2025 PHS P1 Q10

It is given that \(\mathrm{f}(x) = 2\sin\frac{x}{2}\) and \(\mathrm{g}(x) = 3\cos x + 1\), where \(0 \le x \le 2\pi\).

1. State the period of \(\mathrm{f}(x)\).[1]
2. State the smallest value of \(\mathrm{f}(x)\).[1]
3. State the largest value of \(\mathrm{g}(x)\).[1]

4. Sketch on the same axes, the graphs of \(y = \mathrm{f}(x)\) and \(y = \mathrm{g}(x)\) for \(0 \le x \le 2\pi\).

   Label your graphs clearly.

   [4]
   

5. The solutions to the equation \(\mathrm{f}(x) = \mathrm{g}(x)\) for \(0 \le x \le 2\pi\) are \(a\) and \(b\), where \(a < b\).

   State, in terms of \(a\) and \(b\), the range of values of \(x\) for which \(\mathrm{f}(x) > \mathrm{g}(x)\).

   [1]