Explanation

(d)
Given, $\left( \overrightarrow{\mathbf{a}} - \lambda\ \overrightarrow{\mathbf{b}} \right) \bullet \left( \overrightarrow{\mathbf{b}} - 2\overrightarrow{\mathbf{c}} \right) \times \left( \overrightarrow{\mathbf{c}} + 2\overrightarrow{\mathbf{a}} \right) = 0$
$\Longrightarrow \left( \overrightarrow{\mathbf{a}} - \lambda\ \overrightarrow{\mathbf{b}} \right) \bullet \left\{ \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{b}} \times 2\overrightarrow{\mathbf{a}} - 4\left( \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} \right) \right\} = 0$
$\Longrightarrow \ \overrightarrow{\mathbf{a}} \bullet \left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right) + \overrightarrow{\mathbf{a}}.\left( \overrightarrow{\mathbf{b}} \times 2\overrightarrow{\mathbf{a}} \right) - \overrightarrow{\mathbf{a}}.4\left( \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} \right)$
$- \lambda\overrightarrow{\mathbf{b}} \bullet \left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right) - \lambda\overrightarrow{\mathbf{b}} \bullet \left( \overrightarrow{\mathbf{b}} \times 2\overrightarrow{\mathbf{a}} \right) + 4\lambda\ \overrightarrow{\mathbf{b}} \bullet \left( \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} \right) = 0$
$\Longrightarrow \overrightarrow{\mathbf{a}}\left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right) + 4\lambda\overrightarrow{\mathbf{b}}.\left( \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} \right) = 0$
$\Longrightarrow \left\{ \overrightarrow{\mathbf{a}} \bullet \left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right) \right\}(1 + 4\ \lambda) = 0$
$\Longrightarrow \ \lambda = - \frac{1}{4}\ \ \left\lbrack \because\ \overrightarrow{\mathbf{a}}.\left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right) \neq 0,given \right\rbrack$