If`alpha,betaandgamma` are the roots of `X^(3)-3X^(2)+3X+7+0,` find the value of `(alpha =1)/(beta-1)+(beta-1)/(gamma-1)+(gamma-1)/(alpha-1).`

Options

A. `3//omega`
B. `omega^(2)`
C. `2 omega^(2)`
D. `3 omega^(2)`

Answer

D

Explanation

We have, `x^(3) - 3x^(2) + 3x + 7 = 0`
`rArr" "(x-1)^(3)+8=0`
`rArr" "(x-1)^(3)= -8`
`rArr" "x - 1 = -2, -2 omega, -2 omega^(2) rArr x = -1, 1 - 2 omega, 1 - 2 omega^(2)`
Let `alpha = -1, beta = 1 - 2 omega and gamma = 1 - 2 omega^(2)`
Then, `alpha-1 = -2, beta -1 = -2 omega and gamma -1 = - 2 omega^(2)`
`therefore" "(alpha-1)/(beta-1)+(beta-1)/(gamma-1)+(gamma-1)/(alpha-1)=(2)/(2 omega)+(2 omega)/(2 omega^(2))+(2 omega^(2))/(2)=3 omega^(2)`