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
Answer & explanation
Correct answer: option 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 $\alpha \, \beta \, \gamma=-b.$
$Δ=\begin{vmatrix}\alpha & \beta & \gamma \\\beta & \gamma & \alpha \\\gamma & \alpha & \beta \end{vmatrix}$
$Δ = -(\alpha + \beta + \gamma ) ( \alpha^2 + \beta^2 + \gamma^2 -\alpha \beta - \beta \gamma -\gamma \alpha )$
$Δ= - ( \alpha + \beta + \gamma ) \begin{Bmatrix}(\alpha + \beta + \gamma )^2- 3 (\alpha \, \beta + \beta \, \gamma + \gamma \, \alpha ) \end{Bmatrix}$
$Δ= -(-a)\begin{Bmatrix}a^2-0\end{Bmatrix}=a^3$