Explanation

(d)
Since, $\overrightarrow{\mathbf{a}}\mathbf{=}\widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}}\mathbf{+}\widehat{\mathbf{k}}\mathbf{,\ }\overrightarrow{\mathbf{b}} = 2\widehat{\mathbf{i}} - 4\widehat{\mathbf{k}},\ \overrightarrow{\mathbf{c}} = \widehat{\mathbf{i}} + \lambda\widehat{\mathbf{j}} + 3\widehat{\mathbf{k}}$ are coplanar.
$\therefore\left\lbrack \overrightarrow{\mathbf{a}}\mathbf{\ \ }\overrightarrow{\mathbf{b}}\mathbf{\ }\overrightarrow{\mathbf{c}}\ \right\rbrack = 0\ \Rightarrow \ \left| \begin{matrix}
1 & 1 & 1 \\
2 & 0 & - 4 \\
1 & \lambda & 3
\end{matrix} \right| = 0$
$\Rightarrow 4\lambda - 1(6 + 4) + 2\lambda = 0$
$\Rightarrow 6\lambda = 10\ \Rightarrow \ \lambda = \frac{5}{3}$