Edexcel FD2 AS (Further Decision 2 AS) 2024 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. State the value of this flow.
   (1)
2. Explain why arcs CD and CG cannot both be saturated.
   (1)
3. Find the capacity of
   1. cut \(C _ { 1 }\)
   2. cut \(C _ { 2 }\)
4. Write down a flow augmenting route of weight 6 which saturates BF. The flow augmenting route in part (d) is applied to give an increased flow.
5. Prove that this increased flow is maximal.

2. A team of 5 players, A, B, C, D and E, competes in a quiz. Each player must answer one of 5 rounds, \(\mathrm { P } , \mathrm { Q } , \mathrm { R } , \mathrm { S }\) and T . Each player must be assigned to exactly one round, and each round must be answered by exactly one player. Player B cannot answer round Q, player D cannot answer round T, and player E cannot answer round R. The number of points that each player is expected to earn in each round is shown in the table. The team wants to maximise its total expected score.
The Hungarian algorithm is to be used to find the maximum total expected score that can be earned by the 5 players.

1. Explain how the table should be modified.
2. 1. Reducing rows first, use the Hungarian algorithm to obtain an allocation which maximises the total expected score.
   2. Calculate the maximum total expected score.

3. Haruki and Meera play a zero-sum game. The game is represented by the following pay-off matrix for Haruki.

1. Determine whether the game has a stable solution. Option Y for Meera is now removed.
2. Write down the reduced pay-off matrix for Meera.
3. 1. Given that Meera plays Option X with probability \(p\), determine her best strategy.
   2. State the value of the game to Haruki.
   3. State which option(s) Haruki should never play. The number of points scored by Haruki when he plays Option C and Meera plays Option X changes from - 1 to \(k\) Given that the value of the game is now the same for both players,
4. determine the value of \(k\). You must make your method and working clear.

4. Peter sets up a savings plan. He makes an initial deposit of \(\pounds D\) and then pays in \(\pounds M\) at the end of each month. The value of the savings plan, in pounds, is modelled by $$u _ { n + 1 } = 1.025 u _ { n } + 1800$$ where \(n \geqslant 0\) is an integer and \(u _ { n }\) is the total value of the savings plan, in pounds, after \(n\) years.

1. Calculate the value of \(M\) Given that the value of the savings plan after 1 year is \(\pounds 6925\)
2. solve the recurrence relation for \(u _ { n }\)
3. Determine the value of \(D\)
4. Hence determine, using algebra, the number of years it will take for the value of the savings plan to exceed \(\pounds 20000\)