If `alpha,beta,gamma` are roots of the equation `x^(3)+px+q=0` then the value of `|{:(alpha,beta,gamma),(beta,gamma,alpha),(gamma,alpha,beta):}|` is

Options

A. `-a^(3)`
B. `a^(3) -3b`
C. `a^(3)`
D. `a^(2) - 3b`

Answer

C

Explanation

Since, `alpha, beta , gamma` are the roots of the given equations
`:. alpha + beta + gamma = -a, alpha beta + beta gamma + gamma alpha = 0 and alpha beta gamma = -b`
`Delta = |(alpha,beta,gamma),(beta,gamma,alpha),(gamma,alpha,beta)|`
`Delta = - (alpha + beta + gamma) (alpha^(2) + beta^(2) + gamma^(2) - alpha beta - beta gamma - gamma alpha)`
`Delta = - (alpha + beta + gamma) {(alpha + beta + gamma)^(2) - 3(alpha beta + beta gamma + gamma beta)}`
`Delta = - (-a) {(a^(2) - 0)} = a^(3)`