1. A right circular cone has its **volume equal to its curved surface area**. If the height and radius of the cone are respectively **h units** and **r units**, then **show that**: \[ \frac{1}{h^2} + \frac{1}{r^2} = \frac{1}{9} \]

2. The base radius of a solid right circular cone is \(r\) units, its vertical height is \(h\) units, and its slant height is \(l\) units. The cone’s base is attached to the base of a solid right circular cylinder. If the base radius and height of the cylinder are equal to those of the cone, then the total surface area of the combined solid is \((πrl + 2πrh + 2πr^2)\) square units.

3. If a cone has volume \(V\) cubic units, total surface area \(S\) square units, height \(h\) units, and base radius \(r\) units, then show that: \[ S = 2V\left(\frac{1}{h} + \frac{1}{r}\right) \]

4. 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 \[ \left( \frac{1}{h^2} + \frac{1}{r^2} \right) \]

5. If a solid cone and a solid right circular cylinder have the same base radius and same height, and the ratio of their curved surface areas is 5:8, then determine the ratio of their base radius to height.

6. 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

7. 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 \[ h = \frac{\sqrt{S^2 - \pi^2 r^4}}{\pi r} \]

8. 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 \(\frac{1}{h^2} + \frac{1}{r^2}\).

9. "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 \(\frac{1}{h^2} +\frac{1}{r^2}\)."

10. 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\(\pi^2\)

11. 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.

12. A solid hemisphere and a solid right circular cylinder have the same height and the same base radius. Determine the ratio of their curved surface areas.

13. 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 \(\cfrac{1}{h^2}+\cfrac{1}{r^2}\).

14. 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.

15. 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.

16. 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.

17. If a right circular cone has a radius of \(\frac{r}{2}\) units and a slant height of \(2l\) units, then the total surface area is \(2πr(r + l)\).

18. 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 \(r = R\).

19. 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

20. If the radius of a right circular cylinder is doubled and the height is halved, then the curved surface area of the new cylinder will be what fraction of the original cylinder’s curved surface area?

(a) equal (b) double (c) half (d) four times

21. The curved surface area of a solid sphere is equal to the curved surface area of a solid right circular cylinder. The height and diameter of the cylinder are both 12 cm. Write the radius of the sphere.

22. A solid right circular cylinder and two hemispheres have equal base radii. If the two hemispheres are attached to the flat circular ends of the cylinder, then the total surface area of the resulting solid = curved surface area of one hemisphere + curved surface area of the cylinder + curved surface area of the other hemisphere.

23. The bottom part of a solid is in the shape of a hemisphere, and the top part is in the shape of a right circular cone. If the surface areas of both parts are equal, then calculate and write the ratio of the radius to the height of the cone.

24. If the base radius of a right circular cone is 3 cm and the height is 4 cm, then the lateral surface area of the cone will be.

(a) \(10\pi \, cm^2\) (b) \(15\pi \, cm^2\) (c) \(12\pi \, cm^2\) (d) \(18\pi \, cm^2\)

25. If the radius of a right circular cone is \(\cfrac{r}{2}\) units and the slant height is \(2l\) units, then the total surface area is –

(a) \(2πr (l+r)\) square units (b) \(πr\left(l+\cfrac{r}{4}\right)\) square units (c) \(πr(l+r)\) square units (d) \(2πr l\) square units

26. A sphere and a right circular cylinder both have a radius of 3 cm. If their volumes are equal, what is the height of the cylinder?

27. If the height and the base radius of a right circular cone are equal, then the measure of the cone's apex angle will be _____.

28. If \(h, c, v\) represent the height, curved surface area, and volume of a cone, respectively, then prove that \(3\pi v h^3 - c^2 h^2 + 9v^2 = 0\).

29. If the volumes of two solid right circular cylinders are equal and their height ratio is 1:4, then their radius ratio will be—.

(a) \(2:1\) (b) \(\sqrt2:1\) (c) \(1:2\) (d) \(1:\sqrt2\)

30. If the volumes of two solid right circular cylinders are equal and their height ratio is 1:2, then their radius ratio will be –?