Question

Suppose \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 + qx + r = 0 \) (with \( r \ne 0 \)) and they are in A.P. Then the rank of the matrix \[ \begin{pmatrix} \alpha & \beta & \gamma
\beta & \gamma & \alpha
\gamma & \alpha & \beta \end{pmatrix} \] is:

• A. \( 3 \)
• B. \( 2 \)
• C. \( 0 \)
• D. \( 1 \)

Hint:

For rank problems with roots: \begin{itemize} \item Use symmetric root conditions. \item A.P. roots simplify nicely. \item Check row sums for dependence. \end{itemize}

Answer 1

The Correct Option is B