If α, β are zeroes of the polynomial 3x² - 8x + 4, then form a quadratic polynomial in x whose zeroes are (1)/(α) and (1)/(β).

x² - 2x + 3/4, i.e. k(4x² - 8x + 3), k ≠ 0

MathsCBSE Class 10 · Basic · Board exam 2025, set 430/5/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2025: question 26(a)

Set in the CBSE Class 10 board exam of 2025, paper 430/5/1, section C, question 26(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q26(a) · 3 marks · Short answerPolynomials

If $α, β$ are zeroes of the polynomial $3 x^{2} - 8 x + 4$ , then form a quadratic polynomial in $x$ whose zeroes are $\frac{1}{α}$ and $\frac{1}{β}$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/5/1.

Hints

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

Hint 1

Use sum $= - \frac{b}{a}$ and product $= \frac{c}{a}$ for the given polynomial.

Hint 2

Write $\frac{1}{α} + \frac{1}{β}$ as a single fraction in terms of $α + β$ and $α β$ , then use $x^{2} - (sum) x + product$ .

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 $3 x^{2} - 8 x + 4$ : $α + β = - \frac{- 8}{3} = \frac{8}{3}$ and $α β = \frac{4}{3}$ .
1 mark
Sum of new zeroes: $\frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β} = \frac{8/3}{4/3} = 2$
½ mark
Product of new zeroes: $\frac{1}{α} \cdot \frac{1}{β} = \frac{1}{α β} = \frac{3}{4}$
½ mark
Required polynomial $= x^{2} - (sum) x + product = x^{2} - 2 x + \frac{3}{4}$ , or $k (4 x^{2} - 8 x + 3)$ for any non-zero real $k$ .
1 mark

Answer: $x^{2} - 2 x + \frac{3}{4}$ , i.e. $k (4 x^{2} - 8 x + 3)$ , $k \neq = 0$

Where the marks are

Writes $α + β = \frac{8}{3}$ and $α β = \frac{4}{3}$ (1 mark)
Finds sum of reciprocals = 2 (½ mark)
Finds product of reciprocals $= \frac{3}{4}$ (½ mark)
Forms the polynomial $x^{2} - 2 x + \frac{3}{4}$ or $4 x^{2} - 8 x + 3$ (1 mark)

Revise the chapter

Chapter: Polynomials (notes, formulas, where marks are lost and every past question)
Topic: Relationship Between Zeros and Coefficients

Practise this chapterBoard-style questions from Polynomials, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 430/5/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2025: question 26(a)

Hints

Step-by-step solution

Revise the chapter

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