Vardaan Learning Institute

1. The ratio of the length of a vertical pole and its shadow on the ground is $\sqrt{3}:1$. The angle of elevation of the sun is:

1. $30^\circ$
2. $45^\circ$
3. $60^\circ$
4. $90^\circ$

2. A ladder makes an angle of $60^\circ$ with the ground when placed against a wall. If the foot of the ladder is 2m away from the wall, the length of the ladder is:

1. 4 m
2. $4\sqrt{3}$ m
3. $2\sqrt{2}$ m
4. 2 m

3. The angle of depression of a car parked on the road from the top of a 150m high tower is $30^\circ$. The distance of the car from the tower is:

1. $50\sqrt{3}$ m
2. $150\sqrt{3}$ m
3. 150 m
4. 75 m

4. From a point on the ground, 20m away from the foot of a vertical tower, the angle of elevation of the top of the tower is $60^\circ$. The height of the tower is:

1. 20 m
2. $20\sqrt{3}$ m
3. 10 m
4. $40\sqrt{3}$ m

5. If a pole 6m high casts a shadow $2\sqrt{3}$m long on the ground, then the sun's elevation is:

1. $60^\circ$
2. $45^\circ$
3. $30^\circ$
4. $90^\circ$

6. The angle of elevation of the top of a tower from two points at distances a and b ($a>b$) from the base and in the same straight line with it are complementary. The height of the tower is:

1. $\sqrt{a+b}$
2. $\sqrt{ab}$
3. $\sqrt{a-b}$
4. $\sqrt{a/b}$

7. If the length of the shadow of a tower is increasing, then the angle of elevation of the sun:

1. Increases
2. Decreases
3. Remains same
4. None of these

8. A kite is flying at a height of 50m. If the length of string is 100m, then the inclination of string with the ground is:

1. $30^\circ$
2. $45^\circ$
3. $60^\circ$
4. $90^\circ$

9. The angle of elevation of the Sun when the length of the shadow of a vertical pole is equal to its height is:

1. $30^\circ$
2. $45^\circ$
3. $60^\circ$
4. $90^\circ$

10. Assertion (A): If the height of a tower and the distance of the point of observation from its foot, both are increased by 10%, then the angle of elevation of its top remains unchanged.
Reason (R): $\tan \theta = \text{Height} / \text{Base}$.

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is not the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.

11. A kite is flying at a height of 60m 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 $60^\circ$. Find the length of the string, assuming that there is no slack in the string.

12. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle $30^\circ$ with it. The distance between the foot of the tree to the point where the top touches the ground is 8m. Find the height of the tree.

13. The shadow of a tower standing on a level ground is found to be 40m longer when the Sun's altitude is $30^\circ$ than when it is $60^\circ$. Find the height of the tower.

14. An observer 1.5m tall is 28.5m away from a chimney. The angle of elevation of the top of the chimney from her eyes is $45^\circ$. What is the height of the chimney?

15. From the top of a 7m high building, the angle of elevation of the top of a cable tower is $60^\circ$ and the angle of depression of its foot is $45^\circ$. Determine the height of the tower.

16. As observed from the top of a 75m high lighthouse from the sea-level, the angles of depression of two ships are $30^\circ$ and $45^\circ$. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

17. A 1.2m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is $60^\circ$. After some time, the angle of elevation reduces to $30^\circ$. Find the distance travelled by the balloon during the interval.

18. The angles of elevation of the top of a tower from two points at a distance of 4m and 9m from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is 6m.

19. Two pillars of equal height are on either side of the road, which is 80m wide. From a point between them on the road, the angles of elevation of the top of the pillars are $60^\circ$ and $30^\circ$ respectively. Find the height of the pillars and the position of the point.

20. A man 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 $60^\circ$ and the angle of depression of the base of the hill as $30^\circ$. Calculate the distance of the hill from the ship and the height of the hill.

21. The angle of elevation of a cloud from a point $h$ metres above a lake is $\alpha$ and the angle of depression of its reflection in the lake is $\beta$. Prove that the height of the cloud is $\frac{h(\tan \beta + \tan \alpha)}{\tan \beta - \tan \alpha}$.

22. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of $30^\circ$, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be $60^\circ$. Find the time taken by the car to reach the foot of the tower from this point.

23. Case Study: Skyscraper
A boy standing on a horizontal plane finds a bird flying at a distance of 100m from him at an elevation of $30^\circ$. A girl standing on the roof of a 20m high building finds the angle of elevation of the same bird to be $45^\circ$. The boy and the girl are on opposite sides of the bird.

(i) Find the height of the bird from the ground. (1 Mark)
(ii) Find the distance of the bird from the girl. (2 Marks)
(iii) Find the horizontal distance between the boy and the girl. (1 Mark)