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

Let $b_1=\widehat{i}+2\widehat{j}+\widehat{k}$   and   $b_2=4\widehat{i}+5\widehat{j}-3\widehat{k}$
Normal vector to plane
 $$\begin{gathered} b_1\times b_2=\left|\begin{array}{ccc}\widehat{i}& \widehat{j}& \widehat{k}\\ 1& 2& 1\\ 4& 5& -3\end{array}\right| \\ =\widehat{i}(-6-5)-\widehat{j}(-3-4)+\widehat{k}(5-8) \\ =-11\widehat{i}+7\widehat{j}-3\widehat{k} \end{gathered}$$
$\therefore$   Direction cosines of normal to the plane
 $=\frac{-11}{\sqrt{{(-11)}^2+7^2+{(-3)}^2}},\frac{7}{\sqrt{{(-11)}^2+7^2+{\left(3\right)}^2}},\frac{-3}{\sqrt{{(-11)}^2+7^2+{\left(3\right)}^2}}$
 $=\frac{-11}{\sqrt{179}},\frac{7}{\sqrt{179}},\frac{-3}{\sqrt{179}}$