(i) The graph of \(y = 2 \cos x\) is a cosine wave with amplitude 2. It completes one full cycle from \(-\pi\) to \(\pi\). The point of intersection with the \(y\)-axis is at \(x = 0\), giving \(y = 2 \cos(0) = 2\). Thus, the coordinates are \((0, 2)\).

(ii) The coordinates of \(P\) are \(\left( \frac{\pi}{3}, 2 \cos \left( \frac{\pi}{3} \right) \right) = \left( \frac{\pi}{3}, 1 \right)\) and \(Q\) are \((\pi, 2 \cos(\pi)) = (\pi, -2)\). The distance \(PQ\) is given by:

\(PQ = \sqrt{\left( \pi - \frac{\pi}{3} \right)^2 + (1 - (-2))^2} = \sqrt{\left( \frac{2\pi}{3} \right)^2 + 3^2} = \sqrt{\frac{4\pi^2}{9} + 9}\)

\(PQ = \sqrt{\frac{4\pi^2}{9} + 9} = \sqrt{\frac{4\pi^2 + 81}{9}} = \sqrt{\frac{4\pi^2 + 81}{9}} \approx 3.7\)

(iii) The equation of the line through \(P\) and \(Q\) is:

\(y - 1 = \frac{-2 - 1}{\pi - \frac{\pi}{3}} (x - \frac{\pi}{3})\)

\(y - 1 = \frac{-3}{\frac{2\pi}{3}} (x - \frac{\pi}{3})\)

\(y - 1 = -\frac{9}{2\pi} (x - \frac{\pi}{3})\)

Setting \(y = 0\) to find \(h\):

\(0 - 1 = -\frac{9}{2\pi} (h - \frac{\pi}{3})\)

\(-1 = -\frac{9}{2\pi} h + \frac{3}{2}\)

\(-\frac{5}{2} = -\frac{9}{2\pi} h\)

\(h = \frac{5}{9} \pi\)

Setting \(x = 0\) to find \(k\):

\(k - 1 = -\frac{9}{2\pi} (0 - \frac{\pi}{3})\)

\(k - 1 = \frac{3}{2}\)

\(k = \frac{5}{2}\)