Question 4 - A-Level Maths

AQA Further AS Paper 2 Discrete 2018 June — Question 4 6 marks

Exam Board	AQA
Module	Further AS Paper 2 Discrete (Further AS Paper 2 Discrete)
Year	2018
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Network Flows
Type	Calculate cut capacity
Difficulty	Moderate -0.5 This is a straightforward application of standard network flow algorithms requiring identification of cut capacity, sources/sinks, and adding supersource/supersink nodes. While it's Further Maths content, these are routine procedural tasks with minimal problem-solving—students follow learned algorithms directly from the specification. The multi-part structure adds marks but not conceptual difficulty.
Spec	7.04f Network problems: choosing appropriate algorithm

1. Find the value of the cut given by \(\{ A , B , C , D , F , J \} \{ E , G , H \}\).
  4
  1. (ii) State what can be deduced about the maximum flow through the network.
    4
2. 1. List the nodes which are sources of the network. 4
3. (ii) Add a supersource \(S\) to the network. 4
4. 1. List the nodes which are sinks of the network. 4
5. (ii) Add a supersink \(T\) to the network.

Show mark scheme Show mark scheme source

Question 4(a)(i):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
\(37\)	B1	Finds the value of the cut

Question 4(a)(ii):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
\(\text{Max flow} \leq 37\)	B1F	Deduces weak inequality; must use \(\leq\) their value of the cut

Question 4(b)(i):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
\(D\) and \(J\)	B1	Lists both sources CAO

Question 4(b)(ii):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
Supersource \(S\) with directed arcs to \(D\) (weight 28) and \(J\) (weight 14)	B1	Weights at least 28 and at least 14 respectively

Question 4(c)(i):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
\(E\) and \(H\)	B1	Lists both sinks CAO

Question 4(c)(ii):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
Supersink \(T\) with directed arcs from \(E\) (weight 30) and \(H\) (weight 15)	B1	Weights at least 30 and at least 15 respectively

## Question 4(a)(i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $37$ | B1 | Finds the value of the cut |

## Question 4(a)(ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{Max flow} \leq 37$ | B1F | Deduces weak inequality; must use $\leq$ their value of the cut |

## Question 4(b)(i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $D$ and $J$ | B1 | Lists both sources CAO |

## Question 4(b)(ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Supersource $S$ with directed arcs to $D$ (weight 28) and $J$ (weight 14) | B1 | Weights at least 28 and at least 14 respectively |

## Question 4(c)(i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $E$ and $H$ | B1 | Lists both sinks CAO |

## Question 4(c)(ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Supersink $T$ with directed arcs from $E$ (weight 30) and $H$ (weight 15) | B1 | Weights at least 30 and at least 15 respectively |

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Show LaTeX source

4
\begin{enumerate}[label=(\alph*)]
\item (i) Find the value of the cut given by $\{ A , B , C , D , F , J \} \{ E , G , H \}$.\\

4 (a) (ii) State what can be deduced about the maximum flow through the network.\\

4
\item (i) List the nodes which are sources of the network.

4 (b) (ii) Add a supersource $S$ to the network.

4
\item (i) List the nodes which are sinks of the network.

4 (c) (ii) Add a supersink $T$ to the network.
\end{enumerate}

\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete 2018 Q4 [6]}}

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Q1 1 Q2 1 Q3 4 Q4 6 Q5 9 Q6 5 Q7 14