Question 1

A sphere has a radius of 7 cm. What is its surface area? (Use π = 22/7)

Accepted Answer

Step 1: Surface area of sphere = 4πr². Step 2: Substitute r = 7 cm. Step 3: SA = 4 × (22/7) × 7² = 4 × (22/7) × 49 = 4 × 22 × 7 = 616 cm². Therefore, the answer is 616 cm².

Question 2

Two parallel lines are cut by a transversal. If one of the alternate interior angles is 72°, what is the other alternate interior angle?

Accepted Answer

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

Question 3

A cone has a base radius of 7 cm and height of 12 cm. What is its volume? (Use π = 22/7)

Accepted Answer

Step 1: Volume of cone = (1/3)πr²h. Step 2: Substitute r = 7 cm, h = 12 cm. Step 3: V = (1/3) × (22/7) × 7² × 12 = (1/3) × (22/7) × 49 × 12 = (1/3) × 22 × 7 × 12 = (22 × 84)/3 = 616 cm³. Therefore, the answer is 616 cm³.

Question 4

Two lines intersect and form four angles. If one of the angles is 65°, what is the measure of the angle adjacent to it?

Accepted Answer

Step 1: When two lines intersect, adjacent angles are supplementary (they add up to 180°). Step 2: If one angle is 65°, the adjacent angle = 180° - 65° = 115°. Therefore, the answer is 115°.

Question 5

A line makes an angle of 35° with the horizontal. What is the angle it makes with the vertical?

Accepted Answer

Step 1: Horizontal and vertical lines are perpendicular to each other, forming a 90° angle. Step 2: If a line makes 35° with the horizontal, it makes 90° - 35° = 55° with the vertical. Therefore, the answer is 55°.

Question 6

At a point, three rays emanate forming three angles. Two of the angles are 85° and 120°. What is the third angle?

Accepted Answer

Step 1: Three rays emanating from a point form angles that sum to 360° (complete angle around a point). Step 2: Third angle = 360° - 85° - 120° = 155°. Therefore, the answer is 155°.

Question 7

A straight line is divided into two angles by a ray. If one angle is 4 times the other, what is the measure of the larger angle?

Accepted Answer

Step 1: A ray dividing a straight line creates two supplementary angles that sum to 180°. Step 2: Let the smaller angle = x, then the larger angle = 4x. Step 3: x + 4x = 180°, so 5x = 180°, giving x = 36°. Step 4: Larger angle = 4 × 36° = 144°. Therefore, the answer is 144°.

Question 8

Two angles are complementary. If one angle is 28° more than the other, what is the measure of the smaller angle?

Accepted Answer

Step 1: Complementary angles sum to 90°. Step 2: Let the smaller angle = x, then the larger angle = x + 28°. Step 3: x + (x + 28°) = 90°, so 2x + 28° = 90°. Step 4: 2x = 62°, giving x = 31°. Therefore, the smaller angle is 31°.

Question 9

In a triangle, the angles are in the ratio 1:2:3. What is the measure of the largest angle?

Accepted Answer

Step 1: The sum of angles in a triangle is always 180°. Step 2: Let the angles be x, 2x, and 3x. Step 3: x + 2x + 3x = 180°. Step 4: 6x = 180°, so x = 30°. Step 5: The largest angle = 3x = 3(30°) = 90°.

Question 10

Two straight lines intersect at a point. If one of the angles formed is 65°, what is the sum of the two adjacent angles on a straight line?

Accepted Answer

Step 1: When two straight lines intersect, they form two pairs of vertically opposite angles. Step 2: Adjacent angles on a straight line are supplementary (sum to 180°). Step 3: If one angle is 65°, the adjacent angle = 180° - 65° = 115°. Step 4: Sum of two adjacent angles = 65° + 115° = 180°.

Question 11

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

Accepted Answer

Step 1: When a transversal cuts two parallel lines, alternate interior angles are equal. Step 2: If one alternate interior angle is 72°, the other alternate interior angle is also 72°. Step 3: Corresponding angles are also equal when lines are parallel. Step 4: The corresponding angle to the 72° angle is 72°.

Question 12

Three lines meet at a point. The angles formed are in the ratio 2:3:4. What is the measure of the largest angle?

Accepted Answer

Step 1: When three lines meet at a point, the angles around that point sum to 360°. Step 2: Let the angles be 2x, 3x, and 4x. Step 3: 2x + 3x + 4x = 360°. Step 4: 9x = 360°, so x = 40°. Step 5: The largest angle = 4x = 4(40°) = 160°.

Question 13

In a triangle, one exterior angle is 110°. If the two non-adjacent interior angles are in the ratio 3:2, what is the measure of the smaller non-adjacent interior angle?

Accepted Answer

Step 1: By the exterior angle theorem, an exterior angle equals the sum of the two non-adjacent interior angles. Step 2: So the two non-adjacent interior angles sum to 110°. Step 3: Let the angles be 3x and 2x. Step 4: 3x + 2x = 110°, so 5x = 110°, giving x = 22°. Step 5: The smaller angle = 2x = 2(22°) = 44°.

Question 14

Two parallel lines are cut by a transversal. The sum of two co-interior angles is 180°. If one co-interior angle is 35° more than the other, what is the measure of the larger angle?

Accepted Answer

Step 1: Co-interior angles (also called consecutive interior angles) on the same side of a transversal cutting parallel lines sum to 180°. Step 2: Let the smaller angle be x. Then the larger angle is x + 35°. Step 3: x + (x + 35°) = 180°. Step 4: 2x + 35° = 180°, so 2x = 145°, giving x = 72.5°. Step 5: The larger angle = 72.5° + 35° = 107.5°.

Question 15

The curved surface area of a cone is 550 cm². If the 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: 550 = (550/7) × r. Step 4: r = 550 × (7/550) = 7 cm. Therefore, the radius is 7 cm.

Question 16

In a triangle, the exterior angle at vertex B is 110°. If the interior angles at vertices A and C are in the ratio 3:2, find the measure of the smallest interior angle of the triangle (in degrees).

Accepted Answer

Step 1: Exterior angle at B = 110°, so interior angle at B = 180° - 110° = 70°. Step 2: Sum of angles in triangle = 180°. Step 3: Angle A + Angle C = 180° - 70° = 110°. Step 4: Angles A and C are in ratio 3:2, so A = 3k and C = 2k. Step 5: 3k + 2k = 110°, so 5k = 110°, giving k = 22°. Step 6: Angle A = 66°, Angle C = 44°, Angle B = 70°. Step 7: Smallest angle = 44°.

Question 17

Two parallel lines are cut by a transversal. One of the co-interior angles is 40° more than the other. What is the measure of the larger co-interior angle (in degrees)?

Accepted Answer

Step 1: Co-interior angles (also called consecutive interior angles or same-side interior angles) are supplementary when lines are parallel. Step 2: Let one angle = x, then the other = x + 40°. Step 3: x + (x + 40°) = 180°. Step 4: 2x + 40° = 180°. Step 5: 2x = 140°, so x = 70°. Step 6: The larger angle = 70° + 40° = 110°.

Question 18

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. Setting volumes 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.

Question 19

A cylindrical tank has radius 7 m and height 10 m. A conical cavity of height 6 m and base radius 5 m is carved out from the top. What is the volume of the remaining solid (in m³)? [Use π = 22/7]

Accepted Answer

Step 1: Volume of cylinder = πr²h = (22/7)(7)²(10) = (22/7)(49)(10) = 22 × 7 × 10 = 1540 m³. Step 2: Volume of cone = (1/3)πr²h = (1/3)(22/7)(5)²(6) = (1/3)(22/7)(25)(6) = (1/3)(22 × 25 × 6)/7 = (3300)/(3 × 7) = 1100/7 ≈ 157.14 m³. Step 3: Remaining volume = 1540 - 1100/7 = (10780 - 1100)/7 = 9680/7 ≈ 1382.86 m³. Exact: 9680/7 m³ or approximately 1382.86 m³.

Question 20

Two straight lines intersect such that one of the angles formed is 3 times the adjacent angle. If a third line is drawn parallel to one of the original lines, what is the measure of the angle that the third line makes with the other original line (in degrees)?

Accepted Answer

Step 1: When two lines intersect, adjacent angles are supplementary. Let one angle = x, then adjacent angle = 3x. Step 2: x + 3x = 180°, so 4x = 180°, giving x = 45°. Step 3: The two angles are 45° and 135°. Step 4: When a line parallel to one original line intersects the other original line, it makes the same angle as the original parallel line made with that line (corresponding angles). Step 5: The angle formed is either 45° or 135°. Since the question asks for the acute angle typically, the answer is 45°.

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