If `alpha,beta,gamma,sigma` are the roots of the equation `x^4+4x^3-6x^2+7x-9=0,` then the value of `(1+alpha^2)(1+beta^2)(1+gamma^2)(1+sigma^2)` is a. `9` b. `11` c. `13` d. `5`

Options

A. 5
B. 9
C. 11
D. 13

Answer

D

Explanation

Since `alpha, beta, gamma, sigma` are the roots of the given equation. `therefore" "x^(4)+4x^(3)-6x^(2)+7x-9=(x-alpha)(x-beta)(x-gamma)(x-sigma)`
Putting x = I and then x = -I, we get
`1-4i+6+7i-9=(i-alpha)(i-beta)(i-gamma)(i-sigma)`
and, `1+4i+6-7i-9=(-i-alpha)(-i-beta)(-i-gamma)(-i-sigma)`
Multiplying these two, we get
`(-2+3i)(-2-3i)=(1+alpha^(2))(1+beta^(2))(1+gamma^(2))(1+sigma^(2))`
`rArr" "13 = (1+alpha^(2))(1+beta^(2))(1+gamma^(2))(1+sigma^(2))`