Explanation

(d)
Since, $\overrightarrow{\mathbf{a}} \bullet \left( \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} \right) = 0$
$\Longrightarrow \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{c}} = 0$
Similarly, $\overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{a}} = 0$
$and\ \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{b}} = 0$
$\therefore\ \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} = 0$ …(i)
Given, |$\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}}| = 6$
$\Longrightarrow |{\overrightarrow{\mathbf{a}}|}^{2} + |{\overrightarrow{\mathbf{b}}|}^{2} + 2\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} = 36$ ….(ii)
Similarly, $|{\overrightarrow{\mathbf{b}}|}^{2} + |{\overrightarrow{\mathbf{c}}|}^{2} + 2\overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} = 64$ …(iii)
$and\ |{\overrightarrow{\mathbf{c}}|}^{2} + |{\overrightarrow{\mathbf{a}}|}^{2} + 2\overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}} = 100\$….(iv)
On adding Eqs. (ii),(iii) and (iv), we get
$2\left| {\overrightarrow{\mathbf{a}}|}^{2} + 2 \right|{\overrightarrow{\mathbf{b}}|}^{2} + 2|{\overrightarrow{\mathbf{c}}|}^{2} + 2\left( \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}} \right) = 200$
$\Longrightarrow {|\overrightarrow{\mathbf{a}}|}^{2} + |{\overrightarrow{\mathbf{b}}|}^{2} + |{\overrightarrow{\mathbf{c}}|}^{2} = 100$…(v)[from Eqs. (i)]
Now, $|\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}}|^{2} = \left| \overrightarrow{\mathbf{a}} \right|^{2} + |{\overrightarrow{\mathbf{b}}|}^{2} + |{\overrightarrow{\mathbf{c}}|}^{2} + 2$($\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}})$
$\Longrightarrow |\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}}|^{2} = 100$ [from Eqs. (i) and (v)]
$\Longrightarrow |\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}}|^{\ } = 10$