2026 HCI P1 Q7 [Modified]

It is given that

\[\mathrm{f}(x) = \begin{cases} \tan \left( \frac{\pi}{4a} x \right) & \text{for}\hspace{0.5em} -a < x \le a, \\ \sin \left( \frac{\pi}{2a} x \right) & \text{for}\hspace{0.5em} a < x \le 3a, \end{cases}\]

and that \(\mathrm{f}(x) = \mathrm{f}(x - 4a)\) for all real values of \(x\), where \(a\) is a positive real constant.

1. Sketch the graph of \(y = \mathrm{f}(x)\) for \(-4a \le x \le 7a\).[3]

2. By using the sketch in part (a), write down an equation relating \(\mathrm{f}(x)\) and \(\mathrm{f}(-x)\).

   [1]
3. Show that \(\int_{-2a}^{4a} \mathrm{f}(x) \,\mathrm{d}x = \frac{ka}{\pi} (1 + \ln m)\), where \(k\) and \(m\) are constants to be determined.[5]