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CBSE CLASS X

HEIGHT & DISTANCE

Q. 1. A vertical tower stands on a horizontal plane and is surmounted by a vertical flag-staff of height h. at a point on the plane, the angle of elevation of the bottom and the top of the flag staff are α and β respectively. Prove that the height of the tower is.

Q. 2. The angle of elevation and the top of the tower from two points at a distance a and b meters from the base and in the same line with it are complementary. Prove that the height of the tower is  meters.

Q. 3. From a point on the ground 40m away from the foot of the tower, the angle of elevation of the top of the tower is. The angle of elevation of the top of a water tank (on the top of the tower) is. Find the height of tower & depth of tank.

Q. 4. A person, standing on the bank of the river, observes that the angle subtended by a tree on the opposite bank is. When he retreats 20m from the bank, he finds the angle to be. Find the height of the tree and the breadth of the river.

Q. 5. At a point on level ground, the angle of elevation of a vertical tower is found to be such that its tangent is 5/12. On walking 192 m towards the tower, the tangent of angle of elevation is ¾. Find the height of the tower.

Q. 6. Determine the height of a mountain if the elevation of its top at an unknown distance from the base is  and at a distance 10 km off from the mountain, along the same line, the angle of elevation is. (Use tan  = 0.27).

Q. 7. Two stations due south of a leaning tower which leans towards the north are at a distance a and b from its foot. If α , β be the elevations of the top of the tower from these stations, prove that its inclination to the horizontal is given by 

Q. 8. An aeroplane at an altitude of 1200m finds that two ships are sailing towards it in the same direction. The angle of depression of the ships as observed from the aeroplane are  and  respectively. Fins the distance between the two ships.

Q. 9. The shadow of a flag-staff is three times as long as the shadow of the flag staff   when the sun rays meet the ground at an angle of. Find the angle between the sun rays and the ground at the time of longer shadow.

Q. 10. Two pillars of equal height and on either side of a road, which is 100m wide. The angle of elevation of the top of the pillars are  and  at a point on the road between the pillars and the height of each pillars.

Q. 11. As observed from the top of a light house, 100m above sea level, the angle of depression of a ship, sailing directly towards it, changes from  to . Determine the distance traveled by the ship during the period of observation.

Q. 12. The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is. At a point Y, 40m vertically above X, the angle of elevation is. Find the height of the tower PQ and the distance XQ.

Q. 13. From a window 15m high above the ground in a street, the angle of elevation and depression of the top and the foot of  another house on the opposite side of the street are  and  respectively show that the height of the opposite house is 23.66 m. ( Take  = 1.732).

Q. 14. From the top of a building 60m high the angle of depression of the top and the bottom of a tower are observed to be  and. Find the height of the tower.

Q. 15. A man standing on the deck of a ship, which is 10m above water level. He observes the angle of elevation of the top of a hill as  and the angle of depression of the base of the hill as. Calculate the distance of the hill from the ship and the height of the hill.

Q. 16. If the angle of elevation of the jet plane from a point A on the ground is. After a flight of 15 seconds, the angle of elevation changes to. If the jet plane is flying at a constant height of 1500 m, find the speed of the jet plane.

Q. 17. If the angle of elevation of the cloud from appoint h metres above a lake is α and the angle of depression of its reflection in the lake is β, prove that the height of the cloud is   .

Q. 18. The angle of elevation of a cloud from a point 60m above a lake is  and the angle of depression of the reflection of cloud in the lake is. Find the height of the cloud.

Q. 19. A round balloon of radius r subtends an angle α at the eye of the observer while the angle of elevation of its centre is β. Prove that the height of the centre of the balloon is.

Q. 20. The angle of elevation of cliff from a fixed point is θ. After going up a distance of k metres towards the top of the cliff at an angle of , it is found that the angle of elevation is α. Show that the height of the cliff is meters.

Q. 21. At the foot of a mountain the elevation of its summit is , after ascending 1000m towards the mountain up a slope of  inclination, the elevation is found to be . Find the height of the mountain.

Q. 22. The angle of elevation of the top of a hill at the foot of the tower is  and the angle of elevation of the top of the tower from the foot of the hill is . If the tower is 50m high, what is the height of the hill?

Q. 23. From an aeroplane vertically above a straight horizontal road, the angle of depression of two consecutive mile stones on opposite sides of the aeroplane are observed to be α and β. Show that the height in miles of aeroplane above the road is given by.

Q. 24. If the angle of elevation of a cloud from a point h metres above a lake is α and the angle of depression of its reflection in the lake be β, prove that the distance of the cloud from the point of observation is  .

Q. 25. A 1.2 m 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 . After sometime, the angle of elevation reduces to. Find the distance traveled by the balloon during the interval.

Q. 26. 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, 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. Find the time taken by the car to reach the foot of the tower from this point.

Q. 27. A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from  to  as he walks towards the building. Find the distance he walked towards the building.

Q. 28. An aero plane when flying at a height of 4000m from the ground passes vertically above another aero plane at an instant when the angle of elevation of the two aero planes from the same point on the  ground are  and  respectively. Find the vertical distance between the two aero planes at that instant. (Take   = 1.73).

Paper By Mr. Neeraj Pandey