Let $\lambda \in R$, the system of equations 

$\begin{array}{l}\begin{array}{l}2x_1-4x_2+\lambda x_3=1,\\ x_1-6x_2+x_3=2,\\ \lambda x_1-10x_2+4x_3=3\end{array}\end{array}$

is consistent for 

The system of equations are inconsistent, if the determinant of the coefficient matrix is zero and at least one  of ${\Delta }_1,{\Delta }_2,{\Delta }_3$ will be non zero

$\begin{array}{l}\Delta =\left|\begin{array}{ccc}2& -4& \lambda \\ 1& -6& 1\\ \lambda & -10& 4\end{array}\right|=0\\ \Rightarrow 2\left(-24+10\right)+4\left(4-\lambda \right)+\lambda \left(-10+6\lambda \right)=0\\ \Rightarrow -28+16-4\lambda -10\lambda +6{\lambda }^2=0\\ \Rightarrow 6{\lambda }^2-14\lambda -12=0\\ \Rightarrow 3{\lambda }^2-7\lambda -6=0\\ \Rightarrow \left(\lambda -3\right)\left(3\lambda +2\right)=0\\ \Rightarrow \lambda =3,-\frac{2}{3}\end{array}$

$\begin{array}{l}{\Delta }_1=\left|\begin{array}{ccc}1& -4& \lambda \\ 2& -6& 1\\ 3& -10& 4\end{array}\right|=\left|\begin{array}{ccc}1& 0& \lambda \\ 2& 2& 1\\ 3& 2& 4\end{array}\right|=\left|\begin{array}{ccc}1& 0& \lambda \\ 2& 2& -3\\ 3& 2& 0\end{array}\right|=\left|\begin{array}{ccc}1& 0& \lambda \\ -1& 0& -3\\ 3& 2& 0\end{array}\right|=-2\left(\lambda -3\right)\\ {\Delta }_2=\left|\begin{array}{ccc}2& 1& \lambda \\ 1& 2& 1\\ \lambda & 3& 4\end{array}\right|=2\left(5\right)-1\left(4-\lambda \right)+\lambda \left(3-2\lambda \right)=-2{\lambda }^2+4\lambda +6\\ {\Delta }_3=\left|\begin{array}{ccc}2& -4& 1\\ 1& -6& 2\\ \lambda & -10& 3\end{array}\right|=-2\left|\begin{array}{ccc}2& 2& 1\\ 1& 3& 2\\ \lambda & 5& 3\end{array}\right|=-2\left[2\left(-1\right)-2\left(3-2\lambda \right)+1\left(5-3\lambda \right)\right]\end{array}$

observing the above values, option 4 is correct answer