If `alpha, beta. gamma` are the roots of `x^3 + px^2 + q = 0,` where `q=0,` ther `Delta=[(1/alpha,1/beta,1/gamma),(1/beta,1/gamma,1/alpha),(1/gamma,1/alpha,1/beta)]` equals (A) `alpha beta gamma` (B) `alpha +beta + gamma ` (C) 0 (D) none of these

Options

A. `alpha beta gamma`
B. `alpha +beta + gamma `
C. `0`
D. None of these

Answer

C

Explanation

`(d)` We have `betaalpha+gammaalpha+alphabeta=0`
`Delta=(1)/(alpha^(3)beta^(3)gamma^(3))|{:(betagamma,gammaalpha,alphabeta),(gammaalpha,alphabeta,betagamma),(alphabeta,betagamma,gammaalpha):}|`
`=(1)/(alpha^(3)beta^(3)gamma^(3))|{:(betagamma+gammaalpha+alphabeta,gammaalpha,alphabeta),(gammaalpha+alphabeta+betagamma,alphabeta,betagamma),(alphabeta+betagamma+gammaalpha,betagamma,gammaalpha):}|` [using `C_(1)toC_(1)+C_(2)+C_(3)`]
`=(1)/(alpha^(3)beta^(3)gamma^(3))|{:(0,gammaalpha,alphabeta),(0,alphabeta,betagamma),(0,betagamma,gammaalpha):}|=0` [all zero property].