(i) Sketch the graphs of \(y = \csc x\) and \(y = x(\pi - x)\) for \(0 < x < \pi\). The graph of \(y = \csc x\) has vertical asymptotes at \(x = 0\) and \(x = \pi\), and the graph of \(y = x(\pi - x)\) is a downward-opening parabola with roots at \(x = 0\) and \(x = \pi\). The intersection points of these graphs represent the roots of the equation \(\csc x = x(\pi - x)\). By analyzing the graphs, we find exactly two intersection points in the interval \(0 < x < \pi\).

(ii) Start with \(\csc x = x(\pi - x)\), which is \(\frac{1}{\sin x} = x(\pi - x)\). Rearrange to get \(x = \frac{1 + x^2 \sin x}{\pi \sin x}\).

(iii)(a) Use the iterative formula \(x_{n+1} = \frac{1 + x_n^2 \sin x_n}{\pi \sin x_n}\) starting with an initial guess. Iterate until the value stabilizes to 4 decimal places. The iterations converge to \(\alpha = 0.66\).

(iii)(b) From the mark scheme, \(\beta = 2.48\).