Explanation

(c)
Given plane is $\overrightarrow{r} = (1 + \lambda - \mu)\widehat{i} + (2 - \lambda)\widehat{j} + (3 - 2\lambda + 2\mu)\widehat{k}$
$\Rightarrow \ \overrightarrow{r} = \left( \widehat{i} + 2\widehat{j} + 3\widehat{k} \right) + \lambda\left( \widehat{i} - \widehat{j} - 2\widehat{k} \right) + \mu( - \widehat{i} + 2\widehat{k})$
Which is a plane passing through $\overrightarrow{a} = \widehat{i} + 2\widehat{j} + 3\widehat{k}$ and parallel to the vectors $\overrightarrow{b} = \widehat{i} - \widehat{j} - 2\widehat{k}$ and $\overrightarrow{c} = - \widehat{i} + 2\widehat{k}$
Therefore, it is perpendicular to the vector $\overrightarrow{n} = \overrightarrow{b} \times \overrightarrow{c} = - 2\widehat{i} - \widehat{k}$
Hence, equation of plane is $- 2(x - 1) + (0)(y - 2) - \left( \mathcal{z} - 3 \right) = 0$ or $2x + \mathcal{z} = 5$