If `f(x)=a=b x+c x^2a n dalpha,beta,gamma` are the roots of the equation `x^3=1,t h e n|a b c b c a c a b|` is equal to `f(alpha)+f(beta)+f(gamma)` `f(alpha)f(beta)+f(beta)f(gamma)+f(gamma)f(alpha)` `f(alpha)f(beta)f(gamma)` `-f(alpha)f(beta)f(gamma)`

Options

A. `f(alpha) +f(beta) +f(gamma)`
B. `f(alpha) f(beta)+f(beta)f(gamma)+f(gamma) f(alpha)`
C. `f(alpha) f(beta) f(gamma)`
D. `-f(alpha) f(beta)f(gamma)`

Answer

D

Explanation

`|{:(a,,b,,c),(b,,c,,a),(c,,a,,b):}| =- (a^(3) +b^(3) +c^(3) -3abc)`
`=-(a+b+c) (a+b omega^(2) +comega)(a+bomega+comega^(2))`
(where `omega` is cube roots of unity)
`[ :' alpha =1, beta =omega ,gamma=omega^(2)]`
`=-f(alpha)f(beta)f(gamma)`