Explanation

(a)
Given that, $\overrightarrow{\mathbf{OA}} = \widehat{\mathbf{i}} + x\widehat{\mathbf{j}} + 3\widehat{\mathbf{k}}$
$\overrightarrow{\mathbf{OB}} = 3\widehat{\mathbf{i}} + 4\widehat{\mathbf{j}} + 7\widehat{\mathbf{k}}$
and $\overrightarrow{\mathbf{O}}\mathbf{C} = y\widehat{\mathbf{i}} - 2\widehat{\mathbf{j}} - 5\widehat{\mathbf{k}}$
Since $A,B,C$ are collinear. Then $\overrightarrow{\mathbf{A}} = \lambda\ \overrightarrow{\mathbf{BC}}$
$\Rightarrow 2\widehat{\mathbf{i}} + (4 - x)\widehat{\mathbf{j}} + 4\widehat{\mathbf{k}} = \lambda\ \lbrack\ (\ y - 3)\widehat{\mathbf{i}} - 6\widehat{\mathbf{j}} - 12\widehat{\mathbf{k}}\rbrack$
On comparing the coefficient of $\widehat{i},\ \widehat{j}\ and\ \widehat{k}$, we get
$2 = (y - 3)\lambda$ …(i)
$4 - x = - 6\lambda$ ….(ii)
and $4 = - 12\lambda\ \Rightarrow \ \lambda = - \frac{1}{3}$ …..(iii)
On putting the value of $\lambda$ is Eqs. (i) and (ii),we get
$y = - 3$ and $x = 2$