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\includegraphics[max width=\textwidth, alt={}, center]{f462c036-45d3-4679-ad53-4edbf99df76d-14_558_963_260_589} The function \(\mathrm { f } : x \mapsto p \sin ^ { 2 } 2 x + q\) is defined for \(0 \leqslant x \leqslant \pi\), where \(p\) and \(q\) are positive constants. The diagram shows the graph of \(y = \mathrm { f } ( x )\).

1. In terms of \(p\) and \(q\), state the range of f .
2. State the number of solutions of the following equations.
   (a) \(\mathrm { f } ( x ) = p + q\)
   (b) \(\mathrm { f } ( x ) = q\)
   (c) \(\mathrm { f } ( x ) = \frac { 1 } { 2 } p + q\)
3. For the case where \(p = 3\) and \(q = 2\), solve the equation \(\mathrm { f } ( x ) = 4\), showing all necessary working.