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  4. If α ,β and γ are the roots of the equation px3+qx2+r=0, then the value of the determinant | α β β γ γ α β γ γ α α β γ α α β β γ | is

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Q. If $\alpha ,\beta $ and $\gamma $ are the roots of the equation $px^{3}+qx^{2}+r=0,$ then the value of the determinant $\begin{vmatrix} \alpha \beta & \beta \gamma & \gamma \alpha \\ \beta \gamma & \gamma \alpha & \alpha \beta \\ \gamma \alpha & \alpha \beta & \beta \gamma \end{vmatrix}$ is

NTA AbhyasNTA Abhyas 2022

Solution:

The roots of the equation $\textit{px}^{3}+\textit{qx}^{2}+\textit{r}=0$ are $\alpha ,\beta ,\gamma $
i.e. $\alpha \beta +\beta \gamma +\gamma \alpha =0$
$\therefore \begin{vmatrix} \alpha \beta & \beta \gamma & \gamma \alpha \\ \beta \gamma & \gamma \alpha & \alpha \beta \\ \gamma \alpha & \alpha \beta & \beta \gamma \end{vmatrix}$
Applying $\left[\text{C}_{1} \rightarrow \text{C}_{1} + \text{C}_{2} + \text{C}_{3}\right]$
$\begin{vmatrix} \alpha \beta +\beta \gamma +\gamma \alpha & \beta \gamma & \gamma \alpha \\ \alpha \beta +\beta \gamma +\gamma \alpha & \gamma \alpha & \alpha \beta \\ \alpha \beta +\beta \gamma +\gamma \alpha & \alpha \beta & \beta \gamma \end{vmatrix}=0$