| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2010 |
| Session | June |
| Marks | 21 |
| Paper | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Solve with multiple compound angles |
| Difficulty | Challenging +1.2 This is a structured multi-part question requiring compound angle formula expansion, algebraic manipulation, and calculus. Part (a) is algebraically involved but follows standard techniques. Parts (b-c) are routine once the simplified form is obtained. Part (d) requires integration with a substitution but is guided by the previous work. While from AEA, this is more computational than conceptually demanding, placing it moderately above average difficulty. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05l Double angle formulae: and compound angle formulae1.07n Stationary points: find maxima, minima using derivatives1.08d Evaluate definite integrals: between limits |
7.
$$\mathrm { f } ( x ) = \left[ 1 + \cos \left( x + \frac { \pi } { 4 } \right) \right] \left[ 1 + \sin \left( x + \frac { \pi } { 4 } \right) \right] , \quad 0 \leqslant x \leqslant 2 \pi$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { f } ( x )$ may be written in the form
$$f ( x ) = \left( \frac { 1 } { \sqrt { 2 } } + \cos x \right) ^ { 2 } , \quad 0 \leqslant x \leqslant 2 \pi$$
\item Find the range of the function $\mathrm { f } ( x )$.
The graph of $y = \mathrm { f } ( x )$ is shown in Figure 2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-5_426_938_849_591}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
\item Find the coordinates of all the maximum and minimum points on this curve.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-5_432_942_1535_589}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
The region $R$, bounded by $y = 2$ and $y = \mathrm { f } ( x )$, is shown shaded in Figure 3.
\item Find the area of $R$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2010 Q7 [21]}}