Question

If $\alpha ,\beta ,\gamma$ are the roots of the equation $x^3$ + px + q = 0, then the value of the determinant $\left|\begin{array}{ccc}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{array}\right|$ is

• A. 4
• B. 2
• C. 0
• D. -2

Answer 1

The correct option is C

0

Since, $\alpha ,\beta ,\gamma$ are the roots of $x^3$ + px + q = 0

$\therefore \alpha +\beta +\gamma =0$

$Applying{C}_1\to C_1+C_2+C_3,then$

$\left|\begin{array}{ccc}\alpha +\beta +\gamma & \beta & \gamma \\ \alpha +\beta +\gamma & \gamma & \alpha \\ \alpha +\beta +\gamma & \alpha & \beta \end{array}\right|=\left|\begin{array}{ccc}0& \beta & \gamma \\ 0& \gamma & \alpha \\ 0& \alpha & \beta \end{array}\right|=0$