Question

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

• A. p
• B. q
• C. $p^{2} - 2q$
• D. 0

Answer 1

Answer: (D)

0

Answer 2

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

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

\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{matrix} \right|$$ Applying $R_{1} \rightarrow R_{1} + R_{2} + R_{3}$, We get, $\left| \begin{matrix} \alpha + \beta + \gamma & \alpha + \beta + \gamma & \alpha + \beta + \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{matrix} \right|$ = $\left| \begin{matrix} 0 & 0 & 0 \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{matrix} \right| = 0$