MathsCBSE Class 10 · Standard · Board exam 2025, set 30/5/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2025: question 27

Set in the CBSE Class 10 board exam of 2025, paper 30/5/1, section C, question 27, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q27 · 3 marks · Short answerPolynomials

Obtain the zeroes of the polynomial $7 x^{2} + 18 x - 9$ . Hence, write a polynomial each of whose zeroes is twice the zeroes of given polynomial.

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

Split 18x into two terms whose coefficients multiply to $7 \times (- 9) = - 63$ .

Hint 2

Double each zero, then find their sum and product.

Hint 3

A quadratic with known sum S and product P is $k (x^{2} - S x + P)$ .

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.

Split the middle term: $7 x^{2} + 18 x - 9 = 7 x^{2} + 21 x - 3 x - 9 = 7 x (x + 3) - 3 (x + 3) = (7 x - 3) (x + 3)$ . So the zeroes are $\frac{3}{7}$ and $- 3$ .
1 mark
Twice the zeroes: $\frac{6}{7}$ and $- 6$
½ mark
Sum of the new zeroes $= \frac{6}{7} + (- 6) = - \frac{36}{7}$
½ mark
Product of the new zeroes $= \frac{6}{7} \times (- 6) = - \frac{36}{7}$
½ mark
Required polynomial $= x^{2} - (sum) x + product = x^{2} + \frac{36}{7} x - \frac{36}{7}$ , or $7 x^{2} + 36 x - 36$ (or any non-zero multiple of it).
½ mark

Answer: Zeroes are $\frac{3}{7}$ and $- 3$ ; required polynomial $7 x^{2} + 36 x - 36$

Where the marks are

Factorises and finds the zeroes $\frac{3}{7}$ , $- 3$ (1 mark)
Writes the new zeroes $\frac{6}{7}$ , $- 6$ (½ mark)
Sum $= - \frac{36}{7}$ (½ mark)
Product $= - \frac{36}{7}$ (½ mark)
Forms the polynomial $7 x^{2} + 36 x - 36$ (½ mark)

Revise the chapter

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

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 30/5/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2025: question 27

Hints

Step-by-step solution

Revise the chapter

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