A plane passing through the point

$\because$ Plane contains two lines

$\therefore \vec{n}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & 2 \\ 2 & 3 & -1\end{array}\right|$

$=i(2-6)-j(-1-4)+k(3+4)=-4 i+5 j+7 k$

So, equation of plane is

$-4(x-3)+5(y-1)+7(z-1)=0$

$\Rightarrow-4 x+12+5 y-5+7 z-7=0$

$\Rightarrow-4 x+5 y+7 z=0$

This also passes through $(\alpha,-3,5)$

So, $-4 \alpha-15+35=0$

$\Rightarrow-4 \alpha=-20 \Rightarrow \alpha=5$