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  • Let denote the set of all real values of <img alt="\lambda" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clambda" /

Let S denote the set of all real values of \lambda such that the system of equations
$$ \begin{aligned} & \lambda x+y+z=1 \\ & x+\lambda y+z=1 \\ & x+y+\lambda z=1 \end{aligned}
is inconsistent, then  \sum_{\lambda \in S}\left(|\lambda|^2+|\lambda|\right)  is equal to

Option: 1

4


Option: 2

12


Option: 3

6


Option: 4

2


Answers (1)

best_answer

Given system of equation is inconsistent

$$ \begin{aligned} & \Rightarrow \Delta=0 \\ & \left|\begin{array}{lll} \lambda & 1 & 1 \\ 1 & \lambda & 1 \\ 1 & 1 & \lambda \end{array}\right|=0 \\ & \Rightarrow \lambda^3-3 \lambda+2=0 \\ & \Rightarrow(\lambda-1)^2(\lambda+2)=0 \\ & \Rightarrow \lambda=1,-2 \\ & \end{aligned}
But for \lambda=1 all planes are same
Then \lambda=-2
$$ \sum_{\lambda \in s}\left(|\lambda|^2+|\lambda|\right)=4+2=6

Posted by

jitender.kumar

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