MathsCBSE Class 12 · Board exam 2025, set 65/7/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2025: question 29

Set in the CBSE Class 12 board exam of 2025, paper 65/7/1, section C, question 29, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29 · 3 marks · Short answerApplication of Derivatives

Amongst all pairs of positive integers with product as 289, find which of the two numbers add up to the least.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/7/1.

Hints

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

Hint 1

Use the product condition to write the sum in terms of one variable.

Hint 2

Differentiate the sum, set the derivative to zero, and check the sign of the second derivative.

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.

Let the numbers be x and y with $x y = 289$ , so $y = \frac{289}{x}$ . Their sum is $S = x + \frac{289}{x}$ , $x > 0$ .
1 mark
$\frac{d S}{d x} = 1 - \frac{289}{x ^{2}}$ . Setting $\frac{d S}{d x} = 0$ gives $x^{2} = 289$ , so $x = 17$ (taking the positive value).
1½ marks
$\frac{d ^{2} S}{d x ^{2}} = \frac{578}{x ^{3}} > 0$ at $x = 17$ , so S is minimum there. Then $y = \frac{289}{17} = 17$ . The numbers are 17 and 17, with least sum 34.
½ mark

Answer: 17 and 17 (least sum 34)

Where the marks are

Forms $S = x + \frac{289}{x}$ (1 mark)
Solves $S^{'} = 0$ to get $x = 17$ (1½ marks)
Second-derivative test confirms the minimum; numbers are 17, 17 (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 paper2025 board paper, set 65/7/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 29

Hints

Step-by-step solution

Revise the chapter

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