If α, β are zeroes of quadratic polynomial 5x² + 5x + 1, find the value of 1. α² + β² 2. α^-1 + β^-1

1. α²+β² = 35; 2. α^-1+β^-1 = -5

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

CBSE Class 10 Mathematics Standard 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

If $α, β$ are zeroes of quadratic polynomial $5 x^{2} + 5 x + 1$ , find the value of
1. $α^{2} + β^{2}$
2. $α^{- 1} + β^{- 1}$

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

Write down the sum and the product of the zeroes using −b/a and c/a.

Hint 2

Write α² + β² using (α+β)² and αβ. Write 1/α + 1/β as one fraction.

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 $p (x) = 5 x^{2} + 5 x + 1$ : $a = 5$ , $b = 5$ , $c = 1$ .
½ mark
Sum of zeroes: $α + β = - \frac{b}{a} = - \frac{5}{5} = - 1$
½ mark
Product of zeroes: $α β = \frac{c}{a} = \frac{1}{5}$
½ mark
1. Use $α^{2} + β^{2} = (α + β)^{2} - 2 α β$ :
$= (- 1)^{2} - 2 (\frac{1}{5})$
½ mark
$= 1 - \frac{2}{5} = \frac{3}{5}$
½ mark
2. $α^{- 1} + β^{- 1} = \frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β} = \frac{- 1}{1/5} = - 5$
½ mark

Answer: 1. $α^{2} + β^{2} = \frac{3}{5}$ ; 2. $α^{- 1} + β^{- 1} = - 5$

Where the marks are

Identifies the coefficients a = 5, b = 5, c = 1 (1/2 mark)
Finds α + β = −1 (1/2 mark)
Finds αβ = 1/5 (1/2 mark)
Uses the identity for α² + β² (1/2 mark)
Gets α² + β² = 3/5 (1/2 mark)
Gets α⁻¹ + β⁻¹ = −5 (1/2 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 2023-24 (Standard): every question, by section.

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

Hints

Step-by-step solution

Revise the chapter

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