$\alpha, \beta, \gamma$ are the roots of the equation $x^3-10 x^2+7 x+8=0$. Match the following and choose…

$\alpha, \beta, \gamma$ are the roots of the equation $x^3-10 x^2+7 x+8=0$. Match the following and choose the correct answer.

1. $\begin{array}{cccc}\text { A } & \text { B } & \text { C } & \text { D } \\ 5 & 3 & 1 & 2\end{array}$
2. $\begin{array}{cccc}\text { A } & \text { B } & \text { C } & \text { D } \\ 4 & 3 & 1 & 2\end{array}$
3. $\begin{array}{cccc}\text { A } & \text { B } & \text { C } & \text { D } \\ 5 & 3 & 2 & 1\end{array}$
4. $\begin{array}{cccc}\text { A } & \text { B } & \text { C } & \text { D } \\ 5 & 2 & 3 & 1\end{array}$

Solution

Since α, β and γ are the roots of the equation
On squaring equation (i) both sides, we get

$\begin{aligned} & \text { Again now, } \frac{\alpha}{\beta \gamma}+\frac{\beta}{\gamma \alpha}+\frac{\gamma}{\alpha \beta} \\ & =\frac{\alpha^2+\beta^2+\gamma^2}{\alpha \beta \gamma}=\frac{86}{-8} \quad \text { [from Eqs (iii) and (iv)] } \\ & =-\frac{43}{4}\end{aligned}$ From the above discussion we see that option (3) is correct.

Asked in: AP EAMCET 2004

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