# Important Questions for CBSE Class 10 Maths Chapter 9 - Some Applications of Trigonometry 2 Mark Question

## CBSE Class 10 Maths Chapter-9 Some Applications of Trigonometry – Free PDF Download

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CBSE Class 10 Maths Chapter-9 Some Applications of Trigonometry Important Questions

## 2 Mark Questions

1. A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is  (see figure).

Ans.
In right triangle ABC,

AB = 10 m
Hence, the height of the pole is 10 m.

2. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle  with it. 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.

Ans. In right triangle ABC,

m
Again,

AB =  m
Height of the tree = AB + AC
=
m

3. A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m and is inclined at an angle of  to the ground, whereas for elder children, she wants to have a steep slide at a height of 3 m and inclined at an angle of  to the ground. What should be the length of the slide in each case?

Ans. In right triangle ABC,

AC = 3 m
In right triangle PQR,

PR =  m
Hence, the lengths of the slides are 3 m and  m respectively.

4. The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the lower is . Find the height of the tower.

Ans. In right triangle ABC,

AB =  m
=  m

5. A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is Find the length of the string, assuming that there is no slack in the string.

Ans. In right triangle ABC,

AC =  m
Hence the length of the string is  m.

6. A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is From another point 20 m away from this point on the line joining this point to the foot of the tower, the angle of elevation of the top of the tower is  (see figure). Find the height of the tower and the width of the canal.

Ans. In right triangle ABC,

AB =  m ……….(i)
In right triangle ABD,

AB =  m ……….(ii)
From eq. (i) and (ii),
=
3BC = BC + 20
BC = 10 m
From eq. (i), AB =  m
Hence height of the tower is  m and the width of the canal is 10 m.

7. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is  and the angle of depression of its foot is  Determine the height of the tower.

Ans. In right triangle ABD,

BD = 7 m
AE = 7 m
In right triangle AEC,

CE =  m
CD = CE + ED
= CE + AB

m
Hence height of the tower is m.

8. A round balloon of radius r subtends an angle at the eye of the observer while the angle of the elevation of its centre is. Prove that the height of the centre of the balloon is r sincosec

Ans.
In right

In right

9. An aeroplane flying horizontally 1 km above the ground is observed at an elevation of. After 10 seconds, its elevation is observed to. Find the speed of the aeroplane in km/hr.

Ans.
In right

In right

Speed =

10. From a window (h m high above the ground) of a house in a street, the angles of elevation and depression of the top and foot of another house on the opposite side of the street are and respectively, show that the height of the opposite house is

Ans.
Let DE = h m
DC = x m
In right

In right

Hence Proved.

11. A ladder rests against a wall at an angle  to the horizontal. Its foot is pulled away from the wall through a distance ‘a’ so that it slides a distance ‘b’ down the wall making an angle ‘’ with the horizontal. Show that

Ans.
In right

Similarly, in right

Hence Proved.

12. A vertical tower stands on a horizontal plane and surmounted by vertical flagstaff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flagstaff are and respectively, Prove that the height of the tower is

Ans.
Let AB = Height of tower = H
Let BC = Height of flagstaff = h
In right-angled triangle OAB and OAC,

And
Equating value of x, we get

13. The angle of elevation of the top of a tower at a point on the level ground is 30. After walking a distance of 100 m towards the foot of the tower along the horizontal line through the foot of the tower on the same level ground the angle of elevation to the top of the tower is 60, find the height of the tower.

Ans. In ,

In

14. As observed from the top of light house 100 m high above sea level the angle of depression of a ship sailing directly towards it changes from 30 to 60. Determine the distance travelled by the ship during the period of observation. (Use )

Ans.
Let PQ be the light house such that PQ = 100 m. Let A and B be the positions of ship when the angle of depression changes from 30to 60respectively.
Let and BP = y m
From right-angled

From right-angled triangle BPQ,

From (i) and (ii),

15. The angles of elevation of the top of a tower from two points P and Q at distances of a and b respectively from the base and in the same straight line with are complementary. Prove that the height of the tower is  where a > b.

Ans.
Let AB be tower of height h. Let P and Q be the given points in the same straight line with the foot B of the tower.
Let BP = a, BQ = b

From right angled

From right-angled

Multiplying (i) and (ii), we get

16. An aeroplane flying horizontally at a height of 1.5 km above the ground is observed at a certain point on the earth to subtend an angle of 60. After 15 seconds, its angle of elevation is observed to be 30. Calculate the speed of aeroplane in km/hr.

Ans.
Let O be the observation point.
Let A be the position of aeroplane such that and
Let B be the position of aeroplane after 15 seconds.
In right

In right

Distance covered in 15 seconds
Distance covered in 1 second =
Distance covered in 3600 seconds =

17. A man is standing on the deck of a ship which is 25 m above water level. He observes the angle of elevation of the top of a lighthouse as 60 and the angle of depression of the base of the lighthouse as 45. Calculate the height of the lighthouse.

Ans.
H = Height of lighthouse = h + 25 …….(i)
In right

In right

Now

18. An aeroplane when flying at a height of 3125 m form the ground passes vertically below another aeroplane at an instant when the angle of elevation of the two planes from the same point on the ground are 30 and 60 respectively. Find the distance between the two aeroplanes at that instant.

Ans.
Let and be the positions of the two aeroplanes
Let
And OM = y

Also

19. A boy standing on a horizontal plane finds a bird flying at a distance of 100 m from him at an elevation of 30. A girl standing on the roof of 20 m high building finds the angle of elevation of the same bird to be 45. Both the boy and the girl are on opposite side of the bird. Find the distance of the bird from the girl.

Ans.
Positions of bird at A, boy at P and girl at B are as shown in figure.
In

Also BC = DQ = 20m

In

Hence, the bird is 30m away from the girl.

20. At a point on level ground, the angle of a elevation of a vertical tower is found to be such that its tangent is on walking 19.2 m towards the tower, the tangent of the angle of elevation is . Find the height of tower.

Ans.
Suppose height of tower is h meter
In

In

21. From the top of a building 60 m high, the angles of depression of the top and bottom of a vertical lamp post are observed to be 30 and 60 respectively. Find:
(i) the horizontal distance between the building and the lamp post
(ii) the height of the lamp post. [Take ]

Ans.
Suppose height of lamppost is h meter.
In

In right

By eq. (i) and (ii)
(60 – h)= 20

22. A man standing on the deck of a ship which is 10 m above the water level 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. Calculate the distance of the hill from the ship and the height of the hill.

Ans.
Let H = Height of hill

In right

In right

Equating the values of x, we get

From H = 10 + h = 10 + 30 = 40 m
And x = distance of hill from ship = 10m

23. The angle of elevation of a jet-plane from a point P on the ground is 60. After a flight of 15 seconds the angle of elevation (change to 30). If the jet plane is flying at a constant height of, find the speed of the jet plane in km/hour.

Ans.
Let A be the point on the ground E is the position of aeroplane such that and
C is the position of plane after 15 seconds

In right

In

Distance

Speed =

24. A pole 5 m high is fixed on the top of a tower. The angle of elevation of the top of pole observed from a point A on the ground is 60 and the angle of depression of the point A from the top of the tower is 45. Find the height of the tower. (Take )

Ans.
In

In

25. From a window 15 m high above the ground in a street. The angles of elevation and depression of the top and foot of another house on the opposite side of the street are 30 and 45 respectively. Show that the height of the apposite house is 23.66 m. [Take ]

Ans.
Suppose and
Then, AB =
In

In

Then

Height of another house =

26. The angle of elevation of the top of a tower as observed from a point on the ground is and on moving ‘a’ meter towards the tower. The angle of elevation is prove that height of tower is

Ans.
Let AB b tower and height of tower = h m
In

In

27. A TV tower stands vertically on a bank of a canal. From a point on the other bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is 60from a point 20 m away from this point on the same back the angle of elevation of the top of the tower is 30. Find the height of the tower and the width of the canal.

Ans.
Let h be the height of tower and x be the width of the river
In

In

Equating (i) and (ii),

Put in (i),