Explanation

(a)
$\left| \overrightarrow{\mathbf{p}} + \overrightarrow{\mathbf{q}} \right| = 6$
${\Longrightarrow \ \ \ \left| \overrightarrow{\mathbf{p}} + \overrightarrow{\mathbf{q}} \right|}^{2} = 36$
$\Longrightarrow \ p^{2} + q^{2} + 2\ \overrightarrow{\mathbf{p}} \bullet \overrightarrow{\mathbf{q}} = 36$
Similarly, $q^{2} + r^{2} + 2\ \overrightarrow{\mathbf{q}} \bullet \overrightarrow{\mathbf{r}} = 48$
and $r^{2} + p^{2} + 2\ \overrightarrow{\mathbf{r}} \bullet \overrightarrow{\mathbf{p}} = 16$
adding all, we get
$2\left( p^{2} + q^{2} + r^{2} + \overrightarrow{\mathbf{p}} \bullet \overrightarrow{\mathbf{q}} + \overrightarrow{\mathbf{q}} \bullet \overrightarrow{\mathbf{r}} + \overrightarrow{\mathbf{r}} \bullet \overrightarrow{\mathbf{p}} \right)$
$\Longrightarrow 2\left( p^{2} + q^{2} + r^{2} \right) = 100\ \ \ \left( \because\overrightarrow{\mathbf{p}} \bullet \overrightarrow{\mathbf{q}} + \overrightarrow{\mathbf{q}} \bullet \overrightarrow{\mathbf{r}} + \overrightarrow{\mathbf{r}} \bullet \overrightarrow{\mathbf{p}} = 0 \right)$
$\Longrightarrow p^{2} + q^{2} + r^{2} = 50$
$\Longrightarrow |\overrightarrow{\mathbf{p}} + \overrightarrow{\mathbf{q}} + \overrightarrow{\mathbf{r}}|^{2} = 50$
$\Longrightarrow |\overrightarrow{\mathbf{p}} + \overrightarrow{\mathbf{q}} + \overrightarrow{\mathbf{r}}| = 5\sqrt{2}$