The number of values of α for which the system of equations $\begin{array}{l}\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{\alpha },\mathrm{\alpha x}+2\mathrm{\alpha y}+3\mathrm{z}=-1,\mathrm{x}+3\mathrm{\alpha y}+5\mathrm{z}=4\end{array}$is inconsistent, is

$\begin{array}{l}\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{\alpha }\\ \mathrm{\alpha x}+2\mathrm{\alpha y}+3\mathrm{z}=-1\\ \mathrm{x}+3\mathrm{\alpha y}+5\mathrm{z}=4\end{array}$
Has inconsistent solution
$\begin{array}{l}\mathrm{D}=\left|\begin{array}{ccc}1& 1& 1\\ \mathrm{\alpha }& 2\mathrm{\alpha }& 3\\ 1& 3\mathrm{\alpha }& 5\end{array}\right|=0\\ \Rightarrow 1\left(10\alpha -9\alpha \right)-1\left(5\alpha -3\right)+1\left(3{\alpha }^2-2\alpha \right)=0\Rightarrow 3{\alpha }^2-6\alpha +3=0\\ \Rightarrow {\alpha }^2-2\alpha +1=0\\ \Rightarrow (\mathrm{\alpha }-1{)}^2=0\Rightarrow \mathrm{\alpha }=1\end{array}$
For  $\alpha =1$
$\begin{array}{l}{\mathrm{D}}_1=\left|\begin{array}{ccc}1& 1& 1\\ -1& 2& 3\\ 4& 3& 5\end{array}\right|\\ =(10-9)-(-5-12)+(-3-8)\\ =1+17-11\ne 0\end{array}$
For α  = 1 the system of equation has Inconsistent solution