If $\alpha , \beta , \gamma $ are the roots of $x^3 + ax^2 + b = 0, $ then the value of $\begin{vmatrix}\alpha & \beta & \gamma \\\beta & \gamma & \alpha \\\gamma & \alpha & \beta \end{vmatrix},$ is

$a^3$

The correct answer is option (3) : $a^3$

Since $\alpha , \beta , \gamma $ are the roots of the given equation.

$∴\alpha + \beta + \gamma = -a, \alpha \beta + \beta \gamma + \gamma \alpha = 0 $ and $\alpha \, \beta \, \gamma=-b.$

$Δ=\begin{vmatrix}\alpha & \beta & \gamma \\\beta & \gamma & \alpha \\\gamma & \alpha & \beta \end{vmatrix}$

$Δ = -(\alpha + \beta + \gamma ) ( \alpha^2 + \beta^2 + \gamma^2  -\alpha \beta - \beta \gamma -\gamma \alpha )$

$Δ= - ( \alpha + \beta + \gamma ) \begin{Bmatrix}(\alpha + \beta + \gamma )^2- 3 (\alpha \, \beta  + \beta \, \gamma  + \gamma \, \alpha  ) \end{Bmatrix}$

$Δ= -(-a)\begin{Bmatrix}a^2-0\end{Bmatrix}=a^3$