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Question

# The set of all values of λ for which the system of linear equation : 2x1−2x2+x3=λx1,2x1−3x2+2x3=λx2−x1+2x2=λx3 has a non-trivial solution

A
contains two elements
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B
contains more than two elements
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C
is an empty set
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D
is a singleton
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Solution

## The correct option is A contains two elements(2−λ)x1−2x2+x3=02x1−(3+λ)x2+2x3=0−x1+2x2−λx3=0Non-Trivial solution △=0∣∣ ∣∣2−λ−212−3−λ2−12−λ∣∣ ∣∣=0⇒(2−λ){3λ+λ2−4}+2[−2λ+2}+(4−3−λ)=0⇒(6λ+2λ2−8−3λ2−λ3+4λ)−4λ+4+1−λ=0⇒−λ3−λ2−5λ+3=0x3−λ2+2λ2−2λ−3λ+3=0λ2(λ−1)+2λ(λ−1)−3(λ−1)=0(λ−1)=(λ2+2λ−3)=0(λ−1)(λ+3)(λ−1)=0λ=1,1,−3Therefore, thus are only Two elements.

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