AQA Further AS Paper 2 Discrete (Further AS Paper 2 Discrete) 2023 June

1 The graph \(G\) has 8 vertices and 13 edges as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-03_494_392_482_806} Graph \(H\) is a simple-connected subgraph of graph \(G\) Which of the following diagrams could represent graph \(H\) ? Tick ( ✓ ) one box. \includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-03_312_310_1354_351} \includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-03_321_310_1676_351} \includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-03_117_115_1448_822} \includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-03_312_310_2014_351} \includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-03_122_117_1777_822} \includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-03_314_314_2343_349} \includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-03_120_115_2108_822}

2 The diagram below shows a network of pipes with their capacities. \includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-04_691_1155_340_424} A supersource is added to the network. Which nodes are connected to the supersource? Tick ( ✓ ) one box. \(A\) and \(B\) □ \(A\) and \(G\) □ \(G\) and \(H\) \includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-04_104_108_1822_685} \(H\) and \(I\) □

3 Ben is packing eggs into boxes, labelled Town Box or Country Box. Each Town Box must contain 10 chicken eggs and 2 duck eggs. Each Country Box must contain 4 chicken eggs and 8 duck eggs. Ben has 253 chicken eggs and 151 duck eggs. Ben wants to pack as many boxes as possible. Formulate Ben's situation as a linear programming problem, defining any variables you introduce.

4 A community project consists of 10 activities \(A , B , \ldots , J\), as shown in the activity network below. \includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-06_899_1083_367_466} The duration of each activity is shown in days. 4

1. 1. Complete the activity network in the diagram above, showing the earliest start time and latest finish time for each activity. 4
      1. (ii) State the minimum completion time for the community project.
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   2. Write down the critical activities of the network.
      4
   3. Glyn claims that a project's activity network can be used to determine its minimum completion time by adding together the durations of all the project's critical activities. 4
   4. 1. Show that Glyn's claim is false for this community project's activity network.
         4
   5. (ii) Describe a situation in which Glyn's claim would be true.

5

1. The set \(S\) is defined as \(S = \{ 0,1,2,3,4,5 \}\) 5
   1. (i) State the identity element of \(S\) under the operation multiplication modulo 6 5
   2. (ii) An element \(g\) of a set is said to be self-inverse under a binary operation * if $$g * g = e$$ where \(e\) is the identity element of the set. Find all the self-inverse elements in \(S\) under the operation multiplication modulo 6
      5
   3. \(\quad\) The set \(T\) is defined as $$T = \{ a , b , c \}$$ Figure 1 shows a partially completed Cayley table for \(T\) under the commutative binary operation - \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1}

      -	\(a\)	\(b\)	c
      \(a\)	\(a\)	c	b
      \(b\)		\(b\)	\(a\)
      c			c

      \end{table} 5
   4. 1. Complete the Cayley table in Figure 1 5
   5. (ii) Prove that is not associative when acting on the elements of \(T\)

6 Xander and Yvonne are playing a zero-sum game. The game is represented by the pay-off matrix for Xander. \begin{table}[h] \end{table} 6

1. Show that the game has a stable solution.
   6
2. State the play-safe strategy for each player. Play-safe strategy for Xander is \(\_\_\_\_\) Play-safe strategy for Yvonne is \(\_\_\_\_\) 6
3. The game that Xander and Yvonne are playing is part of a marbles challenge. The pay-off matrix values represent the number of marbles gained by Xander in each game. In the challenge, the game is repeated until one player has 24 marbles more than the other player. Explain why Xander and Yvonne must play at least 3 games to complete the challenge.

7 A construction company has built eight wind turbines on a moorland site. The network below shows nodes which represent the site entrance, \(E\), and the wind turbine positions, \(S , T , \ldots , Z\) \includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-12_924_1294_479_356} Each arc represents an access track with its length given in metres.
These 17 tracks were created in order to build the wind turbines. Eight of the tracks are to be retained so that each turbine can be accessed for maintenance, directly or indirectly, from the site entrance. The other nine tracks will be removed. 7

1. 1. To save money the construction company wants to maximise the total length of the eight tracks to be retained. Determine which tracks the construction company should retain.
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      1. (ii) Find the total length of the eight tracks that are to be retained. 7
   2. The total length of the 17 tracks is 14.6 km
      The cost of removing all 17 tracks would be \(\pounds 87,600\) Using your answer to part (a)(ii), calculate an estimate for the cost of removing the nine tracks that will not be retained.
      [0pt] [2 marks]
      7
   3. Comment on why the modelling used in part (b) may not give an accurate estimate for the cost of removing the nine tracks.

8

1. The graph \(G\) has 2 vertices. The sum of the degrees of all the vertices of \(G\) is 6 Draw \(G\) 8
2. 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.