MathsCBSE Class 10 · Standard · Board exam 2026, set 30/5/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard 2026: question 36

Set in the CBSE Class 10 board exam of 2026, paper 30/5/1, 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

An arch of a railway bridge, built on Chenab riverbed, is shown in the diagram. It is a parabolic arch connecting two hills at P and Q. If the parabolic curve is represented by the polynomial $p (x) = - 0.0025 x^{2} - 0.025 x + 136$ .

Observe the diagram and based on above information, answer the following questions:

(i)
Write the co-ordinates of point A.
1 mark
(ii)
Find the span of the arch.
1 mark
(iii)(a)
Write the zeroes of the polynomial using diagram and verify the relationship between sum of zeroes and polynomials.
2 marks
OR
(iii)(b)
Find the values of $p (x)$ at $x = 100$ and $x = - 100$ . Are they same?
2 marks

Figure in the paper: Left: photograph of a steel arch railway bridge over a gorge. Right: schematic with axes X' (left) and X (right), origin O, and Y axis upward. A parabolic arch rises from Q at (–238.5, 0) on the left to P at (228.5, 0) on the right, with its peak A on the Y-axis at the top, under a horizontal 'Railway Track' line. Vertical support lines hang from the arch to the track. A dashed 'Riverbed' line lies below the X-axis. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 30/5/1.

Hints

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

Hint 1

Point A is on the y-axis. What is the value of x there?

Hint 2

The zeroes are the x-coordinates where the curve meets the x-axis, which are points P and Q.

Hint 3

For the sum of zeroes, compare with $- b / a$ , where $a = - 0.0025$ and $b = - 0.025$ .

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) A lies on the y-axis, so $x = 0$ there.
$p (0) = - 0.0025 (0)^{2} - 0.025 (0) + 136 = 136$
So A = $(0, 136)$ .
1 mark
(ii) The arch meets the x-axis at Q(–238.5, 0) and P(228.5, 0).
Span $= QP = 228.5 - (- 238.5) = 467$ units.
1 mark
(iii)(a) The curve cuts the x-axis at P and Q, so the zeroes of $p (x)$ are $228.5$ and $- 238.5$ .
1 mark
(iii)(a) Sum of zeroes $= 228.5 + (- 238.5) = - 10$ .
$- \frac{coefficient of x}{coefficient of x ^{2}} = - \frac{- 0.025}{- 0.0025} = - 10$
Both are equal, so the relationship is verified.
1 mark
(iii)(b) $p (100) = - 0.0025 (100)^{2} - 0.025 (100) + 136 = - 25 - 2.5 + 136 = 108.5$
1 mark
(iii)(b) $p (- 100) = - 0.0025 (100)^{2} - 0.025 (- 100) + 136 = - 25 + 2.5 + 136 = 113.5$
½ mark
(iii)(b) $108.5 \neq = 113.5$ , so $p (100) \neq = p (- 100)$ . The values are not the same.
½ mark

Answer: (i) A(0, 136); (ii) 467 units; (iii)(a) zeroes 228.5 and −238.5, sum = −10 = −b/a, verified; (iii)(b) p(100) = 108.5, p(−100) = 113.5, not same

Where the marks are

(i) A = (0, 136) (1 mark)
(ii) Span = 238.5 + 228.5 = 467 units (1 mark)
(iii)(a) Zeroes 228.5 and −238.5 read from the diagram (1 mark)
(iii)(a) Sum = −10 = −(coefficient of x)/(coefficient of x²), verified (1 mark)
(iii)(b) p(100) = 108.5 (1 mark)
(iii)(b) p(−100) = 113.5 (½ mark)
(iii)(b) Concludes the values are not the same (½ mark)

Revise the chapter

Chapter: Polynomials (notes, formulas, where marks are lost and every past question)
Topic: Relationship Between Zeros and Coefficients

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 30/5/1 (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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