If `alpha, beta and gamma` are the roots of the equation `x^(3) + px + q = 0` (with `p != 0 and p != 0 and q != 0`), the value of the determinant
`|(alpha,beta,gamma),(beta,gamma,alpha),(gamma,alpha,beta)|`, is

Options

A. p
B. q
C. `p^(2) - 2q`
D. none of these

Answer

D

Explanation

It is given that `alpha, beta, gamma` are the roots of the given equations.
`:. alpha + beta + gamma = 0`.
So,
`Delta = |(alpha,beta,gamma),(beta,gamma,alpha),(gamma,alpha,beta)|`
`Delta = |(alpha + beta + gamma,beta,gamma),(alpha + beta + gamma,gamma,alpha),(alpha + beta + gamma,alpha,beta)| " " [ " Using" C_(1) rarr C_(2) + C_(2) + C_(3)]`
`Delta = |(0,beta,gamma),(0,gamma,alpha),(0,alpha,beta)| [ :. alpha + beta + gamma = 0]`