MathsCBSE Class 12 · Sample paper, session 2026-27 · 4 marks · Case-based

CBSE Class 12 Mathematics sample paper 2026-27: question 37

From the official CBSE sample question paper for 2026-27, section E, question 37, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q37 · 4 marks · Case-basedCompetency-basedApplication of Derivatives

Water is essential for life, and rivers are vital water sources across the world. The Yamuna Action Plan (YAP), one of India's largest river restoration projects, was launched to clean the Yamuna, but its effectiveness (WQI) the Water Quality Index must be measured through cost-benefit, economic and environmental analysis. Therefore, comprehensive pollutant treatment, prevention of further contamination, and community awareness are crucial for sustainable river management.
Implementing the same concept to the river Yamuna, if an initial base line WQI is 35, a quadratic expression is formed to measure the impact of investment of ₹ $x$ crore on the WQI i.e., $W (x)$ as
$W (x) = 35 + 1.6 x - 0.02 x^{2}$ , where $x$ is in Rupees in crores
Use the given $W (x)$ to answer the following questions:

(i)
Find the value of $x$ at the stationary point for the function $W (x)$ .
1 mark
(ii)
Find the maximum value of the WQI using the second derivative test?
1 mark
(iii)(a)
If the investment is ₹ 35 crores, find by how much the WQI is short of its maximum achievable value.
2 marks
OR
(iii)(b)
Evaluate the WQI when the investment is ₹60 crore. Determine whether this indicates an enhancement or a diminishing return in our objective when compared with the optimum WQI.
2 marks

Figure in the paper: Photo of the polluted Yamuna river with a railway bridge in the background and garbage on the bank. See it in the official paper.

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

A stationary point is where the first derivative is zero.

Hint 2

Check the sign of $W^{''} (x)$ to decide whether it is a maximum, then substitute back into $W$ .

Hint 3

For part (iii), substitute the given investment into $W (x)$ and compare with the maximum.

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) $W^{'} (x) = 1.6 - 0.04 x$ . Setting $W^{'} (x) = 0$ gives $x = \frac{1.6}{0.04} = 40$ . So the stationary point is at $x = 40$ (₹40 crore).
1 mark
(ii) $W^{''} (x) = - 0.04 < 0$ , so $W$ has a maximum at $x = 40$ by the second derivative test. Maximum WQI $= W (40) = 35 + 64 - 32 = 67$ .
1 mark
(iii)(a) $W (35) = 35 + 1.6 (35) - 0.02 (35)^{2} = 35 + 56 - 24.5 = 66.5$ .
1 mark
(iii)(a) Shortfall from the maximum $= 67 - 66.5 = 0.5$ . So the WQI is short by $0.5$ units.
1 mark
(iii)(b) $W (60) = 35 + 1.6 (60) - 0.02 (3600) = 35 + 96 - 72 = 59$ .
1 mark
(iii)(b) Since $59 < 67$ , raising the investment beyond ₹40 crore to ₹60 crore actually lowers the WQI. This indicates a diminishing return.
1 mark

Answer: (i) $x = 40$ crore; (ii) max WQI $= 67$ ; (iii)(a) short by 0.5 units; (iii)(b) $W (60) = 59$ , diminishing return

Where the marks are

Differentiates and solves $W^{'} (x) = 0$ to get $x = 40$ (1 mark)
Shows $W^{''} < 0$ and finds the maximum $W (40) = 67$ (1 mark)
(iii)(a) Computes $W (35) = 66.5$ (1 mark)
(iii)(a) States the shortfall of 0.5 (1 mark)
(iii)(b) Computes $W (60) = 59$ (1 mark)
(iii)(b) Compares with 67 and concludes diminishing return (1 mark)

Revise the chapter

Chapter: Application of Derivatives (notes, formulas, where marks are lost and every past question)
Topic: Maxima and Minima

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

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CBSE Class 12 Mathematics sample paper 2026-27: question 37

Hints

Step-by-step solution

Revise the chapter

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