| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Discrete (Further AS Paper 2 Discrete) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Route Inspection |
| Type | Basic Chinese Postman (closed route) |
| Difficulty | Moderate -0.5 This is a straightforward Chinese Postman Problem application requiring students to identify odd-degree vertices, find minimum pairings, and calculate total route length. The graph is small (5 vertices), the algorithm is standard, and the calculation is routine. Slightly easier than average because it's a textbook application with no complications, though it does require knowledge of a specific algorithm from Further Maths. |
| Spec | 7.04e Route inspection: Chinese postman, pairing odd nodes7.04f Network problems: choosing appropriate algorithm |
| Billinge | Garswood | Haydock | Orrell | Up Holland | |
| Billinge | - | 2.5 | *** | 4.3 | 4.8 |
| Garswood | 2.5 | - | 3.1 | *** | 5.9 |
| Haydock | *** | 3.1 | - | 6.7 | 7.8 |
| Orrell | 4.3 | *** | 6.7 | - | 2.1 |
| Up Holland | 4.8 | 5.9 | 7.8 | 2.1 | - |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| A cable cannot be laid between these two districts as there may be a river or housing estate in the way | E1 | Gives a plausible reason in context of the question |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Odd nodes: \(B\), \(G\), \(H\) and \(O\) | M1 | Identifies problem as route inspection; nodes \(B\), \(G\), \(H\), \(O\) are odd-degree (PI) |
| \(B\text{-}G: 2.5\); \(B\text{-}H: 5.6\); \(B\text{-}O: 4.3\); \(G\text{-}H: 3.1\); \(G\text{-}O: 6.8\); \(H\text{-}O: 6.7\) | M1 | Finds shortest distance between each pair of odd nodes (at least four correct) |
| Minimum pair is \(B\text{-}O\) and \(G\text{-}H\), extra distance \(= 7.4\); total \(= 37.2 + 7.4 = 44.6\) miles | A1 | Determines minimum total distance; CAO |
| \((44.6 \div 12) \times 60 = 223\) minutes | A1F | Follow through from their minimum total distance using 12 mph average speed |
**Question 4(a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| A cable cannot be laid between these two districts as there may be a river or housing estate in the way | E1 | Gives a plausible reason in context of the question |
---
**Question 4(b):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Odd nodes: $B$, $G$, $H$ and $O$ | M1 | Identifies problem as route inspection; nodes $B$, $G$, $H$, $O$ are odd-degree (PI) |
| $B\text{-}G: 2.5$; $B\text{-}H: 5.6$; $B\text{-}O: 4.3$; $G\text{-}H: 3.1$; $G\text{-}O: 6.8$; $H\text{-}O: 6.7$ | M1 | Finds shortest distance between each pair of odd nodes (at least four correct) |
| Minimum pair is $B\text{-}O$ and $G\text{-}H$, extra distance $= 7.4$; total $= 37.2 + 7.4 = 44.6$ miles | A1 | Determines minimum total distance; CAO |
| $(44.6 \div 12) \times 60 = 223$ minutes | A1F | Follow through from their minimum total distance using 12 mph average speed |4 A communications company is conducting a feasibility study into the installation of underground television cables between 5 neighbouring districts.
The length of the possible pathways for the television cables between each pair of districts, in miles, is shown in the table.
The pathways all run alongside cycle tracks.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
& Billinge & Garswood & Haydock & Orrell & Up Holland \\
\hline
Billinge & - & 2.5 & *** & 4.3 & 4.8 \\
\hline
Garswood & 2.5 & - & 3.1 & *** & 5.9 \\
\hline
Haydock & *** & 3.1 & - & 6.7 & 7.8 \\
\hline
Orrell & 4.3 & *** & 6.7 & - & 2.1 \\
\hline
Up Holland & 4.8 & 5.9 & 7.8 & 2.1 & - \\
\hline
\end{tabular}
\end{center}
4
\begin{enumerate}[label=(\alph*)]
\item Give a possible reason, in context, why some of the table entries are labelled as ***.
4
\item As part of the feasibility study, Sally, an engineer needs to assess each possible pathway between the districts. To do this, Sally decides to travel along every pathway using a bicycle, starting and finishing in the same district. From past experience, Sally knows that she can travel at an average speed of 12 miles per hour on a bicycle.
Find the minimum time, in minutes, that it will take Sally to cycle along every pathway.\\[0pt]
[4 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete Q4 [6]}}