1. If the numerical values of the volume and the curved surface area of a right circular cylinder are equal, then the diameter of the cylinder is ___________ units.

2. The numerical value of the volume and the lateral surface area of a right circular cone are equal. If the height and radius of the cone are \( h \) units and \( r \) units respectively, find the value of \[ \left( \frac{1}{h^2} + \frac{1}{r^2} \right) \]

3. 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

4. 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 \(\frac{AH}{V}\).

5. The numerical value of the volume and the lateral surface area of a right circular cone are equal. If the height of the cone is \( h \) and the radius is \( r \), determine the value of \(\cfrac{1}{h^2}+\cfrac{1}{r^2}\).

6. A right circular cone has equal height and diameter. Determine the ratio of the curved surface area to the base area of the cone.

7. A right circular cone has a base diameter of 21 meters and a height of 14 meters. If the cost of painting is ₹1.50 per square meter, how much will it cost to paint the curved surface area?

8. If the numerical value of the volume of a right circular cone is equal to the numerical value of its curved surface area, what is the radius of the cone?

9. The curved surface area of a solid right circular cone is 1320 square centimeters. If the diameter of the base of the cone is 14 cm, find its height.

10. 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

11. If \(h\), \(s\), and \(v\) represent the height, curved surface area, and volume of a right circular cone respectively, then the value of \(3πVh^3 – s^2h^2 + 9V^2\).

(a) 16\(\pi\) (b) 0 (c) 4\(\pi\) (d) 32\(\pi^2\)

12. The volume and the lateral surface area of a right circular cone are numerically equal. If the height and radius of the cone are \(h\) and \(r\) units respectively, write the value of \(\frac{1}{h^2} + \frac{1}{r^2}\).

13. 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 \(\frac{cr}{v}\).

14. 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.

15. The curved surface area of a right circular cylinder is 264 square meters, and its volume is 924 cubic meters. Find the diameter and height of the cylinder.

16. If the radius of a right circular cylinder is 2 units, then for any height of the cylinder, the numerical values of its volume and curved surface area are equal.

17. If the radius of a right circular cylinder is 2 units, then for any height of the cylinder, the numerical values of its volume and curved surface area are equal.

18. If the volume and curved surface area of a right circular cylinder are numerically equal, then the radius of its base will be 1 unit.

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

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

21. 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 \(\cfrac{cr}{v}\).

22. "When a right circular cone has equal numerical values for its volume and lateral surface area, and its height and radius are \(h\) units and \(r\) units respectively, write the value of \(\frac{1}{h^2} +\frac{1}{r^2}\)."

23. If the slant height of a right circular cone is 7 cm and its total surface area is 147.84 square cm, then find the diameter of the cone.

24. The base perimeter of a right circular cone is 44 cm, and its slant height is 2 decimeters. Find the curved surface area of the cone.

25. The radius of the base of a right circular cone is equal to the radius of a sphere. The volume of the cone (in cubic centimeters) is equal to the curved surface area of the sphere (in square centimeters). What is the height of the cone?

(a) 4 cm (b) 3 cm (c) 8 cm (d) 12 cm

26. A solid sphere has a surface area of 616 square cm. It is melted to form 14 identical right circular cones, each with a height of 2 cm. Find the diameter of the base of each cone.

27. A right circular cone and a cylinder have equal curved surface areas. If the height and radius of the cone are \(h\) and \(r\), and the height and radius of the cylinder are \(H\) and \(r\), then show that: \[ h^2 = (2H + r)(2H - r) \]

28. If a solid hemisphere and a solid cone have equal base diameters and equal heights, find the ratio of their curved surface areas.

29. If the height of a right circular cone is 14 cm and its curved surface area is 264 cm², what is the volume of the cone?

30. A right circular cone has a slant height of 7 cm and a total surface area of 147.84 cm². Find the radius of the base of the cone.