Question 1

The volume of a sphere is 288π cm³. What is its radius (in cm)?

Accepted Answer

Step 1: Volume of sphere = (4/3)πr³. Step 2: (4/3)πr³ = 288π. Step 3: (4/3)r³ = 288. Step 4: r³ = 288 × (3/4) = 216. Step 5: r = ∛216 = 6 cm. Therefore, the answer is 6 cm.

Question 2

A cylinder has a total surface area of 352 cm² and radius of 4 cm. What is its height (in cm)?

Accepted Answer

Step 1: Total Surface Area of cylinder = 2πr(r + h). Step 2: 2 × (22/7) × 4 × (4 + h) = 352. Step 3: (44/7) × 4 × (4 + h) = 352. Step 4: (176/7) × (4 + h) = 352. Step 5: (4 + h) = 352 × (7/176) = 2464/176 = 14. Step 6: h = 14 - 4 = 10 cm. Therefore, the answer is 10 cm.

Question 3

The volume of a cylinder is 1540 cm³ and its height is 10 cm. What is its radius (in cm)?

Accepted Answer

Step 1: Volume of cylinder = πr²h. Step 2: (22/7) × r² × 10 = 1540. Step 3: (220/7) × r² = 1540. Step 4: r² = 1540 × (7/220) = 10780/220 = 49. Step 5: r = √49 = 7 cm. Therefore, the answer is 7 cm.

Question 4

Two lines intersect each other. If one of the angles formed is 65°, what is the measure of the angle adjacent to it on a straight line?

Accepted Answer

Step 1: When two lines intersect, adjacent angles on a straight line are supplementary (sum to 180°). Step 2: If one angle is 65°, the adjacent angle = 180° − 65° = 115°. Therefore, the answer is 115°.

Question 5

A transversal crosses two parallel lines. If one of the corresponding angles is 72°, what is the measure of the co-interior angle on the same side of the transversal?

Accepted Answer

Step 1: When a transversal crosses two parallel lines, corresponding angles are equal. Step 2: Co-interior angles (also called consecutive interior angles) are supplementary, summing to 180°. Step 3: If the corresponding angle is 72°, the co-interior angle = 180° − 72° = 108°. Therefore, the answer is 108°.

Question 6

In a triangle, one angle measures 48° and another measures 67°. What is the measure of the third angle?

Accepted Answer

Step 1: The sum of all angles in a triangle is 180°. Step 2: Third angle = 180° − 48° − 67° = 180° − 115° = 65°. Therefore, the answer is 65°.

Question 7

A cone has a base radius of 6 cm and height of 8 cm. What is its slant height (in cm)?

Accepted Answer

Step 1: For a cone, slant height l is related to radius r and height h by: l² = r² + h². Step 2: l² = 6² + 8² = 36 + 64 = 100. Step 3: l = √100 = 10 cm. Therefore, the answer is 10 cm.

Question 8

The volume of a cylinder is 1540 cm³ and its height is 10 cm. If a cone with the same base radius and height is formed, what is the ratio of the cylinder's volume to the cone's volume?

Accepted Answer

Step 1: Volume of cylinder = πr²h = 1540 cm³, with h = 10 cm. Step 2: πr² × 10 = 1540 → πr² = 154. Step 3: Volume of cone with same base radius and height = (1/3)πr²h = (1/3) × 154 × 10 = 1540/3 cm³. Step 4: Ratio = V_cylinder : V_cone = 1540 : (1540/3) = 1540 × 3 : 1540 = 3 : 1. Therefore, the ratio is 3:1.

Question 9

A cylinder has a radius of 7 cm and height of 10 cm. If the radius is increased by 50% while the height remains the same, by what percentage does the volume increase?

Accepted Answer

Step 1: Original volume = πr²h = π(7)²(10) = 490π cm³. Step 2: New radius = 7 × 1.5 = 10.5 cm. Step 3: New volume = π(10.5)²(10) = 1102.5π cm³. Step 4: Percentage increase = [(1102.5π - 490π) / 490π] × 100 = (612.5 / 490) × 100 = 125%. Therefore, the volume increases by 125%.

Question 10

A cone has a base radius of 6 cm and slant height of 10 cm. What is the ratio of its curved surface area to its total surface area?

Accepted Answer

Step 1: Curved surface area (CSA) = πrl = π(6)(10) = 60π cm². Step 2: Base area = πr² = π(6)² = 36π cm². Step 3: Total surface area (TSA) = CSA + Base area = 60π + 36π = 96π cm². Step 4: Ratio = CSA : TSA = 60π : 96π = 60 : 96 = 5 : 8. Therefore, the ratio is 5:8.

Question 11

A cone is inscribed in a sphere of radius 13 cm such that the base of the cone touches the sphere and the apex is at the centre of the sphere. If the base radius of the cone is 12 cm, what is the height of the cone?

Accepted Answer

Step 1: The apex of the cone is at the centre of the sphere (O). The base of the cone is a circle on the sphere. Step 2: Let the base radius of cone = 12 cm, and height = h. Step 3: Any point on the base circle is at distance 13 cm from the centre O (sphere radius). Step 4: Using Pythagoras: h² + 12² = 13². Step 5: h² + 144 = 169. Step 6: h² = 25, so h = 5 cm. Therefore, the height of the cone is 5 cm.

Question 12

A solid metallic sphere of radius 6 cm is melted and recast into a solid cone with base radius 4 cm. Find the height of the cone (in cm).

Accepted Answer

Step 1: Volume of sphere = (4/3)πr³ = (4/3)π(6)³ = (4/3)π(216) = 288π cm³. Step 2: Volume of cone = (1/3)πr²h. Since volumes are equal: 288π = (1/3)π(4)²h. Step 3: 288π = (1/3)π(16)h → 288 = (16h)/3 → h = (288 × 3)/16 = 864/16 = 54 cm.

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