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,` the

Question

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,` the

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

Monjur Alahi

6 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - C
`(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`