CHAPTER 09 SOME APPLICATIONS OF TRIGONOMETRY

SUBJECT: MATHEMATICS MAX. MARKS : 40

CLASS : X DURATION : 1½ hrs
General Instructions:
(i). All questions are compulsory.
(ii). This question paper contains 20 questions divided into five Sections A, B, C, D and E.
(iii). Section A comprises of 10 MCQs of 1 mark each. Section B comprises of 4 questions of 2 marks
each. Section C comprises of 3 questions of 3 marks each. Section D comprises of 1 question of 5
marks each and Section E comprises of 2 Case Study Based Questions of 4 marks each.
(iv). There is no overall choice.
(v). Use of Calculators is not permitted

SECTION – A
Questions 1 to 10 carry 1 mark each.
1. If 300 m high pole makes an angle of elevation at a point on ground which is 300 m away from
its foot, then the angle of elevation is:
(a) 60° (b) 90° (c) 30° (d) 45°

2. The angle of depression of a bike parked on the road from the top of a 90 m high pole is 60
degrees. The distance of the bike from the pole is:
(a) 20√3 m (b) 90 m (c) 15√3 m (d) 30√3 m

3. A stone is 15√3 m away from a tower 15 m high, then the angle of elevation of the top of the
tower from the stone is:
(a) 45° (b) 60° (c) 30° (d) 90°

4. The ratio of the length of a tower and its shadow is √3 : 1. The altitude of the sun is:
(a) 0° (b) 60° (c) 30° (d) 45°

5. The tops of the poles of height 16 m and 10 m are connected by a wire of length l meters. If the
wire makes an angle of 30° with the horizontal, then l =
(a) 26 m (b) 16 m (c) 12 m (d) 10 m

6. The tops of two poles of heights 20 m and 14 m are connected by a wire. If the wire makes an
angle of 30° with the horizontal, then the length of the wire is
(a) 8 m (b) 10 m (c) 12 m (d) 14 m

7. If the angle of depression of an object from a temple is 30°, and the distance of the object from
the temple is 45 m, then the height of the temple is:
(a) 45√3 m (b) 15√3 m (c) 20 m (d) 20√3 m

8. If two towers of heights h1 and h2 subtend angles of 60° and 30° respectively at the mid-point of
the line joining their feet, then h1 : h2 =
(a) 1 : 2 (b) 1 : 3 (c) 2 : 1 (d) 3 : 1

In the following questions 9 and 10, 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.
9. Assertion (A): If the length of shadow of a vertical pole is equal to its height, then the angle of
elevation of the sun is 45º.
Reason (R): According to Pythagoras theorem, h² = l² + b², where h = hypotenuse, l = length and
b = base.

10. Assertion (A): The ladder 20 m long makes an angle 60° with the wall, then the height of the
point where the ladder touches the wall is 15 m.
Adjacent Side
Reason (R): For an angle θ, cos  
Hypotenuse

SECTION – B
Questions 11 to 14 carry 2 marks each.
11. The angle of depression of a car standing on the ground, from the top of a 85 m high tower is
45º. Find the distance of the car from the base of the tower.

12. A pole casts a shadow of length 2√3 m on ground, when the sun’s elevation is 60°. Find the
height of the pole.

13. The figure shows the observation of point C from point A. Find the angle of depression from A.

14. The shadow of a flagstaff is three times as long as the shadow of the flagstaff when the sunrays
meet the ground at an angle of 60°. Find the angle between the sunrays and the ground at the
time of longer shadow.

SECTION – C
Questions 15 to 17 carry 3 marks each.

15. A man rowing a boat away from a lighthouse 150 m high takes 2 minutes to change the angle of
elevation of the top of lighthouse from 45° to 30°. Find the speed of the boat. (Use 3 = 1.732)
0.53 m/s
16. A man on the deck of a ship, 12 m above water level, observes that the angle of elevation of the
top of a cliff is 60° and the angle of depression of the base of the cliff is 30°. Find the distance of
the cliff from the ship and the height of the cliff. [Use 3 = 1.732] 12 rt 3, 48

17. As observed from the top of a 100 m high light house from the sea-level, the angles of
depression of two ships are 30⁰ and 45⁰. If one ship is exactly behind the other on the same side
of the light house, find the distance between the two ships [Use √3 = 1.732]
157.73
SECTION – D
Questions 18 carry 5 marks.
18. At a point A, 20 metres above the level of water in a lake, the angle of elevation of a cloud is
30°. The angle of depression of the reflection of the cloud in the lake, at A is 60°. Find the
distance of the cloud from A.
 SECTION – E (Case Study Based Questions)
Questions 19 to 20 carry 4 marks each.

19. A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m
from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is
60°. After 30 seconds, the angle of elevation reduces to 30° (see the below figure).

Based on the above information, answer the following questions. (Take √3 =1.732) 101.85
(i) Find the distance travelled by the balloon during the interval. (2) 3.36
(ii) Find the speed of the balloon. (2)
OR
(ii) If the elevation of the sun at a given time is 30°, then find the length of the shadow cast by a
tower of 150 feet height at that time. (2)
259.8
20. Anita purchased a new building for her business. Being in the prime location, she decided to
make some more money by putting up an advertisement sign for a rental ad income on the roof
of the building.

From a point P on the ground level, the angle of elevation of the roof of the building is 30° and
the angle of elevation of the top of the sign board is 45°. The point P is at a distance of 24 m
from the base of the building.
On the basis of the above information, answer the following questions:
(i) Find the height of the building (without the sign board). 8 rt 3 (2)
OR
Find the height of the building (with the sign board) 24 (2)
(ii) Find the height of the sign board. 10 (1)
(iii) Find the distance of the point P from the top of the sign board. (1) 24 rt2