Find the zeroes of the polynomial 4x² - 4x + 1 and verify the relationship between the zeroes and the coefficients.

Zeroes: 1/2, 1/2; sum = 1 = -b/a, product = 1/4 = c/a (verified)

MathsCBSE Class 10 · Basic · Board exam 2024, set 430/2/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2024: question 29(b)

Set in the CBSE Class 10 board exam of 2024, paper 430/2/1, section C, question 29(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29(b) · 3 marks · Short answerPolynomials

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

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/2/1.

Hints

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

Hint 1

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

Hint 2

The expression is a perfect square.

Hint 3

Compare the sum with $- b / a$ and the product with $c / a$ .

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.

$4 x^{2} - 4 x + 1 = 4 x^{2} - 2 x - 2 x + 1 = 2 x (2 x - 1) - 1 (2 x - 1) = (2 x - 1) (2 x - 1)$
1 mark
Setting $(2 x - 1)^{2} = 0$ gives the zeroes $\frac{1}{2}$ and $\frac{1}{2}$ .
1 mark
Sum of zeroes $= \frac{1}{2} + \frac{1}{2} = 1$ , and $- \frac{coefficient of x}{coefficient of x ^{2}} = - \frac{( - 4 )}{4} = 1$ . Verified.
½ mark
Product of zeroes $= \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ , and $\frac{constant term}{coefficient of x ^{2}} = \frac{1}{4}$ . Verified.
½ mark

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

Where the marks are

Factorises as $(2 x - 1) (2 x - 1)$ (1 mark)
Zeroes $\frac{1}{2}, \frac{1}{2}$ (1 mark)
Verifies the sum of zeroes (½ mark)
Verifies the product of zeroes (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 paper2024 board paper, set 430/2/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2024: question 29(b)

Hints

Step-by-step solution

Revise the chapter

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