1. If the curved surface area of a right circular cylinder is \(S\), its volume is \(V\), and its radius is \(r\), then the value of \(\cfrac{Sr}{V}\) is -

(a) 2 (b) 4 (c) 8 (d) none of the above

2. If the curved surface area of a right circular cylinder is \(c\) square units, the base radius is \(r\) units, and the volume is \(v\) cubic units, then the value of \(\frac{cr}{v}\) is = _____.

3. If the surface area of a sphere is \(A\) and its volume is \(V\), then find the value of \(\cfrac{A^3}{V^2}\).

4. If the curved surface area of a right circular cylinder is \(c\) square units, the radius of the base is \(r\) units, and the volume is \(v\) cubic units, then find the value of \(\frac{cr}{v}\).

5. If the curved surface area of a solid sphere is \(s\) and its volume is \(v\), then what is the value of \(\frac{s^3}{v^2}\)?

6. If the curved surface area of a right circular cylinder is \(C\) square units, the radius of the base is \(r\) units, and the volume is \(V\) cubic units, then determine the value of \(\cfrac{Cr}{V}\).

7. If the curved surface area of a right circular cylinder is \(c\) square units, the radius of the base is \(r\) units, and the volume is \(v\) cubic units, then find the value of \(\cfrac{cr}{v}\).

8. If the volume of a hemisphere is \(v\) and the radius is \(r\), then \(r = \sqrt{\frac{3v}{2\pi}}\)

9. If the surface area of a solid sphere is \(S\), and its volume is \(V\), then find the value of \(\cfrac{S^3}{V^2}\), without substituting the value of \(\pi\).

10. If the base area of a right circular cone is \(A\), its height is \(H\), and its volume is \(V\), then express \(H\) in terms of \(V\) and \(A\).

(a) \(H=\cfrac{3V}{A}\) (b) \(H=\cfrac{V}{A}\) (c) \(H=\cfrac{V}{3A}\) (d) \(H=\cfrac{3V}{2A}\)

11. 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 \(V\).

12. The volume of a sphere is directly proportional to the cube of its radius. If three solid spheres with radii of 3 cm, 4 cm, and 5 cm are melted to form a new solid sphere, and there is no loss in volume during melting, then find the radius of the new sphere.

13. If the volume of a right circular cone is \(V\), the area of its base is \(A\), and its height is \(H\), then find the value of \(\left(\frac{AH}{V}\right)^2\).

(a) 1 (b) 4 (c) 9 (d) 3

14. If the numerical values of the volume and curved surface area of a sphere are equal, then its radius is _____ units.

15. The radius of a right circular cylinder is \(r\) units, and its height \(R\) units is \(\frac{2}{3}\) times the diameter of a sphere of radius \(R\). If their volumes are equal, show that \(r = R\).

16. If the curved surface area of a solid sphere is \(S\) and the volume is \(V\), then write the value of \( \cfrac{S^3}{V^2} \) (without substituting the value of \(π\)).

17. The volume of a sphere is in direct variation with the cube of its radius. If three solid spheres of radii 3 cm, 4 cm, and 5 cm are melted to form a single new solid sphere, and if there is no change in volume due to melting, then determine the diameter of the new sphere using the principle of variation.

18. If a sphere's volume and surface area have equal numerical values, then the numerical value of its radius is ——.

19. 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

20. A sphere and a right circular cone have the same radius \(r\) and equal volume. What is the height of the cone?

(a) \(\cfrac{r}{3}\) (b) \(\cfrac{r}{4}\) (c) \(\cfrac{3r}{4}\) (d) \(\cfrac{4r}{3}\)

21. If the radius of a cone is \(r\) units, the height is \(h\) units, and the curved surface area is \(S\) square units, then prove that \[ h = \frac{\sqrt{S^2 - \pi^2 r^4}}{\pi r} \]

22. If the curved surface area of a right circular cylinder is 264 square meters and its volume is 924 cubic meters, then what is the radius of the base?

23. The volume of a right circular cone is \(V\) cubic units. If the area of the base is \(A\) square units and the height is \(H\) units, then find the value of \(\frac{AH}{V}\).

24. If the radius of a sphere is doubled, then its volume will be twice the volume of the original sphere.

25. The volume of a sphere is directly proportional to the cube of its radius. Three solid spheres with diameters of \(1\frac{1}{2}\) m, 2 m, and \(2\frac{1}{2}\) m are melted and recast into a single solid sphere. Using the concept of proportionality, determine the diameter of the new sphere.

26. The volume of a cone is jointly proportional to the square of the radius of the base and its height. If the ratio of the radii of two cones is 2:3 and their heights are in the ratio 5:4, then find the ratio of their volumes.

27. If the volume and curved surface area of a sphere are numerically equal, what is its radius?

(a) 5 units (b) 3 units (c) 2 units (d) 7 units

28. 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.

29. If a right circular cone has volume \(V\) cubic units and base area \(A\) square units, then the height is _____.

30. If the radius of a right circular cone is halved and its height is doubled, then the volume of the cone will be — of its original volume.