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If $\alpha,\beta$ are roots of $x^2+px+q=0$ and $\omega$ is cube root of unity,then the value of $(\omega\alpha+\omega^2\beta)(\omega^2\alpha+\omega \beta)$ is

$\begin{array}{1 1}(A)\;p^2-q&(B)\;p^2-2q\\(C)\;p^2-3q&(D)\;\text{None of these}\end{array} $

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$\alpha+\beta=-p$
$\alpha\beta=q$
$(\omega \alpha+\omega^2\beta)(\omega^2\alpha+\omega \beta)$
$\Rightarrow \omega^3\alpha^2+\omega^2\alpha\beta+\omega^4\alpha\beta+\omega^3\beta^2$
$\Rightarrow \alpha^2+\beta^2+\alpha\beta(\omega^2+\omega)=\alpha^2+\beta^2-\alpha\beta$
$\Rightarrow (\alpha+\beta)^2-3\alpha \beta$
$\Rightarrow p^2-3q$
Hence (C) is the correct answer.
answered Apr 10, 2014 by sreemathi.v
 

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