MathsCBSE Class 10 · Basic · Board exam 2024, set 430/3/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2024: question 37

Set in the CBSE Class 10 board exam of 2024, paper 430/3/1, 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

NSS (National Service Scheme) aims to connect the students to the community and to involve them in problem solving process. NSS symbol is based on the 'Rath' wheel of the Konark Sun Temple situated in Odisha. The wheel signifies the progress cycle of life. The diagrammatic representation of the symbol is given below. Observe the figure given above. The diameters of inner circle are equally placed. Given that $O P = 21$ cm, $O S = 10$ cm.

Based on the above information, answer the following questions:

(i)
Find $m ∠ R O S$ .
1 mark
(ii)
Find the perimeter of sector $O P Q$ .
1 mark
(iii)
(a) Find the area of shaded region $P QR S$ .
2 marks
OR
(iii)
(b) Find the area of shaded region $A C B$ i.e. the segment $A C B$ .
2 marks

Figure in the paper: A large outer circle with centre O and a smaller inner circle with centre O. Diameters of the inner circle are drawn at equal angular spacing, forming spokes of the wheel. Points on the outer circle are labelled P (on the right), Q (upper right), and the inner circle has points R, S, A, B and C. OP is a radius of the outer circle of length 21 cm and OS is a radius of the inner circle of length 10 cm. Sector OPQ is shaded at the upper right and the region PQRS is shaded between the two circles. The segment ACB of the inner circle is shaded on the left side, bounded by chord AB and arc ACB. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/3/1.

Hints

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

Hint 1

Count how many equal angles the 4 diameters make at O.

Hint 2

The perimeter of a sector is two radii plus the arc length $\frac{θ}{360} \times 2 π r$ . The region PQRS is the outer sector minus the inner sector with the same angle.

Hint 3

For the segment ACB, find $∠ A O B$ (it covers two of the equal sectors), then subtract the area of $△ O A B$ from the area of the sector.

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) The 4 equally spaced diameters divide the full angle into 8 equal parts.
$∠ R O S = \frac{36 0 ^{\circ}}{8} = 4 5^{\circ}$
1 mark
(ii) Perimeter of sector $= 2 r + \frac{θ}{36 0 ^{\circ}} \times 2 π r$
$= 21 + 21 + \frac{45}{360} \times 2 \times \frac{22}{7} \times 21$
$= 42 + 16.5 = 58.5$ cm
1 mark
(iii)(a) Area of $P QR S$ = area of sector $O P Q$ − area of sector $O S R$
$= \frac{45}{360} \times \frac{22}{7} \times (2 1^{2} - 1 0^{2})$
1 mark
(iii)(a) $= \frac{1}{8} \times \frac{22}{7} \times 341 = \frac{3751}{28}$ cm² $\approx 133.96$ cm²
1 mark
(iii)(b) Chord $A B$ spans two $4 5^{\circ}$ sectors, so $∠ A O B = 9 0^{\circ}$ .
Area of segment = area of sector $O A C B$ − area of $△ O A B$
$= \frac{90}{360} \times \frac{22}{7} \times 1 0^{2} - \frac{1}{2} \times 10 \times 10$
1 mark
(iii)(b) $= \frac{550}{7} - 50 = \frac{200}{7}$ cm² $\approx 28.57$ cm²
1 mark

Answer: (i) 45°; (ii) 58.5 cm; (iii)(a) 3751/28 cm² ≈ 133.96 cm²; (iii)(b) 200/7 cm² ≈ 28.57 cm²

Where the marks are

(i) $∠ R O S = 36 0^{\circ} /8 = 4 5^{\circ}$ (1 mark)
(ii) Perimeter $= 42 + 16.5 = 58.5$ cm (1 mark)
(iii)(a) Writes area as difference of sectors, $\frac{45}{360} \cdot \frac{22}{7} (2 1^{2} - 1 0^{2})$ (1 mark)
(iii)(a) Gets $\frac{3751}{28}$ cm² ≈ 133.96 cm² (1 mark)
(iii)(b) Sector of $9 0^{\circ}$ minus right triangle, correct expression (1 mark)
(iii)(b) Gets $\frac{200}{7}$ cm² ≈ 28.57 cm² (1 mark)

Revise the chapter

Chapter: Areas Related to Circles (notes, formulas, where marks are lost and every past question)
Topic: Area of a Segment 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 paper2024 board paper, set 430/3/1 (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 2024: question 37

Hints

Step-by-step solution

Revise the chapter

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