If `alpha` and `beta` be the roots of equation `x^(2) + 3x + 1 = 0` then the value of `((alpha)/(1 + beta))^(2) + ((beta)/(1 + alpha))^(2)` is equal to

Options

A. `18`
B. `19`
C. `20`
D. `21`

Answer

A

Explanation

`(a)` `alpha+beta=-3` , `alpha beta=1`.
Also `alpha^(2)+3alpha+1=0` and `beta^(2)+3beta+1=0`
`implies` where `alpha^(2)=-(3alpha+1)` and `beta^(2)=-(3beta+1)`
`y=(alpha^(2))/((1+beta)^(2))+(beta^(2))/((alpha+1)^(2))`
`= (alpha^(2))/(1+2beta+beta^(2))+(beta^(2))/(1+2alpha+alpha^(2))`
`=((-(3alpha+1))/(-beta))+((-(1+3beta))/(-alpha))`
`=(1+3alpha)/(beta)+(1+3beta)/(alpha)`
`=(alpha(1+3alpha)+beta(1+3beta))/(alpha beta)`
`=3(alpha^(2)+beta^(2))+(alpha+beta)` (as `alpha beta=1`)
`=3[9-2]+(-3)=21-3=18`