1. If the length of the diagonal of a cube is \(\sqrt{12}\) cm, what is its volume?

(a) 18 cubic cm (b) 8 cubic cm (c) 6 cubic cm (d) 16 cubic cm

2. 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

3. If each side of a rhombus is 5 cm and one of its diagonals is 4 cm, find the length of the other diagonal.

4. If the length of a cube’s diagonal is \(4\sqrt{3}\) cm, then its volume is—

(a) 60 cubic cm (b) 64 cubic cm (c) 72 cubic cm (d) 80 cubic cm

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

6. If a cube has a side length of \(a\) units and a diagonal length of \(d\) units, then the relationship between \(a\) and \(d\) is—?

(a) \(\sqrt2a=d\) (b) \(\sqrt3a=d\) (c) \(a=\sqrt3d\) (d) \(a=\sqrt2d\)

7. The sum of the length, width, and height of a rectangular box is 24 cm, and the length of its diagonal is 15 cm. What is the total surface area of the rectangular box?

(a) 360 square cm (b) 221 square cm (c) 351quare cm (d) 256 quare cm

8. The diagonal of a cuboid is √725 cm and its volume is 3000 cubic cm. The total surface area of the cuboid is 1300 square cm. Find the length, breadth, and height of the cuboid.

9. Here’s the English translation of your sentence: "A right rectangular prism has its length twice the breadth and its height half the breadth. If the total surface area of the prism is 448 square centimeters, find its volume."

10. The areas of the three adjacent faces of a right rectangular prism are \(a\), \(b\), and \(c\) square units. Find the length of the space diagonal (the longest diagonal) of the prism.

11. 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.

12. 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.

13. The sum of the length, breadth, and height of a rectangular box is 24 cm, and the length of its diagonal is 15 cm. What is the total surface area of the box?

14. 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) \(d^2 = \cfrac{s}{2}\)

15. A rod of length \(6\sqrt{3}\) cm can be placed inside a cube along its longest diagonal. What is the volume of the cube?

16. If the diagonal of a rectangle is 10 cm and its area is 62.5 square cm, then the sum of the length and breadth of the rectangle is —

(a) 12 cm (b) 15 cm (c) 20 cm (d) 25 cm

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. If the length of the diagonal of each face of a cube is \(6\sqrt{2}\) cm, what is the total surface area of the cube?

19. If the length of each face diagonal of a cube is \(5\sqrt{2}\) cm, then what is the length of the space diagonal of the cube?

(a) \(3\sqrt5\) cm (b) \(4\sqrt3\) (c) \(5\sqrt3\) (d) \(6\sqrt3\)

20. 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 \(s^3 = d^2\).

21. The sum of the length, width, and height of a right rectangular prism is 25 cm, and its total surface area is 264 square cm. Determine the length of its diagonal.

22. The volume of a right circular cone is 180\(\pi\) cubic cm, and its height is 15 cm. Find the length of the base diameter.

23. If the total length of the edges of a solid cube is 36 cm, then its volume will be –?

(a) 9 cubic cm (b) 54 cubic cm (c) 27 cubic cm (d) 36 cubic cm

24. If the height of a right circular cylinder is 14 cm and its curved surface area is 264 square cm, then find its volume.

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

26. If the length of a diagonal of a cube is \(4\sqrt{3}\) cm, calculate and write the total surface area of the cube.

27. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, what is the relationship between \(s\) and \(d\)?

(a) s=6d\(^2\) (b) 3s=7d (c) s\(^3\)=d2\(^2\) (d) d\(^2\)=s/2

28. From a solid cube with side length \(x\) units, the largest possible solid sphere is carved out. Find the diameter of the sphere.

(a) \(x\) units (b) \(x\) units (c) \(\cfrac{x}{2}\) unitsts (d) \(4x\) units

29. A rod of length \( 4\sqrt{3} \) cm is placed inside a cube along its longest diagonal. What is the volume of the cube?

30. Let the length of the space diagonal of the cube be \(x\), the length of a face diagonal be \(y\), and the edge of the cube be \(a\). Then, \[ x = a\sqrt{3} \Rightarrow a = \cfrac{x}{\sqrt{3}} \quad \text{— (i)} \] And, \[ y = a\sqrt{2} \Rightarrow a = \cfrac{y}{\sqrt{2}} \quad \text{— (ii)} \] Comparing equations (i) and (ii): \[ \cfrac{x}{\sqrt{3}} = \cfrac{y}{\sqrt{2}} \Rightarrow x = \cfrac{\sqrt{3}}{\sqrt{2}} \cdot y \Rightarrow x = \cfrac{\sqrt{6}}{2} \cdot y \] \(\therefore\) The length of the space diagonal of a cube is \(\cfrac{\sqrt{6}}{2}\) times the length of its face diagonal.

(a) undefined (b) positive (c) negative (d) zero