| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Discrete (Further AS Paper 2 Discrete) |
| Year | 2020 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Graph Theory Fundamentals |
| Type | Euler's formula application |
| Difficulty | Challenging +1.2 This question requires applying Euler's formula (V - E + F = 2) to a quadratic equation and using the Eulerian property (all vertices have even degree), but the algebraic manipulation is straightforward and the concepts are standard Further Maths content. The constraint that x must be even provides a simple check, making this a moderately above-average question that tests understanding rather than requiring novel insight. |
| Spec | 7.02l Planar graphs: planarity, subdivision, contraction7.02m Euler's formula: V + R = E + 2 |
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| \(3 x + 6\) | \(x ^ { 2 } + 8 x\) | \(2 x ^ { 2 } + 2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| As graph is planar and connected: \(v - e + f = 2\) | M1 | Uses Euler's formula for connected planar graphs |
| \(3x + 6 - (x^2 + 8x) + 2x^2 + 2 = 2\) | M1 | Forms a correct unsimplified equation in \(x\) using Euler's formula |
| \(x^2 - 5x + 6 = 0\), giving \(x = 3\) or \(x = 2\) | A1 | Solves quadratic and gives two solutions |
| An Eulerian graph cannot have a vertex of degree 3, therefore \(x = 2\) | R1F | Deduces \(x = 2\); only even values valid as \(P\) is Eulerian (FT their odd/even values of \(x\)) |
## Question 4:
| Answer/Working | Mark | Guidance |
|---|---|---|
| As graph is planar and connected: $v - e + f = 2$ | M1 | Uses Euler's formula for connected planar graphs |
| $3x + 6 - (x^2 + 8x) + 2x^2 + 2 = 2$ | M1 | Forms a correct unsimplified equation in $x$ using Euler's formula |
| $x^2 - 5x + 6 = 0$, giving $x = 3$ or $x = 2$ | A1 | Solves quadratic and gives two solutions |
| An Eulerian graph cannot have a vertex of degree 3, therefore $x = 2$ | R1F | Deduces $x = 2$; only even values valid as $P$ is Eulerian (FT their odd/even values of $x$) |
---4 The connected planar graph $P$ is Eulerian and has at least one vertex of degree $x$.
Some of the properties of $P$ are shown in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | }
\hline
\begin{tabular}{ c }
Number of \\
vertices \\
\end{tabular} & \begin{tabular}{ c }
Number of \\
edges \\
\end{tabular} & \begin{tabular}{ c }
Number of \\
faces \\
\end{tabular} \\
\hline
$3 x + 6$ & $x ^ { 2 } + 8 x$ & $2 x ^ { 2 } + 2$ \\
\hline
\end{tabular}
\end{center}
Deduce the value of $x$.\\
Fully justify your answer.\\
\includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-07_2488_1716_219_153}
\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete 2020 Q4 [4]}}