2025 Nov TZ1 P2 Q5

Consider the function \(g(x) = a\cos{(bx)} - 3\), where \(x \in \mathbb{R}\) and \(a, b \in \mathbb{R}\).
The following diagram shows part of the graph of \(g\).

The graph of \(g\) has a local minimum at \( (0, -8) \) and a local maximum at \( (16, 2)\).

1. Find the value of

   1. \(a\) ;

   2. \(b\), where \(b > 0\).

      [3]

2. Write down the smallest positive value of the constant \(k\) such that \(g(x + k) = g(x)\) for all \(x\).

   [1]

3. The function \(g(x)\) can be written in the form \(f(x) = p\sin{b(x - q)} -3\) where \(p, q \in \mathbb{Z}^+\).

   1. Find the smallest positive value of \(q\).

   2. For this value of \(q\), write down the value of \(p\).

      [3]