Explanation

(b)
Given, $\overrightarrow{\mathbf{a}} + 2\overrightarrow{\mathbf{b}} + 4\overrightarrow{\mathbf{c}} = \overrightarrow{\mathbf{0}}$
Now, $\overrightarrow{\mathbf{a}} \times \left( \overrightarrow{\mathbf{a}} + 2\overrightarrow{\mathbf{b}} + 4\overrightarrow{\mathbf{c}} \right) = \overrightarrow{\mathbf{0}}$
$\Longrightarrow 2\left( \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} \right) + 4\left( \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{c}} \right) = \overrightarrow{\mathbf{0}}$
$\Longrightarrow \ \frac{\left( \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} \right)}{4} = \frac{\left( \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} \right)}{2}\ \ \ \ \ \ldots.(i)$
Again, $\overrightarrow{\mathbf{b}} \times (\overrightarrow{\mathbf{a}} + 2\overrightarrow{\mathbf{b}} + 4\overrightarrow{\mathbf{c}}$)=$\ \overrightarrow{\mathbf{0}}$
$\Longrightarrow \ \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}} + 4\left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right) = \overrightarrow{\mathbf{0}}$
$\Longrightarrow \ \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} = (\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})/4$ …(ii)
From Eqs. (i) and (ii)
$(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})/4 = \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} = (\overrightarrow{\mathbf{c}}$×$\overrightarrow{\mathbf{a}})/2 = \overrightarrow{\mathbf{p}}$
$\therefore\ \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} = 4\overrightarrow{\mathbf{p}},\ \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} = \overrightarrow{\mathbf{p}}$
$and\ \ \ \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} = 2\overrightarrow{\mathbf{p}}$
$\therefore\left( \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} \right) +$($\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}) + \ (\overrightarrow{\mathbf{c}}$×$\overrightarrow{\mathbf{a}})$=$4\overrightarrow{\mathbf{p}} + \overrightarrow{\mathbf{p}} + 2\overrightarrow{\mathbf{p}}$
$= 7\overrightarrow{\mathbf{p}} = 7(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})$
$\therefore\lambda = 7$