + (-\cos x - \sin x)^{2} = 2 \)

Question. If \( \cos \theta_{1} + \cos \theta_{2} + \cos \theta_{3} + \cos \theta_{4} + \cos \theta_{5} = 5 \) then \( \sin \theta_{1} + \sin \theta_{2} + \sin \theta_{3} + \sin \theta_{4} + \sin \theta_{5} = \)
(a) 3
(b) 2
(c) 1
(d) 0
Answer: (d) 0
Put \( \theta_{1} = \theta_{2} = \theta_{3} = 0^{\circ} \)

Question. \( 9\cos^{2} x + 4\sin^{2} x = 5 \)
\( \implies \) \( \tan x = \)
(a) \( \pm 1 \)
(b) \( \pm 2 \)
(c) \( \pm 3 \)
(d) \( \pm 4 \)
Answer: (b) \( \pm 2 \)
Divide the given equation by \( \cos^{2} x \) both sides.

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

Question. If \( \cot(\alpha + \beta) = 0 \) then \( \sin(\alpha + 2\beta) = \)
(a) \( \sin \alpha \)
(b) \( \cos \alpha \)
(c) \( \sin \beta \)
(d) \( \cos \beta \)
Answer: (d) \( \cos \beta \)
\( \cot(\alpha + \beta) = 0 \)
\( \implies \) \( \alpha + \beta = \frac{\pi}{2} \)
\( \implies \) \( \sin(\alpha + 2\beta) = \sin(90^{\circ} + \beta) = \cos \beta \)

Question. If \( \tan \alpha + \cot \alpha = 2 \) then \( \sqrt{\tan \alpha} + \sqrt{\cot \alpha} = \)
(a) \( \sqrt{2} \)
(b) \( 2\sqrt{2} \)
(c) 2
(d) \( 4\sqrt{2} \)
Answer: (c) 2
Put \( \alpha = \frac{\pi}{4} \)

Question. \( 5\sin x + 4\cos x = 3 \)
\( \implies \) \( 4\sin x - 5\cos x = \)
(a) 4
(b) \( 4\sqrt{2} \)
(c) \( 3\sqrt{2} \)
(d) \( \sqrt{2} \)
Answer: (b) \( 4\sqrt{2} \)
\( a\sin x + b\cos x = c \)
\( \implies \) \( b\sin x - a\cos x = \pm\sqrt{a^{2}+b^{2}-c^{2}} \)

Question. If \( k = (\sec A + \tan A)(\sec B + \tan B)(\sec C + \tan C) = (\sec A - \tan A)(\sec B - \tan B)(\sec C - \tan C) \) then \( k = \)
(a) 0
(b) \( \pm 1 \)
(c) \( \pm 3 \)
(d) \( \pm 4 \)
Answer: (b) \( \pm 1 \)
\( K^{2} = (1+\sin A)(1+\sin B)(1+\sin C)(1-\sin A)(1-\sin B)(1-\sin C) \)
\( K^{2} = (1-\sin^{2} A)(1-\sin^{2} B)(1-\sin^{2} C) \)
\( K^{2} = \cos^{2} A \cdot \cos^{2} B \cdot \cos^{2} C \)
\( K = \pm \cos A \cos B \cos C \)

Question. The value of \( \cot^{2} \alpha \left( \frac{\sec \alpha - 1}{1 + \sin \alpha} \right) + \sec^{2} \alpha \left( \frac{\sin \alpha - 1}{1 + \sec \alpha} \right) \) is
(a) 0
(b) 1
(c) 2
(d) -2
Answer: (a) 0
Put \( \alpha = 45^{\circ} \); verify the options.

Question. \( \sec^{2} \theta + \tan \theta = 13 \)
\( \implies \) \( \tan \theta = \)
(a) 4
(b) 3
(c) -3
(d) 5
Answer: (b) 3
\( 1 + \tan^{2} \theta + \tan \theta = 13 \)
\( \implies \) \( \tan \theta = 3 \text{ or } -4 \)

Question. \( \log(\tan 1^{\circ}) \log(\tan 2^{\circ}) \log(\tan 3^{\circ}) \dots \log(\tan 45^{\circ}) = \)
(a) 0
(b) 1
(c) -1
(d) 1/2
Answer: (a) 0
\( \log(\tan 45^{\circ}) = \log 1 = 0 \)

Question. If \( \sin(\alpha + \beta) = 1, \sin(\alpha - \beta) = \frac{1}{2} \) then \( \tan(\alpha + 2\beta) \tan(2\alpha + \beta) = \)
(a) 1
(b) -1
(c) 0
(d) 2
Answer: (a) 1
\( \alpha + \beta = 90^{\circ}, \alpha - \beta = 30^{\circ} \)
\( \implies \) \( \alpha = 60^{\circ}, \beta = 30^{\circ} \)

Question. If \( x = r \cos \alpha \cos \beta \cos \gamma \); \( y = r \cos \alpha \cos \beta \sin \gamma \); \( z = r \sin \alpha \cos \beta \); \( \mu = r \sin \beta \) then \( x^{2} + y^{2} + z^{2} + \mu^{2} = \)
(a) \( r \)
(b) \( 2r \)
(c) \( r^{2} \)
(d) \( 4r^{2} \)
Answer: (c) \( r^{2} \)
Squaring and adding

Question. \( \sec^{2} A \sec^{2} B - \sec^{2} A \tan^{2} B - \tan^{2} A \sec^{2} B + \tan^{2} A \tan^{2} B = \)
(a) 1
(b) 0
(c) -1
(d) 2
Answer: (a) 1
Put A = B = 45°

Question. \( 2\cos^{2} B - 1 = \tan^{2} A \), then \( \cos A \cos B = \)
(a) \( \pm 1/2 \)
(b) \( \pm 1/3 \)
(c) \( \pm 1/\sqrt{2} \)
(d) \( \pm 1 \)
Answer: (c) \( \pm 1/\sqrt{2} \)
\( 2\cos^{2} B = 1 + \tan^{2} A \)

Question. \( \sin^{4} \theta + 2\sin^{2} \theta \left( 1 - \frac{1}{\text{cosec}^{2} \theta} \right) + \cos^{4} \theta = \)
(a) 1
(b) 0
(c) 1/2
(d) -1
Answer: (a) 1
\( \sin^{4} \theta + \cos^{4} \theta = 1 - 2\sin^{2} \theta \cos^{2} \theta \)

Question. If \( x\cot^{2} 120^{\circ} + 4\sin^{2} 150^{\circ} = 3 \), then \( x = \)
(a) 2
(b) 7
(c) 6
(d) 5
Answer: (c) 6
\( x \cdot \frac{1}{3} + 4 \cdot \frac{1}{4} = 3 \)
\( \implies \) \( x = 6 \)

Question. In a right angled triangle ABC, \( \angle C = 90^{\circ} \), then \( \cos^{2} A + \cos^{2} B = \)
(a) 2
(b) 1
(c) 1/2
(d) 3/4
Answer: (b) 1
\( A+B = 90^{\circ} \)
\( \implies \) \( \cos^{2} A + \cos^{2} B = 1 \)

Question. If \( x = \sin 130^{\circ} + \cos 130^{\circ} \) then
(a) \( x < 0 \)
(b) \( x = 0 \)
(c) \( x > 0 \)
(d) \( x \ge 0 \)
Answer: (c) \( x > 0 \)
\( x = \sin 50^{\circ} - \cos 50^{\circ} \)

Question. \( \cot \frac{\pi}{20} \cdot \cot \frac{3\pi}{20} \cdot \cot \frac{5\pi}{20} \cdot \cot \frac{7\pi}{20} \cdot \cot \frac{9\pi}{20} \cdot \cot \frac{15\pi}{20} = \)
(a) 1
(b) -1
(c) \( \sqrt{3} \)
(d) \( -\sqrt{3} \)
Answer: (b) -1
\( A+B = 90^{\circ} \)
\( \implies \) \( \cot A \cot B = 1 \)

Question. If \( x = a\sec^{n} \theta; y = b\tan^{n} \theta \) then \( \left( \frac{x}{a} \right)^{\frac{2}{n}} - \left( \frac{y}{b} \right)^{\frac{2}{n}} = \)
(a) 0
(b) -1
(c) 1
(d) 2
Answer: (c) 1
\( \sec^{2} \theta - \tan^{2} \theta = 1 \)

Question. If \( x = h + p\sec \alpha, y = k + q\text{cosec} \alpha \) then \( \left( \frac{p}{x-h} \right)^{2} + \left( \frac{q}{y-k} \right)^{2} = \)
(a) 1
(b) -1
(c) 0
(d) 1/2
Answer: (a) 1
\( \sin^{2} \theta + \cos^{2} \theta = 1 \)

Question. If A, B, C are angles of a triangle such that A is obtuse then
(a) \( \tan A \tan B > 1 \)
(b) \( \tan B \tan C < 1 \)
(c) \( \tan C \tan A > 1 \)
(d) \( \tan A \tan B \tan C > 1 \)
Answer: (b) \( \tan B \tan C < 1 \)
A is obtuse
\( \implies \) \( B+C < 90^{\circ} \)
\( \implies \) \( \tan B \tan C < 1 \)