1. If a solid hemisphere and a solid cone have equal base areas and equal heights, then what is the ratio of their volumes?

(a) 1:3 (b) 1:2 (c) 2:3 (d) 3:4

2. If two cones have equal base radii and their heights are in the ratio 2:3, then the ratio of their volumes will be ______.

3. If a solid hemisphere and a solid cone have equal base diameters and equal heights, find the ratio of their curved surface areas.

4. If a right circular cone and a hemisphere have equal volumes and equal bases, then their height ratio will be 2:1.

5. A right circular cone and a hemisphere have equal base areas and volumes. Determine the ratio of their heights.

6. A solid hemisphere and a solid cone have equal base diameters and equal heights. Calculate and write the ratio of their volumes and the ratio of their curved surface areas.

7. If the diameters of two cones are equal and the ratio of their slant heights is 5:7, then what is the ratio of their curved surface areas?

(a) 25:7 (b) 25:49 (c) 5:49 (d) 5:7

8. If the radii of two cones are in the ratio 2:3 and their heights are in the ratio 5:3, then what will be the ratio of their volumes?

(a) 4:9 (b) 9:4 (c) 27:20 (d) 20:27

9. A solid hemisphere and a solid cone have the same base radius and height. What is the ratio of their volumes?

(a) 3:2 (b) 1:3 (c) 2:3 (d) None of the above

10. A solid sphere and a solid cone have the same diameter, and their total surface areas are equal. What is the ratio of the cone's height to its diameter?

11. Two friends start a partnership business by investing ₹40,000 and ₹50,000 respectively. They agree that 50% of the profit will be shared equally, and the remaining 50% will be divided in the ratio of their investments. If the first friend’s share of the profit is ₹800 less than the second friend’s share, then what is the first friend’s profit?

12. If a solid hemisphere and a solid cone have the same base radius and height, what is the ratio of their volumes?

(a) 3:2 (b) 1:3 (c) 2:3 (d) None of the above

13. In a joint business, if 6 times A's capital, 8 times B's capital, and 10 times C's capital are all equal, then what will be the ratio of their capitals?

(a) 15 : 20 : 12 (b) 20 : 12 : 15 (c) 20 : 15 : 12 (d) 15 : 12 : 17

14. The radius of a hemisphere and the height of a cone are equal. If their volumes are numerically equal, then what is the ratio of the radius of the hemisphere to the radius of the cone?

15. If two triangles have their bases on the same straight line and share the same vertex (the opposite vertex), then the ratio of their areas is equal to the ratio of the lengths of their bases.

16. If two solid right circular cones have equal volume and their heights are in the ratio 1:2, then what is the ratio of the lengths of their radii?

(a) \(1:\sqrt2\) (b) \(1:2\) (c) \(\sqrt2 : 1\) (d) \(2:1\)

17. If the volumes of two vertical circular cylinders are equal, and their heights are in the ratio 4:9, then the ratio of their radii will be –

(a) 3:2 (b) 2:3 (c) 4:9 (d) 8:9

18. If the volumes of two vertical solid cylinders are equal, and their heights are in the ratio 1:2, then the ratio of the lengths of their radii will be –

(a) 1: √2 (b) √2:1 (c) 1:2 (d) 2:1

19. If the volumes of two solid right circular cylinders are equal and their heights are in the ratio 1:2, what will be the ratio of the lengths of their radii?

(a) 1:√2 (b) √2:1 (c) 1:2 (d) 2:1

20. Two right circular cones of equal height have their base diameters in the ratio 2 : 3. The ratio of their volumes will be —

(a) 3:4 (b) 4:5 (c) 5:4 (d) 4:9

21. A hemisphere and a cone have the same base and height. Determine the ratio of their volumes and the ratio of their curved surface areas.

22. If a hemisphere and a cone have the same base area and the same height, find the ratio of their curved surface areas.

23. A right circular cylinder and a cone have equal bases, and the ratio of their volumes is 3:2. Prove that the height of the cone is half the height of the cylinder.

24. A right circular cylinder and a right circular cone have equal base radii and equal heights. The ratio of their volumes = ___________.

25. If the volumes of two cones are in the ratio \(1:4\) and their radii are in the ratio \(4:5\), then what will be the ratio of their heights?

(a) 1:5 (b) 5:4 (c) 5:16 (d) None of the above

26. A cone and a sphere have equal volumes. If their radii are the same, find the ratio of the height of the cone to the diameter of its base.

27. If the height and the base radius of a right circular cone are equal, then the measure of the cone's apex angle will be _____.

28. A solid hemisphere and a solid cone have the same base diameter and height. Determine the ratio of their volumes and the ratio of their curved surface areas.

29. The ratio of the base radii of a right circular cylinder and a right circular cone is 3:4, and the ratio of their heights is 2:3. What will be the ratio of the volumes of the cylinder and the cone?

30. If the volumes of two solid right circular cylinders are equal and their height ratio is 1:4, then their radius ratio will be—.

(a) \(2:1\) (b) \(\sqrt2:1\) (c) \(1:2\) (d) \(1:\sqrt2\)