Explanation

(a)
Let the roots of $x^{2} - 6x + a = 0$ be $\alpha,\ 4\beta$ and that of $x^{2} - cx + 6 = 0$ be $\alpha,\ 3\beta$
$\therefore\ \ \alpha + 4\beta = 6$ and $4\alpha\beta = a$
And $\alpha + 3\beta = c$ and $3\alpha\beta = 6$
$\Rightarrow \ \ \frac{a}{6} = \frac{4}{3}\ \ \ \Rightarrow \ \ \ a = 8$
$\therefore\ \ \ x^{2} - 6x + 8 = 0\ \ \ \Rightarrow \ \ \ x = 2,\ 4$
And $x^{2} - cx + 6 = 0\ \ \Rightarrow \ \ 2^{2} - 2c + 6 = 0\ \Rightarrow c = 5$
$\therefore\ \ x^{2} - 5x + 6 = 0$
$\Rightarrow x = 2,\ 3$
Hence, common root is 2