Official solution
If \[\theta \] be the angle between the given line and plane, then
\[\sin \theta = {{1 \times 1 + 2 \times 2 + \lambda \times 3} \over {\sqrt {{1^2} + {2^2} + {\lambda ^2}} .\sqrt {{1^2} + {2^2} + {3^2}} }}\]
\[ = {{5 + 3\lambda } \over {\sqrt {14} .\sqrt {5 + {\lambda ^2}} }}\]
But it is given that
\[\theta = {\cos ^{ - 1}}\sqrt {{5 \over {14}}} \Rightarrow \sin \theta = {3 \over {\sqrt {14} }}\]
\[\therefore\] \[{{5 + 3\lambda } \over {\sqrt {14} \sqrt {5 + {\lambda ^2}} }} = {3 \over {\sqrt {14} }} \Rightarrow \lambda = {2 \over 3}\]