MathsCBSE Class 10 · Basic · Board exam 2025, set 430/1/1 · 5 marks · Long answer

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

Set in the CBSE Class 10 board exam of 2025, paper 430/1/1, 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

The difference of the squares of two positive numbers is 180. The square of the smaller number is 8 times the greater number. Find the two numbers.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/1/1.

Hints

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

Hint 1

Use two variables and write one equation for each sentence.

Hint 2

Replace $y^{2}$ in the first equation using the second, which gives a quadratic in the greater number only.

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 greater number be $x$ and the smaller number be $y$ . Then $x^{2} - y^{2} = 180$ ...(1)
1 mark
Also $y^{2} = 8 x$ ...(2)
1 mark
Substitute (2) in (1): $x^{2} - 8 x = 180 \Rightarrow x^{2} - 8 x - 180 = 0$ .
1 mark
$x^{2} - 18 x + 10 x - 180 = 0 \Rightarrow (x - 18) (x + 10) = 0$ , so $x = 18$ or $x = - 10$ . Reject $x = - 10$ because the numbers are positive, so $x = 18$ .
1 mark
$y^{2} = 8 \times 18 = 144 \Rightarrow y = 12$ (positive). The numbers are 18 and 12. Check: $324 - 144 = 180$ .
1 mark

Answer: 18 and 12

Where the marks are

Forms $x^{2} - y^{2} = 180$ (1 mark)
Forms $y^{2} = 8 x$ (1 mark)
Gets the quadratic $x^{2} - 8 x - 180 = 0$ (1 mark)
Solves to get $x = 18$ and rejects $- 10$ (1 mark)
Finds the smaller number 12 and states both numbers (1 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 430/1/1 (Basic): 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 Basic 2025: question 32(a)

Hints

Step-by-step solution

Revise the chapter

More questions from Quadratic Equations