Let $\lambda \in \mathrm{R}$. The system of linear equations
$2 x_{1}-4 x_{2}+\lambda x_{3}=1$
$x_{1}-6 x_{2}+x_{3}=2$
$\lambda x_{1}-10 x_{2}+4 x_{3}=3$
is inconsistent for :
Correct Option: 1
$D=\left|\begin{array}{ccc}2 & -4 & \lambda \\ 1 & -6 & 1 \\ \lambda & -10 & 4\end{array}\right|$
$=2(3 \lambda+2)(\lambda-3)$
$\mathrm{D}_{1}=-2(\lambda-3)$
$\mathrm{D}_{2}=-2(\lambda+1)(\lambda-3)$
$\mathrm{D}_{3}=-2(\lambda-3)$
When $\lambda=3$, then
$\mathrm{D}=\mathrm{D}_{1}=\mathrm{D}_{2}=\mathrm{D}_{3}=0$
$\Rightarrow$ Infinite many solution
when $\lambda=-\frac{2}{3}$ then $\mathrm{D}_{1}, \mathrm{D}_{2}, \mathrm{D}_{3}$ none of them
is zero so equations are inconsistant
$\therefore \lambda=-\frac{2}{3}$
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