(a) The graph of \(y = a \cos(bx) + c\) shows a cosine wave with a maximum value of 8 and a minimum value of -2. The amplitude \(a\) is half the distance between the maximum and minimum values, so \(a = \frac{8 - (-2)}{2} = 5\).

The period of the cosine function is \(\frac{2\pi}{b}\). From the graph, the period is \(\pi\), so \(\frac{2\pi}{b} = \pi\), giving \(b = 2\).

The vertical shift \(c\) is the average of the maximum and minimum values, so \(c = \frac{8 + (-2)}{2} = 3\).

(b)(i) For \(a \cos(bx) + c = \frac{6}{\pi} x\), substitute \(a = 5\), \(b = 2\), \(c = 3\) to get \(5 \cos(2x) + 3 = \frac{6}{\pi} x\). The number of solutions is the number of intersections of the line \(y = \frac{6}{\pi} x\) with the cosine graph in the interval \(0 \leq x \leq 2\pi\), which is 3.

(b)(ii) For \(a \cos(bx) + c = 6 - \frac{6}{\pi} x\), substitute \(a = 5\), \(b = 2\), \(c = 3\) to get \(5 \cos(2x) + 3 = 6 - \frac{6}{\pi} x\). The number of solutions is the number of intersections of the line \(y = 6 - \frac{6}{\pi} x\) with the cosine graph in the interval \(0 \leq x \leq 2\pi\), which is 2.