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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
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B
q
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C
p22q
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D
None of these
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Solution

The correct option is B None of these
Since, $$\alpha, \beta$$ 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 $$C_{1} \rightarrow C_{1} + C_{2} + C_{3})$$

$$= \begin{vmatrix}0 & \beta & \gamma\\ 0 & \gamma & \alpha\\ 0 & \alpha & \beta\end{vmatrix} = 0$$.

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