level – 1

1. A hollow spherical metallic ball has an external diameter of 8 cm and is 1 cm thick. Find the volume of metal used. (Take π = 3)
(a) 120 cm³
(b) 132 cm³
(c) 148 cm³
(d) 160 cm³

2. The sum of the radii of two solid spheres is 10 cm. The sum of their volumes is 1408 cm³. (Take π = 22/7) What is the product of their radii?
(a) 12
(b) 16
(c) 20
(d) 24

3. If the radius of a solid sphere is tripled, its volume becomes:
(a) 3 times
(b) 9 times
(c) 27 times
(d) 81 times

4. The radii of two spheres are in the ratio 4 : 3. Their volumes are in the ratio:
(a) 16 : 9
(b) 64 : 27
(c) 12 : 9
(d) 8 : 3

5. The total surface area of a solid hemisphere is 300π cm². The volume of the hemisphere is:
(a) 400π cm³
(b) 500π cm³
(c) 600π cm³
(d) 800π cm³

6. The largest sphere is carved out of a cube of side 6 cm. The volume of the sphere (in cm³) is:
(a) 36π
(b) 48π
(c) 72π
(d) 96π

7. The surface areas of two spheres are in the ratio 9 : 16. The ratio of their volumes is:
(a) 27 : 64
(b) 81 : 256
(c) 3 : 4
(d) 9 : 16

8. A sphere and a hemisphere have volumes in the ratio 4 : 1. The ratio of their radii is:
(a) 2 : 1
(b) 4 : 1
(c) 1 : 2
(d) 3 : 2

9. A solid sphere of radius 4 cm is melted and recast into 8 solid spheres of equal volume. The radius (in cm) of each small sphere is:
(a) 1
(b) 2
(c) 3
(d) 4

10. The total surface area of a sphere is 36π square units. The volume of the sphere is:
(a) 18π
(b) 24π
(c) 36π
(d) 48π

11. The volume of a right circular cone and a sphere are equal. If their radii are equal, then the ratio of the height of the cone to the radius of the sphere is:
(a) 2 : 1
(b) 3 : 1
(c) 4 : 1
(d) 1 : 2

12. A cylindrical vessel of internal radius 7 cm contains some water. A solid sphere of radius 7 cm is gently placed inside the vessel and the water just covers the sphere. If the sphere fits exactly in the vessel, the depth of water in the vessel before the sphere was placed is:
(a) 7 cm
(b) 14 cm
(c) 21 cm
(d) 28 cm

13. Four solid metallic spheres of diameters 4 cm, 6 cm, 8 cm and 10 cm are melted and recast into a new solid sphere. The diameter of the new sphere is:
(a) 12 cm
(b) 14 cm
(c) 16 cm
(d) 18 cm

14. Three solid metallic balls of radii 1 cm, 2 cm and 3 cm are melted and moulded into a single solid ball. The radius of the new ball is:
(a) 3 cm
(b) 4 cm
(c) 5 cm
(d) 6 cm

15. Three solid spheres of a metal with radii 2 cm, 5 cm and 7 cm are melted to form another solid sphere. The radius of the new sphere is:
(a) 6 cm
(b) 7 cm
(c) 8 cm
(d) 9 cm

16. A solid sphere of radius 3 cm is completely immersed in water contained in a cylindrical vessel of base radius 3 cm. The rise in the level of water in the cylinder is:
(a) 3 cm
(b) 4 cm
(c) 5 cm
(d) 6 cm

17. Eight solid spheres of equal size are formed by melting a solid cylinder of diameter 14 cm and height 3 cm. The diameter of each sphere is:
(a) 3 cm
(b) 4 cm
(c) 5 cm
(d) 6 cm

18. By melting a solid metal sphere of diameter 10 cm, three smaller spheres are formed whose diameters are in the ratio 1 : 2 : 2. The radius (in cm) of the smallest sphere is:
(a) 1 cm
(b) 2 cm
(c) 3 cm
(d) 4 cm

19. A solid metallic sphere of radius 2 decimetres is melted to form a circular sheet of uniform thickness 2 mm. The diameter of the sheet so formed is:
(a) 4 m
(b) 6 m
(c) 8 m
(d) 10 m

20. A metal wire of length 44 m and diameter 2 mm is melted to form a solid sphere. The radius of the sphere (in cm) is:
(a) 3 cm
(b) 4 cm
(c) 5 cm
(d) 6 cm

21. A solid metallic spherical ball of diameter 8 cm is melted and recast into a cone whose base has diameter 16 cm. The height of the cone is:
(a) 2 cm
(b) 3 cm
(c) 4 cm
(d) 6 cm

22. The diameter of an iron ball is 14 cm. It is melted and recast into a solid cylinder of height 28/3 cm. The diameter of the base of the cylinder is:
(a) 7 cm
(b) 10 cm
(c) 14 cm
(d) 21 cm

23. A solid spherical metal ball of diameter 21 cm is melted and converted into a cylindrical wire of diameter 7 cm. The length of the wire is:
(a) 18 cm
(b) 21 cm
(c) 24 cm
(d) 27 cm

24. A solid sphere is melted and recast into a right circular cone whose base radius is twice the radius of the sphere. The ratio of the height to the radius of the cone formed is:
(a) 1 : 2
(b) 2 : 1
(c) 3 : 2
(d) 4 : 1

25. A solid copper sphere of diameter 14 cm is drawn into a wire of diameter 7 mm. The length of the wire (in metres) is:
(a) 120 m
(b) 140 m
(c) 160 m
(d) 180 m

26. A rectangular block of metal has dimensions 14 cm, 33 cm and 24 cm. The block is melted and recast into a sphere. The radius of the sphere is (Take π = 22/7):
(a) 7 cm
(b) 10.5 cm
(c) 14 cm
(d) 21 cm

27. A solid iron ball of diameter 6 cm is coated with copper. If the volume of copper is equal to the volume of iron, then the thickness of the copper coating is approximately [given ∛2 = 1.26]:
(a) 0.7 cm
(b) 0.8 cm
(c) 0.9 cm
(d) 1.0 cm

28. A spherical ball of radius 3 cm is dropped into a cylindrical vessel. The volume of water (in cm³) required to just immerse the ball completely is:
(a) 36π
(b) 72
(c) 108
(d) 144

29. Assume that each drop of water is spherical and has a diameter of 1/5 cm. A conical glass has its height equal to the diameter of its rim. If 8000 such drops completely fill the glass, then the height of the glass (in cm) is:
(a) 4 cm
(b) 6 cm
(c) 8 cm
(d) 10 cm

30. The internal and external radii of a hollow spherical shell are 4 cm and 7 cm respectively. The shell is melted and recast into a solid cylinder of height 4 cm. The radius (in cm) of the cylinder is:
(a) 6
(b) 7
(c) 8
(d) 9

31. The radii of two solid metallic spheres are 2 cm and 5 cm respectively. They are melted together to form a hollow sphere. If the external radius of the hollow sphere is 7 cm, then the thickness (in cm) of the hollow sphere is:
(a) 1
(b) 2
(c) 3
(d) 4

32. The ratio of the weights of two solid spheres is 16 : 81. The ratio of weights per 1 cc (densities) of the materials of the two spheres is 2 : 1. The ratio of the radii of the two spheres is:
(a) 1 : 2
(b) 2 : 3
(c) 3 : 4
(d) 4 : 5

33. A solid metallic sphere of radius 4 cm is melted and recast into small spheres, each of radius 2 mm. The total number of small spheres formed is:
(a) 800
(b) 1000
(c) 1600
(d) 2000

34. A solid sphere of radius 3.5 cm is melted and drawn into a cylindrical wire of radius 0.35 cm. The length (in cm) of the wire is:
(a) 700
(b) 800
(c) 900
(d) 1000

level – 2

1. If the radius of a sphere is increased by 3 cm, its surface area increases by 792 cm². The original radius (in cm) of the sphere is: (Take π = 22/7)
(a) 4
(b) 5
(c) 6
(d) 7

2. The diameters of two hollow spheres made from the same metal sheet are 14 cm and 21 cm respectively. The ratio of the areas of metal sheets required to make the two spheres is:
(a) 2 : 3
(b) 4 : 9
(c) 9 : 16
(d) 16 : 25

3. A solid metallic sphere of radius 6 cm is melted to form 27 equal small solid spheres. The ratio of the surface area of the original sphere to that of one small sphere is:
(a) 3 : 1
(b) 6 : 1
(c) 9 : 1
(d) 12 : 1

4. The area of the largest circle that can be drawn inside a square of side 14 cm is:
(a) 154 cm²
(b) 196 cm²
(c) 100 cm²
(d) 308 cm²

5. The diameter of a sphere is three times the diameter of another sphere. The curved surface area of the larger sphere and the volume of the smaller sphere are numerically equal. The numerical value of the radius of the larger sphere is:
(a) 18
(b) 24
(c) 27
(d) 36

level – 3

1. The maximum number of spherical balls, each of diameter 2 m, that can be made from a rectangular block of lead measuring 12 m × 8 m × 4 m is:
(a) 91
(b) 192
(c) 384
(d) 768

2. A metallic sphere of radius 7 cm is melted and recast into small cones, each of radius 3.5 cm and height 4 cm. The number of cones formed is:
(a) 12
(b) 14
(c) 16
(d) 28

ANSWERS