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Let \lambda \; \epsilon\; R. the system of linear equations  2x_{1}-4x_{2}+\lambda x_{3}=1 x_{1}-6x_{2}+ x_{3}=2 \lambda x_{1}-10x_{2}+4 x_{3}=3 is inconsistent for :
Option: 1 exactly one negative value of \lambda.
Option: 2 exactly one positive value of \lambda.
Option: 3 every value of \lambda.
Option: 4 exactly two value of \lambda.

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\begin{array}{l} \mathrm{D}=\left|\begin{array}{ccc} 2 & -4 & \lambda \\ 1 & -6 & 1 \\ \lambda & -10 & 4 \end{array}\right| =2(3 \lambda+2)(\lambda-3) \end{array}

\begin{array}{l} \mathrm{D}_{1}=-2(\lambda-3) \\ \mathrm{D}_{2}=-2(\lambda+1)(\lambda-3) \\ \mathrm{D}_{3}=-2(\lambda-3) \end{array}

\begin{aligned} &\text { When } {\lambda=3}, \text { then }\\ &\mathrm{D}=\mathrm{D}_{1}=\mathrm{D}_{2}=\mathrm{D}_{3}=0\\ &\Rightarrow \text { Infinite many solution } \end{aligned}

\begin{aligned} &\text { when }\lambda=-\frac{2}{3} \text { then } \mathrm{D}_{1}, \mathrm{D}_{2}, \mathrm{D}_{3}. \text { none of them is zero so equations are inconsistant }\\ &\therefore \lambda=-\frac{2}{3} \end{aligned}

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