**( 3 Marks Questions )**

**Q.1Â Â Â Â 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 60Â° and angle of depression of the base of the hill as 30Â°. Find the distance of the hill from the ship and height of the hill.**

**[AI 2006]**

In

m

In

So, Height of the hill m

Distance of the hill from the ship m .

**Q.2Â Â Â Â The angle of elevation of the top of a tower from a point A on the ground is 30Â°. On moving a distance of 20 meters towards the foot of the tower to a point B, the angle of elevation increases to 60Â°. Find the height of the tower and distance of the tower from the point A. .**

**[Delhi 2006C]**

* Sol.Â Â Â Â Â Â *Let height of the tower be h and BQ

In

â€¦â€¦â€¦(1)

In

[Using equation (1)]

From equation (1)

m

So, Height of the tower m.

Distance of tower from point A m

**Q.3 A pole 5 m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point â€˜Aâ€™ on the ground is 60Â° and the angle of depression of the point â€˜Aâ€™ from the top of the tower is 45Â°. Find the height of the tower. **

**[AI 2007]**

* Sol. Â Â Â * Let PQ is the tower of height h, PR be pole and AQ = x

In

â€¦â€¦â€¦â€¦ (1)

In

[Using equation (1)]

1} \right)} \over {{{(\sqrt 3 )}^2} - {{(1)}^2}}} = {{5 \times 2.73} \over 2} = 6.825m" />

m [Using equation (1)]

So, height of the tower m

**Q.4Â Â Â Â A statue 1.46 m tall stand on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60Â° and from the same point, the angle of elevation of the top of the pedestal is 45Â°. Find the height of the pedestal. **

**[AI 2008]**

* Sol.*Â Â Â Â Let height of pedestal = h and distance between point of observation & pedestal = x

In

â€¦â€¦â€¦ (1)

In

[Using equation (1)]

**Q.5Â Â Â Â A person standing on the bank of a river observes that the angle of the elevation of the top of a tree standing on the opposite bank is 60Â°. When he moves 40 m away from the bank, he finds the angle of elevation to be 30Â°. Find the height of the tree and the width of the river. **

**[Delhi 2008]**

* Sol.*Â Â Â Â Let height of the tree = h, Width of the river = x

In

â€¦â€¦ (1)

In

[Using equation (1)]

m

On putting x=20 in(1),We get

m

**Q.6Â Â Â Â An aeroplane when flying at a height of 3125 m from the ground passes vertically below another plane at an instant when the angles of elevation of the two planes from the same point on the ground are 30Â° and 60Â° respectively. Find the distance between the two planes at that instant.**

**[AI 2009]**

* Sol.*Â Â Â Â Let A & D are two aeroplanes and AB m

In

In

(Substituting value of BC)

m

**Q.7Â Â Â Â The angle of elevation of the top of a building from the foot of a tower is 30Â° and the angle of elevation of the top of the tower from the foot of the building is 60Â°. If the tower is 50 m high. Find the height of the building. **

**[Foreign 2009]**

* Sol*.Â Â Â Â Let AB be the building and PQ be the tower

In

...............(1)

In

[Using (1)]

m

**Q.8Â Â Â Â 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 60Â° and the angle of depression of the base of the cliff is 30Â°. Find the distance of the cliff from the ship and the height of the cliff. **

**[AI 2010]**

* Sol.*Â Â Â Â Let AB be the cliff and PQ be the ship

In

m

In

[Because PM=x]

m

Height of the cliff = AM + BM m

**Q.9Â Â Â Â From the top of a tower 100 m high, a man observes two cars on the opposite sides of the tower with angles of depression 30Â° and 45Â° respectively. Find the distance between the cars. **

**[Delhi 2011]**

* Sol.*Â Â Â Â Let PQ be the tower and from A and B two man observe the tower.

In

m

In

m

m

**Q.10Â Â Â Â A ladder of length 6 m makes an angle of 45Â° with the floor while leaning against one wall of a room. If the foot of the ladder is kept fixed on the floor and it is made to lean against the opposite wall of the room, it makes an angle of 60Â° with the floor. Find the distance between these two walls of the room.**

**[Foreign 2011]**

* Sol.*Â Â Â Â Â Let AB & CD be two opposite walls & the ladder is fixed at point O.

In

m

In

m

Distance between two walls = BO + DO m

**Q.11Â Â Â Â The horizontal distance between two poles is 15 m. The angle of depression of the top of first pole as seen from the top of second pole is 30Â°. If the height of the second pole is 24 m, find the height of the first pole. **

**[Delhi 2013]**

* Sol.*Â Â Â Â Â Let AB & CD be two poles

Let AB = x

Therefore MD = AB = x

And CM = CD â€“ MD

Now,AM = BD = 15 m

In

m

Â **( 5 Marks Questions )**

Â **Q.1Â Â The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is 60Â°. At a point Y, 40 m vertically above X, the angle of elevation is 45Â°. Find the height of the tower PQ and the distance XQ.**

**[Foreign 2006]**

Let height of the tower (PQ) = h

PM = YX

And MQ = PQ â€“ PM

Let PX = MY = x

In

â€¦â€¦.. (1)

In

â€¦â€¦â€¦â€¦.(2)

From equation (1) & (2)

...........(3)

[Using (3)]

**Q.2Â Â Â Â A man on the top of a 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 from 30Â° to 45Â° how soon after this, will the car reach the tower ?**

**[AI 2006C]**

**Sol.**

Let the tower be PQ and from point A, its angle of elevation is 30Â° after 12 min from point B its angle of elevation is 45Â°.

Let the height of the tower be h & the distance between BQ = x

* *

Let the speed of the car be y m/min

[Because ]

In

â€¦â€¦â€¦.. (1)

In

â€¦â€¦â€¦â€¦. (2)

From equation (1) & (2)

min

**Q.3Â Â Â Â A boy standing on a horizontal plane finds a bird flying at a distance of 100 m from him at an elevation of 30Â°. A girl standing on the roof of 20 m high building, finds the angle of elevation of the same bird to be 45Â°. Both the boy and the girl are on opposite sides of the bird. Find the distance of bird from the girl.**

**[Delhi 2007]**

Let CD = X

In

In

Now, in

Therefore, distance of bird from the girl = 42.42 m.

**Q.4Â Â Â Â The angle of elevation of a jet fighter from a point A on the ground is 60Â°. After a flight of 15 seconds, the angle of elevation changes to 30Â°. If the jet is flying at a speed of 720 km/hr, find the constant height. **

**[AI 2008]**

**Sol.**

Speed of jet fighter

Distance covered in 15 sec = 200 Ã— 15 = 3000 m

QC = PB = 3000 m

AC = AQ + QC = y + 3000

In

â€¦â€¦â€¦â€¦ (1)

In

â€¦â€¦â€¦. (2)

From equation (1) & (2)

On putting in (1),We get

= 2598 m

Â **Q.5 Â The angle of elevation of an aeroplane from a point A on the ground is 60Â°. After a flight of 30 seconds, the angle of elevation changes to 30Â°. If the plane is flying at a constant height of 3600 m, find the speed in km / hr of the plane.**

**[Foreign 2008]**

* Sol.Â Â Â Â *Let OS = x And SQ = y

In

............(1)

In

..............(2)

On putting in (2) ,We get

3600 + y = 10800

km/hr

**Q.6Â Â Â A straight highway leads to the foot of the tower. A man standing at the top of the tower observes a car at angle of depression of 30Â°, which is approaching the foot of the tower with a uniform speed, 6 second later the angle of depression of the car is found to be 60Â°. Find the time taken by the car to reach the foot of the tower from this point.**

**[Foreign 2008]**

Let height of tower PQ be h

A is initial point & B is final point

In

â€¦â€¦â€¦â€¦. (1)

In

â€¦â€¦â€¦â€¦.(2)

From equation (1) and (2)

Therefore,Time taken = seconds

**Q.7Â Â Â Â A pole of height 5 m is fixed on the top of a tower. The angle of elevation of the top of the pole as observed from a point A on the ground is 60Â° and the angle of depression of the point A from the top of the tower is 45Â°. Find the height of the tower. **

**[Delhi 2009]**

* Sol.*Â Â Â Â Let height of the tower be h

In

â€¦â€¦â€¦(1)

In

â€¦â€¦â€¦â€¦ (2)

From equation (1) & (2)

**Q.8Â Â Â Â 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 30Â° and 60Â° respectively. Find**

** (i) The horizontal distance between the building and the lamp post.**

** (ii) The height of the lamp post, **

**[AI 2009]**

* Sol.*Â Â Â Â Let PQ be building and AB be lamp post.

In

In

MQ = PQ â€“ PM = 60 â€“ 20 = 40 m

AB = MQ = 40 m

**Q.9Â Â Â Â Â 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 that instants is 60Â°. After some time the angle of elevation reduces to 30Â°. Find the distance travelled by the balloon during the internal.**

**[AI 2009]**

* Sol.*Â Â Â Â Let initially balloon is at R and its final position is at P.

In

In

RP = ST

Therefore ,Distance covered by balloon

**Q.10Â Â Â Â A tower stand 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 60Â°, 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 30Â°. Find the height of the tower and the width of the canal.**

**[Foreign 2009]**

* Sol.*Â Â Â Â Â Â Let PQ be the tower and BQ be x width of canal

Suppose PQ = h

And BQ = x

In

â€¦â€¦â€¦ (1)

In

â€¦â€¦â€¦â€¦. (2)

From equation (1) & (2)

On putting x=10 in (1),We get,

Therefore, height of tower = 17.3 m

And Width of canal = 10 m

**Q.11Â Â Â Â The angle of elevation of a cloud from a point 60 m above a lake is 30Â° and the angle of depression of the reflection of the cloud in the lake is 60Â°. Find the height of the cloud from the surface of the lake.**

**[AI 2010]**

**Sol.**

Let the height of the cloud C from the lake be hm. A is position of the point 60 m above the lake. D is the reflection of the cloud in lake.

Let AE = x and CF = h

CE = h â€“ 60

And DE = 60 + h

In

â€¦â€¦â€¦â€¦â€¦ (1)

In

â€¦â€¦â€¦ (2)

From equation (1) & (2)

Therefore, height of the cloud = 120 m

**Q.12Â Â Â Â The angle of elevation of the top of a vertical tower from a point on the ground is 60Â°. From another point 10 m vertically above the first, its angle of elevation is 30Â°. Find the height of the tower.**

**[AI 2012]**

* Sol.*Â Â Â Â Â Â Â Let height of the tower be h

In

â€¦â€¦â€¦.. (1)

In

â€¦â€¦â€¦â€¦â€¦ (2)

From equation (1) & (2)

**Q.13Â Â Â The angles of elevation and depression of the top and bottom of a light-house from the top of a 60 m high building are 30Â° and 60Â° respectively. Find
(i) the difference between the heights of the light-house and the building.
(ii) the distance between the light-house and the building.**

**[AI 2012]**

* Sol. Â Â Â *Let PQ be the light house and AB be the building

MQ = AB = 60 m

Suppose BQ = y

And PM = x

In

[Because AM = BQ]

â€¦â€¦â€¦â€¦. (1)

In

From equation (1)

Therefore,

(i) The difference between the height of the light house and the building = 20 m

(ii) The distance between the light house and the building m

**Q.14Â Â Â Â From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60Â° and the angle of depression of its foot is 30Â°. Determine the height of the tower.**

**[Foreign 2013]**

* Sol.Â Â Â Â Â Â *Let AB be the building & PQ be the cable tower

In

In

[Because AM = x]

Â **( ****Multiple Choice Questions )
**

Â **Q.1 At some time of the day, the length of the shadow of a tower is equal to its height. Then the Sunâ€™s altitude at that time is - (a) 30Â° ** **(b) 60Â° (c) 90Â° (d) 45Â°**

**[Foreign 2011]**

[Because AB = BC]

**Q.2 A kite is flying at a height of 30 m from the ground. The length of string from the kite to the ground is 60 m. Assuming that here is no slack in the string, the angle of elevation of the kite at the ground is - (a) 45Â° Â Â Â Â **

**[AI 2012]**

**Q.3Â Â Â Â The angle of depression of a car, standing on the ground, from the top of a 75 m high tower, is 30Â°. The distance of the car from the base of the tower (in m) is : (a) (b) (c) (d) 150 [Delhi 2013]**

* Sol.*Â Â Â Â Â Â Let AB be the tower and the car be at C

**Q.4Â Â Â A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60Â° with the wall, then the height of the wall is(a) **

(c)

**[AI 2013]**

* Sol.*Â Â Â Â Let AB be the wall and AC be the ladder

**Q.5Â Â Â Â If at some time, the length of the shadow of a tower is times its height, then the angle of elevation of the Sun, at that time, is(a) 15Â°Â Â Â Â (b) 30Â°Â Â Â (c) 45Â°Â Â Â Â (d) 60Â°**

**[Foreign 2013]**

* Sol.*Â Â Â Â Let tower be h and angle of elevation from point C be Q.