Find the zeroes of the quadratic polynomial 4s² - 4s + 1 and verify the relationship between the zeroes and the coefficients.

Zeroes: 12, 12; sum = 1 = -b/a, product = 14 = c/a (verified)

MathsCBSE Class 10 · Basic · Sample paper, session 2023-24 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic sample paper 2023-24: question 27

From the official CBSE sample question paper for 2023-24, 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

Find the zeroes of the quadratic polynomial $4 s^{2} - 4 s + 1$ and verify the relationship between the zeroes and the coefficients.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

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

Hint 1

Split the middle term: find two numbers whose product is $4 \times 1$ and whose sum is $- 4$ .

Hint 2

The polynomial is a perfect square. Then compare $α + β$ with $- \frac{b}{a}$ and $α β$ with $\frac{c}{a}$ .

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.

$4 s^{2} - 4 s + 1 = 4 s^{2} - 2 s - 2 s + 1 = 2 s (2 s - 1) - 1 (2 s - 1) = (2 s - 1) (2 s - 1)$
So the zeroes are $α = \frac{1}{2}$ and $β = \frac{1}{2}$ .
1 mark
Here $a = 4$ , $b = - 4$ , $c = 1$ .
Sum: $α + β = \frac{1}{2} + \frac{1}{2} = 1$ and $- \frac{b}{a} = - \frac{- 4}{4} = 1$
1 mark
Product: $α β = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ and $\frac{c}{a} = \frac{1}{4}$ .
Hence the relationship between the zeroes and the coefficients is verified.
1 mark

Answer: Zeroes: $\frac{1}{2}, \frac{1}{2}$ ; sum $= 1 = - \frac{b}{a}$ , product $= \frac{1}{4} = \frac{c}{a}$ (verified)

Where the marks are

Factorises and finds the zeroes $\frac{1}{2}, \frac{1}{2}$ (1 mark)
Verifies sum of zeroes $= - \frac{b}{a}$ (1 mark)
Verifies product of zeroes $= \frac{c}{a}$ (1 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 paperSample paper 2023-24 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic sample paper 2023-24: question 27

Hints

Step-by-step solution

Revise the chapter

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