All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)
Equation of plane OABC: r $= \lambda$a $+$ $\mu$b, for $\lambda, \mu \in \mathbb{R}$
Since c is on plane OABC, c $= \lambda$a $+$ $\mu$b for $\lambda, \mu$ are constants

(ii)
Using the ratio theorem formula, $\vec{ON}=\frac{4\mathbf{a}+3\mathbf{c}}{7}$

(iii)
Area ONC = Area OMC
$\frac{1}{2} |{\frac{4\mathbf{a}+3\mathbf{c}}{7} \times \mathbf{c}}| = \frac{1}{2}|{\frac{1}{2}\mathbf{b} \times \mathbf{c}}|$

$\frac{1}{7} |{4 \mathbf{a} \times \mathbf{c} + 3\mathbf{c} \times \mathbf{c}}| = \frac{1}{2}|{\mathbf{b} \times (\lambda \mathbf{a} + \mu \mathbf{b})}|$

$\frac{2}{7} |{4\mathbf{a} \times (\lambda \mathbf{a} + \mu \mathbf{b})}| = |{\mathbf{b} \times \lambda \mathbf{a} + \mathbf{b} \times \mu \mathbf{b}}|$

$\frac{2}{7} |{4\mathbf{a} \times \lambda \mathbf{a} + 4\mathbf{a} \times \mu \mathbf{b}}| = \frac{1}{2}|{\mathbf{b} \times \lambda \mathbf{a}}|$

$\frac{8\mu}{7} |{\mathbf{a} \times \mathbf{b}}| = \lambda|{\mathbf{a} \times \mathbf{b}}|$

$\frac{8\mu}{7} = \lambda$