AQA Further AS Paper 2 Discrete (Further AS Paper 2 Discrete) 2018 June

1 The table shows some of the outcomes of performing a modular arithmetic operation. Which pair are operations that could each be represented by the table?
Tick ( ✓ ) one box.
\includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-02_109_111_1338_497} Addition \(\bmod 6\) and multiplication \(\bmod 5\)
\includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-02_108_109_1471_497} Addition mod 6 and multiplication \(\bmod 6\)
\includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-02_113_109_1603_497} Addition mod 4 and multiplication \(\bmod 5\)
\includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-02_107_109_1742_497} Addition mod 4 and multiplication mod 6

2 The binary operation ⊗ is given by
\(a \otimes b = 3 a ( 5 + b ) ( \bmod 8 )\)
where \(a , b \in \mathbb { Z }\)
Given that \(2 \otimes x = 6\), which of the integers below is a possible value of \(x\) ?
Circle your answer.
[0pt] [1 mark]
0123

3 Alex and Sam are playing a zero-sum game. The game is represented by the pay-off matrix for Alex. 3

1. Explain why the value of the game is 2
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2. Identify the play-safe strategy for each player.
   Each pipe is labelled with its upper capacity in \(\mathrm { cm } ^ { 3 } \mathrm {~s} ^ { - 1 }\)
   \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-04_620_940_450_550}

4

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

5 A group of friends want to prepare a meal. They start preparing the meal at 6:30 pm Activities to prepare the meal are shown in Figure 1 below. \begin{table}[h] \end{table} 5

1. 1. Use Figure 2 shown below to complete Figure 1 above. 5
2. (ii) Complete Figure 2 showing the earliest start time and latest finishing time for each activity. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{5a826f8b-4751-4589-ad0a-109fc5c821f2-06_700_1650_1781_194}
   \end{figure} 5
3. 1. State the activity which must be started first so that the meal is served in the shortest possible time. Fully justify your answer.
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4. (ii) Determine the earliest possible time at which the preparation of the meal can be completed.
   Question 5 continues on the next page 5
5. The group of friends want to cook spring rolls so that they are served at the same time as the rest of the meal. This requires the additional activities shown in Figure 3. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3}

   Label	Activity	Duration	Immediate predecessors
   J	Switch on and heat oven		-
   K	Put spring rolls in oven and cook
   \(L\)	Transfer spring rolls to serving dish

   \end{table} It takes 15 seconds to switch on the oven. The oven must be allowed to heat up for 10 minutes before the spring rolls are put in the oven. It takes 15 seconds to put the spring rolls in the oven.
   The spring rolls must cook in the hot oven for 8 minutes.
   It takes 30 seconds to transfer the spring rolls to a serving dish.
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6. 1. Complete Figure 3 above. 5
7. (ii) Determine the latest time at which the oven can be switched on in order for the spring rolls to be served at the same time as the rest of the meal.
   [0pt] [2 marks]
   \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-09_2488_1716_219_153}

6 An animal sanctuary has a rainwater collection site. The manager of the sanctuary is installing a pipe system to connect the rainwater collection site to five other sites in the sanctuary. Each site does not need to be connected directly to the rainwater collection site. There are nine possible routes between the sites that are suitable for water pipes. The distances, in metres, of the nine possible routes are given in the table below. Water pipe costs 60 pence per metre. Find the minimum cost of connecting all the sites to the rainwater collection site. Fully justify your answer.
\(7 \quad\) A linear programming problem has the constraints $$\begin{aligned} 1 \leq x & \leq 6
1 \leq y & \leq 6
y & \geq x
x + y & \leq 11 \end{aligned}$$

7

1. 1. Complete Figure 4 to identify the feasible region for the problem. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{5a826f8b-4751-4589-ad0a-109fc5c821f2-12_922_940_849_552}
      \end{figure} 7
2. (ii) Determine the maximum value of \(5 x + 4 y\) subject to the constraints.
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3. The simple-connected graph \(G\) has seven vertices. The vertices of \(G\) have degree \(1,2,3 , v , w , x\) and \(y\)
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4. 1. Explain why \(x \geq 1\) and \(y \geq 1\)
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5. (ii) Explain why \(x \leq 6\) and \(y \leq 6\)
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6. (iii) Explain why \(x + y \leq 11\)
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7. (iv) State an additional constraint that applies to the values of \(x\) and \(y\) in this context.
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8. The graph \(G\) also has eight edges. The inequalities used in part (a)(i) apply to the graph \(G\). 7
9. 1. Given that \(v + w = 4\), find all the feasible values of \(x\) and \(y\).
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10. (ii) It is also given that the graph \(G\) is semi-Eulerian. On Figure 5, draw \(G\). Figure 5