Math Ch 9: Some Applications of Trigonometry PYQs

A kite is flying at a height of 150 m from the ground. It is attached to a string inclined at an angle of $30^{\circ}$ to the horizontal. The length of the string is:

[1]

A. $100\sqrt{3}\text{ m}$ B. $300\text{ m}$ C. $150\sqrt{2}\text{ m}$ D. $150\sqrt{3}\text{ m}$

Amrita stood near the base of a lighthouse, gazing up at its towering height. She measured the angle of elevation to the top and found it to be $60^{\circ}$. Then, she climbed a nearby observation deck, 40 metres higher than her original position and noticed the angle of elevation to the top of lighthouse to be $45^{\circ}$.

Based on the above given information, answer the following questions:

(i)	If CD is h metres, find the distance BD in terms of 'h'.
(ii)	Find distance BC in terms of 'h'.
(iii)	(a) Find the height CE of the lighthouse [Use $\sqrt{3}=1.73$]
	OR
(iii)	(b) Find distance AE, if $AC=100\text{ m}$.

[4]

A pole 6m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point P on the ground is $60^{\circ}$ and the angle of depression of the point P from the top of the tower is $45^{\circ}$ Find the height of the tower and the distance of point P from the foot of the tower. (Use $\sqrt{3}=1.73)$

[5 marks]

(a) The angle of elevation of the top of a tower 24 m high from the foot of another tower in the same plane is $ 60^{\circ} $. The angle of elevation of the top of second tower from the foot of the first tower is $ 30^{\circ} $. Find the distance between two towers and the height of the other tower. Also, find the length of the wire attached to the tops of both the towers.

(b) A spherical balloon of radius r subtends an angle of $ 60^{\circ} $ at the eye of an observer. If the angle of elevation of its centre is $ 45^{\circ} $ from the same point, then prove that height of the centre of the balloon is $ \sqrt{2} $ times its radius.

[5 marks]

(a) The angle of elevation of the top of a building from the foot of the tower is $30^{\circ}$ and the angle of elevation of the top of the tower from the foot of the building is $60^{\circ}$. If the tower is 50 m high, then find the height of the building.

OR

(b) From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are $30^{\circ}$ and $45^{\circ}$ respectively. If the bridge is at a height of 3 m from the banks, then find the width of the river.

[3 marks]

Case Study - 2
Gadisar Lake is located in the Jaisalmer district of Rajasthan. It was built by the King of Jaisalmer and rebuilt by Gadsi Singh in 14th century. The lake has many Chhatris. One of them is shown below:

Observe the picture. From a point A hm above from water level, the angle of elevation of top of Chhatri (point B) is $45^{\circ}$ and angle of depression of its reflection in water (point C) is $60^{\circ}$. If the height of Chhatri above water level is (approximately) 10 m, then
(a) draw a well-labelled figure based on the above information;
(b) find the height (h) of the point A above water level. (Use $\sqrt{3}=1.73$)

[4 marks]

In the given figure, which of the following angles represents the angle of depression?

[1 marks]

A. x B. y C. z D. a

Case Study - 1

An injured bird was found on the roof of a building. The building is 15 m high. A fireman was called to rescue the bird. The fireman used an adjustable ladder to reach the roof. He placed the ladder in such a way that the ladder makes an angle of $60^{\circ}$ with the ground in order to reach the roof.

Based on the above information, answer the following questions :

(i) Find the length of the ladder used by the fireman to reach the roof.
(ii) Find the distance of the point on the ground at which the ladder was fixed from the bottom of the building.
(iii) (a) In order to avoid skidding, the fireman placed the ladder in such a way that the bottom of the ladder touches the base of the wall which is opposite to the building, making an angle of $30^{\circ}$ with the ground. Draw a neat diagram to represent the above situation and hence find the width of the road between the building and the wall.
OR
(b) Find the length of the ladder used by the fireman in this case.

[4 marks]

(A) The shadow of a tower standing on a level ground is found to be $40$ m longer when the Sun's altitude is $30^{\circ}$ than when it was $60^{\circ}$. Find the height of the tower and the length of original shadow. (use $\sqrt{3}=1.73$)

(B) The angles of depression of the top and the bottom of an $8$ m tall building from the top of a multi-storeyed building are $30^{\circ}$ and $45^{\circ}$ respectively. Find the height of the multi-storeyed building and the distance between the two buildings. (use $\sqrt{3}=1.73$)

[5 marks]

The angle subtended by a vertical pole of height 100 m at a point on the ground $ 100\sqrt{3} $ m from the base is, has measure of

IMAGE

[1 mark]

A. $ 90^{\circ} $ B. $ 60^{\circ} $ C. $ 45^{\circ} $ D. $ 30^{\circ} $

(a) A TV tower stands vertically on the 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 $ 60^{\circ} $. From another point 20 m away from the point on the line joining this point to the foot of the tower, the angle of elevation of the top of the tower is $ 30^{\circ} $. Find the height of the tower.

(b) An aeroplane when flying at a height of 4000 m from the ground passes vertically above another aeroplane at an instant when the angles of elevation of the two planes from the same point on the ground are $ 60^{\circ} $ and $ 45^{\circ} $ respectively. Find the vertical distance between the aeroplanes at that instant. (Use $ \sqrt{3}=1.73 $)

[5 marks]

Math Ch 9: Some Applications of Trigonometry PYQs | CBSE Class 10

Sign in to continue