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  4. If α, β γ are the roots of the equation x 3+ px + q =0, then the value of the determinant |α β γ β γ α γ α β|=

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

Determinants

Solution:

$\operatorname{det} S C \quad D=\alpha^3+\beta^3+\gamma^3-3 \alpha \beta \gamma$
Suppose given $\alpha+\beta+\gamma=0$
as coefficient of $x^2=0$
Now $\operatorname{det}(\alpha+\beta+\gamma)\begin{vmatrix}1 & 1 & 1 \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta\end{vmatrix}=0 \Rightarrow( D )$