| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Discrete (Further AS Paper 2 Discrete) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Graph Theory Fundamentals |
| Type | Degree sum calculations |
| Difficulty | Moderate -0.5 Part (a) is a straightforward application of the handshaking lemma requiring students to draw a simple graph with 2 vertices and degree sum 6 (a multigraph with 3 edges). Part (b) involves using Euler's formula and properties of Eulerian graphs, which is more routine manipulation than deep insight. While this is Further Maths content, these are standard textbook exercises in graph theory requiring recall and basic application rather than problem-solving. |
| Spec | 7.02b Graph terminology: tree, simple, connected, simply connected |
| Answer | Marks | Guidance |
|---|---|---|
| Graph with 2 vertices and 3 edges (two vertices connected by three parallel edges) | B1 (AO 1.1b) | Draws a graph with 2 vertices and 3 edges |
| Answer | Marks | Guidance |
|---|---|---|
| Euler's formula for connected planar graphs: \(v - e + f = 2\) | B1 (AO 1.2) | Recalls Euler's formula for connected planar graphs |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = 5\) or \(x = 4\) | M1 (AO 3.1a) | Forms an equation in \(x\) using Euler's formula for connected planar graphs. Allow one sign error |
| \(x = 5\) or \(x = 4\) | A1 (AO 1.1b) | Solves the quadratic equation, giving both correct solutions |
| For an Eulerian graph, all vertices have even degree, therefore \(x = 4\) | E1 (AO 1.1b) | States that all vertices of an Eulerian graph have even degree |
| \(x = 4\) | M1 (AO 2.2a) | Deduces that \(x = 4\) |
| Answer | Marks | Guidance |
|---|---|---|
| Therefore, total vertex degree \(= 2e = 2 \times 84 = 168\) | A1 (AO 3.2a) | Finds the correct sum of degrees for \(P\) |
## Question 8(a):
Graph with 2 vertices and 3 edges (two vertices connected by three parallel edges) | B1 (AO 1.1b) | Draws a graph with 2 vertices and 3 edges
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## Question 8(b):
Euler's formula for connected planar graphs: $v - e + f = 2$ | B1 (AO 1.2) | Recalls Euler's formula for connected planar graphs
$18(x-1) - x(5x+1) + 4x(x-2) = 2$
$18x - 18 - 5x^2 - x + 4x^2 - 8x = 2$
$-x^2 + 9x - 20 = 0$
$x^2 - 9x + 20 = 0$
$(x-5)(x-4) = 0$
$x = 5$ or $x = 4$ | M1 (AO 3.1a) | Forms an equation in $x$ using Euler's formula for connected planar graphs. Allow one sign error
$x = 5$ or $x = 4$ | A1 (AO 1.1b) | Solves the quadratic equation, giving both correct solutions
For an Eulerian graph, all vertices have even degree, therefore $x = 4$ | E1 (AO 1.1b) | States that all vertices of an Eulerian graph have even degree
$x = 4$ | M1 (AO 2.2a) | Deduces that $x = 4$
When $x = 4$:
$e = x(5x+1) = 4 \times 21 = 84$
Each edge has 2 ends, so each edge contributes 2 to the total vertex degree.
Therefore, total vertex degree $= 2e = 2 \times 84 = 168$ | A1 (AO 3.2a) | Finds the correct sum of degrees for $P$8
\begin{enumerate}[label=(\alph*)]
\item The graph $G$ has 2 vertices.
The sum of the degrees of all the vertices of $G$ is 6
Draw $G$
8
\item The planar graph $P$ is Eulerian, with at least one vertex of degree $x$, where $x$ is a positive integer.
Some of the properties of $P$ are shown in the table below.
Question number
Additional page, if required. Write the question numbers in the left-hand margin.
Question number
Additional page, if required. Write the question numbers in the left-hand margin.
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete 2023 Q8 [7]}}