AQA Further Paper 3 Discrete (Further Paper 3 Discrete) Specimen

1 Which of the following graphs is not planar? Circle your answer.
[0pt] [1 mark]
\includegraphics[max width=\textwidth, alt={}, center]{88669bc0-9d3f-431a-8939-8aef2682412b-02_186_301_1000_520}
\includegraphics[max width=\textwidth, alt={}, center]{88669bc0-9d3f-431a-8939-8aef2682412b-02_218_224_986_884}
\includegraphics[max width=\textwidth, alt={}, center]{88669bc0-9d3f-431a-8939-8aef2682412b-02_241_236_982_1201}
\includegraphics[max width=\textwidth, alt={}, center]{88669bc0-9d3f-431a-8939-8aef2682412b-02_241_241_982_1521}

2 The set \(\{ 1,2,4,8,9,13,15,16 \}\) forms a group under the operation of multiplication modulo 17. Which of the following is a generator of the group? Circle your answer.
[0pt] [1 mark] 491316

3 Deva Construction Ltd undertakes a small building project. The activity network for this project is shown below in Figure 1, where each activity's duration is given in hours. \begin{figure}[h] \end{figure} 3

1. Complete the activity network for the building project. 3
2. Deva Construction Ltd is able to reduce the duration of a single activity to 1 hour by using specialist equipment. State, with a reason, which activity should have its duration reduced to 1 hour in order to minimise the completion time for the building project.
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3. State one limitation in the building project used by Deva Construction Ltd. Explain how this limitation affects the project.
   [0pt] [2 marks]

4 Optical fibre broadband cables are being installed between 5 neighbouring villages. The distance between each pair of villages in metres is shown in the table. The company installing the optical fibre broadband cables wishes to create a network connecting each of the 5 villages using the minimum possible length of cable. Find the minimum length of cable required.
[0pt] [3 marks]

5 The binary operation * is defined as $$a * b = a + b + 4 ( \bmod 6 )$$ where \(a , b \in \mathbb { Z }\). 5

1. Show that the set \(\{ 0,1,2,3,4,5 \}\) forms a group \(G\) under *.
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2. Find the proper subgroups of the group \(G\) in part (a).
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3. Determine whether or not the group \(G\) in part (a) is isomorphic to the group \(K = \left( \langle 3 \rangle , \times _ { 14 } \right)\)
   [0pt] [3 marks]

6
The network shows a system of pipes, where \(S\) is the source and \(T\) is the sink.
The lower and upper capacities, in litres per second, of each pipe are shown on each arc.
\includegraphics[max width=\textwidth, alt={}, center]{88669bc0-9d3f-431a-8939-8aef2682412b-09_649_1399_580_424} 6

1. There is a feasible flow from \(S\) to \(T\). 6
2. 1. Explain why arc \(A D\) must be at its lower capacity.
      [0pt] [1 mark] 6
3. (ii) Explain why arc \(B E\) must be at its upper capacity.
   [0pt] [1 mark] 6
4. Explain why a flow of 11 litres per second through the network is impossible.
   [0pt] [1 mark] 6
5. The network in Figure 2 shows a second system of pipes, where \(S\) is the source and \(T\) is the sink. The lower and upper capacities, in litres per second, of each pipe are shown on each edge. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{88669bc0-9d3f-431a-8939-8aef2682412b-10_760_1372_680_470}
   \end{figure} Figure 3 shows a feasible flow of 17 litres per second through the system of pipes. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{88669bc0-9d3f-431a-8939-8aef2682412b-10_750_1371_1811_466}
   \end{figure} 6
6. 1. Using Figures 2 and 3, indicate on Figure 4 potential increases and decreases in the flow along each arc.
      [0pt] [2 marks] \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{88669bc0-9d3f-431a-8939-8aef2682412b-11_749_1384_457_426}
      \end{figure} 6
7. (ii) Use flow augmentation on Figure 4 to find the maximum flow from \(S\) to \(T\).
   You should indicate any flow augmenting paths clearly in the table below and modify the potential increases and decreases of the flow on Figure 4.
   [0pt] [3 marks]

   Augmenting Path	Flow

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8. (iii) Prove the flow found in part (d) (ii) is maximum.
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9. (iv) Due to maintenance work, the flow through node \(E\) is restricted to 9 litres per second.
   [0pt] Interpret the impact of this restriction on the maximum flow through the system of pipes. [2 marks]

7 A company repairs and sells computer hardware, including monitors, hard drives and keyboards. Each monitor takes 3 hours to repair and the cost of components is \(\pounds 40\). Each hard drive takes 2 hours to repair and the cost of components is \(\pounds 20\). Each keyboard takes 1 hour to repair and the cost of components is \(\pounds 5\). Each month, the business has 360 hours available for repairs and \(\pounds 2500\) available to buy components. Each month, the company sells all of its repaired hardware to a local computer shop. Each monitor, hard drive and keyboard sold gives the company a profit of \(\pounds 80 , \pounds 35\) and \(\pounds 15\) respectively. The company repairs and sells \(x\) monitors, \(y\) hard drives and \(z\) keyboards each month. The company wishes to maximise its total profit. 7

1. Find five inequalities involving \(x , y\) and \(z\) for the company's problem.
   [0pt] [3 marks]
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2. 1. Find how many of each type of computer hardware the company should repair and sell each month.
      7
3. (ii) Explain how you know that you had reached the optimal solution in part (b) (i).
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4. (iii) The local computer shop complains that they are not receiving one of the types of computer hardware that the company repairs and sells. Using your answer to part (b) (i), suggest a way in which the company's problem can be modified to address the complaint.
   [0pt] [1 mark]

8 John and Danielle play a zero-sum game which does not have a stable solution. The game is represented by the following pay-off matrix for John. Find the optimal mixed strategy for John.