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BITSAT 2016 Paper

Section: Mathematics

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Question : 98 of 150

Marks: +1, -0

If

\alpha,\beta

are the roots of the equations

x^{2}-2x-1=0

, then what is the value of

\alpha^{2}\beta^{-2}+\alpha^{-2}\beta^{2}

–2
0
30
34

Solution:

Since, D and E are the roots of the equation

x^{23}-2x-1=0

,then

Sum of the roots

\alpha+\beta=2

and

product of the root

\alpha\beta=-1

Since,

(\alpha+\beta)=\alpha^{2}+\beta^{2}+2\alpha\beta

\Rightarrow 4=\alpha^{2}+\beta^{2}-2

\Rightarrow \alpha^{2}+\beta^{2}=6

Now

\alpha^{2}\beta^{-2}+\alpha^{-2}\beta^{2} = \frac{\alpha^{2}}{\beta^{2}} + \frac{\beta^{2}}{\alpha^{2}} = \frac{\alpha^{4}+\beta^{4}}{(\alpha\beta)^{2}}

\Rightarrow (\alpha^{2}+\beta^{2})^{2}=6^{2}

\Rightarrow \alpha^{4}+\beta^{4}+2\alpha^{2}\beta^{2}=36

\Rightarrow \alpha^{4}+\beta^{4}+2=36

\Rightarrow \alpha^{4}+\beta^{4}=34

.....(i)

\Rightarrow \frac{\alpha^{4}+\beta^{4}}{(\alpha\beta)^{2}} = \frac{34}{(-1)^{2}} = 34

[Putting value of

\alpha^{4}+\beta^{4}=34

from Equation (i)]

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