If `alpha, beta, gamma` are the roots fo `x^(3)-x^(2)+ax+b=0` and `beta, gamma, delta` are the roots of `x^(3)-4x^(2)+mx+n=0`. If `alpha, beta, gamma` and `delta` are in A.P. with common difference `d` then

Options

A. `a=m`
B. `a=m-5`
C. `n=b-a-2`
D. `b=m+n-3`

Explanation

`:' a, beta, gamma, delta` are in AP with common difference `d` then
`beta=alpha +d, gamma =alpha +2x` and `delta=alpha+3d`…i
Given `a, beta, gamma` are the roots of `x^(3)-x^(2)+ax+b=0` then
`alpha+beta+gamma=1`………….ii
`alpha beta+beta gamma+gamma alpha =a`...........iii
`alpha beta gamma=-b`...........iv
Also `beta, gamma, delta` are the roots of `x^(3)-4x^(2)+mx+n=0` then
`beta+gamma+dleta=4`...........v
`beta gamma+gamma delta+delta beta=m`.......vi
`beta gamma delta=-n`..........vii
From eqs i and ii we get
`3 alpha +3 d=1`............viii
and from Eqs i and v we get
`3 alpha +6d=4` .ix
From eqs viii and ix we get
`d=1, alpha=-2/3`
NOw from Eq i we get
`beta=1/3, gamma=4/3` and `delta=7/3`
From eqs iii, iv, vi and vii we get
`a=-2/3, b=8/27,m=13/3,n=-28/27`
`:.a=m-5,n=b-a-2` and `b=m+n-3`