11 A curve has equation \(y = 3 \cos 2 x + 2\) for \(0 \leqslant x \leqslant \pi\).
- State the greatest and least values of \(y\).
- Sketch the graph of \(y = 3 \cos 2 x + 2\) for \(0 \leqslant x \leqslant \pi\).
- By considering the straight line \(y = k x\), where \(k\) is a constant, state the number of solutions of the equation \(3 \cos 2 x + 2 = k x\) for \(0 \leqslant x \leqslant \pi\) in each of the following cases.
- \(k = - 3\)
- \(k = 1\)
- \(k = 3\)
Functions \(\mathrm { f } , \mathrm { g }\) and h are defined for \(x \in \mathbb { R }\) by
$$\begin{aligned}
& \mathrm { f } ( x ) = 3 \cos 2 x + 2
& \mathrm {~g} ( x ) = \mathrm { f } ( 2 x ) + 4
& \mathrm {~h} ( x ) = 2 \mathrm { f } \left( x + \frac { 1 } { 2 } \pi \right)
\end{aligned}$$
- Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) on to \(y = \mathrm { g } ( x )\).
- Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) on to \(y = \mathrm { h } ( x )\). [2]
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.