Question

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

• A. $p$
• B. $q$
• C. $p^2-2q$
• D. None of these

Answer 1

The correct option is D None of these

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

$S=\alpha +\beta +\gamma =0$
By Applying $R_1\to R_1+R_2+R_3$ we get

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

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

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