Explanation

(b)
The line is $\frac{x + 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{2}$ and the plane is $2x - y + \sqrt{\lambda}\ z + 4 = 0$
If $\theta$ be the angle between the line and the plane, then $90{^\circ} - \theta$ is the angle between the line and normal to the plane
$\Rightarrow \cos(90{^\circ} - \theta) = \frac{(1)(2) + (2)( - 1) + (2)\left( \sqrt{\lambda} \right)}{\sqrt{1 + 4 + 4}\sqrt{4 + 1 + \lambda}}$
$\Rightarrow \sin\theta = \frac{2 - 2 + 2\sqrt{\lambda}}{3\sqrt{5 + \lambda}}\ \Rightarrow \frac{1}{3} = \frac{2\sqrt{\lambda}}{3\sqrt{5 + \lambda}}$
$\Rightarrow \sqrt{5 + \lambda} = 2\sqrt{\lambda}$
$\Rightarrow 5 + \lambda = 4\lambda$
$\Rightarrow 3\lambda = 5$
$\Rightarrow \ \lambda = \frac{5}{3}$