CBSE Class 10 Introduction to Trigonometry Sure Shot Questions Set 08

MCQ

Question. The two legs AB & AC of right angled ∆ ABC are in the ratio 1:3, what will be the value of sin C ?
(a) \( \sqrt{10} \)
(b) \( \frac{1}{\sqrt{10}} \)
(c) \( \frac{3}{\sqrt{10}} \)
(d) \( \frac{1}{2} \)
Answer: (b) \( \frac{1}{\sqrt{10}} \)

Question. If \( \sin A + \sin^2 A = 1 \), then find the value of \( \cos^2 A + \cos^4 A \)
(a) 1
(b) \( \frac{1}{2} \)
(c) 2
(d) 3
Answer: (a) 1

Question. The value of \( \frac{4 - \sin^2 45^\circ}{\cot A \tan 60^\circ} \) is 3.5 , what is the value of A
(a) \( 30^\circ \)
(b) \( 45^\circ \)
(c) \( 60^\circ \)
(d) \( 90^\circ \)
Answer: (c) \( 60^\circ \)

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

Question. If \( \cos A = \frac{4}{5} \), then the value of \( \tan A \) is
(a) \( \frac{3}{5} \)
(b) \( \frac{3}{4} \)
(c) \( \frac{4}{3} \)
(d) \( \frac{5}{3} \)
Answer: (b) \( \frac{3}{4} \)

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

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

Question. Given that \( \sin a = \frac{\sqrt{3}}{2} \) and \( \cos b = 0 \), then the value of \( b - a \) is
(a) \( 0^\circ \)
(b) \( 90^\circ \)
(c) \( 60^\circ \)
(d) \( 30^\circ \)
Answer: (d) \( 30^\circ \)

Question. If \( \Delta ABC \) is right angled at C , then the value of \( \sec(A + B) \) is
(a) 0
(b) 1
(c) \( \frac{2}{\sqrt{3}} \)
(d) not defined
Answer: (d) not defined

Question. If \( \sin \theta + \cos \theta = \sqrt{2} \cos \theta \), (\( \theta \neq 90^\circ \)) then the value of \( \tan \theta \) is
(a) \( \sqrt{2} - 1 \)
(b) \( \sqrt{2} + 1 \)
(c) \( \sqrt{2} \)
(d) \( -\sqrt{2} \)
Answer: (a) \( \sqrt{2} - 1 \)

Question. If \( \Delta ABC \) is right angled at C , then the value of \( \cos( A + B) \) is
(a) 0
(b) 1
(c) \( \frac{1}{2} \)
(d) \( \frac{\sqrt{3}}{2} \)
Answer: (a) 0

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

Question. If \( 4 \tan \theta = 3 \), then \( \frac{4 \sin \theta - \cos \theta}{4 \sin \theta + \cos \theta} \) is equal to
(a) \( \frac{2}{3} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{1}{2} \)
(d) \( \frac{3}{4} \)
Answer: (c) \( \frac{1}{2} \)

Question. If \( \sin \theta - \cos \theta = 0 \), then the value of \( \sin^4 \theta + \cos^4 \theta \) is
(a) 1
(b) \( \frac{3}{4} \)
(c) \( \frac{1}{2} \)
(d) \( \frac{1}{4} \)
Answer: (c) \( \frac{1}{2} \)

Question. If \( x = p \sec \theta \) and \( y = q \tan \theta \), then
(a) \( x^2 - y^2 = p^2 q^2 \)
(b) \( x^2 q^2 - y^2 p^2 = pq \)
(c) \( x^2 q^2 - y^2 p^2 = \frac{1}{pq} \)
(d) \( x^2 q^2 - y^2 p^2 = p^2 q^2 \)
Answer: (d) \( x^2 q^2 - y^2 p^2 = p^2 q^2 \)

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

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

Question. If \( \sin \theta = \frac{5}{13} \), then the value of \( \tan \theta \) is .......... .
(a) \( \frac{5}{13} \)
(b) \( \frac{5}{12} \)
(c) \( \frac{12}{13} \)
(d) \( \frac{8}{13} \)
Answer: (b) \( \frac{5}{12} \)

Question. The value of the \( (\tan^2 60^\circ + \sin^2 45^\circ) \) is .......... .
(a) \( \frac{1}{2} \)
(b) \( \frac{3}{2} \)
(c) 1
(d) \( \frac{7}{2} \)
Answer: (d) \( \frac{7}{2} \)

Question. \( \sin^2 60^\circ - 2 \tan 45^\circ - \cos^2 30^\circ = ? \)
(a) 2
(b) –2
(c) 1
(d) –1
Answer: (b) –2

Question. What happens to value of \( \cos \theta \) when increases from \( 0^\circ \) to \( 90^\circ \).
(a) \( \cos \theta \) decreases from 1 to 0.
(b) \( \cos \theta \) increases from 0 to 1.
(c) \( \cos \theta \) increases from \( \frac{1}{2} \) to 1
(d) \( \cos \theta \) decreases from 1 to \( \frac{1}{2} \)
Answer: (a) \( \cos \theta \) decreases from 1 to 0.

Question. \( \tan^4 \theta + \tan^2 \theta = ? \)
(a) \( \sec^2 \theta - 2 \sec^4 \theta \)
(b) \( 2 \sec^2 \theta - \sec^4 \theta \)
(c) \( \sec^2 \theta - \sec^4 \theta \)
(d) \( \sec^4 \theta - \sec^2 \theta \)
Answer: (d) \( \sec^4 \theta - \sec^2 \theta \)

Question. \( \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} = ? \)
(a) \( \sin \theta - \cos \theta \)
(b) \( \sec \theta - \tan \theta \)
(c) \( \sec \theta + \tan \theta \)
(d) \( \sin \theta + \cos \theta \)
Answer: (b) \( \sec \theta - \tan \theta \)

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

Question. If \( \tan 5 \theta = 1 \) then \( \theta \) is equal to
(a) \( 9^\circ \)
(b) \( 90^\circ \)
(c) \( 45^\circ \)
(d) \( 30^\circ \)
Answer: (a) \( 9^\circ \)

Question. If \( \tan(A + B) = \sqrt{3} \) and \( \tan(A - B) = \frac{1}{\sqrt{3}} \), A > B , then the value of A is ......... .
(a) \( 45^\circ \)
(b) \( 60^\circ \)
(c) \( 90^\circ \)
(d) \( 30^\circ \)
Answer: (a) \( 45^\circ \)

Question. The value of \( (1 + \tan^2 \theta) (1 - \sin \theta) (1 + \sin \theta) = \)
(a) 0
(b) 1
(c) 2
(d) \( \frac{1}{2} \)
Answer: (b) 1

Question. If \( \tan A = \cot B \), then the value of \( (A + B) \) is
(a) \( 90^\circ \)
(b) \( 120^\circ \)
(c) \( 60^\circ \)
(d) \( 180^\circ \)
Answer: (a) \( 90^\circ \)

Assertion and Reasoning Questions

Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion: If \( \cos A + \cos^2 A = 1 \) then \( \sin^2 A + \sin^4 A = 1 \).
Reason: \( \sin^2 A + \cos^2 A = 1 \), for any value of A
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

Question. Assertion: The value of \( 2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ \) is 2.
Reason: value of \( \tan 45^\circ = 1, \cos 30^\circ = \sqrt{3}/2 \) and \( \sin 60^\circ = \sqrt{3}/2 \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Question. Assertion: If \( x = 2 \sin^2 \theta \) and \( y = 2 \cos^2 \theta + 1 \) then the value of \( x + y = 3 \).
Reason: For any value of \( \theta, \sin^2 \theta + \cos^2 \theta = 1 \)
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Question. Assertion: sinA is the product of sin & A.
Reason: The value of \( \sin \theta \) increases as \( \theta \) increases.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (d) Assertion (A) is false but reason (R) is true.

Question. Assertion: In a right \( \Delta ABC \), right angled at B, if \( \tan A = 1 \), then \( 2 \sin A \cdot \cos A = 1 \)
Reason: cosecA is the abbreviation used for cosecant of angle A.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

SHORT ANSWER QUESTIONS (TYPE I)

Question. If \( \tan \alpha = \frac{5}{12} \), find the value of \( \sec \alpha \). (Hint :using identity \( \sec^2 \alpha = \tan^2 \alpha + 1 \))
Answer: \( \sec \alpha = 13/12 \)

Question. If \( \sec^2 \theta (1 + \sin \theta) (1 - \sin \theta) = k \), find the value of k
Answer: \( \sec^2 \theta (1 + \sin \theta) (1 - \sin \theta) = \sec^2 \theta (1 - \sin^2 \theta) = \sec^2 \theta \cos^2 \theta = 1 \)

Question. If \( \sin \theta = \frac{1}{3} \), then find the value of \( 2 \cot^2 \theta + 2 \).
Answer: \( 2 \cot^2 \theta + 2 = 2(\cot^2 \theta + 1) = 2 \csc^2 \theta = 2 \cdot 3^2 = 18 \)

Question. What is the value of \( \theta \), if \( \sqrt{3} \sin \theta = \cos \theta \)
Answer: \( \theta = 30^\circ \)

CASE STUDY QUESTIONS

Case Study -1
Golf is a game played in an open field where the golfer plays his golf ball into a hole by using different types of clubs (golf instruments). In golf, a golfer plays a number of holes in a given order. 18 holes played in an order controlled by the golf course design, normally make up a game.
On your approach shot to the ninth green, the Global Positioning System (GPS) your cart is equipped with tells you the pin is 120 meter away.

Question. The distance plate states the straight line distance to the hole is 60 meter. Relative to a straight line between the plate and the hole, at what acute angle should you hit the shot?
Answer: 60°

Question. What is the value of the tangent of the above angle?
Answer: \( \sqrt{3} \)

Question. What is the length of the side opposite to the angle \( \theta \) in the given picture ?
Answer: 103.9 m

Case Study-II
A heavy-duty ramp is used to winch heavy appliances from street level up to a warehouse loading dock. If the ramp is 2 meter high and the incline is 4 meter long. (Use \( \sqrt{3} = 1.73 \))

Question. What angle does the dock make with the street?
Answer: 30°

Question. How long is the base of the ramp? ( In round figure)
Answer: 3.5 m

Question. If the ramp is inclined at the angle of \( 45^\circ \), what is the height of the ramp ? Use \( \sqrt{2} = 1.41 \)
Answer: 2.82 m