α and β are zeroes of a quadratic polynomial px² + qx + 1. Form a quadratic polynomial whose zeroes are (2)/(α) and (2)/(β).

MathsCBSE Class 10 · Standard · Board exam 2025, set 30/4/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2025: question 30

Set in the CBSE Class 10 board exam of 2025, paper 30/4/1, section C, question 30, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30 · 3 marks · Short answerQuadratic Equations

$α$ and $β$ are zeroes of a quadratic polynomial $p x^{2} + q x + 1$ . Form a quadratic polynomial whose zeroes are $\frac{2}{α}$ and $\frac{2}{β}$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/4/1.

Hints

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

Hint 1

Start with $α + β$ and $α β$ from the coefficients.

Hint 2

A quadratic with given zeroes is $x^{2} - (sum) x + (product)$ .

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.

For $p x^{2} + q x + 1$ : $α + β = - \frac{q}{p}$ , $α β = \frac{1}{p}$
½ mark
Sum of new zeroes $= \frac{2}{α} + \frac{2}{β} = \frac{2 ( α + β )}{α β} = 2 \times (- \frac{q}{p}) \times p = - 2 q$
1 mark
Product of new zeroes $= \frac{2}{α} \times \frac{2}{β} = \frac{4}{α β} = 4 p$
1 mark
Required polynomial $= x^{2} - (sum) x + product = x^{2} + 2 q x + 4 p$ (or any non-zero multiple $k (x^{2} + 2 q x + 4 p)$ ).
½ mark

Answer: $x^{2} + 2 q x + 4 p$

Where the marks are

Writes sum and product of zeroes $- \frac{q}{p}$ and $\frac{1}{p}$ (½ mark)
Sum of new zeroes $= - 2 q$ (1 mark)
Product of new zeroes $= 4 p$ (1 mark)
Forms $x^{2} + 2 q x + 4 p$ (½ mark)

Revise the chapter

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

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 30/4/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2025: question 30

Hints

Step-by-step solution

Revise the chapter

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