Q57 / 160 · TS EAMCET 2007 · Mathematics

If $\alpha, \beta, \gamma$ are the roots of $x^3-2 x^2+3 x-4=0$, then the value of $\alpha^2 \beta^2+\beta^2 \gamma^2+\gamma^2 \alpha^2$ is

  • Correct answer
    A
    $-7$
  • B
    $-5$
  • C
    $-3$
  • D
    $0$

Solution

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-2 x^2+3 x-4=0$, then $\begin{aligned} & \alpha+\beta+\gamma=\frac{2}{1}=2 \\ & \alpha \beta+\beta \gamma+\gamma \alpha=3 \\ & \text { and } \\ & \alpha \beta \gamma=4 \\ & \end{aligned}$ We know that $\alpha^2 \beta^2+\beta^2 \gamma^2+\gamma^2 \alpha^2$ $\begin{aligned} & =(\alpha \beta+\beta \gamma+\gamma \alpha)^2-2(\alpha \beta \gamma)(\alpha+\beta+\gamma) \\ & =(3)^2-2(4)(2) \\ & =9-16 \\ & =-7\end{aligned}$

Hence, the correct answer is option: 'A'.

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