Case Study 1: During a theatre drama, a backdrop of building arches was used. The shape of the curve shown below can be represented by the polynomial p(x) = -x² + 2x + 8, where x is the length (in feet) on stage level. Based on the figure given above, answer the following questions : (i) Determine the height of the arch. (ii)(a) Find zeroes of the polynomial p(x). Which points on the graph represent the zeroes? (ii)(b) Find the span of the arch on the stage floor. (iii) Write the coordinates of the point of intersection of the above curve with the y-axis.

MathsCBSE Class 10 · Standard · Board exam 2026, set 30/3/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard 2026: question 36

Set in the CBSE Class 10 board exam of 2026, paper 30/3/1, 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-basedPolynomials

Case Study 1: During a theatre drama, a backdrop of building arches was used. The shape of the curve shown below can be represented by the polynomial $p (x) = - x^{2} + 2 x + 8$ , where x is the length (in feet) on stage level. Based on the figure given above, answer the following questions :

(i)
Determine the height of the arch.
1 mark
(ii)(a)
Find zeroes of the polynomial p(x). Which points on the graph represent the zeroes?
2 marks
OR
(ii)(b)
Find the span of the arch on the stage floor.
2 marks
(iii)
Write the coordinates of the point of intersection of the above curve with the y-axis.
1 mark

Figure in the paper: Left: photograph of a stage backdrop with three arches. Right: graph of an arch with axes X'OX (horizontal) and Y'OY (vertical, Y = p(x)). The arch is a curve from B on the negative x-axis to A on the positive x-axis, with its top point C(1, y) on the curve; the curve meets the y-axis above O. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 30/3/1.

Hints

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

Hint 1

The top of the arch is at x = 1. Substitute it into p(x).

Hint 2

Zeroes are where p(x) = 0. Factorise $x^{2} - 2 x - 8$ .

Hint 3

The span is the distance between the two zeroes. The y-axis crossing is where x = 0.

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 highest point of the arch is $C (1, y)$ , so the height is $p (1)$ :
$p (1) = - (1)^{2} + 2 (1) + 8 = - 1 + 2 + 8 = 9$
So the height of the arch is 9 feet.
1 mark
(ii)(a) Factorise:
$p (x) = - x^{2} + 2 x + 8 = - (x^{2} - 2 x - 8) = - (x - 4) (x + 2)$
1 mark
(ii)(a) Setting $p (x) = 0$ gives $x = 4$ or $x = - 2$ . So the zeroes are $- 2$ and $4$ .
½ mark
(ii)(a) The zeroes are where the curve meets the x-axis, i.e. the points $B (- 2, 0)$ and $A (4, 0)$ .
½ mark
(ii)(b) Factorise: $p (x) = - (x - 4) (x + 2)$ .
½ mark
(ii)(b) The zeroes are $- 2$ and $4$ , so the arch meets the floor at $B (- 2, 0)$ and $A (4, 0)$ .
½ mark
(ii)(b) Span: $A B = 4 - (- 2) = 6$ feet.
1 mark
(iii) On the y-axis, $x = 0$ :
$p (0) = - (0)^{2} + 2 (0) + 8 = 8$
½ mark
(iii) So the curve meets the y-axis at $(0, 8)$ .
½ mark

Answer: (i) 9 feet; (ii)(a) zeroes −2 and 4, shown by points B and A; (ii)(b) span 6 feet; (iii) (0, 8)

Where the marks are

(i) Evaluates $p (1) = 9$ , height 9 feet (1 mark)
(ii)(a) Factorises $p (x) = - (x - 4) (x + 2)$ (1 mark)
(ii)(a) States zeroes $- 2$ , $4$ (½ mark)
(ii)(a) Identifies points B and A (½ mark)
(ii)(b) Factorises (½ mark) and finds zeroes (½ mark)
(ii)(b) Span $A B = 6$ feet (1 mark)
(iii) Evaluates $p (0) = 8$ (½ mark) and writes $(0, 8)$ (½ mark)

Revise the chapter

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

Practise this chapterBoard-style questions from Polynomials, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 30/3/1 (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 2026: question 36

Hints

Step-by-step solution

Revise the chapter

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