MathsCBSE Class 12 · Board exam 2026, set 65/1/1 · 4 marks · Case-based

CBSE Class 12 Mathematics 2026: question 36

Set in the CBSE Class 12 board exam of 2026, paper 65/1/1, 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-basedApplication of Derivatives

An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹ 300 per subscriber for unlimited free deliveries.

The company wishes to increase the annual subscription fee. It is predicted that, for every increase of ₹ 1, ten subscribers will discontinue. Assume that the company increased the annual fee by ₹ $x$ .

Based on the given information, answer the following questions :

(i)
How many subscribers will discontinue after an increase of ₹ $x$ in annual fee ?
1 mark
(ii)
If R( $x$ ) denotes the total revenue collected after the increase of ₹ $x$ in subscription fee, express R( $x$ ) as a function of $x$ .
1 mark
(iii)(a)
Find the value of $x$ for which R( $x$ ) is maximum.
2 marks
OR
(iii)(b)
Find the sub-intervals of (0, 5000) in which R( $x$ ) is increasing and decreasing.
2 marks

Figure in the paper: Illustration of an online delivery app and a delivery person, alongside a graph with 'Subscription Fee' on x-axis and 'No. of Subscribers' on y-axis showing a straight line decreasing. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/1/1.

Hints

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

Hint 1

Revenue = (number of subscribers left) × (fee per subscriber).

Hint 2

Expand $R (x)$ and differentiate. Where does $R^{'} (x)$ become zero?

Hint 3

Check the sign of $R^{''} (x)$ for the maximum, or the sign of $R^{'} (x)$ on either side of the critical point for the intervals.

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) Each ₹1 increase loses 10 subscribers, so an increase of ₹ $x$ makes $10 x$ subscribers discontinue.
1 mark
(ii) Subscribers left $= 5000 - 10 x$ and the new fee $= ₹ (300 + x)$ .
So $R (x) = (5000 - 10 x) (300 + x) = 1500000 + 2000 x - 10 x^{2}$ .
1 mark
(iii)(a) Differentiate: $R^{'} (x) = 2000 - 20 x$ .
½ mark
(iii)(a) Set $R^{'} (x) = 0$ : $2000 - 20 x = 0 \Rightarrow x = 100$ .
½ mark
(iii)(a) Second derivative: $R^{''} (x) = - 20 < 0$ at $x = 100$ .
½ mark
(iii)(a) So $R (x)$ is maximum at $x = 100$ , i.e. when the fee is raised by ₹100. The maximum revenue is $R (100) = 4000 \times 400 = ₹16, 00, 000$ .
½ mark
(iii)(b) $R^{'} (x) = 2000 - 20 x$ .
½ mark
(iii)(b) $R^{'} (x) = 0 \Rightarrow x = 100$ , which splits $(0, 5000)$ into $(0, 100)$ and $(100, 5000)$ .
½ mark
(iii)(b) For $0 < x < 100$ , $R^{'} (x) > 0$ , so $R (x)$ is increasing on $(0, 100)$ (or $(0, 100]$ ).
½ mark
(iii)(b) For $100 < x < 5000$ , $R^{'} (x) < 0$ , so $R (x)$ is decreasing on $(100, 5000)$ (or $[100, 5000)$ ).
½ mark

Answer: (i) $10 x$ ; (ii) $R (x) = (5000 - 10 x) (300 + x)$ ; (iii)(a) $x = 100$ ; (iii)(b) increasing on $(0, 100)$ , decreasing on $(100, 5000)$

Where the marks are

(i) States $10 x$ subscribers discontinue (1 mark)
(ii) Writes $R (x) = (5000 - 10 x) (300 + x)$ (1 mark)
(iii)(a) Finds $R^{'} (x) = 2000 - 20 x$ (½ mark)
(iii)(a) Solves $R^{'} (x) = 0$ to get $x = 100$ (½ mark)
(iii)(a) Shows $R^{''} (x) = - 20 < 0$ (½ mark)
(iii)(a) Concludes $R$ is maximum at $x = 100$ (½ mark)
(iii)(b) Finds $R^{'} (x)$ (½ mark)
(iii)(b) Gets the critical point $x = 100$ (½ mark)
(iii)(b) States increasing on $(0, 100)$ with reason $R^{'} > 0$ (½ mark)
(iii)(b) States decreasing on $(100, 5000)$ with reason $R^{'} < 0$ (½ 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 paper2026 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 36

Hints

Step-by-step solution

Revise the chapter

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