Answer
(iii) is correct.
Work Step by Step
we have $r(x)=\dfrac{4}{x(x-2)^2}$
This shows a one single root and one repeated root.
Applying the partial fraction decomposition rule.
Therefore,
$\dfrac{4}{x(x-2)^2}=\dfrac{A}{x}+\dfrac{B}{x-2}+\dfrac{C}{(x-2)^2}$
Hence, option (iii) is correct.