If α and β are zeroes of a polynomial 6x²-5x+1 then form a quadratic polynomial whose zeroes are α² and β².

36x²-13x+1 (or any non-zero multiple of it)

MathsCBSE Class 10 · Standard · Sample paper, session 2024-25 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard sample paper 2024-25: question 28

From the official CBSE sample question paper for 2024-25, section C, question 28, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28 · 3 marks · Short answerPolynomials

If $α$ and $β$ are zeroes of a polynomial $6 x^{2} - 5 x + 1$ then form a quadratic polynomial whose zeroes are $α^{2}$ and $β^{2}$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

Use the relations between the zeroes and the coefficients to find $α + β$ and $α β$ .

Hint 2

Use $α^{2} + β^{2} = (α + β)^{2} - 2 α β$ , then form $x^{2} - (sum) x + product$ .

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 $6 x^{2} - 5 x + 1$ : $α + β = - \frac{b}{a} = \frac{5}{6}$ and $α β = \frac{c}{a} = \frac{1}{6}$ .
1 mark
Sum of the new zeroes: $α^{2} + β^{2} = (α + β)^{2} - 2 α β = \frac{25}{36} - \frac{2}{6} = \frac{25 - 12}{36} = \frac{13}{36}$ .
1 mark
Product of the new zeroes: $α^{2} β^{2} = (α β)^{2} = (\frac{1}{6})^{2} = \frac{1}{36}$ .
½ mark
Required polynomial: $k [x^{2} - (sum) x + product] = k (x^{2} - \frac{13}{36} x + \frac{1}{36})$ . Taking $k = 36$ , the polynomial is $36 x^{2} - 13 x + 1$ .
½ mark

Answer: $36 x^{2} - 13 x + 1$ (or any non-zero multiple of it)

Where the marks are

Writes $α + β = \frac{5}{6}$ and $α β = \frac{1}{6}$ (1 mark)
Finds $α^{2} + β^{2} = \frac{13}{36}$ (1 mark)
Finds $α^{2} β^{2} = \frac{1}{36}$ (½ mark)
Forms the polynomial $36 x^{2} - 13 x + 1$ (½ 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 paperSample paper 2024-25 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard sample paper 2024-25: question 28

Hints

Step-by-step solution

Revise the chapter

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