2024 CCHY P1 Q9

The function \(\mathrm{f}\) is defined, for \(0^{\circ} \le x \le 180^{\circ}\), by \(\mathrm{f}(x) = c + a \cos bx\), where \(a\) and \(b\) are positive integers and \(c\) is an integer. Given that the amplitude of \(\mathrm{f}\) is \(4\) and the period of \(\mathrm{f}\) is \(120^{\circ}\),

1. state the value of \(a\) and of \(b\),[2]

Given that the minimum value of \(\mathrm{f}\) is \(-5\),

2. find the value of \(c\),[1]
3. sketch the graph of \(y = \mathrm{f}(x)\) for \(0^{\circ} \le x \le 180^{\circ}\).[3]
4. A line \(y = k\) is drawn on the graph of \(y = \mathrm{f}(x)\). State the range of values of \(k\) that will give \(3\) intersections.[1]