$\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ \alpha & 2 \alpha & 3 \\ 1 & 3 \alpha & 5\end{array}\right|$

$$ \begin{aligned} &=1(10 \alpha-9 \alpha)-1(5 \alpha-3)+1\left(3 \alpha^{2}-2 \alpha\right) \\\\ &=\alpha-5 \alpha+3+3 \alpha^{2}-2 \alpha \\\\ &=3 \alpha^{2}-6 \alpha+3 \end{aligned} $$

For inconsistency $\Delta=0$ i.e. $\alpha=1$

Now check for $\alpha=1$

$$ x+y+z=1\quad\quad...(i) $$

$x+2 y+3 z=-1\quad\quad...(ii)$

$x+3 y+5 z=4\quad\quad...(iii)$

By (ii) $\times 2-$ (i) $\times 1$

$$ x+3 y+5 z=-3 $$

so equations are inconsistent for $\alpha=1$