Given, equation of plane
$r=\left(2\widehat{i}+\widehat{k}\right)+\lambda \widehat{i}+\mu \left(\widehat{i}+2\widehat{j}-3\widehat{k}\right)\dots$ (i)
$\mathrm{H}\mathrm{e}\mathrm{r}\mathrm{e}$ , plane (i) passing through a (let) $=2\widehat{i}+\widehat{k}$ and parallel to vector b (let) $=\widehat{i}$ and $c=\widehat{i}+2\widehat{j}-3\widehat{k}$
We know that equation of plane passing through a point a and parallel to non-parallel vectors b and c is $r.\left(b\times c\right)=a.\left(b\times c\right)=\left[abc\right]$
Now, $\left[abc\right]=\left|\begin{array}{ccc}2& 0& 1\\ 1& 0& 0\\ 1& 2& -3\end{array}\right|$
$=2\left(0\right)-0+1\left(2-0\right)=2$
and $b\times c=\left|\begin{array}{ccc}\widehat{i}& \widehat{j}& \widehat{k}\\ 1& 0& 0\\ 1& 2& -3\end{array}\right|=3\widehat{i}+2\widehat{k}$
$\therefore r\left(3\widehat{i}+2\widehat{k}\right)=2$
$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}$ , $\alpha =2$