Let the direction ratio of the line be, $\mathrm{a}, \mathrm{b}, \mathrm{c}$
This line is contained by both plane,
$3 x+y+2 z=7$ and $x+2 y+3 z=5$
$\Rightarrow 3 \mathrm{a}+\mathrm{b}+2 \mathrm{c}=0$
and $a+2 b+3 c=0$
Solving these two equations
$\frac{a}{\left|\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right|}=\frac{-b}{\left|\begin{array}{ll}3 & 2 \\ 1 & 3\end{array}\right|}=\frac{c}{\left|\begin{array}{ll}3 & 1 \\ 1 & 2\end{array}\right|}=k$
Let
$\Rightarrow \frac{\mathrm{a}}{-1}=\frac{-\mathrm{b}}{7}=\frac{\mathrm{c}}{5}=\mathrm{k}$
$\mathrm{a}=-\mathrm{k}, \mathrm{b}=-7 \mathrm{k}, \mathrm{c}=5 \mathrm{k}$
Direction cosines are
$\frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}}, \frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}}, \frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}}$
$\sqrt{a^{2}+b^{2}+c^{2}}=\sqrt{(-k)^{2}+(-7 k)^{2}+(5 k)^{2}}$
$=\sqrt{\mathrm{k}^{2}+49 \mathrm{k}^{2}+25^{2}}=\mathrm{k} \sqrt{75}$
So, Direction cosines are $\left(\frac{-\mathrm{k}}{\mathrm{k} \sqrt{75}}, \frac{-7 \mathrm{k}}{\mathrm{k} \sqrt{75}}, \frac{5 \mathrm{k}}{\mathrm{k} \sqrt{75}}\right)$
$=\left(-\frac{1}{\sqrt{75}},-\frac{7}{\sqrt{75}}, \frac{5}{\sqrt{75}}\right)$