| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Network Flows |
| Type | Find missing flow values |
| Difficulty | Moderate -0.3 This is a standard network flows question requiring application of the labelling procedure (Ford-Fulkerson algorithm) with straightforward flow conservation to find missing values. While multi-part with several marks, it follows a routine algorithmic procedure taught in D2 with no novel problem-solving required, making it slightly easier than average A-level difficulty. |
| Spec | 7.02p Networks: weighted graphs, modelling connections |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use conservation of flow at nodes to find \(a\), \(b\), \(c\) | M1 | |
| At node \(G\): flow in \(= \) flow out; \(a\) determined | A1 | |
| \(b\) and \(c\) found correctly | A1 | |
| Initial flow value stated | B1 | |
| Identify augmenting path(s) | M1 | |
| Correct augmentation shown | A1 | |
| Maximum flow found with value stated | A1 | |
| Cut identified to verify maximality | M1 A1 |
# Question 5:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use conservation of flow at nodes to find $a$, $b$, $c$ | M1 | |
| At node $G$: flow in $= $ flow out; $a$ determined | A1 | |
| $b$ and $c$ found correctly | A1 | |
| Initial flow value stated | B1 | |
| Identify augmenting path(s) | M1 | |
| Correct augmentation shown | A1 | |
| Maximum flow found with value stated | A1 | |
| Cut identified to verify maximality | M1 A1 | |
I can see these are exam questions, but the images provided show only the **question pages** (pages 7, 8, and 9), not the mark scheme. I cannot extract mark scheme content from these pages as no mark scheme is visible.
What I can see are:
- **Question 5 (continued):** Parts (a)–(f) on network flows (max flow, labelling procedure)
- **Question 6:** Assignment problem (linear programming formulation + Hungarian Algorithm modification)
- **Question 7:** Dynamic programming scheduling problem
To get the actual mark scheme, you would need to provide the **mark scheme document** which is a separate file/paper. These question papers do not contain the marking guidance, M1/A1/B1 allocations, or examiner notes.
If you can share the mark scheme pages, I would be happy to extract and format that content for you.5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{d28d78c1-052d-4350-a7e3-284380e3bbab-6_663_1363_242_351}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a capacitated directed network. The number on each arc is its capacity.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{d28d78c1-052d-4350-a7e3-284380e3bbab-6_665_1363_1117_351}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows an initial flow through the same network.
\begin{enumerate}[label=(\alph*)]
\item State the values of flows $a , b$ and $c$, and the value of the initial flow.
\item By entering values along HG, HT and FG, complete the labelling procedure on Diagram 1 in the answer book.
\item Find the maximum flow through the network. You must list each flow-augmenting route you use, together with its flow.
\item State the value of the maximum flow through the network.
\item Show your maximum flow on Diagram 2 in the answer book.
\item Prove that your flow is maximal.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D2 2011 Q5 [15]}}