AQA Further Paper 3 Discrete (Further Paper 3 Discrete) 2021 June

1 Which of the following statements about critical path analysis is always true? Tick ( \(\checkmark\) ) one box. All activity networks have exactly one critical path. □ All critical activities have a non-zero float. □ The first activity in a critical path has an earliest start time of zero. □ A delay on a critical activity may not delay the project. □

2 The network below represents a system of pipes. The numbers on each arc represent the lower and upper capacity for each pipe.
\includegraphics[max width=\textwidth, alt={}, center]{59347089-ea4a-4ee6-b40e-1ab78aa7cdc3-03_616_1415_447_310} Find the value of the cut \(\{ A , B , C , D , E \} \{ F , G , H , I \}\).
Circle your answer. 56586370

3 A mining company wants to open a new mine in an area where the ground contains a precious metal. The mining company has carried out a survey of the area. The network below shows nodes which represent the entrance to the new mine, \(X\), and the 8 ventilation shafts, \(A , B , \ldots , H\), which have been installed to prevent the build up of dangerous gases underground.
\includegraphics[max width=\textwidth, alt={}, center]{59347089-ea4a-4ee6-b40e-1ab78aa7cdc3-04_846_1228_623_404} Each arc represents a possible underground tunnel which could be mined.
The weight on each arc represents the estimated amount of precious metal in that possible underground tunnel in tonnes. Due to geological reasons, the mining company can only create 8 underground tunnels. All 8 ventilation shafts must be accessible from the entrance of the mine. 3

1. 1. The mining company wants to maximise the amount of precious metal it can extract from the new mine. Determine the tunnels the mining company should use.
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2. (ii) Estimate the maximum amount of precious metal the mining company can extract from the new mine. 3
3. Comment on why the maximum amount of precious metal the mining company can extract from the new mine may be different from your answer to part (a)(ii).
   [0pt] [2 marks]
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4. Before the mining company begins work on the new mine, a government survey prevents the mining company drilling the tunnel represented by \(C F\). Determine the effect, if any, the government survey has on your answers to part (a)(i) and part (a)(ii).

4 Derrick, a tanker driver, is required to deliver fuel to 6 different service stations \(A , B\), \(C , D , E\) and \(F\). Derrick needs to begin and finish his delivery journey at the refinery \(O\).
The distances, in miles, between the 7 locations which have a direct road between them are shown in the network below.
\includegraphics[max width=\textwidth, alt={}, center]{59347089-ea4a-4ee6-b40e-1ab78aa7cdc3-06_921_1440_628_303} Derrick spends 30 minutes at each service station to complete the fuel delivery.
When driving, the tanker travels at an average speed of 40 miles per hour.
The minimum total time that it takes Derrick to travel to and deliver fuel to all 6 service stations, starting and finishing at the refinery, is \(T\) minutes. 4

1. Using the nearest neighbour algorithm starting from the refinery, find an upper bound for \(T\)
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2. Before setting off to make his fuel deliveries, Derrick is notified that, due to a low bridge, the road represented by CE is not suitable for tankers to travel along. State, with a reason, the effect this new information has on your answer to part (a).
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5

1. Describe the conditions necessary for a set of elements, \(S\), under a binary operation * to form a group.
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2. In the multiplicative group of integers modulo 13, the group \(G\) is defined as $$G = \left( \langle 10 \rangle , \times _ { 13 } \right)$$ 5
3. 1. Explain why \(G\) is an abelian group.
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4. (ii) Find the order of \(G\).
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5. State the identity element of \(G\) and prove it is an identity element. Fully justify your answer.
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6. Find all the proper non-trivial subgroups of \(G\), giving your answers in the form \(\left( \langle g \rangle , \times _ { 13 } \right)\), where \(g\) is an integer less than 13

6

1. A connected planar graph has \(( x + 1 ) ^ { 2 }\) vertices, \(( 25 + 2 x - 2 y )\) edges and \(( y - 1 ) ^ { 2 }\) faces, where \(x > 0\) and \(y > 0\) Find the possible values for the number of vertices, edges and faces for the graph.
   [0pt] [6 marks]
   LL
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2. Explain why \(K _ { 6 }\), the complete graph with 6 vertices, is not planar. Fully justify your answer.

7 Avon and Roj play a zero-sum game. The game is represented by the following pay-off matrix for Avon. 7 (c) (i) Find the optimal mixed strategy for Avon.
7 (c) (ii) Find the value of the game for Avon.
7 (d) Roj thinks that his best outcome from the game is to play strategy \(\mathbf { R } _ { \mathbf { 2 } }\) each time. Avon notices that Roj always plays strategy \(\mathbf { R } _ { \mathbf { 2 } }\) and Avon wants to use this knowledge to maximise his expected pay-off from the game. Explain how your answer to part (c)(i) should change and find Avon's maximum expected pay-off from the game.
\includegraphics[max width=\textwidth, alt={}, center]{59347089-ea4a-4ee6-b40e-1ab78aa7cdc3-16_2490_1735_219_139}