If `alpha`, `beta`, `gamma` are roots of the equation `x^(2)(px+q)=r(x+1)`, then the value of determinant `|{:(1+alpha,1,1),(1, 1+beta,1),(1,1,1+gamma):}|` is

Options

A. `alphabetagamma`
B. `1+(1)/(alpha)+(1)/(beta)+(1)/(gamma)`
C. `0`
D. none of these

Answer

C

Explanation

`(c )` The given equation is `px^(3)+qx^(2)-rx-r=0`
`alpha+beta+gamma=(-q)/(p)`, `alphabeta+betagamma+gammaalpha=-(r )/(p)`, `alphabetagamma=(r )/(p)`
`D=(1+alpha)(1+beta)(1+gamma)+1+1-(1+alpha)-(1+beta)-(1+gamma)`
`=alphabetagamma+alphabeta+alphagamma+betagamma`
`=(r )/(p)-(r )/(p)=0`