Q101 / 180 · COMEDK 2014 · Mathematics
If $\alpha, \beta, \gamma$ are the roots of $x^{3}+a^{2} x+b=0$, then the value of $\left|\begin{array}{lll}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta\end{array}\right|$ is
- A$-a^{3}$
- B$a^{3}-3 b$
- C$a^{3}$
- Correct answerD0
Solution
Given, equation is $x^{3}+a^{2} x+b=0$
Since, $\alpha, \beta, \gamma$ are its roots.
$\therefore$ Sum of roots $=\alpha+\beta+\gamma=0 \quad \text{...(i)}$
Now, $\left|\begin{array}{lll}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta\end{array}\right|$
Using operation $C_{1} \rightarrow C_{1}+C_{2}+C_{3}$, we get $\left|\begin{array}{lll}(\alpha+\beta+\gamma) & \beta & \gamma \\ (\alpha+\beta+\gamma) & \gamma & \alpha \\ (\alpha+\beta+\gamma) & \alpha & \beta\end{array}\right|=\left|\begin{array}{lll}0 & \beta & \gamma \\ 0 & \gamma & \alpha \\ 0 & \alpha & \beta\end{array}\right|=0 \quad$ [Using Eq. (i)]
Hence, the correct answer is option: 'D'.