2025 NOV TZ1 P1 Q6

For \(x \in \mathbb{R}\), the functions \(f\) and \(g\) are defined by

\(f(x) = -x^2 + 4x + p\) and
\(g(x) = x^2 + qx - 1\),

where \(p\) and \(q\) are non-zero constants.
The graph of \(y = f(x)\) has a vertex at \(\mathrm{A}\) and the graph of \(y = g(x)\) has a vertex at \(\mathrm{B}\)
Points \(\mathrm{A}\) and \(\mathrm{B}\) both lie on the line \(y = 2x - 1\).

1. Show that \(p = -1\).

   [3]

2. Find the value of \(q\).

   [5]