Question

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

A
p
B
q
C
p22q
D
None of these

Solution

## The correct option is B None of theseSince, $$\alpha, \beta$$ and $$\gamma$$ are the roots of the equation$$x^{3} + px + q = 0$$$$\therefore \alpha + \beta + \gamma = 0$$Now, $$\begin{vmatrix}\alpha & \beta & \gamma\\ \beta & \gamma & \alpha\\ \gamma & \alpha & \beta\end{vmatrix} =\begin{vmatrix}\alpha + \beta + \gamma & \beta & \gamma\\ \alpha + \beta + \gamma & \gamma & \alpha\\ \alpha + \beta + \gamma & \alpha & \beta\end{vmatrix}$$(using $$C_{1} \rightarrow C_{1} + C_{2} + C_{3})$$$$= \begin{vmatrix}0 & \beta & \gamma\\ 0 & \gamma & \alpha\\ 0 & \alpha & \beta\end{vmatrix} = 0$$.Maths

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