MathsCBSE Class 10 · Standard · Sample paper, session 2026-27 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard sample paper 2026-27: question 32

From the official CBSE sample question paper for 2026-27, section D, question 32, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32 · 5 marks · Long answerCompetency-basedQuadratic Equations

In a rectangular park measuring 50 m $\times$ 40 m, the gram panchayat plans to construct a rectangular swimming pool in the middle, surrounded by a uniform-width grass strip throughout the park. Find the dimensions of the swimming pool, if the area of the grassy strip is $1184 m^{2}$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2026-27.

Hints

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

Hint 1

Let the width of the grass strip be $x$ . The strip is on both sides, so the pool loses $2 x$ from each dimension.

Hint 2

Area of strip = area of park − area of pool. Form a quadratic in $x$ , solve it, and check which root keeps both pool dimensions positive.

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.

Figure: ABCD is the park, 50 m by 40 m. PQRS is the pool in the middle, with a grass strip of width $x$ m on all four sides.
Length of pool $= (50 - 2 x)$ m and breadth of pool $= (40 - 2 x)$ m.
½ mark
Area of grass strip = Area of park − Area of pool:
$50 \times 40 - (50 - 2 x) (40 - 2 x) = 1184$ .
1 mark
$2000 - (2000 - 180 x + 4 x^{2}) = 1184$
$\Rightarrow 180 x - 4 x^{2} = 1184$
$\Rightarrow 4 x^{2} - 180 x + 1184 = 0$
$\Rightarrow x^{2} - 45 x + 296 = 0$ .
1 mark
$(x - 8) (x - 37) = 0 \Rightarrow x = 8$ or $x = 37$ .
½ mark
If $x = 37$ , the breadth $40 - 74 = - 34$ m, which is negative. So reject $x = 37$ .
Hence $x = 8$ .
1 mark
Length of pool $= 50 - 2 (8) = 34$ m.
½ mark
Breadth of pool $= 40 - 2 (8) = 24$ m.
(Check: $2000 - 34 \times 24 = 2000 - 816 = 1184 m^{2}$ .)
The swimming pool is 34 m × 24 m.
½ mark

Answer: Swimming pool: 34 m × 24 m (strip width 8 m)

Where the marks are

Writes the pool's dimensions as $(50 - 2 x)$ m and $(40 - 2 x)$ m (½ mark)
Forms the area equation $2000 - (50 - 2 x) (40 - 2 x) = 1184$ (1 mark)
Reduces to $x^{2} - 45 x + 296 = 0$ (1 mark)
Factorises to get $x = 8, 37$ (½ mark)
Rejects $x = 37$ with a reason and takes $x = 8$ (1 mark)
Finds length 34 m (½ mark)
Finds breadth 24 m (½ mark)

Revise the chapter

Chapter: Quadratic Equations (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Quadratic Equations, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2026-27 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard sample paper 2026-27: question 32

Hints

Step-by-step solution

Revise the chapter

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