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Question

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

A
is an empty set
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B
is a singleton set
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C

contains two elements

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D
contains more than two elements
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Solution

## The correct option is C contains two elements Given system of linear equations 2x1−2x2+x3=λx1 ⇒(2−λ)x1−2x2+x3=0 …(i) 2x1−3x2+2x3=λx2 ⇒2x1−(3+λ)x2+2x3=0 …(ii) −x1+2x2=λx3 ⇒−x1+2x2−λx3=0 …(iii) Since, the system has non - trival solution. ∴∣∣ ∣∣2−λ−212−(3+λ)2−12−λ∣∣ ∣∣=0 ⇒(2−λ)(3λ+λ2−4)+2(−2λ+2)+1(4−3)−λ=0 ⇒(2−λ)(λ2+3λ−4)+4(1−λ)+(1−λ)=0 ⇒(2−λ)(λ+4)(λ−1)+5(1−λ)=0 ⇒(λ−1)[(2−λ)(λ+4)−5]=0 ⇒(λ−1)(λ2+2λ−3)=0 ⇒(λ−1)[(λ−1)(λ+3)]=0 ⇒(λ−1)2(λ+3)=0 ⇒λ=1,1,−3 Hence, λ contains two elements.

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