1. 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

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

3. 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

4. If the length of the diagonal of a cube is \(5\sqrt{3}\) cm, find its volume.

5. 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?

6. 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?

7. 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

8. 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?

9. 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) \(6\sqrt3\)

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

11. 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

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

13. 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) \(a=\sqrt2d\)

14. If a triangle similar to one with sides 4 cm, 6 cm, and 8 cm has its largest side measuring 6 cm, what is the length of the smallest side of that triangle?

(a) 4 cm (b) 3 cm (c) 2 cm (d) 5 cm

15. 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

16. 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

17. 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

18. The volume of a cube is equal to the area of one of its faces. What is the length of its edge?

(a) 4 units (b) 5 units (c) 6 units (d) 8 units

19. If each face diagonal of a cube is \(8\sqrt{2}\) cm long, then what is the length of the space diagonal of the cube?

(a) \(8\sqrt3\) cm (b) \(3\sqrt8\) cm (c) \(3\sqrt2\) cm (d) \(2\sqrt3\) cm

20. If the length of a cuboid is increased by 10%, the width by 20%, and the height is decreased by 30%, what is the percentage change in its volume?

(a) 6.5 increased (b) 7.6 increased (c) 6.5decreased (d) 7.6 decreased

21. If the total surface area of a cube is 216 square centimeters, what is its volume?

(a) 216 tcubic cm (b) 212 cubic cm (c) 316 cubic cm (d) 256

22. If the length, breadth, and height of a right rectangular prism are in the ratio 3:2:1 and its total surface area is 88 square centimeters, then what is its volume?

(a) 120 cubic cm (b) 64 cubic cm (c) 48 cubic cm (d) 24 cubic cm

23. The length of the diagonal of a rectangular field is 15 meters, and its length is 3 meters more than its width. What is the length of the rectangular field?

(a) 9 meter (b) 12 meter (c) 15 meter (d) 18 meter

24. If the length of each edge of a cube is increased by 50%, by what percentage will its volume increase?

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

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

27. From a cuboid measuring 16 cm × 4 cm × 2 cm, several cubes with edge length 2 cm are cut out. What is the ratio of the total surface area of the original cuboid to the total surface area of all the cubes?

(a) 11:48 (b) 12:50 (c) 13:36 (d) 14:28

28. If each edge of a cube is decreased by 10%, what is the percentage decrease in its total volume?

(a) 36.2% (b) 27.1% (c) 30.5% (d) 20.6%

29. 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

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