1. If the curved surface area of a solid sphere and three times its volume are numerically equal, then the diameter of the sphere is _____ units.

2. If the curved surface area of a solid sphere is numerically equal to three times its volume, then the numerical value of the diameter of the sphere is —.

(a) 1 units (b) 2 units (c) 3 units (d) 4 units

3. 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 _____.

4. If the radius of a hemisphere is \(2r\) units, then the total surface area will be _____.

5. Two identical solid hemispheres, each with a base radius of \(r\) units, are joined together along their flat surfaces. The total surface area of the resulting solid body will be \(6πr^3\) square units.

6. The volume and total surface area of a solid hemisphere are numerically equal. Then the diameter of the hemisphere will be –

(a) 4:5 units (b) 6 units (c) 9 units (d) 3 units

7. 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

8. If the total surface area of a solid sphere is 462 square centimeters, what is its diameter?

(a) 7 cm (b) 14 cm (c) 21 cm (d) 3.5 cm

9. If the diameter of a solid sphere is doubled, its volume becomes four times greater.

10. A solid sphere and a solid cone have the same diameter, and their total surface areas are equal. What is the ratio of the cone's height to its diameter?

11. If the total surface area of a hemisphere is \(36\pi\) square centimeters, then its radius will be 3 cm.

12. 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

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. If the radius of a solid sphere is doubled, then its volume will become three times.

15. If the radius of a solid hemisphere is \(3r\) units, then its total surface area is _____.

16. 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

17. 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 _____.

18. If the surface area of a hemisphere is \( 27\pi \) square cm, then its radius will be 3 cm.

19. If the numerical values of the volume and surface area of a solid sphere are equal, then what is the numerical value of its diameter?

20. If the largest possible solid sphere is cut from a solid cube with an edge length of \(x\) units, then the diameter of the sphere will be – ?

(a) \(x\) unit (b) \(\cfrac{x}{2}\) unit (c) \(2x\) unit (d) \(4x\) unit

21. If the curved surface area of a solid sphere is numerically equal to three times its volume, find the radius of the sphere.

(a) 1 units (b) 2 units (c) 3 units (d) 4 units

22. If the radius of a solid hemisphere is 2r units, then the total surface area is _______ \( πr^2 \) square units.

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. From a solid hemisphere with radius \( r \) units, the volume of the largest solid cone that can be cut out will be _____ .

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. 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

27. The curved surface area of a right circular cone is \(\sqrt{10}\) times the area of its base. What is the ratio of the height of the cone to the diameter of its base?

28. If the radius of a sphere is increased by 3%, by what percentage will its surface area increase?

(a) 6.09% (b) 7% (c) 5.06% (d) 9%

29. A solid vertical cylindrical cone has both its diameter and height equal to 21 cm. Find the volume of the largest possible sphere that can be obtained from this cone. Also, determine the ratio of the volumes of the cone and the sphere.

30. If the curved surface area of a sphere is \(S\) square units and its volume is \(V\) cubic units, then determine the relationship between \(S\) and \(V\).