Q53 / 155 · WBJEE 2025 · Mathematics
Suppose, $\alpha, \beta, \gamma$ are the roots of the equation $\mathrm{x}^3+\mathrm{qx}+\mathrm{r}=0($ with $\mathrm{r} \neq 0)$ and they are in A.P. Then the rank of the matrix $\left(\begin{array}{lll}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta\end{array}\right)$ is
- A$3$
- B$2$
- C$0$
- Correct answerDNone of the above
Solution
Here, $\alpha+\beta+\gamma=0$
$\left|\begin{array}{lll}
\alpha & \beta & \gamma \\
\beta & \gamma & \alpha \\
\gamma & \alpha & \beta
\end{array}\right|=-\left(\alpha^3+\beta^3+\gamma^3-3 \alpha \beta \gamma\right)=0(\because \alpha+\beta+\gamma=0)$
Given, $2 \beta=\alpha+\gamma \Rightarrow 3 \beta=\alpha+\beta+\gamma \Rightarrow \beta=0 \Rightarrow r=0$ (But given $r \neq 0$ )
Hence, the correct answer is option: 'D'.