CBSE Class 10 Mathematics Some Applications of Trigonometry Worksheet Set 06

SECTION A

Choose and write the correct option in the following questions.

Question. A boy is standing at the top of the tower and another boy is at the ground at some distance from the foot of the tower, then the angle of elevation and depression between the boys when both boys look at each other will be
(a) equal
(b) angle of elevation will be greater
(c) cannot be predicted
(d) angle of depression will be greater
Answer: (a) equal

Question. The angle of elevation of a ladder leaning against a wall is 60° the foot of the ladder is 12.4 m away from the wall. The length of the ladder is
(a) 14.8 m
(b) 6.2 m
(c) 12.4 m
(d) 24.8 m
Answer: (d) 24.8 m

Solve the following questions.

Question. An observer 1.5 m tall is 20.5 m away from a tower 22 m high. Determine the angle of elevation of the top of the tower from the eye of the observer.
Answer: 45°

Question. If the angle of elevation of top of a tower from a point on the ground which is \( 20\sqrt{3} \) m away from the foot of the tower is 30°, find the height of the tower.
Answer: 20 m

SECTION B

Solve the following questions.

Question. In the \( \Delta ABC \) shown below \( \angle x : \angle y = 1 : 2 \). What is the value of \( \tan x \)? 
Answer: \( \frac{1}{\sqrt{3}} \)

Question. Find an acute angle \( \theta \), when \( \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} = \frac{1 - \sqrt{3}}{1 + \sqrt{3}} \).
Answer: 60°

Question. Evaluate: \( \frac{\cos 60^\circ - \cot 45^\circ + \csc 30^\circ}{\sec 60^\circ + \tan 45^\circ - \sin 30^\circ} \)
Answer: -1

Question. If \( A = 30^\circ \) and \( B = 30^\circ \), verify that \( \sin (A + B) = \sin A \cos B + \cos A \sin B \).
Answer: 1

Solve the following questions.

Question. If the length of the ladder placed against a wall is twice the distance between the foot of the ladder and the wall. Find the angle made by the ladder with the horizontal.
Answer: 60°

Question. The shadow of a tower standing on a level plane is found to be 50 m longer when Sun's elevation is 30° than when it is 60°. Find the height of the tower. 
Answer: \( 25\sqrt{3} \) m

Question. The angles of elevation of the top of a tower from two points distant \( s \) and \( t \) from its foot are 30° and 60°. Prove that the height of the tower is \( \sqrt{st} \). 
Answer: height \( = \sqrt{st} \)

Solve the following questions.

Question. A man standing on the deck of a ship, which is 10 m above water level, observes the angle of elevation of the top of a hill as 60° and the angle of depression of the base of hill as 30°. Find the distance of the hill from the ship and the height of the hill. 
Answer: \( 10\sqrt{3} \) m, 40 m

Question. An observer finds the angle of elevation of the top of a tower from certain point is 30°. If the observer moves 20 m towards the tower, the angle of elevation of the top increases by 15°. Find the height of the tower. 
Answer: \( 10(\sqrt{3} + 1) \) m

Question. The angles of depression of the top and bottom of a 50 m high building from the top of a tower are 45° and 60° respectively. Find the height of the tower and the horizontal distance between the tower and the building. (Use \( \sqrt{3} = 1.73 \)). 
Answer: 118.25 m, 68.25 m

Solve the following questions.

Question. A man observes a car from the top of a tower, which is moving towards the tower with a uniform speed. If the angle of depression of the car changes from 30° to 45° in 12 minutes, find the time taken by the car now to reach the tower.
Answer: 16 minutes 23 seconds

Solve the following questions.

Question. The angles of depression of the top and bottom of a building 50 metres high as observed from the top of a tower are 30° and 60°, respectively. Find the height of the tower and also the horizontal distances between the building and the tower.
Answer: \( 75 \) m, \( 25\sqrt{3} \) m

Question. An aeroplane is flying at a height of 300 m above the ground. Flying at this height, the angles of depression of the top of a hill as 30° and 45° respectively. Find the width of the river. [Use \( \sqrt{3} = 1.732 \)] 
Answer: 473.2 m