CBSE Class 10 Introduction to Trigonometry Sure Shot Questions Set 09

Question. If \( \text{cosec } \theta = u \) and \( \cos \theta = v \), then \( \cot \theta \) is equal to
(a) \( \frac{u}{v} \)
(b) \( \frac{v}{u} \)
(c) \( u^2v \)
(d) \( uv \)
Answer: (d) \( uv \)

Question. If \( 3 \cot A = 4 \cos A \), then the relation between \( \sec A \) and \( \tan A \) is
(a) \( 4 \sec A = 3 \tan A \)
(b) \( 3 \sec^2 A - 4 \tan^2 A = 0 \)
(c) \( 4 \sec^2 A - 3 \tan^2 A = 0 \)
(d) \( 3 \sec A - 4 \tan A = 0 \)
Answer: (d) \( 3 \sec A - 4 \tan A = 0 \)

Question. If \( \cos \theta = \frac{a}{b} \), then \( \cot \theta \) is equal to
(a) \( \frac{a}{\sqrt{b^2 + a^2}} \)
(b) \( \frac{b}{\sqrt{a^2 - b^2}} \)
(c) \( \frac{a}{\sqrt{b^2 - a^2}} \)
(d) None of the options
Answer: (c) \( \frac{a}{\sqrt{b^2 - a^2}} \)

Question. The positive minimum value of \( \text{cosec } \theta \) is
(a) 0
(b) 1
(c) 2
(d) \( \frac{1}{2} \)
Answer: (b) 1

Question. If \( -x \tan 45^\circ \sin 60^\circ + \cos 60^\circ \cdot \cot 45^\circ = \frac{4}{5} \), then the value of \( x \) is
(a) \( \frac{\sqrt{3}}{10} \)
(b) \( -\frac{\sqrt{3}}{5} \)
(c) \( \frac{\sqrt{3}}{5} \)
(d) \( -\frac{\sqrt{3}}{10} \)
Answer: (b) \( -\frac{\sqrt{3}}{5} \)

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

Question. If \( \triangle ABC \) is right angled at \( C \), then the value of \( \sin(A + B) \) is
(a) 1
(b) 0
(c) -1
(d) \( \frac{1}{2} \)
Answer: (a) 1

Question. If \( \cot \theta = \frac{4}{3} \), then the value of \( \frac{2 \cos \theta - \sin \theta}{\sin \theta + 3 \cos \theta} \) is
(a) \( \frac{1}{3} \)
(b) \( \frac{3}{5} \)
(c) \( \frac{5}{3} \)
(d) None of the options
Answer: (a) \( \frac{1}{3} \)

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

Question. The value of \( \frac{1}{1 + \cos \theta} + \frac{1}{1 - \cos \theta} - 2 \) is
(a) \( 2 \sec^2 \theta \)
(b) \( 2 \cot^2 \theta \)
(c) \( 2 \sec \theta \)
(d) \( \sec \theta \cdot \tan \theta \)
Answer: (b) \( 2 \cot^2 \theta \)

Question. If \( m = \cos \theta - \sin \theta \) and \( n = \cos \theta + \sin \theta \), then the value of \( \sec^2 \theta \) is
(a) \( \frac{(m + n)^2}{2} \)
(b) \( \frac{(m + n)^2}{4} \)
(c) \( \frac{4}{(m + n)^2} \)
(d) None of the options
Answer: (c) \( \frac{4}{(m + n)^2} \)

Question. The value of \( 1 + \frac{\tan^2 \alpha}{1 + \sec \alpha} \) is
(a) \( \sec \alpha \)
(b) \( \text{cosec } \alpha \)
(c) \( \cos \alpha \)
(d) \( \cot \alpha \)
Answer: (a) \( \sec \alpha \)

Question. If \( a = 4 \cos^2 \theta \) and \( b = 3 - 4 \sin^2 \theta \), then \( a - b \) is equal to
(a) 2
(b) -1
(c) 1
(d) -3
Answer: (c) 1

Question. The value of \( (\cot \theta \sec \theta)^2 - (\cos \theta \text{cosec } \theta)^2 \) is
(a) 0
(b) -1
(c) 1
(d) 2
Answer: (c) 1