CBSE Class 10 Maths HOTs Introduction to Trigonometry Set 04

Question. The value of \( \frac{2 \tan 30^\circ}{1 - \tan^2 30^\circ} \) equals to :
(a) \( \cos 60^\circ \)
(b) \( \sin 60^\circ \)
(c) \( \tan 60^\circ \)
(d) \( \sin 30^\circ \)
Answer: (c) \( \tan 60^\circ \)

Question. If \( \sin \alpha = \frac{1}{2} \) and \( \alpha \) is acute, then \( (3 \cos \alpha - 4 \cos^3 \alpha) \) is equal to :
(a) 0
(b) \( \frac{1}{2} \)
(c) \( \frac{1}{6} \)
(d) \( -1 \)
Answer: (a) 0

Question. If \( \cot A + \frac{1}{\cot A} = 1 \) the value of \( \cot^2 A + \frac{1}{\cot^2 A} \) is:
(a) 1
(b) 2
(c) -1
(d) -2
Answer: (c) -1

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

Question. If \( 2 \sin 2\theta = \sqrt{3} \), then the value of \( \theta \) is :
(a) \( 90^\circ \)
(b) \( 30^\circ \)
(c) \( 45^\circ \)
(d) \( 60^\circ \)
Answer: (b) \( 30^\circ \)

Question. If \( x \cos A = 1 \) and \( \tan A = y \), then \( x^2 - y^2 \) is equal to :
(a) \( \tan A \)
(b) 1
(c) 0
(d) \( -\tan A \)
Answer: (b) 1

Question. \( \cos^4 A - \sin^4 A \) is equal to :
(a) \( 2 \cos^2 A + 1 \)
(b) \( 2 \cos^2 A - 1 \)
(c) \( 2 \sin^2 A - 1 \)
(d) \( 2 \sin^2 A + 1 \)
Answer: (b) \( 2 \cos^2 A - 1 \)

Question. The value of the expression \( [(\sec^2 \theta - 1) (1 - \csc^2 \theta)] \) is :
(a) -1
(b) 1
(c) 0
(d) \( \frac{1}{2} \)
Answer: (b) 1

Question. If \( \cos A + \cos^2 A = 1 \), then \( \sin^2 A + \sin^4 A \) is:
(a) -1
(b) 0
(c) 1
(d) 2
Answer: (c) 1

Question. If \( a \cos \theta + b \sin \theta = 4 \) and \( a \sin \theta - b \cos \theta = 3 \), then \( a^2 + b^2 \) is
(a) 7
(b) 12
(c) 25
(d) None
Answer: (c) 25

Question. If \( \csc^2 \theta (1 + \cos \theta) (1 - \cos \theta) = \lambda \), then the value of \( \lambda \) is
(a) 0
(b) \( \cos^2 \theta \)
(c) 1
(d) -1
Answer: (c) 1

Question. If \( \sin \theta = 1/2 \), then the value of \( (\sin \theta - \csc \theta) \) is
(a) \( \frac{3}{2} \)
(b) \( -\frac{3}{2} \)
(c) \( \frac{\sqrt{3}}{2} \)
(d) \( -\frac{\sqrt{3}}{2} \)
Answer: (b) \( -\frac{3}{2} \)

Question. The value of \( \sin^2 30^\circ - \cos^2 30^\circ \) is:
(a) \( -\frac{1}{2} \)
(b) \( \frac{\sqrt{3}}{2} \)
(c) \( \frac{3}{2} \)
(d) \( \frac{2}{3} \)
Answer: (a) \( -\frac{1}{2} \)

Question. The expression of \( \sin A \) in terms of \( \cot A \) is :
(a) \( \frac{\sqrt{1 + \cot^2 A}}{\cot A} \)
(b) \( \frac{1 + \cot^2 A}{\cot A} \)
(c) \( \frac{1}{\sqrt{1 + \cot^2 A}} \)
(d) \( \frac{\sqrt{1 - \cot^2 A}}{\cot A} \)
Answer: (c) \( \frac{1}{\sqrt{1 + \cot^2 A}} \)

Question. If \( \tan 2A = \cot (A - 18^\circ) \), then the value of \( A \) is
(a) \( 18^\circ \)
(b) \( 36^\circ \)
(c) \( 24^\circ \)
(d) \( 27^\circ \)
Answer: (b) \( 36^\circ \)

Question. Which of the following is not defined ?
(a) \( \cos 0^\circ \)
(b) \( \tan 45^\circ \)
(c) \( \sec 90^\circ \)
(d) \( \sin 90^\circ \)
Answer: (c) \( \sec 90^\circ \)

Question. If \( \sin \theta = \frac{1}{3} \), then the value of \( 2 \cot^2 \theta + 2 \) is :
(a) 6
(b) 9
(c) 18
(d) 4
Answer: (c) 18

Question. The value of \( \frac{\tan 45^\circ}{\sin 30^\circ + \cos 60^\circ} \) is:
(a) \( \frac{1}{\sqrt{2}} \)
(b) 1
(c) \( \frac{1}{2} \)
(d) \( \sqrt{2} \)
Answer: (b) 1

Question. \( (\sec A + \tan A) (1 - \sin A) \) is equal to :
(a) \( \sec A \)
(b) \( \sin A \)
(c) \( \csc A \)
(d) \( \cos A \)
Answer: (d) \( \cos A \)

Question. The value of \( [\sin^2 20^\circ + \sin^2 70^\circ - \tan^2 45^\circ] \) is :
(a) 0
(b) 1
(c) 2
(d) -1
Answer: (a) 0

Question. If \( \sin (A - B) = \frac{1}{2} \) and \( \cos (A + B) = \frac{1}{2} \), then the value of \( B \) is:
(a) \( 45^\circ \)
(b) \( 60^\circ \)
(c) \( 15^\circ \)
(d) \( 0^\circ \)
Answer: (c) \( 15^\circ \)

Question. If \( 3 \cos \theta = 1 \), then the value of \( \csc \theta \) is:
(a) \( 2\sqrt{2} \)
(b) \( 3/2\sqrt{2} \)
(c) \( 2\sqrt{3}/3 \)
(d) \( 4/3\sqrt{2} \)
Answer: (b) \( 3/2\sqrt{2} \)

Question. If \( \triangle PQR \) is right angled at R, then the value of \( \cos (P + Q) \) is:
(a) 1
(b) 0
(c) \( \frac{1}{2} \)
(d) \( \sqrt{3}/2 \)
Answer: (b) 0

Question. Given that \( \sin \alpha = \frac{1}{2} \) and \( \cos \beta = \frac{1}{2} \) then the value of \( \alpha + \beta \) is:
(a) \( 0^\circ \)
(b) \( 90^\circ \)
(c) \( 30^\circ \)
(d) \( 60^\circ \)
Answer: (b) \( 90^\circ \)

Question. The value of \( \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \dots \tan 89^\circ \) is:
(a) 0
(b) 1
(c) 2
(d) \( \frac{1}{2} \)
Answer: (b) 1

Question. Given that \( \sin A = \frac{1}{2} \) and \( \cos B = \frac{1}{\sqrt{2}} \), then the value of \( A + B \) is:
(a) \( 30^\circ \)
(b) \( 45^\circ \)
(c) \( 75^\circ \)
(d) \( 15^\circ \)
Answer: (c) \( 75^\circ \)

Question. The value of \( 5 \tan^2 \theta - 5 \sec^2 \theta \) is :
(a) 1
(b) -5
(c) 0
(d) 5
Answer: (b) -5

Question. \( \left( \frac{\cos A}{\cot A} + \sin A \right) \) is:
(a) \( \cot A \)
(b) \( 2 \sin A \)
(c) \( 2 \cos A \)
(d) \( \sec A \)
Answer: (b) \( 2 \sin A \)

Question. If \( x = 2 \sin^2 \theta \), \( y = 2 \cos^2 \theta + 1 \) then the value of \( x + y \) is :
(a) 2
(b) 3
(c) \( \frac{1}{2} \)
(d) 1
Answer: (b) 3

Question. If \( \theta = 45^\circ \), the value of \( \csc^2 \theta \) is
(a) \( \frac{1}{\sqrt{2}} \)
(b) 1
(c) \( \frac{1}{2} \)
(d) 2
Answer: (d) 2

Question. If \( \csc \theta - \cot \theta = \frac{1}{3} \), the value of \( (\csc \theta + \cot \theta) \) is:
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. \( 5 \csc^2 \theta - 5 \cot^2 \theta \) is equal to:
(a) 5
(b) 1
(c) 0
(d) -5
Answer: (a) 5

Question. If \( \tan A = \frac{5}{12} \), find the value of \( (\sin A + \cos A) \times \sec A \) :
(a) \( \frac{17}{12} \)
(b) \( \frac{12}{17} \)
(c) \( \frac{5}{13} \)
(d) \( \frac{13}{5} \)
Answer: (a) \( \frac{17}{12} \)

Question. If \( \sin \theta = \cos \theta \), then the value of \( \theta \) is :
(a) \( 0^\circ \)
(b) \( 45^\circ \)
(c) \( 60^\circ \)
(d) \( 30^\circ \)
Answer: (b) \( 45^\circ \)

Question. If \( x = 3 \sec^2 \theta - 1 \), \( y = \tan^2 \theta - 2 \) then \( x - 3y \) is equal to
(a) 3
(b) 4
(c) 8
(d) 5
Answer: (c) 8

Question. If \( \sec \theta - \tan \theta = \frac{1}{3} \), the value of \( (\sec \theta + \tan \theta) \) is:
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. The value of \( \cos 60^\circ \sin 30^\circ + \sin 60^\circ \cos 30^\circ \) is :
(a) \( \frac{1}{4} \)
(b) \( \frac{3}{4} \)
(c) 1
(d) \( \frac{2}{4} \)
Answer: (c) 1

Question. If \( \cot \theta = \frac{7}{8} \), the value of \( \frac{(1 + \cos \theta)(1 - \cos \theta)}{(1 - \sin \theta)(1 + \sin \theta)} \) is :
(a) \( \frac{49}{64} \)
(b) \( \frac{8}{7} \)
(c) \( \frac{64}{49} \)
(d) \( \frac{7}{8} \)
Answer: (a) \( \frac{49}{64} \)

Question. If \( \cos (20 + \theta) = \sin 30^\circ \), then the value of \( \theta \) is :
(a) \( 20^\circ \)
(b) \( 50^\circ \)
(c) \( 30^\circ \)
(d) \( 40^\circ \)
Answer: (d) \( 40^\circ \)

Question. The value of \( \sin \theta \cos(90^\circ - \theta) + \cos \theta \sin(90^\circ - \theta) \) is:
(a) 1
(b) 0
(c) 2
(d) -1
Answer: (a) 1