CBSE Class 10 Some Applications of Trigonometry Sure Shot Questions Set 05

MCQ

Question. The height of tower is-
(a) \( 50\sqrt{3} \)
(b) \( 50/\sqrt{3} \)
(c) \( 50 + \sqrt{3} \)
(d) \( 50 - \sqrt{3} \)
Answer: (b) \( 50/\sqrt{3} \)

Question. A plane is observed to be approaching the airport. It is at a distance of 12 km from the point of observation and makes an angle of elevation of 60°. The height above the ground of the plane is
(a) \( 6\sqrt{3} \text{ m} \)
(b) \( 4\sqrt{3} \text{ m} \)
(c) \( 3\sqrt{3} \text{ m} \)
(d) \( 2\sqrt{3} \text{ m} \)
Answer: (a) \( 6\sqrt{3} \text{ m} \)

Question. Two poles are 25 m and 15 m high and the line joining their tops makes an angle of 45° with the horizontal. The distance between these poles is
(a) 5 m
(b) 8 m
(c) 9 m
(d) 10 m
Answer: (d) 10 m

Question. The shadow of a tower is equal to its height at 10:45 a.m. The sun’s altitude is
(a) 30°
(b) 45°
(c) 60°
(d) 90°
Answer: (b) 45°

Question. When the length of shadow of a vertical pole is equal to \( \sqrt{3} \) times of its height, the angle of elevation of the Sun’s altitude is
(a) 30°
(b) 45°
(c) 60°
(d) 15°
Answer: (a) 30°

Assertion and Reasoning questions

DIRECTION: In the following questions (Q1-10), a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) if both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) if both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) if Assertion (A) is true but reason (R) is false.
(d) if Assertion (A) is false but reason (R) is true.

Question. Two poles of equal heights are standing opposite to each other, on either side of the road, which is 80m wide. From a point between them on the road, the angles of elevation of top of the poles are 60° and 30° respectively. Find the height of the poles.
Assertion (A) : First pole is nearer.
Reason (R): Either one of angle of elevation is required
(a) A
(b) B
(c) C
(d) D
Answer: (d) D

Question. An Aeroplane is flying horizontally 4000 m above the ground and is going away from an observer on the level ground. At a certain instant the observer finds that the angle of elevation of the plane is 45°. After 15 seconds, its elevation from the same point changes to 30°. Find the speed of the aeroplane in km/h.
Assertion (A): Distance between the plane and observer is required.
Reason (R): Time 15 second is sufficient to calculate
(a) A
(b) B
(c) C
(d) D
Answer: (d) D

Question. A man is standing on the deck of a ship which is 8 m above water level. He observes the angle of elevation of the top of a hill as 60° and the angle of depression of the base of the hill as 30°.
Assertion (A): The height of hill from the ship can be measured.
Reason (R): When the ship is in move
(a) A
(b) B
(c) C
(d) D
Answer: (c) C

Short Answer Type – I  

Question. A 25-foot-tall flagpole casts a 25-foot shadow. What is the angle that the sun hits the flagpole?
Answer: 45°

Question. Mark is flying a kite and realizes that 300 feet of string are out. The angle of the string with the ground is 45°. How high is Mark's kite above the ground?
Answer: \( 150\sqrt{2} \text{ ft} \)

Question. The angle of elevation of the top of the building from the foot of the tower is 30° and the angle of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.
Answer: 50/3 m

Question. The shadow of a tower standing on a level ground is found to be 40 m longer when Suns altitude is 30° than when it was 60°. Find the height of the tower.
Answer: \( 20\sqrt{3} \text{ m} \)

Short Answer Type – II  

Question. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle of 30° with the ground. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree
Answer: \( 20/\sqrt{3} \text{ m} \)

Question. A man on the top of the vertical tower observes a car moving at a uniform speed coming directly towards it. If it takes 12 minutes for the angle of depression to change its measure from 30° to 45°, how soon after this, will the car reach the tower?
Answer: \( (18 + 6\sqrt{3}) \text{ min} \)

Question. The ratio of the height of a tower and the length of its shadow on the ground is \( \sqrt{3} : 1 \). What is the angle of elevation of the sun?
Answer: 60°

Question. A vertical tower stands on horizontal plane and is surmounted by a vertical flagstaff of height h m. At a point on the plane, the angle of elevation of the bottom of the flagstaff is \( \alpha \) and that of the top of the flagstaff is \( \beta \). Prove that the height of the tower is \( \frac{h \tan \alpha}{\tan \beta - \tan \alpha} \)
Answer: NA

Question. From a point on a bridge across a river the angle of depression of the banks on opposite sides of the river are 30° and 45° respectively. If the bridge is at the height of 30 m from the banks, Find the width of the river.
Answer: \( 30(\sqrt{3} + 1) \text{ m} \)

Long Answer Type  

Question. Two poles of equal heights are standing opposite to each other, on either side of the road, which is 80m wide. From a point between them on the road, the angles of elevation of top of the poles are 60° and 30° respectively. Find the height of the poles.
Answer: \( 20\sqrt{3} \text{ m} \)

Question. At the foot of a mountain, the elevation of its summit is 45°. After ascending 1000 m towards the mountain up a slope of 30° inclination, the elevation is found to be 60°. Find the height of the mountain
Answer: \( \frac{1000}{\sqrt{3} - 1} \text{ m} \)

Question. The angles of depression of the top and bottom of 8 m tall building from the top of a multi storeyed building are 30° and 45° respectively. Find the height of the multi-storeyed building and the distance between the two buildings.
Answer: \( (12 + 4\sqrt{3}) \text{ m} \)

Question. The horizontal distance between two buildings is 140 m. The angle of depression of the top of the first building when seen from the top of the second building is 30°. If the height of the first building is 60 m, find the height of the second building
Answer: \( 12(\sqrt{3} + 1) \text{ m}, 4\sqrt{3}(\sqrt{3} + 1) \text{ m} \)

Question. The shadow of a tower at a time is three times as long as its shadow when the angle of elevation of the sun is 60°. Find the angle of elevation of the sun at the time of the longer shadow
Answer: 30°

CASE STUDY BASED QUESTIONS

A flag pole ‘h’ metres is on the top of the hemispherical dome of radius ‘r’ metres. A man is standing 7 m away from the dome. Seeing the top of the pole at an angle 45° and moving 5 m away from the dome and seeing the bottom of the pole at an angle 30°.

Question. Find (i) the height of the pole (ii) radius (height) of the dome (iii) Is it possible to see the pole at the angle of 60° (iv) If the height of pole is increased, the angle elevation will …..
Answer: (i) 7 (ii) \( 6(\sqrt{3} + 1) \) (iii) NO (iv) Decrease

A man is watching a boat speeding away from the top of a tower. The boat makes an angle of depression of 60° with the man’s eye when at a distance of 200 m from the tower. After 10 seconds, the angle of depression becomes 45°.

Question. i. What is the approximate speed of the boat (in km / hr), assuming that it is sailing in still water? ii. How far is the boat when the angle is 45° iii. What is the height of tower is iv. As the boat moves away from the tower, angle of elevation will decrease/ increase?
Answer: i. 53 km/hr approx ii. 147m approx iii. \( 200\sqrt{3} \text{ m} \) iv. decrease

Two trees are standing on flat ground. The angle of elevation of the top of Both the trees from a point X on the ground is 60°. If the horizontal distance between X and the smaller tree is 8 m and the distance of the top of the two trees is 20 m, calculate –

Question. i. the distance between the point X and the top of the smaller tree. ii. the horizontal distance between the two trees. iii. Height of big tree …………. iv. Height of small tree is ……….
Answer: i. 16 m ii. 10m iii. \( 18\sqrt{3} \text{ m} \) iv. \( 8\sqrt{3} \text{ m} \)

Radio towers are used for transmitting a range of communication services including radio and television. The tower will either act as an antenna itself or support one or more antennas on its structure, including microwave dishes. They are among the tallest human made structures. There are 2 main types: guyed and Self supporting structures. On a similar concept, a radio station tower was built in two sections A and B. Tower is supported by wires from a point O. Distance between the base of the tower and point O is 36 m. From point O, the angle of elevation of the top of section B is 30° and the angle of elevation of the top of section A is 45°.

Question. i. What is the height of the section B? ii. What is the height of the section A? iii. What is the length of the wire structure from the point O to the top of section A? iv. What is the angle of depression from top of tower to point O ?
Answer: i. (a) ii. (c) iii. (d) iv. (b)

Navy officer Mr. Colin is tasked with planning a coup on the enemy at a certain date. Currently he is inspecting the area standing on top of the cliff. Agent Dev is on a chopper in the sky. When Mr. Colin looks down below the cliff towards the sea, he has Bhawani and Amar in boats. Amar is behind the Bhawani boat.

Question. (i) Which of the following is a pair of angle of elevation?
(a) \( (\angle a, \angle e) \)
(b) \( (\angle b, \angle e) \)
(c) \( (\angle c, \angle d) \)
(d) \( (\angle a, \angle f) \)
Answer: (b) \( (\angle b, \angle e) \)

Question. (ii) Which of the following is a pair of angle of depression?
(a) \( (\angle a, \angle e) \)
(b) \( (\angle b, \angle e) \)
(c) \( (\angle c, \angle d) \)
(d) \( (\angle a, \angle f) \)
Answer: (c) \( (\angle c, \angle d) \)

Question. (iii) If angle of depression of Colin to Bhawani is 30°, what is the distance of Amar boat from the Bhawani boat?
(a) \( \frac{\sqrt{3}}{2}h \)
(b) \( \frac{h}{\sqrt{3}} \)
(c) \( \frac{2h}{\sqrt{3}} \)
(d) \( \sqrt{3}h \)
Answer: (b) \( \frac{h}{\sqrt{3}} \)

Question. (iv) If angle of depression of Dev to Colin is 60°, what is the height of Dev from base of hill ?
(a) h
(b) 2h
(c) 3h
(d) 4h
Answer: (d) 4h