MathsCBSE Class 10 · Standard · Board exam 2025, set 30/6/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard 2025: question 36

Set in the CBSE Class 10 board exam of 2025, paper 30/6/1, section E, question 36, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q36 · 4 marks · Case-basedCompetency-basedAreas Related to Circles

Anurag purchased a farmhouse which is in the form of a semicircle of diameter 70 m. He divides it into three parts by taking a point P on the semicircle in such a way that $∠ P A B = 3 0^{\circ}$ as shown in the following figure, where O is the centre of semicircle.

In part I, he planted saplings of Mango tree, in part II, he grew tomatoes and in part III, he grew oranges. Based on given information, answer the following questions.

(i)
What is the measure of $∠ P O A$ ?
1 mark
(ii)
Find the length of wire needed to fence entire piece of land.
1 mark
(iii)
(a) Find the area of region in which saplings of Mango tree are planted.
2 marks
OR
(iii)
(b) Find the length of wire needed to fence the region III.
2 marks

Figure in the paper: A semicircle with diameter AB (A at left, B at right, O centre at the midpoint of the base). P is on the arc, joined to A and to B. Angle PAB is marked $3 0^{\circ}$ . Region I is the triangle PBA side near B (between PB and arc), region II is the lower part between AB, AP and the regions, and region III is the upper-left segment between AP and the arc. Symbols show mango saplings (stars in III), tomatoes (faces in II) and oranges (circles, marked I/II/III as indicated). See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/6/1.

Hints

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

Hint 1

The angle at the centre is twice the angle at the circumference on the same arc. Use this to find $∠ P O B$ first.

Hint 2

The fence for the whole land runs along the curved boundary and the diameter. Region I is a segment: take the sector minus the triangle, and note that triangle POB is equilateral.

Hint 3

For region III you need chord AP and arc AP. $∠ A P B$ is an angle in a semicircle.

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 angle subtended by arc PB at the centre is twice the angle at the circumference: $∠ P O B = 2∠ P A B = 6 0^{\circ}$ . Since AOB is a straight line, $∠ P O A = 18 0^{\circ} - 6 0^{\circ} = 12 0^{\circ}$ .
1 mark
(ii) Radius $r = 35$ m. Fence = semicircular arc + diameter $= π r + 2 r = \frac{22}{7} \times 35 + 70 = 110 + 70 = 180$ m.
1 mark
(iii)(a) Mango region I is the segment cut off by chord PB. Its area = area of sector POB ( $6 0^{\circ}$ ) − area of $△ P O B$ . $△ P O B$ is equilateral because $O P = O B = 35$ m and $∠ P O B = 6 0^{\circ}$ . So area $= \frac{60}{360} \times \frac{22}{7} \times 3 5^{2} - \frac{3}{4} \times 3 5^{2}$ .
1 mark
$= (\frac{1925}{3} - \frac{1225 3}{4})$ m $^{2} \approx 641.67 - 530.42 \approx 111.2$ m $^{2}$ .
1 mark
(iii)(b) $∠ A P B = 9 0^{\circ}$ [angle in a semicircle]. In right $△ A P B$ , $\frac{A P}{A B} = cos 3 0^{\circ}$ , so $A P = 70 \times \frac{3}{2} = 353$ m.
1 mark
Region III is bounded by chord AP and arc AP, and arc AP subtends $12 0^{\circ}$ at O. Arc AP $= \frac{120}{360} \times 2 \times \frac{22}{7} \times 35 = \frac{220}{3}$ m.
½ mark
Wire needed $= (\frac{220}{3} + 353)$ m $\approx 73.33 + 60.62 \approx 133.9$ m.
½ mark

Answer: (i) 120°; (ii) 180 m; (iii)(a) (1925/3 − 1225√3/4) m² ≈ 111.2 m²; (iii)(b) (220/3 + 35√3) m ≈ 133.9 m

Where the marks are

(i) $∠ P O A = 12 0^{\circ}$ (1 mark)
(ii) Fence = arc + diameter = 180 m (1 mark)
(iii)(a) Sets up sector POB minus equilateral triangle POB (1 mark)
(iii)(a) Area $= \frac{1925}{3} - \frac{1225 3}{4}$ m² ≈ 111.2 m² (1 mark)
(iii)(b) $A P = 353$ m (1 mark)
(iii)(b) Arc AP $= \frac{220}{3}$ m set up (½ mark)
(iii)(b) Total length ≈ 133.9 m (½ 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 30/6/1 (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 36

Hints

Step-by-step solution

Revise the chapter

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