1. 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) \(6\sqrt3\)

2. 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?

3. What multiple of the length of a face diagonal of a cube is the length of the space diagonal of the cube?

(a) √3 (b) √2 (c) \(\cfrac{\sqrt6}{2}\) (d) None of the above

4. 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

5. 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?

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

7. 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

8. ABCD is a rectangle. O is the point where the diagonals intersect. If AB = 4 cm and OD = 2.5 cm, then what is the length of BC?

(a) 4 cm (b) 1.5 cm (c) 3 cm (d) 2 cm

9. 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

10. 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

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. 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?

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

14. 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

15. 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 \[ s = 6d^2 \]

16. If the length of each edge of a cube is doubled, then by what percentage will the total surface area of the cube increase?

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

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

19. In a rectangular prism, the ratio of length, width, and height is \(4:5:3\). If the length of one diagonal is \(35\sqrt2\) cm, what is the total surface area?

20. A cube can accommodate a longest diagonal of \(6\sqrt{3}\) cm. What is the volume of the cube?

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

22. The space diagonal of a cube is 4√3 cm. Find the total surface area of the cube.

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

24. 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

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

26. If the length of the diagonal of a cube is \(\sqrt{48}\), then what is its volume?

27. 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) \(a=\sqrt2d\)

28. AB and CD are two parallel straight lines. AD and BC intersect each other at point O. If OA = 2 cm, OB = 3 cm, and OD = 4 cm, then what is the length of OC?

(a) 6 cm (b) 4 cm (c) 4.8 cm (d) 4.2 cm

29. If one diagonal of a rhombus with side length 17 cm is 16 cm, what is the length of the other diagonal?

(a) 30 cm (b) 25 cm (c) 32 cm (d) 18 cm

30. In the rhombus PQRS, if ∠Q = 60° and RS = 6 cm, what is the length of the diagonal PR?

(a) \(6\sqrt3\) cm (b) \(\sqrt3\) cm (c) \(\cfrac{1}{\sqrt3}\) cm (d) 6 cm