Question 1

Two complementary angles are in the ratio 1:5. What is the measure of the smaller angle?

Accepted Answer

Step 1: Complementary angles sum to 90°. Step 2: Let the angles be x and 5x. Step 3: x + 5x = 90°. Step 4: 6x = 90°, so x = 15°. Step 5: The smaller angle = 15°. Therefore, the answer is 15°.

Question 2

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

Accepted Answer

Step 1: The horizontal and vertical directions 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 3

Two supplementary angles are in the ratio 3:7. What is the measure of the larger angle?

Accepted Answer

Step 1: Supplementary angles sum to 180°. Step 2: Let the angles be 3x and 7x. Step 3: 3x + 7x = 180°. Step 4: 10x = 180°, so x = 18°. Step 5: The larger angle = 7x = 7(18°) = 126°. Therefore, the answer is 126°.

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°, then the adjacent angle = 180° - 65° = 115°. Therefore, the answer is 115°.

Question 5

The radius of a circle is 7 cm. What is its circumference? (Use π = 22/7)

Accepted Answer

Step 1: Apply the circumference formula C = 2πr. Step 2: Substitute r = 7 cm and π = 22/7. Step 3: C = 2 × (22/7) × 7 = 2 × 22 = 44 cm. Therefore, the circumference is 44 cm.

Question 6

A wheel has a radius of 35 cm. How many complete revolutions will it make to cover a distance of 2200 m? (Use π = 22/7)

Accepted Answer

Step 1: Calculate the circumference of the wheel: C = 2πr = 2 × (22/7) × 35 = 2 × 22 × 5 = 220 cm. Step 2: Convert distance to cm: 2200 m = 220000 cm. Step 3: Number of revolutions = Distance ÷ Circumference = 220000 ÷ 220 = 1000 revolutions. Therefore, the wheel makes 1000 complete revolutions.

Question 7

A circular garden has a radius of 10.5 m. What is the cost of fencing it at ₹8 per metre? (Use π = 22/7)

Accepted Answer

Step 1: Calculate the circumference using C = 2πr. Step 2: C = 2 × (22/7) × 10.5 = 2 × 22 × 1.5 = 66 m. Step 3: Total cost = 66 × 8 = ₹528. Therefore, the fencing cost is ₹528.

Question 8

Two circles have radii 5 cm and 3 cm respectively. What is the difference in their areas? (Use π = 22/7)

Accepted Answer

Step 1: Calculate area of first circle: A₁ = (22/7) × 25 = 550/7 cm². Step 2: Calculate area of second circle: A₂ = (22/7) × 9 = 198/7 cm². Step 3: Difference = (550 - 198)/7 = 352/7 cm². Step 4: 352/7 = 50.29 cm² (approximately). For cleaner calculation: (22/7) × (25 - 9) = (22/7) × 16 = 352/7 ≈ 50.29 cm². Using π ≈ 3.14 for approximation: difference ≈ 3.14 × 16 ≈ 50.24 cm².

Question 9

A circle has a diameter of 14 m. Find its area. (Use π = 22/7)

Accepted Answer

Step 1: Find the radius from diameter: r = d/2 = 14/2 = 7 m. Step 2: Apply area formula A = πr². Step 3: A = (22/7) × 7² = (22/7) × 49 = 22 × 7 = 154 m². Therefore, the area is 154 m².

Question 10

The circumference of a circle is 44 cm. What is its radius? (Use π = 22/7)

Accepted Answer

Step 1: Use the circumference formula C = 2πr. Step 2: Substitute C = 44 and π = 22/7: 44 = 2 × (22/7) × r. Step 3: 44 = (44/7) × r. Step 4: r = 44 × (7/44) = 7 cm. Therefore, the radius is 7 cm.

Question 11

A transversal intersects two parallel lines. If one of the alternate interior angles is 72°, what is the measure of 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 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 sum of all angles around that point is 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°. Therefore, the answer is 160°.

Question 13

A cone has a base radius of 5 cm and slant height of 13 cm. What is the total surface area of the cone? [Use π = 22/7]

Accepted Answer

Step 1: Total surface area of cone = πr(r + l), where r = radius, l = slant height. Step 2: TSA = (22/7) × 5 × (5 + 13) = (22/7) × 5 × 18 = (22 × 5 × 18)/7 = 1980/7 = 282⁶⁄₇ cm². Alternatively: TSA = πr² + πrl = (22/7) × 25 + (22/7) × 5 × 13 = (550/7) + (1430/7) = 1980/7 = 282⁶⁄₇ cm².

Question 14

The curved surface area of a cone is 550 cm² and its slant height is 25 cm. Find the radius of the base. [Use π = 22/7]

Accepted Answer

Step 1: Curved surface area of cone = πrl, where r = radius, l = slant height. Step 2: 550 = (22/7) × r × 25. Step 3: 550 = (550/7) × r. Step 4: r = 550 × (7/550) = 7 cm.

Question 15

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) = 101871/88 = 1157.625. Wait, let me recalculate: 4851 × 21 = 101871; 101871 ÷ 88 = 1157.625. This doesn't give a clean cube root. Let me verify: if r = 10.5, then r³ = 1157.625. ∛(1157.625) ≈ 10.5. Actually: (4/3) × (22/7) × r³ = 4851 → r³ = 4851 × (21/88) = 4851 × 21/88. Let me try r = 10.5: (4/3) × (22/7) × (10.5)³ = (88/21) × 1157.625 = 4851. Yes! So r = 10.5 cm.

Question 16

A cylindrical tank has radius 3.5 m and height 8 m. How many litres of water can it hold? [Use π = 22/7; 1 m³ = 1000 litres]

Accepted Answer

Step 1: Volume of cylinder = πr²h = (22/7) × (3.5)² × 8 = (22/7) × 12.25 × 8. Step 2: Convert 3.5 = 7/2, so (3.5)² = 49/4 = 12.25. Step 3: Volume = (22/7) × (49/4) × 8 = (22 × 49 × 8)/(7 × 4) = (22 × 7 × 8)/4 = (1232)/4 = 308 m³. Step 4: In litres = 308 × 1000 = 308,000 litres.

Question 17

Two lines intersect such that one of the angles formed is 65°. If a third line is drawn parallel to one of the original lines, what is the angle between the third line and the second original line (corresponding angle)?

Accepted Answer

Step 1: When two lines intersect, one angle is 65°. Step 2: The vertically opposite angle is also 65°, and adjacent angles are 180° − 65° = 115°. Step 3: When a third line is drawn parallel to one of the original lines, corresponding angles are equal. Step 4: The angle between the third line and the second original line equals the angle between the first and second original lines. Step 5: Since one angle of intersection is 65°, the corresponding angle formed by the parallel line is also 65°.

Question 18

Two adjacent angles on a straight line are in the ratio 2:7. What is the measure of the larger angle?

Accepted Answer

Step 1: Adjacent angles on a straight line are supplementary (sum to 180°). Step 2: Let the angles be 2k and 7k. Step 3: 2k + 7k = 180°. Step 4: 9k = 180°. Step 5: k = 20°. Step 6: The larger angle = 7k = 7 × 20° = 140°.

Question 19

Three lines meet at a point. If the angles formed are in the ratio 1:2:3, what is the measure of the smallest angle?

Accepted Answer

Step 1: Three lines meeting at a point form angles that sum to 360°. Step 2: Let the angles be k, 2k, and 3k. Step 3: k + 2k + 3k = 360°. Step 4: 6k = 360°. Step 5: k = 60°. Step 6: The smallest angle = k = 60°.

Question 20

A transversal cuts two parallel lines. If one of the co-interior angles is 108°, what is the other co-interior angle?

Accepted Answer

Step 1: Co-interior angles (also called consecutive interior angles or same-side interior angles) are supplementary when a transversal cuts parallel lines. Step 2: They sum to 180°. Step 3: If one angle is 108°, the other = 180° − 108° = 72°.

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