The direction cosines of the normal to the plane containing the lines having direction ratios 1, 2, 1 and 4, 5, -3 are

Let $\overline{a}=\overline{i}+2\overline{j}+\overline{k}$
$\overline{b}=4\overline{i}+5\overline{j}-3\overline{k}$ 
$\overline{a}\times \overline{b}=\left|\begin{array}{ccc}\overline{i}& \overline{j}& \overline{k}\\ 1& 2& 1\\ 4& 5& -3\end{array}\right|$ 
$=\overline{i}\left(-11\right)-\overline{j}\left(-3-4\right)+\overline{k}\left(5-8\right)$ 
$\frac{\overline{a}\times \overline{b}}{\left|\overline{a}\times \overline{b}\right|}=\frac{-11\overline{i}+7\overline{j}-3\overline{k}}{\sqrt{121+49+9}}=\frac{-11\overline{i}+7\overline{j}-3\overline{k}}{\sqrt{179}}$
$\therefore$ DC’s  of normal $\left(\frac{-11}{\sqrt{179}},\frac{7}{\sqrt{179}},\frac{-3}{\sqrt{179}}\right)$