Answer the following. (a) If α, β are zeroes of the polynomial 8x² - 5x - 1, then form a quadratic polynomial in x whose zeroes are (2)/(α) and (2)/(β). (b) Find the zeroes of the polynomial p(x) = 3x² + x - 10 and verify the relationship between zeroes and its coefficients.

x²+10x-32 (or k(x²+10x-32), k≠0)

MathsCBSE Class 10 · Basic · Board exam 2025, set 430/6/1 · 3 marks · Short answer

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

Set in the CBSE Class 10 board exam of 2025, paper 430/6/1, 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

Answer the following.

(a)
If $α, β$ are zeroes of the polynomial $8 x^{2} - 5 x - 1$ , then form a quadratic polynomial in $x$ whose zeroes are $\frac{2}{α}$ and $\frac{2}{β}$ .
0 marks
(b)
Find the zeroes of the polynomial $p (x) = 3 x^{2} + x - 10$ and verify the relationship between zeroes and its coefficients.
0 marks

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/6/1.

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

Express $\frac{2}{α} + \frac{2}{β}$ and $\frac{2}{α} \cdot \frac{2}{β}$ 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 $8 x^{2} - 5 x - 1$ : $α + β = - \frac{- 5}{8} = \frac{5}{8}$ and $α β = \frac{- 1}{8}$ .
1 mark
Sum of new zeroes: $\frac{2}{α} + \frac{2}{β} = \frac{2 ( α + β )}{α β} = \frac{2 \times \frac{5}{8}}{- \frac{1}{8}} = - 10$ .
½ mark
Product of new zeroes: $\frac{2}{α} \times \frac{2}{β} = \frac{4}{α β} = \frac{4}{- \frac{1}{8}} = - 32$ .
½ mark
Required polynomial $= x^{2} - (sum) x + product = x^{2} + 10 x - 32$ , or $k (x^{2} + 10 x - 32)$ for any non-zero real $k$ .
1 mark

Answer: $x^{2} + 10 x - 32$ (or $k (x^{2} + 10 x - 32)$ , $k \neq = 0$ )

Where the marks are

Writes $α + β = \frac{5}{8}$ and $α β = - \frac{1}{8}$ (½ + ½ mark)
Sum of new zeroes $= - 10$ (½ mark)
Product of new zeroes $= - 32$ (½ mark)
Forms polynomial $x^{2} + 10 x - 32$ (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 430/6/1 (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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