Explanation

(d)
$\because\ |\overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}}\mathbf{\ }\left| \mathbf{=} \right|\overrightarrow{\mathbf{a}}\mathbf{-}\overrightarrow{\mathbf{b}}|$
$\Rightarrow \ |\ \overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}}\left| \ ^{2} = \right|\overrightarrow{\mathbf{a}}\mathbf{-}\overrightarrow{\mathbf{b}}\ \left. \ \right|^{2}$
$\Rightarrow a^{2} + b^{2} + 2\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} = a^{2} + b^{2} - 2\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}}$
$\Rightarrow 4\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} = 0$
$\Rightarrow \ \overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} = 0$
$\therefore$ Angle between $\overrightarrow{\mathbf{a}}\ and\ \overrightarrow{\mathbf{b}}$ is $\frac{\pi}{2}$.