MathsCBSE Class 10 · Standard · Board exam 2025, set 30/1/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard 2025: question 37

Set in the CBSE Class 10 board exam of 2025, paper 30/1/1, 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

Case Study – 2
A brooch is a decorative piece often worn on clothing like jackets, blouses or dresses to add elegance. Made from precious metals and decorated with gemstones, brooches come in many shapes and designs.

One such brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in the figure.

Based on the above given information, answer the following questions :

(i)
Find the central angle of each sector.
1 mark
(ii)
Find the length of the arc ACB.
1 mark
(iii)(a)
Find the area of each sector of the brooch.
2 marks
OR
(iii)(b)
Find the total length of the silver wire used.
2 marks

Figure in the paper: Photo of 25 decorative brooches, and a circle divided by 5 diameters into 10 equal sectors with decorative swirls. Points A, C and B are marked on the circumference at the upper right, with C between A and B; arc ACB spans one sector. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/1/1.

Hints

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

Hint 1

10 equal sectors share the full $36 0^{\circ}$ at the centre.

Hint 2

One sector's arc is one-tenth of the circumference, and its area is one-tenth of the circle's area.

Hint 3

The wire makes the boundary circle and 5 full diameters.

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.

Radius $r = \frac{35}{2}$ mm. The 5 diameters divide the circle into 10 equal sectors.
(i) Central angle of each sector $= \frac{36 0 ^{\circ}}{10} = 3 6^{\circ}$ .
1 mark
(ii) Arc ACB is the arc of one sector, so it is $\frac{1}{10}$ of the circumference:
$\frac{1}{10} \times 2 \times \frac{22}{7} \times \frac{35}{2} = \frac{110}{10} = 11$ mm.
1 mark
(iii)(a) Area of a sector $= \frac{θ}{36 0 ^{\circ}} π r^{2} = \frac{1}{10} \times \frac{22}{7} \times \frac{35}{2} \times \frac{35}{2}$
1 mark
(iii)(a) $= \frac{1}{10} \times \frac{3850}{4} = \frac{385}{4} = 96.25 mm^{2}$ .
1 mark
(iii)(b) Wire used = circumference + 5 diameters $= 2 \times \frac{22}{7} \times \frac{35}{2} + 5 \times 35$
1 mark
(iii)(b) $= 110 + 175 = 285$ mm.
1 mark

Answer: (i) 36°; (ii) 11 mm; (iii)(a) 96.25 mm², (iii)(b) 285 mm

Where the marks are

(i) Central angle $36 0^{\circ} /10 = 3 6^{\circ}$ (1 mark)
(ii) Arc length $= \frac{1}{10} \times 2 π r = 11$ mm (1 mark)
(iii)(a) Correct sector-area expression (1 mark)
(iii)(a) Area $= 385/4 = 96.25$ mm² (1 mark)
(iii)(b) Expression: circumference + $5 \times 35$ (1 mark)
(iii)(b) Total wire 285 mm (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 30/1/1 (Standard): 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 Standard 2025: question 37

Hints

Step-by-step solution

Revise the chapter

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