Question 1

The volume of a sphere is 288π cm³. What is its radius?

Accepted Answer

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

Question 2

A hemisphere has a radius of 7 cm. What is its total surface area (including the flat circular base)?

Accepted Answer

Step 1: Total surface area of hemisphere = 3πr² (curved surface 2πr² plus flat base πr²). Step 2: Substitute r = 7 cm. Step 3: TSA = 3 × (22/7) × 7² = 3 × (22/7) × 49 = 3 × 22 × 7 = 462 cm². Therefore, the total surface area is 462 cm².

Question 3

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 4

A transversal intersects two parallel lines. If one of the corresponding angles is 72°, what is the measure of the other corresponding angle?

Accepted Answer

Step 1: When a transversal intersects two parallel lines, corresponding angles are equal. Step 2: If one corresponding angle is 72°, the other corresponding angle is also 72°. Therefore, the answer is 72°.

Question 5

Three angles of a triangle are in the ratio 2:3:4. What is the measure of the largest angle?

Accepted Answer

Step 1: Let the three angles be 2x, 3x, and 4x. Step 2: Sum of angles in a triangle = 180°, so 2x + 3x + 4x = 180°. Step 3: 9x = 180°, therefore x = 20°. Step 4: The largest angle = 4x = 4 × 20° = 80°. Therefore, the answer is 80°.

Question 6

The curved surface area of a cone is 550 cm² and its slant height is 25 cm. What is the radius of the base of the cone? (Use π = 22/7)

Accepted Answer

Step 1: Curved surface area of cone = πrl, where r is radius and l is slant height. Step 2: 550 = (22/7) × r × 25. Step 3: r = (550 × 7)/(22 × 25) = 3850/550 = 7 cm. Therefore, the radius is 7 cm.

Question 7

The volume of a sphere is 4851 cm³. What is its radius? (Use π = 22/7)

Accepted Answer

Step 1: Volume of sphere = (4/3)πr³ = 4851. Step 2: (4/3) × (22/7) × r³ = 4851. Step 3: (88/21) × r³ = 4851. Step 4: r³ = 4851 × 21/88 = 1157.625 ≈ 1157.63. Step 5: r ≈ 10.5 cm. Therefore, the radius is 10.5 cm.

Question 8

A cone and a cylinder have the same base radius of 5 cm. The volume of the cone is 100π cm³. If the cylinder's height equals twice the cone's height, what is the volume of the cylinder (in cm³)?

Accepted Answer

Step 1: Volume of cone = (1/3)πr²h_c = (1/3)π(5)²h_c = 100π. Step 2: (1/3)(25)h_c = 100 → h_c = 12 cm. Step 3: Height of cylinder h_cyl = 2 × 12 = 24 cm. Step 4: Volume of cylinder = πr²h_cyl = π(5)²(24) = 600π cm³.

Question 9

The curved surface area of a cone is 60π cm² and its slant height is 10 cm. A sphere is inscribed in this cone such that it touches the base and the lateral surface. Find the radius of the sphere (in cm).

Accepted Answer

Step 1: Curved surface area of cone = πrl = 60π, where l = 10. So πr(10) = 60π → r = 6 cm (base radius). Step 2: From l² = r² + h², we get 100 = 36 + h² → h = 8 cm. Step 3: For a sphere inscribed in a cone touching base and lateral surface, radius R = (r × h)/(r + l) = (6 × 8)/(6 + 10) = 48/16 = 3 cm.

Question 10

A hollow sphere of outer radius 8 cm and inner radius 6 cm is melted and recast into a solid cone of base radius 4 cm. Find the height of the cone (in cm).

Accepted Answer

Step 1: Volume of hollow sphere = (4/3)π(R³ - r³) = (4/3)π(8³ - 6³) = (4/3)π(512 - 216) = (4/3)π(296) = (1184π/3) cm³. Step 2: Volume of cone = (1/3)πr²h = (1/3)π(4)²h = (16πh/3) cm³. Step 3: Equate: (1184π/3) = (16πh/3) → 1184 = 16h → h = 74 cm.

Question 11

A solid metallic sphere of radius 6 cm is melted and recast into a solid cylinder of radius 4 cm. Find the height of the cylinder (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 cylinder = πr²h = π(4)²h = 16πh cm³. Step 3: Since volume is conserved: 16πh = 288π. Step 4: h = 288/16 = 18 cm.

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