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Question

# Let λ and α be real. Find the set of all values of λ for which the system of linear equationλx+(sinα)y+(cosα)z=0x+(cosα)y+(sinα)z=0−x+(sinα)y−(cosα)z=0 has a non-trivial solution. For λ=1, find all values of α

A
λ=sin2α+cos2α;α=mπ or α=kπ+π/4(m,kεI)
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B
λ=sinα+cosα;α=mπ or α=kπ+π/4(m,kεI)
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C
λ=sinα+cos2α;α=mπ or α=kπ+π/4(m,kεI)
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D
λ=sin2α+cosα;α=mπ or α=kπ+π/4(m,kεI)
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Solution

## The correct option is A λ=sin2α+cos2α;α=mπ or α=kπ+π/4(m,kεI)Given λx+(sinα)y+(cosα)z=0,x+(cosα)y+(sinα)z=0 and −x+(sinα)y−(cosα)z=0has non trivial solution∴△=0⇒∣∣ ∣∣λsinαcosα1cosαsinα−1sinα−cosα∣∣ ∣∣=0⇒λ(−cos2α−sin2α)−sinα(−cosα+sinα)+cosα(sinα+cosα)⇒−λ+sinαcosα+sinαcosα−sin2α+cos2α=0⇒λ=cos2α+sin2α(∵−√a2+b2≤asinθ+bcosθ≤√a2+b2)∴−√2≤λ≤√2 ...(1)When λ=1 cos2α+sin2α=1⇒1√2cos2α+1√2sin2α=1√2⇒cos(2α−π4)=cos(π4)∴2α−π4=2nπ±π4⇒2α=2nπ−π4+π4 and 2α=2nπ+π4+π4∴α=nπ and α=nπ+π4

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