1. 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°

2. 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.

3. There is a flagpole on top of a monument. When the sun’s elevation angle is 30°, the shadow length of the flagpole is 3√3 meters. > What is the height of the flagpole? > If the height of the monument is 15 meters, then what will be the total shadow length of the monument including the flagpole?

4. When the angle of elevation of the sun is 45°, the length of the shadow of a vertical pillar is a certain value. When the angle of elevation decreases to 30°, the length of the shadow becomes 60 meters longer than before. Determine the height of the pillar.

5. As the sun's angle of elevation increases from 30° to 60°, the length of the monument's shadow decreases by \(50\sqrt{3}\) meters. What is the height of the monument?

6. 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.

7. The English translation is: "When the sun's angle of elevation increases from 45° to 60°, the length of the shadow of a pole decreases by 3 meters. Determine the height of the pole. (Take \(\sqrt{3} = 1.732\))"

8. When the angle of elevation of the Sun is 45°, the height of a post and the length of its shadow will be ________.

9. 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

10. 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°

11. 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.

12. 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.]

13. 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.

14. 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.

15. 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.

16. A girl of height 90 cm is walking away from the base of a lamp post at a speed of 1.2 m per second. If the height of the lamp post is 3.6 m, then what will be the length of her shadow after 4 seconds?

17. When the sun's elevation angle increases from 45° to 60°, the length of a pole's shadow decreases by 3 meters. Find the height of the pole.

(a) 7.1 meter (b) 9 meter (c) 10 meter (d) 11 meter

18. O is the center of a circle where AB and CD are two chords of equal length. If the perpendicular distance from O to the chord AB is 4 cm, then the perpendicular distance from O to the chord CD will also be 4 cm.

(a) 2cm (b) 6cm (c) 8cm (d) 4cm

19. A rectangular container has a base in the shape of a rectangle with a length of 60 cm and a width of 45 cm. The container has a height of 20 cm and is half-filled with water. Determine the side length of a metallic cube that, when placed in the container, will cause the water level to reach the brim.

20. A rectangle has a length of \( a \) units and a breadth of \( b \) units. When the rectangle is wrapped to form a right circular cylinder, the circumference of the base of the cylinder is equal to the length of the rectangle. The lateral surface area of the cylinder will be _____.

21. If the sun's elevation angle increases from 45° to 60°, the length of a pole's shadow decreases by x meters. Determine the height of the pole, assuming √3 = 1.732, rounded to three decimal places.

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

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. When the angle of elevation of the sun is 30°, the length of the shadow of a pillar is 9 meters. Find and write the height of the pillar.

25. If the length of a shadow at the base of a post is √3 times the height of the post, then the angle of elevation of the Sun is:

(a) 30° (b) 45° (c) 60° (d) none of the above

26. When the angle of elevation of the Sun is ____________ than 45°, the length of a pillar’s shadow is less than the height of the pillar.

27. 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.

28. 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.

29. When the sun's elevation angle increases from 45° to 60°, the length of a post's shadow decreases by 3 meters. Determine the height of the post.

30. A girl with a height of 90 cm is walking at a speed of 1.2 meters/second. If the lamp is 3.6 meters high, then after 4 seconds, what will be the length of the girl's shadow?

(a) 4.8m (b) 2.4m (c) 1.6m (d) 0.8m