| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2017 |
| Session | Specimen |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Geometric properties using vectors |
| Difficulty | Challenging +1.8 This AEA question requires systematic vector manipulation in a regular pentagon, finding relationships between vectors through geometric constraints, and deriving an exact trigonometric value. While methodical rather than requiring deep insight, it demands careful multi-step reasoning with vectors, solving a quadratic from geometric constraints, and connecting to trigonometry—significantly above standard A-level but accessible with persistence. |
| Spec | 1.05g Exact trigonometric values: for standard angles1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10g Problem solving with vectors: in geometry |
3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-08_609_631_264_724}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-08_172_168_781_1548}
\end{center}
Figure 1 shows a regular pentagon $O A B C D$. The vectors $\mathbf { p }$ and $\mathbf { q }$ are defined by $\mathbf { p } = \overrightarrow { O A }$ and $\mathbf { q } = \overrightarrow { O D }$ respectively.
Let $k$ be the number such that $\overrightarrow { D B } = k \overrightarrow { O A }$.
\begin{enumerate}[label=(\alph*)]
\item Write down $\overrightarrow { A C }$ in terms of $\mathbf { p } , \mathbf { q }$ and $k$ as appropriate.
\item Show that $\overrightarrow { C D } = - \mathbf { p } - \frac { 1 } { k } \mathbf { q }$
\item Hence find the value of $k$
By considering triangle $D B C$, or otherwise,
\item find the exact value of $\sin 54 ^ { \circ }$
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2017 Q3 [12]}}