1. What should be the angle of elevation of the sun so that the length of the shadow cast behind a vertical pole is \(\cfrac{1}{\sqrt{3}}\) times the height of the pole? (a) 30° (b) 45° (c) 90° (d) 60°

2. If the length of a pillar's shadow is equal to its height, the angle of elevation of the sun will be- (a) 30° (b) 60° (c) 45° (d) 90°

3. When the sun's altitude angle decreases from 60° to 30°, the length of the shadow of a vertical rod increases by 40 meters. What is the height of the rod? (a) \(10\sqrt3\) meter (b) \(15\sqrt3\) meter (c) \(5\sqrt3\) meter (d) \(20\sqrt3\) meter

4. A ship travels 10 km north from a certain point, then 10 km west. What is the ship’s direct (straight-line) distance from the starting point? (a) \(20\) km (b) \(10\sqrt2\) km (c) \(2\sqrt{10}\) km (d) 100 km

5. When the sun's elevation angle decreases from 60° to 30°, the length of the shadow of a vertical rod increases by 40 meters. What is the height of the rod? (a) \(10\sqrt3\) meter (b) \(15\sqrt3\) meter (c) \(5\sqrt3\) meter (d) None of the above

6. If \( \sin\theta + \csc\theta = 2 \), then what is the value of \( \sin^2\theta + \csc^2\theta \)? (a) 1 (b) 2 (c) 4 (d) \cfrac{3}{4}

7. A 30-meter high pole and a pillar are located on the same horizontal plane. The foot of the pillar is seen from the top of the pole at a depression angle of 30°, and the top of the pillar is seen from the base of the pole at an elevation angle of 60°. What is the height of the pillar and the distance between the pole and the pillar?

8. Two poles are placed 120 meters apart, and the height of one pole is double that of the other. If the angles of elevation to the tops of the two poles from the midpoint of the line joining their bases are complementary, find the height of the shorter pole.

9. From a point A on the ground, the angle of elevation to the top of a vertical pillar is 30°. After moving 20 meters toward the base of the pillar and reaching point B, the angle of elevation increases to 60°. Find the height of the pillar and the distance from point A to the pillar.

10. The heights of two pillars are 180 meters and 60 meters respectively. If the angle of elevation to the top of the second pillar from the base of the first pillar is 30°, then find the angle of elevation to the top of the first pillar from the base of the second pillar.

11. From a point 50 meters away from the base of an incomplete pillar, the angle of elevation to its top is 30°. How much taller must the pillar be so that the angle of elevation to its new top from the same point becomes 45°?

12. From two points located on the same side and along the same horizontal line passing through the base of a vertical pillar, the angles of elevation to the top of the pillar are respectively \(\theta\) and \(\phi\). If the height of the pillar is \(h\), find the distance between the two points.

13. Here is the English translation: > Two pillars of equal height are located directly opposite each other at points A and B on either side of a 120-meter wide road. From point C, on the line joining their bases, the angles of elevation to the tops of the pillars at A and B are 60° and 30°, respectively. Find the length of AC.

14. The distance between two pillars is 150 meters. One is three times taller than the other. From the midpoint of the line segment connecting the bases of the two pillars, the angles of elevation to the tops of the pillars are complementary. Find the height of the shorter pillar.

15. If the angles of depression from a lighthouse to two ships located along the same straight line are 60° and 30°, and the nearer ship is 150 meters away from the lighthouse, then what is the distance of the farther ship from the lighthouse?

16. From the roof of a house, the angles of depression to the top and base of a lamp post are 30° and 60°, respectively. Find the ratio of the height of the house to the height of the lamp post.

17. The ratio of the heights of two pillars is 1:3. If the angle of elevation from the base of the shorter pillar to the top of the taller pillar is 60°, then find the angle of elevation from the base of the taller pillar to the top of the shorter pillar.

18. The deck of a ship is 10 meters above sea level. Standing on the deck, Apu observes the top of a lighthouse at an angle of elevation of 60°, and its base at an angle of depression of 30°. Find the distance from the ship to the lighthouse and the height of the lighthouse.

19. Durga was standing on a railway overbridge that is 5√3 meters high. She observed the engine of a moving passenger train at a depression angle of 30° on one side of the bridge. Two seconds later, she saw the same engine at a depression angle of 60° on the other side of the bridge. Durga's position was vertically above the railway track, which is assumed to be a straight line. Find the speed of the train.

20. Standing in the middle of the field, Habu sees a flying bird first at an elevation angle of 30° to the north, and after 2 minutes, at an elevation angle of 60° to the south. If the bird is flying along a straight path at a constant height of \(50\sqrt{3}\) meters, then what is its speed?

21. A person standing on a railway overbridge 5√3 meters high first observes the engine of a train at a depression angle of 30° on one side of the bridge. After 2 seconds, he sees the engine at a depression angle of 45° on the other side of the bridge. What is the speed of the train?

22. Mohit observed a flying bird first at an elevation angle of 30° to the north, and then at an elevation angle of 60° to the south. If the bird was flying along a straight path at a height of \(50\sqrt{3}\) meters, calculate its speed in kilometers per hour.

23. When the sun's angle of elevation changes from 45° to 60°, the length of the shadow of a telegraph pole changes by 4 meters. Determine the length of the shadow when the angle of elevation is 30°.

24. From a point on the roof of a five-storey building, the angle of elevation to the top of a monument is 60°, and the angle of depression to its base is 30°. If the height of the building is 16 meters, find the height of the monument and the distance from the building to the monument.

25. From a point 150 meters away from the base of an incomplete pillar, the angle of elevation to its top is 45°. How much additional height must be added to the pillar so that, from the same point, the angle of elevation to the new top becomes 60°?

26. Standing in an open field, Habul sees a flying bird at an angle of elevation of 30° towards the north, and after 2 minutes, he sees it at an angle of elevation of 60° towards the south. If the bird is flying along a straight horizontal path at a height of \(50\sqrt{3}\) meters, then calculate the speed of the bird.

27. The base of a triangle is \(16\sqrt{3}\) cm, and the two angles adjacent to the base are 30° and 60°. What is the height of the triangle?

28. If the sun's elevation angle is _____, then the length of the pillar's shadow will be equal to the height of the pillar.

29. The distance between two pillars is 150 meters. One pillar's height is three times the other. The angles of elevation to their tops from the midpoint of the line joining their bases are complementary. Find the height of the smaller pillar.

30. From the roof of a five-storey building that is 18 meters high, the angle of elevation to the top of a monument is 45°, and the angle of depression to the base of the monument is 60°. What is the height of the monument?

31. From a point on a horizontal line at the same level as the base of a chimney, a person walks 50 meters towards the chimney. As a result, the angle of elevation to the top of the chimney increases from 30° to 60°. Find the height of the chimney.

32. From a ghat on one side of a 600-meter wide river, two boats set off toward the opposite bank in two different directions. If the first boat moves at an angle of 30° with the riverbank, and the second boat moves at a 90° angle to the path of the first boat, then what will be the distance between the two boats after they reach the opposite bank?

33. A three-storey building has a flagpole of 3.6 meters mounted on its roof. From a point on the street, the angles of elevation to the top and bottom of the flagpole are 50° and 45°, respectively. Find the height of the building. [Assume: \(\tan 50^\circ = 1.2\)]

34. There is a palm tree on the bank of a river. Directly opposite it, on the other side of the river, a pole is fixed into the ground. If someone walks 7\(\sqrt{3}\) meters along the riverbank from the pole toward the tree, the base of the tree appears to form a 60° angle with the riverbank at that point. Determine the width of the river.

35. From a corner point on the roof of a house that is 30\(\sqrt{3}\) meters high, the angles of depression to the top and bottom of a lamp post are 30° and 60°, respectively. Determine the height of the lamp post.

36. Two pillars are 120 meters apart, with one pillar's height being twice that of the other. At the midpoint of the line segment connecting their bases, the angles of elevation to the tops of the pillars are complementary. Determine the heights of the pillars.

37. Let the smaller pillar be AB \(= x\) meters and the larger pillar be CD \(= 2x\) meters.
The midpoint of the base connection BD is O; the angles of elevation from O to the tops of the pillars are \(\angle\)AOB = \(\theta\) and \(\angle\)COD = 90\(^o-\theta\).
Since O is the midpoint of BD, we have BO = OD = \(\frac{120}{2}\) meters = 60 meters.

From \( \triangle \)ABO, we get:
\(\cfrac{AB}{BO} = \tan \theta\)
Or, \(\cfrac{x}{60} = \tan\theta\) --------(i)

From \( \triangle \)COD, we get:
\(\cfrac{CD}{OD} = \tan(90^o - \theta)\)
Or, \(\cfrac{2x}{60} = \cot\theta\) --------(ii)

Multiplying equations (i) and (ii), we get:
\(\cfrac{x}{60} \times \cfrac{2x}{60} = \tan\theta \times \cot\theta\)
Or, \(\cfrac{2x^2}{60 \times 60} = 1\)
Or, \(x^2 = \cfrac{60 \times \cancel{60}30}{\cancel{2}}\)
Or, \(x = 30\sqrt2\)

\(\therefore\) The height of the smaller pillar is \(30\sqrt2\) meters.
And the height of the larger pillar is \(30\sqrt2 \times 2\) meters \(= 60\sqrt2\) meters.
(Proved).

38. A 5-meter-high house is located on one side of a park. From both the roof and the base of the house, the base and the top of a palm tree on the opposite side of the park are seen at angles of depression and elevation of 30° and 60°, respectively. What is the height of the palm tree? What is the distance between the house and the palm tree?

39. A point is located \(h\) meters above a lake. From this point, the angle of elevation to a cloud is \(α\), and the angle of depression to its reflection in the lake is \(β\). Prove that the distance from the point to the cloud is \[ \cfrac{2h \sec α}{\tan β − \tan α} \]

40. The heights of two pillars are 180 meters and 60 meters respectively. If the angle of elevation from the base of the second pillar to the top of the first pillar is 60°, find the angle of elevation from the base of the first pillar to the top of the second pillar.

41. If the ratio of the length of a shadow of a pillar to the height of the pillar is √3:1, find the angle of elevation of the sun.

42. The heights of two pillars are \(h_1\) meters and \(h_2\) meters respectively. If the angle of elevation from the base of the second pillar to the top of the first pillar is 60°, and the angle of elevation from the base of the first pillar to the top of the second pillar is 45°, show that \(h_1^2 = 3h_2^2\).

43. The heights of two pillars are 45 meters and 15 meters respectively. From the base of the second pillar, the angle of elevation to the top of the first pillar is 60°. Determine the angle of elevation to the top of the second pillar from the base of the first pillar.

44. From the top of a building 60 meters high, the angles of depression to the top and bottom of a lamp post are 30° and 60°, respectively. Find: (i) The horizontal distance between the building and the lamp post (ii) The height of the lamp post

45. When the sun's elevation angle increases from 45° to 60°, the length of the shadow of a pole decreases by 3 meters. Find the height of the pole. [Take √3 = 1.732 and calculate the approximate value up to three decimal places.]

46. When the sun's angle of elevation is 45°, the length of the shadow of a vertical pillar on a horizontal plane is a certain value. When the angle of elevation becomes 30°, the shadow length increases by 60 meters compared to the previous case. Find the height of the pillar.

47. A passenger on a plane flying directly above a straight road observes two consecutive pillars located 1 kilometer apart on that road at angles of depression of 60° and 30°, respectively. Find the height of the plane above the road at that moment.

48. When the sun’s angle of elevation changes from 45° to 60°, the shadow of a telegraph pole changes by 4 meters. Find the length of the shadow of the same pole when the angle of elevation is 30°.

49. From the roof and the base of a 16-meter high building, the angles of elevation to the top of a temple are 45° and 60°, respectively. Find the height of the temple and its horizontal distance from the building. (The base of the building and the temple lie on the same horizontal plane.)

50. Two points on the ground lie along the same straight line with the base of a vertical pillar. From these two points, the angles of elevation to the top of the pillar are complementary. If the distances from the two points to the base of the pillar are 9 meters and 16 meters respectively, and both points are on the same side of the pillar, find the height of the pillar.