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

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

3. 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.

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

5. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) will be –

(a) \(s=6d^2\) (b) \(3s=7d\) (c) \(s^3=d^2\) (d) \(d^2 = \cfrac{s}{2}\)

6. If a solid hemisphere has a radius of \(2r\) units, then its total surface area is ____ \(\pi r^2\) square units.

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

8. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) is \(s^3 = d^2\).

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

10. 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) \(18\pi \, cm^2\)

11. If the volume of a cube is \(V\) cubic centimeters, the total surface area is \(S\) square centimeters, and the length of the diagonal is \(d\) centimeters, then prove that \(Sd = 6\sqrt{3}V\).

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

13. 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 \(V\).

14. The length of a rectangular field is \(\frac{3}{2}\) times its breadth. If the length is reduced by 1200 cm and the breadth is increased by 1200 cm, the field becomes a square. Find the area of the field.

15. If the area of a square is \(A_1\) square units, and the area of the square drawn on its diagonal is \(A_2\) square units, then the ratio \(A_1 : A_2\) will be —

(a) 1:2 (b) 2:1 (c) 1:4 (d) 1:4

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

17. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) is \[ s = 6d^2 \]

18. 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

19. If the length of a cube's diagonal is \(4\sqrt{3}\) cm, determine its total surface area.

20. A solid cuboid has a length, width, and height ratio of \(4:3:2\), and its total surface area is \(468\) square cm. Determine the volume of the cuboid.

21. 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 _____.

22. 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 \( \cfrac{ar}{v} \).

23. 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 \(\cfrac{Cr}{V}\).

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

25. 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.

26. 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 = _____.

27. 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

28. If the radius of a sphere is increased by 2 cm, its curved surface area increases by 352 square centimeters. What was the original radius of the sphere?

(a) 5 cm (b) 6 cm (c) 7 cm (d) 5.6 cm

29. If the length of the diagonal of a cube is \(8\sqrt{3}\) cm, then what is the length of its edge?

(a) 8 cm (b) 4 cm (c) 5 cm (d) 7 cm

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