Zeroes of the quadratic polynomial x² - 3x + 2 are α and β. Construct a quadratic polynomial whose zeroes are 2α + 1 and 2β + 1.

x² - 8x + 15 (or k(x² - 8x + 15), k ≠ 0)

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

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

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

Q29(a) · 3 marks · Short answerPolynomials

Zeroes of the quadratic polynomial $x^{2} - 3 x + 2$ are $α$ and $β$ . Construct a quadratic polynomial whose zeroes are $2 α + 1$ and $2 β + 1$ .

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

First find $α + β$ and $α β$ from the coefficients.

Hint 2

Express the sum and the product of $2 α + 1$ and $2 β + 1$ in terms of $α + β$ and $α β$ .

Hint 3

A quadratic with a given sum S and product P of zeroes is $x^{2} - S x + P$ .

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.

For $x^{2} - 3 x + 2$ : $α + β = - \frac{b}{a} = 3$
½ mark
$α β = \frac{c}{a} = 2$
½ mark
Sum of the new zeroes: $(2 α + 1) + (2 β + 1) = 2 (α + β) + 2 = 2 (3) + 2 = 8$
½ mark
Product of the new zeroes: $(2 α + 1) (2 β + 1) = 4 α β + 2 (α + β) + 1 = 4 (2) + 2 (3) + 1 = 15$
1 mark
Required polynomial is $x^{2} - (sum) x + product = x^{2} - 8 x + 15$ , or $k (x^{2} - 8 x + 15)$ for any non-zero constant $k$ .
½ mark

Answer: $x^{2} - 8 x + 15$ (or $k (x^{2} - 8 x + 15)$ , $k \neq = 0$ )

Where the marks are

$α + β = 3$ (½ mark)
$α β = 2$ (½ mark)
New sum = 8 (½ mark)
New product = 15 (1 mark)
Writes the polynomial $x^{2} - 8 x + 15$ (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(a)

Hints

Step-by-step solution

Revise the chapter

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