MathsCBSE Class 10 · Standard · Board exam 2025, set 30/5/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2025: question 32(a)

Set in the CBSE Class 10 board exam of 2025, paper 30/5/1, section D, question 32(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(a) · 5 marks · Long answerQuadratic Equations

A 2-digit number is seven times the sum of its digits and two (2) more than five times the product of its digits. Find the number.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/5/1.

Hints

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

Hint 1

Write the number as 10 × (ten's digit) + (unit's digit).

Hint 2

The first condition gives a simple relation between the two digits; substitute it into the second.

Hint 3

Solve the resulting quadratic and reject any root that cannot be a digit.

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 unit's digit be $x$ and the ten's digit be $y$ . Then the number $= 10 y + x$ .
1 mark
Given $10 y + x = 7 (x + y) \Rightarrow 3 y = 6 x \Rightarrow y = 2 x$ ...(1)
1 mark
Also $10 y + x = 5 x y + 2$ ...(2)
½ mark
Substituting (1) in (2): $20 x + x = 10 x^{2} + 2 \Rightarrow 10 x^{2} - 21 x + 2 = 0$
1 mark
$(x - 2) (10 x - 1) = 0 \Rightarrow x = 2$ or $x = \frac{1}{10}$ . A digit must be a whole number, so $x = 2$ .
½ mark
$y = 2 x = 4$
½ mark
The required number is 42. (Check: $7 (4 + 2) = 42$ and $5 \times 4 \times 2 + 2 = 42$ .)
½ mark

Answer: 42

Where the marks are

Sets up the digits and the number $10 y + x$ (1 mark)
Forms $10 y + x = 7 (x + y)$ and gets $y = 2 x$ (1 mark)
Forms $10 y + x = 5 x y + 2$ (½ mark)
Obtains $10 x^{2} - 21 x + 2 = 0$ (1 mark)
Solves to get $x = 2$ , rejecting $\frac{1}{10}$ (½ mark)
Finds $y = 4$ (½ mark)
States the number 42 (½ mark)

Revise the chapter

Chapter: Quadratic Equations (notes, formulas, where marks are lost and every past question)
Topic: Word Problems on Quadratic Equations

Practise this chapterBoard-style questions from Quadratic Equations, one at a time, marked against the answer key and marking points.Back to the full paper2025 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 2025: question 32(a)

Hints

Step-by-step solution

Revise the chapter

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