1. A metallic sphere is cut in such a way that the curved surface area of the remaining sphere becomes exactly half of the original sphere's curved surface area. Find the ratio of the volume of the removed part to the volume of the remaining part.

2. A metallic sphere's outer surface is cut in such a way that the curved surface area of the remaining part becomes exactly half of the original sphere's curved surface area. Determine the ratio of the volume of the removed portion to that of the remaining portion.

3. The ratio of the total surface area to the curved surface area of a solid hemisphere will be –.

(a) 2:1 (b) 1:2 (c) 1:3 (d) 3:1

4. If the volume ratio of two solid spheres is 1:8, the ratio of their curved surface areas will be -

(a) 1:2 (b) 2:1 (c) 1:4 (d) 1:16

5. If the volume of a cube is \(V\) cubic centimeters, the total surface area is \(S\) square centimeters, and the length of the diagonal is \(d\) centimeters, then prove that \(Sd = 6\sqrt{3}V\).

6. The total surface area of a solid hemisphere and the curved surface area of a solid sphere are equal. What is the ratio of the radius of the hemisphere to the radius of the sphere?

7. If the ratio of the volumes of two cubes is 8:27, then what will be the ratio of their total surface areas?

8. If the ratio of the volumes of two cubes is 1:27, then the ratio of their total surface areas will be —

(a) 1:3 (b) 1:8 (c) 1:9 (d) 1:18

9. From a cuboid measuring 16 cm × 4 cm × 2 cm, several cubes with edge length 2 cm are cut out. What is the ratio of the total surface area of the original cuboid to the total surface area of all the cubes?

(a) 11:48 (b) 12:50 (c) 13:36 (d) 14:28

10. The total surface area of a right square prism is equal to half the square of its diagonal. Show that the prism is a cube.

11. The curved surface area of a cone is \(\sqrt{5}\) times the area of its base. What will be the ratio of its radius to height?

(a) 1:2 (b) 1:3 (c) 1:4 (d) 1:8

12. If the volumes of two spheres are in the ratio 1:8, show that the ratio of their curved surface areas is 1:4.

13. If the ratio of the curved surface areas of two solid spheres is \(1:9\), then their volume ratio will be?

(a) 1:3 (b) 1:27 (c) 1:81 (d) 1:243

14. The ratio of the total surface area of a sphere and a hemisphere with a unit radius will be 4:3.

15. If the ratio of the volumes of two spheres is 27:8, what will be the ratio of their curved surface areas?

16. If the ratio of the volumes of two cubes is 4:343, determine the ratio of their total surface areas.

17. The curved surface area of a right circular cone is \(\sqrt{5}\) times its base area. The ratio of its height to radius will be _____.

18. The ratio of the curved surface area to the total surface area of a solid hemisphere is _____.

19. If the ratio of the volumes of two cubes is 1:27, then the ratio of their total surface areas will be _____.

20. If the ratio of the volumes of two cubes is 1:27, then what is the ratio of their total surface areas?

(a) 1:3 (b) 1:8 (c) 1:9 (d) 1:18

21. If the ratio of the curved surface areas of two hemispheres is 4:9, then the ratio of their radii will be 2:3.

22. The total surface area of a solid hemisphere is equal to the curved surface area of a solid sphere. Write the ratio of the radii of the hemisphere and the sphere.

23. If the ratio of the total surface areas of two solid spheres is 1:4, what will be the ratio of their volumes? Calculate and write it down.

24. "The volume of a sphere varies directly with the cube of its radius, and the surface area of a sphere varies directly with the square of its radius. Prove that the square of the volume of a sphere will be directly proportional to the cube of its surface area."

25. If the total surface area of a sphere is \( A \) and its volume is \( V \), what is the relationship between them ?

(a) \(A^3=36πV^2\) (b) \(A^3=36V^3 \) (c) \(A^2=4V^2\) (d) \(A=3V\)

26. The total surface area of a cube is 216 square centimeters. Find the volume of the cube.

27. The ratio of the volumes of two cubes is 4:125. What is the ratio of the surface areas of the two cubes?

(a) 2:5 (b) 4:25 (c) 4:5 (d) 2:25

28. The volume ratio of two solid spheres is 4:27. The ratio of their curved surface areas is-

(a) 4:9 (b) 2:3 (c) 16:35 (d) 8:27

29. If the ratio of the radii of two vertical solid circular cylinders is 2:3, and the ratio of their heights is 5:3, then the ratio of their curved surface areas will be –

(a) 2:5 (b) 8:7 (c) 10:9 (d) 16:9

30. If a right circular cone and a sphere have the same radius, and the numerical value of the cone's volume is equal to the numerical value of the sphere's curved surface area, then what is the height of the cone?

(a) 10 units (b) 11 units (c) 12 units (d) 13 units