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Question

# Let λ and α be real. Let S denote the set of all values of λ for which the system of linear equationsλx+(sinα)y+(cosα)z=0x+(cosα)y+(sinα)z=0−x+(sinα)y−(cosα)z=0has a non-trivial solution then S contains

A
(1,1)
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B
[2,1]
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C
[1,2]
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D
(2,2)
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Solution

## The correct options are A (−1,1) B [−√2,−1] C [1,√2]The given system of linear equations will have a non-trivial solution if∣∣ ∣∣λsinαcosα1cosαsinα−1sinα−cosα∣∣ ∣∣=0Expanding the determinant along C1 we getλ(−cos2α−sin2α)−(−sinαcosα−sinαcosα)−(sin2α−cos2α)=0⇒−λ+sin2α+cos2α=0⇒λ=sin2α+cos2α=√2sin(π/4+2α)⇒−√2≤λ≤√2Hence, possible options are A,B and C.

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