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Question

The set of all values of λ for which the system of linear equations 2x12x2+x3=λx1,2x13x2+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
2x12x2+x3=λx1
(2λ)x12x2+x3=0 …(i)
2x13x2+2x3=λx2
2x1(3+λ)x2+2x3=0 …(ii)
x1+2x2=λx3
x1+2x2λx3=0 …(iii)
Since, the system has non - trival solution.
∣ ∣2λ212(3+λ)212λ∣ ∣=0
(2λ)(3λ+λ24)+2(2λ+2)+1(43)λ=0
(2λ)(λ2+3λ4)+4(1λ)+(1λ)=0
(2λ)(λ+4)(λ1)+5(1λ)=0
(λ1)[(2λ)(λ+4)5]=0
(λ1)(λ2+2λ3)=0
(λ1)[(λ1)(λ+3)]=0
(λ1)2(λ+3)=0
λ=1,1,3
Hence, λ contains two elements.

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