Explanation

(c)
Given, $\overrightarrow{\mathbf{a}} = \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}},\overrightarrow{\mathbf{b}} = \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}} + 2\widehat{\mathbf{k}}$
$and\ \overrightarrow{\mathbf{c}} = x\ \widehat{\dot{\mathbf{i}}} + (x - 2)\widehat{\dot{\mathbf{j}}} - \widehat{\mathbf{k}}$
Since, $\overrightarrow{\mathbf{c}}$ lies in the plane of vectors $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ therefore $\overrightarrow{\mathbf{a}},\ \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ are coplanar.
$\therefore\ \left| \begin{matrix}
1 & 1 & 1 \\
1 & - 1 & 2 \\
x & (x - 2) & - 1
\end{matrix} \right| = 0$
$\Longrightarrow 1(1 - 2x + 4) - 1( - 1 - 2x) + 1(x - 2 + x) = 0$
⟹ $5 - 2x + 1 + 2x + 2x - 2 = 0$
⟹ $x = - 2$