MathsCBSE Class 12 · Sample paper, session 2025-26 · 4 marks · Case-based

CBSE Class 12 Mathematics sample paper 2025-26: question 37

From the official CBSE sample question paper for 2025-26, 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

Case Study -2

LED bulbs are energy-efficient because they use significantly less electricity than traditional bulbs while producing the same amount of light. They convert more energy into light rather than heat, reducing waste. Additionally, their long lifespan means fewer replacements, saving resources and money over time.

A company manufactures a new type of energy-efficient LED bulb. The cost of production and the revenue generated by selling x bulbs (in an hour) are modelled as $C (x) = 0.5 x^{2} - 10 x + 150$ and $R (x) = - 0.3 x^{2} + 20 x$ respectively, where $C (x)$ and $R (x)$ are both in ₹.

To maximize the profit, the company needs to analyze these functions using calculus. Use the given models to answer the following questions:

I
Derive the profit function $P (x)$
1 mark
II
Find the critical points of $P (x)$ .
1 mark
III A
Determine whether the critical points correspond to a maximum or a minimum profit by using the second derivative test.
2 marks
OR
III B
Identify the possible practical value of $x$ (i.e., the number of bulbs that can realistically be produced and sold) that can maximize the profit, if the resources available and the expenditure on machines allows to produce minimum 10 but not more than 18 bulbs per hour. Also calculate the maximum profit.
2 marks

Figure in the paper: Photograph of a factory assembly line where workers in blue uniforms, safety glasses and gloves assemble LED bulbs on conveyor belts. See it in the official paper.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

Profit equals revenue minus cost; take care with the signs when subtracting the bracket.

Hint 2

Critical points come from $P^{'} (x) = 0$ ; the sign of $P^{''}$ tells you maximum or minimum.

Hint 3

In III B, $x$ must be a whole number in $[10, 18]$ . Check whether $P$ is increasing or decreasing on this interval.

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. Profit is revenue minus cost:
$P (x) = R (x) - C (x) = (- 0.3 x^{2} + 20 x) - (0.5 x^{2} - 10 x + 150)$ .
So $P (x) = - 0.8 x^{2} + 30 x - 150$ .
1 mark
II. Differentiate: $P^{'} (x) = - 1.6 x + 30$ .
For critical points, $P^{'} (x) = 0 \Rightarrow x = \frac{30}{1.6} = 18.75$ .
So $x = 18.75$ is the only critical point.
1 mark
III A. Differentiate again: $P^{''} (x) = - 1.6$ , so $P^{''} (18.75) = - 1.6$ .
1 mark
III A (contd.) Since $P^{''} (18.75) < 0$ , by the second derivative test $x = 18.75$ gives a maximum profit.
1 mark
III B. Production must satisfy $10 \leq x \leq 18$ , and $x$ must be a whole number of bulbs.
On $[10, 18]$ we have $P^{'} (x) = - 1.6 x + 30 > 0$ , because $x < 18.75$ . So $P$ is increasing there.
Hence the maximum profit occurs at the largest allowed value, $x = 18$ bulbs.
1 mark
III B (contd.) Maximum profit:
$P (18) = - 0.8 (18)^{2} + 30 (18) - 150 = - 259.2 + 540 - 150 = 130.80$ .
So the maximum profit is ₹130.80.
1 mark

Answer: I. $P (x) = - 0.8 x^{2} + 30 x - 150$ . II. $x = 18.75$ . III A. $P^{''} = - 1.6 < 0$ , so maximum profit. III B. $x = 18$ bulbs, maximum profit ₹130.80.

Where the marks are

Obtains $P (x) = - 0.8 x^{2} + 30 x - 150$ (1 mark)
Solves $P^{'} (x) = - 1.6 x + 30 = 0$ to get $x = 18.75$ (1 mark)
III A: finds $P^{''} (x) = - 1.6$ (1 mark) and concludes maximum since $P^{''} < 0$ (1 mark)
III B: identifies $x = 18$ as the practical value (1 mark) and computes the maximum profit ₹130.80 (1 mark)

Revise the chapter

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

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 2025-26: every question, by section.

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CBSE Class 12 Mathematics sample paper 2025-26: question 37

Hints

Step-by-step solution

Revise the chapter

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