| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Discrete (Further AS Paper 2 Discrete) |
| Year | 2022 |
| Session | June |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Graph Theory Fundamentals |
| Type | Planarity by redrawing |
| Difficulty | Moderate -0.5 Part (a) is direct substitution into Euler's formula V - E + F = 2, requiring only recall. Part (b) involves redrawing a graph to show planarity with vertices already positioned, which is a standard textbook exercise in graph theory requiring pattern recognition rather than novel insight. Both parts are routine for Further Maths students studying Decision Mathematics. |
| Spec | 7.02l Planar graphs: planarity, subdivision, contraction7.02m Euler's formula: V + R = E + 2 |
| \(A\) | \(B\) | \(C\) |
| \(\bullet\) | \(\bullet\) | \(\bullet\) |
| \(D \bullet\) | \(E \bullet\) | \(\bullet F\) |
| \(\bullet\) | \(\stackrel { \theta } { H }\) | \(\bullet\) |
| Answer | Marks | Guidance |
|---|---|---|
| Uses Euler's formula \(v - e + f = 2\), or obtains \(f = 13\) without using Euler's formula | M1 | |
| For graph \(J\): \(9 - 20 + f = 2\), so \(f = 13\) | A1 | CSO |
| Answer | Marks |
|---|---|
| Draws \(J\) correctly in planar form (diagram with vertices \(A, B, C, D, E, F, G, H, I\)) | B1 |
## Question 5(a):
Uses Euler's formula $v - e + f = 2$, or obtains $f = 13$ without using Euler's formula | M1 |
For graph $J$: $9 - 20 + f = 2$, so $f = 13$ | A1 | CSO
---
## Question 5(b):
Draws $J$ correctly in planar form (diagram with vertices $A, B, C, D, E, F, G, H, I$) | B1 |
---5
\begin{enumerate}[label=(\alph*)]
\item A connected planar graph has 9 vertices, 20 edges and $f$ faces.
Use Euler's formula for connected planar graphs to find $f$
5
\item The graph $J$, shown in Figure 1, has 9 vertices and 20 edges.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{ecbeedf5-148e-40ad-b8a2-a7aa3db4a115-09_778_760_440_641}
\end{center}
\end{figure}
By redrawing the graph $J$ using Figure 2, show that $J$ is planar.
\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\begin{tabular}{ l l l }
$A$ & $B$ & $C$ \\
$\bullet$ & $\bullet$ & $\bullet$ \\
$D \bullet$ & $E \bullet$ & $\bullet F$ \\
$\bullet$ & $\stackrel { \theta } { H }$ & $\bullet$ \\
\end{tabular}
\end{center}
\end{table}
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete 2022 Q5 [3]}}