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EAMCET Engineering 2005 Solved Paper

Section: Mathematics

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Question : 97 of 160

Marks: +1, -0

If

α, β, γ

are the roots of

x^3+2 x^2-3 x-1=0

α^{-2}+β^{-2}+γ^{-2}=

12
13
14
15

Solution:

Given that,

α, β, γ

, are the roots of the equation

x^3+2 x^2-3 x-1=0

\Rightarrow \;\; \alpha+\beta+\gamma=-2

. . . (i)

\alpha \beta+\beta \gamma+\gamma \alpha=-3

. . . (ii)

and

\;\; \alpha \beta \gamma=1

. . . (iii)

On squaring equation (ii), we get

\;\alpha^2 \beta^2+\beta^2 \gamma^2+\gamma^2 \alpha^2+2 \alpha \beta \gamma(\alpha+\beta+\gamma)=9

\;\Rightarrow \alpha^2 \beta^2+\beta^2 \gamma^2+\gamma^2 \alpha^2=9-2 \;\text{ (1) }\;(-2)=13

Now

\;\alpha^{-2}+\beta^{-2}+\gamma^{-2}=\;\frac{\beta^2 \gamma^2+\gamma^2 \alpha^2+\alpha^2 \beta^2}{(\alpha \beta \gamma)^2}

\;\alpha^{-2}+\beta^{-2}+\gamma^{-2}=\;\frac{13}{1}=13

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