Explanation

(d)
$\left( 3\overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}} \right) \bullet \left( \overrightarrow{\mathbf{a}} - 4\overrightarrow{\mathbf{b}} \right) = 3\ \left| \overrightarrow{\mathbf{a}} \right|^{2} - 11\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} - 4\left| \overrightarrow{\mathbf{b}} \right|^{2}$
$= 3 \bullet 36 - 11 \bullet 6 \bullet 8\cos{\pi - 4 \bullet 64 > 0}$
$\therefore$ Angle between $\overrightarrow{\mathbf{a}}\ and\ \overrightarrow{\mathbf{b}}$ is acute angle.
$\therefore$ The longer diagonal is given by
$\overrightarrow{\alpha} = \left( 3\overrightarrow{\mathbf{a}}\mathbf{+}\overrightarrow{\mathbf{b}} \right) + \left( \overrightarrow{\mathbf{a}} - 4\overrightarrow{\mathbf{b}} \right) = 4\overrightarrow{\mathbf{a}} - 3\overrightarrow{\mathbf{b}}$
Now, $\left| \overrightarrow{\alpha} \right|^{2} = \left| 4\overrightarrow{\mathbf{a}} - 3\overrightarrow{\mathbf{b}} \right|^{2}$
$= 16\left| \overrightarrow{\mathbf{a}} \right|^{2} + 9\left| \overrightarrow{\mathbf{b}} \right|^{2} - 24\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}}$
$= 16 \bullet 36 + 9 \bullet 64 - 24 \bullet 6 \bullet 8\cos\pi$
$= 16 \times 144$
$\left| 4\overrightarrow{\mathbf{a}} - 3\overrightarrow{\mathbf{b}} \right| = 48$