IMO Practice Test: Surface Areas and Volumes | Class 9 Mathematics

Question 1 of 6 hard

A cube and a sphere have the same volume. The ratio of their surface areas is:

1:1

π:6

6:π

(6/π)¹/³

Explanation: Volume equality: a³ = (4/3)πr³ → r = a[3/(4π)]¹/³. Surface area ratio = 6a² / 4πr² = (6a²)/(4πa²[3/(4π)]²/³) = (6/4π) × (4π/3)²/³ = (6/π)¹/³? Standard result: ratio = (6/π)¹/³

Question 2 of 6 hard

A hollow sphere has outer radius 10 cm and inner radius 8 cm. Find the volume of material used.

1620 cm³

1950 cm³

2030 cm³

2450 cm³

Explanation: Volume = (4/3)π(10³-8³) = (4/3)π(1000-512) = (4/3)π×488 = (1952/3)π ≈ (1952/3)×3.14 ≈ 2043 cm³ (≈2030)

Question 3 of 6 hard

A cone and a hemisphere have equal bases and equal volumes. The ratio of their heights is:

1:2

2:1

3:1

1:3

Explanation: Volume cone = (1/3)πr²h_c, Volume hemisphere = (2/3)πr³. Equate: (1/3)πr²h = (2/3)πr³ → h = 2r. Hemisphere has height = r, so ratio h_c : h_h = 2r : r = 2:1

Question 4 of 6 hard

A cylindrical pipe has inner diameter 4 cm and outer diameter 6 cm. Its length is 14 cm. Find the volume of metal in the pipe.

220 cm³

440 cm³

660 cm³

880 cm³

Explanation: Inner r=2, outer r=3. Volume = πh(R²-r²) = (22/7)×14×(9-4) = 44×5 = 220 cm³

Question 5 of 6 hard

How many 2 cm cubes can be cut from a 10 cm cube?

25

50

100

125

Explanation: Volume of big cube = 1000 cm³, volume of small cube = 8 cm³, number = 1000/8 = 125

Question 6 of 6 hard

A sphere of radius 3 cm is melted and recast into a cylinder of radius 2 cm. Find the height of the cylinder.

6 cm

9 cm

12 cm

15 cm

Explanation: Volume sphere = (4/3)π×27 = 36π cm³. Volume cylinder = π×4×h = 4πh. Equate: 4πh = 36π → h = 9 cm

IMO Practice Test — Surface Areas and Volumes