Edexcel FD2 AS (Further Decision 2 AS) 2020 June

1. \begin{figure}[h] \end{figure} Figure 1 shows a capacitated, directed network of pipes. The number on each arc represents the capacity of the corresponding pipe. The numbers in circles represent a feasible flow from S to T .

1. 1. Find the value of \(x\).
   2. Find the value of \(y\).
2. List the saturated arcs. Two cuts, \(C _ { 1 }\) and \(C _ { 2 }\), are shown in Figure 1.
3. Find the capacity of
   1. \(C _ { 1 }\)
   2. \(\mathrm { C } _ { 2 }\)
4. Write down a flow-augmenting route, using the arc CF, that increases the flow by two units. Given that the flow through the network is increased by two units using the route found in (d), (e) prove that this new flow is maximal.

2. Four workers, A, B, C and D, are each to be assigned to one of four tasks, P, Q, R and S. Each worker must be assigned to one task, and each task must be done by exactly one worker. Worker C cannot be assigned to task Q. The amount, in pounds, that each worker would earn when assigned to each task is shown in the table below. The Hungarian algorithm is to be used to find the maximum total amount that can be earned by the four workers.

1. Explain how the table should be modified so that the Hungarian algorithm may be applied.
2. Modify the table so that the Hungarian algorithm may be applied.
3. Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total earnings. You should explain how any initial row and column reductions were made and also how you determined if the table was optimal at each stage.

3. Two teams, A and B , each have three team members. One member of Team A will compete against one member of Team B for 10 rounds of a competition. None of the rounds can end in a draw. Table 1 shows, for each pairing, the expected number of rounds that the member of Team A will win minus the expected number of rounds that the member of Team B will win. These numbers are the scores awarded to Team A. This competition between Teams A and B is a zero-sum game. Each team must choose one member to play. Each team wants to choose the member who will maximise its score. \begin{table}[h] \end{table}

1. 1. Find the number of rounds that Team A expects to win if Team A chooses Mischa and Team B chooses Paul.
   2. Find the number of rounds that Team B expects to win if Team A chooses Noel and Team B chooses Qaasim. Table 1 models this zero-sum game.
2. 1. Find the play-safe strategies for the game.
   2. Explain how you know that the game is not stable.
3. Determine which team member Team B should choose if Team B thinks that Team A will play safe. Give a reason for your answer. At the last minute, Rashid is ill and is therefore unavailable for selection by Team B.
4. Find the best strategy for Team B, defining any variables you use.

4. A sequence \(\left\{ u _ { n } \right\}\), where \(n \geqslant 1\), satisfies the recurrence relation $$2 u _ { n } = u _ { n - 1 } - k n ^ { 2 } \text { where } 4 u _ { 2 } - u _ { 0 } = 27 k ^ { 2 }$$ and \(k\) is a non-zero constant.
Show that, as \(n\) becomes large, \(u _ { n }\) can be approximated by a quadratic function of the form \(a n ^ { 2 } + b n + c\) where \(a , b\) and \(c\) are constants to be determined. Please check the examination details below before entering your candidate information
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□ \section*{Thursday 14 May 2020} You may not need to use all of these tables. 3. \begin{table}[h] \end{table} 4. .