The number of real values $\mathrm{\lambda }$, such that the system of linear equations

$\begin{array}{l}2\mathrm{x}-3\mathrm{y}+5\mathrm{z}=9\\ \mathrm{x}+3\mathrm{y}-\mathrm{z}=-18\\ 3\mathrm{x}-\mathrm{y}+\left({\mathrm{\lambda }}^2-\left|\mathrm{\lambda }\right|\right)\mathrm{z}=16\end{array}$

has no solution, is :-

Augmented matrix is

$$\begin{gathered} \left|\begin{array}{ccc}1& 3& -1\\ 2& -3& 5\\ 3& -1& {\lambda }^2-\left|\lambda \right|\end{array}\right|=0 \\ {R}_3\Rightarrow R_3-3\left(R_1\right),R_2\Rightarrow R_2-2\left(R_1\right)= \\ \left|\begin{array}{ccc}1& 3& -1\\ 0& -9& 7\\ 0& -10& {\lambda }^2-\left|\lambda \right|+3\end{array}\right|=0 \\ -9\left({\lambda }^2-\left|\lambda \right|+3\right)+70=0 \\ 9{\lambda }^2+9\left|\lambda \right|-63=0 \\ {\lambda }^2+\left|\lambda \right|-7=0 \\ \left|\lambda \right|=\frac{-1+\sqrt{29}}{2}\Rightarrow \lambda has 2 values \end{gathered}$$