MathsCBSE Class 10 · Basic · Board exam 2025, set 430/3/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2025: question 38

Set in the CBSE Class 10 board exam of 2025, paper 430/3/1, section E, question 38, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q38 · 4 marks · Case-basedCompetency-basedCoordinate Geometry

Case Study 3

There is a semicircular park in Aman's society. He wishes to plant saplings along the boundary of the park. There is a borewell at the centre O of the park along the diameter AB as shown in the figure below.

Based on the above information, answer the following questions:

(i)
Find the coordinates of point O.
1 mark
(ii)
Find the radius of the semicircular park.
1 mark
(iii)
(a) One sapling is kept at point C(12, y). Find the coordinates of C.
2 marks
OR
(iii)
(b) One sapling is kept at point P along AB so that $P A = \frac{1}{3} P B$ . Find the coordinates of P.
2 marks

Figure in the paper: A semicircular park with diameter AB horizontal at the top, where A = (2, 3) at the left end and B = (22, 3) at the right end. The arc of the semicircle lies below AB. O is the centre on AB, with a borewell drawn at O. P is a point on AB between A and O where a sapling is planted. C(12, y) is the lowest point of the arc, where a sapling is planted. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/3/1.

Hints

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

Hint 1

The centre of a circle is the mid-point of any diameter.

Hint 2

The radius is half of AB. Any point on the arc is at a distance equal to the radius from O, so use the distance formula for OC.

Hint 3

Rewrite $P A = \frac{1}{3} P B$ as a ratio PA : PB, then apply the section formula.

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) O is the centre, so it is the mid-point of diameter AB with A(2, 3) and B(22, 3).
$O = (\frac{2 + 22}{2}, \frac{3 + 3}{2}) = (12, 3)$
1 mark
(ii) $A B = (22 - 2)^{2} + (3 - 3)^{2} = 20$ , so the radius $= \frac{20}{2} = 10$ units.
(Equivalently, $O A = (12 - 2)^{2} + 0^{2} = 10$ .)
1 mark
(iii)(a) C lies on the boundary of the park, so $O C$ = radius = 10.
½ mark
(iii)(a) $(12 - 12)^{2} + (y - 3)^{2} = 10 \Rightarrow (y - 3)^{2} = 100 \Rightarrow y - 3 = \pm 10 \Rightarrow y = 13$ or $y = - 7$ .
1 mark
(iii)(a) The coordinates of C are $(12, 13)$ or $(12, - 7)$ . In the figure the arc lies below AB, so C is $(12, - 7)$ .
½ mark
(iii)(b) $P A = \frac{1}{3} P B \Rightarrow P A : P B = 1 : 3$ , so P divides AB internally in the ratio $1 : 3$ .
½ mark
(iii)(b) By the section formula with $m_{1} = 1$ , $m_{2} = 3$ , A(2, 3) and B(22, 3):
$P = (\frac{1 \times 22 + 3 \times 2}{1 + 3}, \frac{1 \times 3 + 3 \times 3}{1 + 3})$
1 mark
(iii)(b) $= (\frac{28}{4}, \frac{12}{4}) = (7, 3)$
½ mark

Answer: (i) O(12, 3); (ii) radius = 10 units; (iii)(a) C(12, 13) or (12, −7), with (12, −7) as drawn; (iii)(b) P(7, 3)

Where the marks are

(i) Mid-point of AB gives O(12, 3) (1 mark)
(ii) Radius = 10 (1 mark)
(iii)(a) Uses OC = radius = 10 (½ mark)
(iii)(a) Solves $(y - 3)^{2} = 100$ to get $y = 13$ or $- 7$ (1 mark)
(iii)(a) States C as (12, 13) or (12, −7) (½ mark)
(iii)(b) Ratio PA : PB = 1 : 3 (½ mark)
(iii)(b) Correct section formula substitution (1 mark)
(iii)(b) P = (7, 3) (½ mark)

Revise the chapter

Chapter: Coordinate Geometry (notes, formulas, where marks are lost and every past question)
Topic: Section Formula (Internal Division)

Practise this chapterBoard-style questions from Coordinate Geometry, one at a time, marked against the answer key and marking points.Back to the full paper2025 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 2025: question 38

Hints

Step-by-step solution

Revise the chapter

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