SECTION C — Short Answer Type Questions [3 Marks Each]
- 3 Marks Q1. A tower stands vertically on the ground. From a point on the ground, which is $15\text{ m}$ away from the foot of the tower, the angle of elevation of the top of the tower is found to be $60^\circ$. Find the height of the tower.
- 3 Marks Q2. An observer $1.5\text{ m}$ tall is $28.5\text{ m}$ away from a chimney. The angle of elevation of her eyes to the top of the chimney is $45^\circ$. What is the height of the chimney?
- 3 Marks Q3. A kite is flying at a height of $60\text{ m}$ 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 $60^\circ$. Find the length of the string, assuming that there is no slack.
- 3 Marks Q4. A $1.2\text{ m}$ tall girl spots a balloon moving with the wind in a horizontal line at a height of $88.2\text{ m}$ 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
- 3 Marks Q5. A straight highway leads to the foot of a tower. A man standing at the top of the tower obs erves a car at an a ngle 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 poi nt.
- 3 Marks Q6. The angles of elevation of the top of a tower from two points at a distance of $4\text{ m}$ and $9\text{ m}$ from the base of the tower and in the same straight line with it are complementary. Prove that the height of the t ower is $6\text{ m}$.
- 3 Marks Q7. A vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height $5\text{ m}$. At a point on the plane, the angle of elevation of the bottom of the flagstaff is $30^\circ$ and that of the top of the flagstaff is $60^\circ$. Find the height of the tower.
- 3 Marks Q8. The shadow of a tower standing on a level plane is found to be $50\text{ m}$ longer when the sun's elevation is $30^\circ$ than when it is $60^\circ$. Find the height of the tower.
- 3 Marks Q9. From the top of a $7\text{ m}$ 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.
- 3 Marks Q10. As observed from the top of a $75\text{ m}$ 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
- 3 Marks Q11. A pole $6\text{ m}$ high casts a shadow $2\sqrt{3}\text{ m}$ long on the ground. Find the sun's elevation angle.
- 3 Marks Q12. A ladder rests against a vertical wall at an inclination of $60^\circ$ to the ground. If the foot of the ladder is $2.5\text{ m}$ away from the wall, find the length of the ladder and how high up the wall it reaches.
- 3 Marks Q13. A tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground making an angle of $30^\circ$ with it. If the distance between the foot of the tree to the point where the top touches the ground is $8\text{ m}$, find the height of the tree before it broke.
- 3 Marks Q14. A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank is $60^\circ$. When he moves $40\text{ m}$ away from the bank, he finds the angle of elevation to be $30^\circ$. Find the height of the tree and the width of the river.
- 3 Marks Q15. A bridge across a river makes an angle of $45^\circ$ with the river bank. If the length of the bridge across the river is $150\text{ m}$, find the breadth of the river
- 3 Marks Q16. The upper part of a tree broken by the wind makes an angle of $30^\circ$ with the ground and the distance from the root to the point where the top touches the ground is $10\text{ m}$. Find the original height of the tree.
- 3 Marks Q17. From the top of a hill, the angles of depression of two consecutive kilometer stones on the east side of the hill are found to be $30^\circ$ and $45^\circ$ respectively. Find the height of the hill.
- 3 Marks Q18. A tower of height $h$ is situated on the bank of a river. The angle of elevation of the top of the tower from a point on the opposite bank is $60^\circ$. From a point $20\text{ m}$ further away from the bank on the same straight line, the angle of elevation is $30^\circ$. Find the height of the tower and the width of the river
- 3 Marks Q19. A boat is sailing towards a lighthouse of height $150\text{ m}$. If the angle of elevation of the top of the lighthouse changes from $30^\circ$ to $45^\circ$ as the boat travels, find the distance covered by the boat during this time.
- 3 Marks Q20. A man on the deck of a ship $12\text{ m}$ above water level observes that the angle of elevation of the top of a cliff is $60^\circ$ and the angle of depression of the base of the cliff is $30^\circ$. Find the height of the cliff.
- 3 Marks Q21. A vertical lamp-post stands on a horizontal plane. From a point $P$ on the ground, the angle of elevation of the top of the lamp-post is $30^\circ$. Moving $10\text{ m}$ closer to the post along the ground, the angle of elevation becomes $60^\circ$. Find the height of the lamp-post and its initial distance from point $P$.
- 3 Marks Q22. The angle of elevation of a cloud from a point $h$ meters 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 $h\left(\frac{\tan \beta + \tan \alpha}{\tan \beta - \tan \alpha}\right)$.
- 3 Marks Q23. 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 $50\text{ m}$ high, find the height of the building.
- 3 Marks Q24. From the top of a cliff $50\text{ m}$ high, the angles of depression of the top and bottom of a tower are observed to be $30^\circ$ and $60^\circ$ respectively. Find the height of the tower.
- 3 Marks Q25. The horizontal distance between two poles is $15\text{ m}$. The angle of depression of the top of the first pole as seen from the top of the second pole is $30^\circ$. If the height of the second pole is $24\text{ m}$, find the height of the first pole.
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English: These model solutions are structured to global academic standards for self-assessment and practice purposes. Students are advised to verify with official textbooks for board examinations.
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