Explanation

(d)
Since, given vectors are coplanar so it can be written as
$\overrightarrow{\mathbf{a}} + \lambda\ \overrightarrow{\mathbf{b}} + 3\overrightarrow{\mathbf{c}} = x\left( - 2\overrightarrow{\mathbf{a}} + 3\overrightarrow{\mathbf{b}} - 4\overrightarrow{\mathbf{c}} \right) + y\left( \overrightarrow{\mathbf{a}} - 3\overrightarrow{\mathbf{b}} + 5\overrightarrow{\mathbf{c}} \right)$
On comparing the coefficient of $\overrightarrow{\mathbf{a}},\ \overrightarrow{\mathbf{b}}\$and $\overrightarrow{\mathbf{c}}$ on both sides, we get
$- 2x + y = 1;\ \ 3x - 3y = \lambda\$and $- 4x + 5y = 3$
On solving, we get
$x = - \frac{1}{3},\ y = \frac{1}{3},\ \lambda = - 2$