Question

If $\alpha ,\beta$ and $\gamma$ are the roots of the equation $p{x}^3+q{x}^2+r=0,$ then the value of the determinant $\left|\begin{array}{ccc}\alpha \beta & \beta \gamma & \gamma \alpha \\ \beta \gamma & \gamma \alpha & \alpha \beta \\ \gamma \alpha & \alpha \beta & \beta \gamma \end{array}\right|$ is

• A. pq
• B. qr
• C. 0
• D. pr

Answer 1

Answer: (C)

The roots of the equation ${\text{px}}^3+{\text{qx}}^2+\text{r}=0$ are $\alpha ,\beta ,\gamma$
i.e. $\alpha \beta +\beta \gamma +\gamma \alpha =0$
$\therefore \left|\begin{array}{ccc}\alpha \beta & \beta \gamma & \gamma \alpha \\ \beta \gamma & \gamma \alpha & \alpha \beta \\ \gamma \alpha & \alpha \beta & \beta \gamma \end{array}\right|$
Applying $\left[{\text{C}}_1\to {\text{C}}_1+{\text{C}}_2+{\text{C}}_3\right]$
$\left|\begin{array}{ccc}\alpha \beta +\beta \gamma +\gamma \alpha & \beta \gamma & \gamma \alpha \\ \alpha \beta +\beta \gamma +\gamma \alpha & \gamma \alpha & \alpha \beta \\ \alpha \beta +\beta \gamma +\gamma \alpha & \alpha \beta & \beta \gamma \end{array}\right|=0$