If `cos^4 alpha +k and sin^4 alpha+k` are the roots of `x^2+lambda(2 x+1)=0 and sin^2 alpha+1 and cos^2 alpha +1` are the roots of `x^2 +8 x+4=0,` then the sum of the possible values of `lambda` is

Options

A. `2`
B. `-1`
C. `1`
D. `3`

Answer

C

Explanation

`(c )` For the `1^(st)` equation `(cos^(4)alpha+k)-(sin^(4)alpha+k)=sqrt(4lambda^(2)-4lambda)`
i.e., `cos^(4)alpha-sin^(4)alpha=sqrt(4lambda^(2)-4lambda)`
For the `2^(nd)` equation `cos^(2)alpha-sin^(2)alpha=sqrt(64-16)=sqrt(48)`
But `cos^(4)alpha-sin^(4)alpha=(cos^(2)alpha-sin^(2)alpha)(cos^(2)alpha+sin^(2)alpha)`
`=cos^(2)alpha-sin^(2)alpha`
`implies 4lambda^(2)-4lambda=48`
`implies lambda=4` or `-3`
Sum `=1`