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Question

# Let S denote the set of all values of λ for which the system of linear equationsλx+(sinα)y+(cosα)z=0,x+(cosα)y+(sinα)z=0,−x+(sinα)y−(cosα)z=0,has a non-trivial solution ∀α∈R, 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]For non-trivial solution, ∣∣ ∣∣λsinαcosα1cosαsinα−1sinα−cosα∣∣ ∣∣=0Expanding along C1⇒λ(−cos2α−sin2α)−1(−sinαcosα−sinαcosα)−1(sin2α−cos2α)=0⇒−λ+sin2α+cos2α=0⇒λ=sin2α+cos2α=√2sin(π4+2α)Since, −1≤sin(π4+2α)≤1⇒−√2≤λ≤√2

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