(i) To sketch the graphs of \(y = 2 \sin x\) and \(y = \cos 2x\) for \(0 \leq x \leq \pi\):

- The graph of \(y = 2 \sin x\) will have a maximum value of 2 and a minimum value of 0 within the interval. It completes half a cycle from 0 to \(\pi\).

- The graph of \(y = \cos 2x\) completes a full cycle from 0 to \(\pi\), oscillating between 1 and -1.

(ii) The number of solutions to \(2 \sin x = \cos 2x\) corresponds to the number of intersection points of the two graphs within the interval \(0 \leq x \leq \pi\). From the sketch, there are 2 points of intersection.