We have,
$2 \mathbf{a}+3 \mathbf{b}-\mathbf{c}, \mathbf{a}-2 \mathbf{b}+3 \mathbf{c}, 3 \mathbf{a}+\lambda \mathbf{b}-2 \mathbf{c}$ and $\mathbf{a}-\mathbf{6 b}+\mathbf{6} \mathbf{c}$ are coplanar.
$\therefore \quad\left|\begin{array}{ccc}1-2 & -2-3 & 3+1 \\ 3-2 & \lambda-3 & -2+1 \\ 1-2 & -6-3 & 6+1\end{array}\right|=0$
$\Rightarrow \quad\left|\begin{array}{ccc}-1 & -5 & 4 \\ 1 & \lambda-3 & -1 \\ -1 & -9 & 7\end{array}\right|=0$

$$
\begin{array}{lccc}
\Rightarrow & -1 & (7 \lambda-21-9)+5(7-1)+4(-9+\lambda-3)=0 \\
\Rightarrow & -7 \lambda+30+30-48+4 \lambda=0 \\
\Rightarrow & & 3 \lambda=12 \\
\Rightarrow & \lambda=4
\end{array}
$$
Direction cosine of $\lambda \hat{\mathbf{i}}-2 \lambda \hat{\mathbf{j}}+\hat{\mathbf{k}}$ is $\frac{\lambda}{\sqrt{5 \lambda^2+1}}, \frac{-2 \lambda}{\sqrt{5 \lambda^2+1}}, \frac{1}{\sqrt{5 \lambda^2+1}}$
Put $\lambda=4$
$\therefore$ Direction cosine is $\frac{4}{9}, \frac{8}{9}, \frac{1}{9}$.