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

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

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 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.

5. 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 _____.

6. 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.

7. 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°

8. 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.

9. 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?

10. 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°.

11. 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°.

12. 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 \(\sqrt{3} = 1.732\))"

13. 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.

14. When the sun's angle of elevation changes from 30° to 60°, the length of a post's shadow becomes half.

15. 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

16. The volume of a cube is equal to the area of one of its faces. What is the length of its edge?

(a) 4 units (b) 5 units (c) 6 units (d) 8 units

17. 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.]

18. 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°?

19. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) will be –

(a) \(s=6d^2\) (b) \(3s=7d\) (c) \(s^3=d^2\) (d) \(d^2 = \cfrac{s}{2}\)

20. A right-angled triangular prism has a base area of 100 square centimeters, and the areas of the two adjacent faces to the base are 40 square centimeters and 160 square centimeters. Its volume will be—

(a) 800 cubic cm (b) 880 cubic cm (c) 890 cubic cm (d) 900 cubic cm

21. 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°?

22. If the radius of a right circular cylinder is 2 units, then for any height, the numerical values of the cylinder’s volume and curved surface area will be equal.

23. When the sun's elevation angle increases from 30° to 60°, the length of a post's shadow — decreases.

24. If the total surface area and the volume of a cube are numerically equal, then the length of its edge will be -- units.

25. A rectangular sheet of paper has length \(l\) units and width \(b\) units. When rolled along its length, it forms a cylindrical tube whose circumference is equal to the length of the paper. The curved surface area of the cylinder is ___ square units.

26. 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

27. 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

28. 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.

29. If the total length of the edges of a solid cube is 36 cm, then its volume will be –?

(a) 9 cubic cm (b) 54 cubic cm (c) 27 cubic cm (d) 36 cubic cm

30. When the elevation angle of the Sun increases from 45° to 60°, the length of a pole's shadow decreases by 3 meters. Determine the height of the pole (√3 = 1.732).