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

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

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

4. 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

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

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

8. If the volume of a right circular cone is \(X\) cubic units, the area of its base is \(Y\) square units, and its height is \(Z\) units, then what is the value of \(\frac{YZ}{X}\)?

9. 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\(\pi^2\)

10. 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, determine the value of \( \cfrac{Cr}{V} \).

11. If the curved surface area of a right circular cylinder is \( a \) square units, the radius of its base is \( r \) units, and the volume is \( v \) cubic units, then determine the value of \( \cfrac{ar}{v} \).

12. 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

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 determine the value of \(\cfrac{Cr}{V}\).

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

15. 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

16. 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) \(H=\cfrac{3V}{2A}\)

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

18. 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.

19. 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 \(2πr(r + l)\).

20. If the numerical values of the volume and lateral surface area of a cone are equal, then the relationship between its height and base radius is?

(a) \(r^2-h^2=2^2\) (b) \(2r^2+h^2=3^2\) (c) \(r^2+h^2=3^2\) (d) none of the above

21. 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

22. 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.

23. 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.

24. If a right circular cone has volume \(V\) cubic units, base area \(A\) square units, and height \(H\) units, then determine the value of \(\cfrac{AH}{3V}\).

25. 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.

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

27. 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.

28. 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 = _____.

29. Translate: "If a right circular cylinder has lateral surface area \(c\) square units, base radius \(r\) units, and volume \(V\) cubic units, then find the height (value) of the cylinder." Would you like me to solve it too? I’m ready when you are.

30. If the volume of a cone is \(x\), the base area is \(y\), and the height is \(z\), then the value of \(\cfrac{x}{yz}\) will be 3.