MathsCBSE Class 10 · Basic · Board exam 2025, set 430/5/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2025: question 37

Set in the CBSE Class 10 board exam of 2025, paper 430/5/1, section E, question 37, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q37 · 4 marks · Case-basedCompetency-basedAreas Related to Circles

A farmer has put up a decorative windmill in his farm in which there are eight blades of equal width and equally placed in a circular arrangement. A circular wire goes through them. The diagram shows two blades OAB and OPQ in a quarter circle with centre O. $∠ A O B = ∠ P O Q = 3 0^{\circ}$ , $O A = 28$ cm, $O C = 21$ cm. O is the centre of both the circles.

(i)
Determine the measure of $∠ B O P$ .
1 mark
(ii)
Find length of arc CD.
1 mark
(iii)
(a) Find the area of region CABD.
2 marks
OR
(iii)
(b) Find perimeter of region CABD.
2 marks

Figure in the paper: Left: a windmill with eight equal blades arranged in a circle around a central hub, joined by a circular wire. Right: concentric circles with centre O. Radii OA (outer, 28 cm, pointing down) and OR (to the right). Points A, B on outer circle and C, D on inner circle along the radius OA-side; OB, OD and OP, OQ are drawn. Angles 30° are marked between OA and OB, and between OP and OQ (with OR as reference). Points on outer arc: A, B, P, Q, R; inner circle points: C, D. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/5/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

The quarter circle AOR is 90°. Subtract the two blade angles and share what is left equally between the two gaps.

Hint 2

Arc length = (θ/360°) × 2πr, using the radius of the circle on which the arc lies.

Hint 3

Region CABD is a sector of radius 28 cm minus a sector of radius 21 cm. Its boundary has two arcs and two straight pieces, each 28 − 21 cm.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

(i) $∠ A O R = 9 0^{\circ}$ (quarter circle). The blades OAB and OPQ take up $3 0^{\circ} + 3 0^{\circ} = 6 0^{\circ}$ . The blades are equally placed, so the remaining $3 0^{\circ}$ splits into two equal gaps, $∠ B O P$ and $∠ QO R$ . Hence $∠ B O P = \frac{1}{2} (9 0^{\circ} - 6 0^{\circ}) = 1 5^{\circ}$ .
1 mark
(ii) Arc CD lies on the inner circle ( $r = 21$ cm) and subtends $3 0^{\circ}$ at O. Length of arc $C D = \frac{θ}{36 0 ^{\circ}} \times 2 π r = \frac{30}{360} \times 2 \times \frac{22}{7} \times 21 = 11$ cm.
1 mark
(iii)(a) Area of region CABD = area of sector OAB − area of sector OCD $= \frac{30}{360} \times \frac{22}{7} \times (2 8^{2} - 2 1^{2})$
1 mark
(iii)(a) $= \frac{1}{12} \times \frac{22}{7} \times (784 - 441) = \frac{1}{12} \times \frac{22}{7} \times 343 = \frac{1078}{12} \approx 89.83 cm^{2}$
1 mark
(iii)(b) Length of arc $A B = \frac{30}{360} \times 2 \times \frac{22}{7} \times 28 = \frac{44}{3} \approx 14.67$ cm.
1 mark
(iii)(b) Perimeter of CABD = arc AB + arc CD + CA + DB. Here $C A = D B = 28 - 21 = 7$ cm, so the perimeter $= 14.67 + 11 + 7 + 7 = 39.67$ cm.
1 mark

Answer: (i) 15°; (ii) 11 cm; (iii)(a) ≈ 89.83 cm²; (iii)(b) ≈ 39.67 cm

Where the marks are

(i) ∠BOP = ½(90° − 60°) = 15° (1 mark)
(ii) Arc CD = (30/360) × 2 × (22/7) × 21 = 11 cm (1 mark)
(iii)(a) Writes area as difference of two sectors: (30/360) × (22/7) × (28² − 21²) (1 mark)
(iii)(a) Area ≈ 89.8 cm² (1 mark)
(iii)(b) Arc AB = 44/3 ≈ 14.67 cm (1 mark)
(iii)(b) Perimeter = 14.67 + 11 + 2 × 7 = 39.67 cm (1 mark)

Revise the chapter

Chapter: Areas Related to Circles (notes, formulas, where marks are lost and every past question)
Topic: Area of a Sector of a Circle

Practise this chapterBoard-style questions from Areas Related to Circles, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 430/5/1 (Basic): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Basic 2025: question 37

Hints

Step-by-step solution

Revise the chapter

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