Questions FD2 AS (32 questions)

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Edexcel FD2 AS 2018 June Q1
  1. Four workers, A, B, C and D, are to be assigned to four tasks, P, Q, R and S. Each worker must be assigned to exactly one task and each task must be done by only one worker. The time, in hours, that each worker takes to complete each task is shown in the table below.
\cline { 2 - 5 } \multicolumn{1}{c|}{}PQRS
A7.53.589.5
B5277.5
C43.53.58
D653.54
Reducing rows first, use the Hungarian algorithm to obtain an allocation which minimises the total time. You must explain your method and show the table after each stage.
Edexcel FD2 AS 2018 June Q2
2. (a) Explain what the term 'zero-sum game' means. Two teams, A and B , are to face each other as part of a quiz.
There will be several rounds to the quiz with 10 points available in each round.
For each round, the two teams will each choose a team member and these two people will compete against each other until all 10 points have been awarded. The number of points that Team A can expect to gain in each round is shown in the table below.
\cline { 3 - 5 } \multicolumn{2}{c|}{}Team B
\cline { 3 - 5 } \multicolumn{2}{c|}{}PaulQaasimRashid
\multirow{3}{*}{Team A}Mischa563
\cline { 2 - 5 }Noel417
\cline { 2 - 5 }Olive458
The teams are each trying to maximise their number of points.
(b) State the number of points that Team B will expect to gain each round if Team A chooses Noel and Team B chooses Rashid.
(c) Explain why subtracting 5 from each value in the table will model this situation as a zero-sum game.
(d) (i) Find the play-safe strategies for the zero-sum game.
(ii) Explain how you know that the game is not stable. At the last minute, Olive becomes unavailable for selection by Team A.
Team A decides to choose its player for each round so that the probability of choosing Mischa is \(p\) and the probability of choosing Noel is \(1 - p\).
(e) Use a graphical method to find the optimal value of \(p\) for Team A and hence find the best strategy for Team A. For this value of \(p\),
(f) (i) find the expected number of points awarded, per round, to Team A,
(ii) find the expected number of points awarded, per round, to Team B.
Edexcel FD2 AS 2018 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{905f2578-e4b2-4d4d-8455-298170fd824b-4_781_1159_365_551} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 models the flow of fluid through a system of pipes from a source, S , to a sink, T . The weights on the arcs show the capacities of the corresponding pipes in litres per minute. Two cuts \(C _ { 1 }\) and \(C _ { 2 }\) are shown.
  1. Find the capacity of
    1. cut \(C _ { 1 }\)
    2. cut \(C _ { 2 }\)
  2. Using only the capacities of cuts \(C _ { 1 }\) and \(C _ { 2 }\) state what can be deduced about the maximum possible flow through the system.
  3. On Diagram 1 in the answer book, show how a flow of 120 litres per minute from S to T can be achieved. You do not need to apply the labelling procedure to find this flow.
  4. Prove that 120 litres per minute is the maximum possible flow through the system. A new pipe is planned from S to A . Let the capacity of this pipe be \(x\) litres per minute.
  5. Find, in terms of \(x\) where necessary, the maximum possible flow through the new system.
Edexcel FD2 AS 2018 June Q4
4. A village has an expected population growth rate (birth rate minus death rate) of \(r \%\) per year. In addition, \(N\) people are expected to move into the village each year. The expected population of the village is modelled by $$u _ { n + 1 } = 1.02 u _ { n } + 50$$ where \(u _ { n }\) is the expected population of the village \(n\) years from now.
  1. State
    1. the value of \(r\),
    2. the value of \(N\). Given that the population 1 year from now is expected to be 560
  2. solve the recurrence relation for \(u _ { n }\)
  3. Hence determine, using algebra, the number of years from now when the model predicts that the population of the village will first be greater than 3000
    (Total for Question 4 is 10 marks)
    TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS END
Edexcel FD2 AS 2019 June Q1
  1. Three workers, A, B and C, are each to be assigned to one of four tasks, P, Q, R and S.
Each worker must be assigned to at most one task, and each task must be done by at most one worker.
The amount, in pounds, that each worker will earn while assigned to each task is shown in the table below.
\cline { 2 - 5 } \multicolumn{1}{c|}{}PQRS
A32403742
B29323541
C37333940
The Hungarian algorithm is to be used to find the maximum total amount that can be earned by the three workers.
  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 earnings.
    2. Explain how any initial row and column reductions were made and also how you determined if the table was optimal at each stage.
Edexcel FD2 AS 2019 June Q2
2. (a) Find the general solution of the recurrence relation $$u _ { n + 1 } = 3 u _ { n } + 2 ^ { n } \quad n \geqslant 1$$ (b) Find the particular solution of this recurrence relation for which \(u _ { 1 } = u _ { 2 }\)
Edexcel FD2 AS 2019 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bbdfa492-6578-484a-a0b5-fcdb78020b83-03_801_1728_269_166} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Alexa is monitoring a system of pipes through which fluid can flow from the source, S , to the sink, T . Currently, fluid is flowing through the system from S to T . Alexa initialises the labelling procedure for this system, and the excess capacities and potential backflows are shown on the arrows either side of each arc, as shown in Figure 1.
  1. State the value of the initial flow.
  2. Explain why arcs DF and DG can never both be full to capacity.
  3. Obtain the capacity of the cut that passes through the \(\operatorname { arcs } \mathrm { AC } , \mathrm { AD } , \mathrm { BD } , \mathrm { DE } , \mathrm { EG }\) and EJ .
  4. Use the labelling procedure to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow.
    (3)
  5. Use your answers to part (d) to find a maximum flow pattern for this system of pipes and draw it on Diagram 1 in the answer book.
  6. Prove that the answer to part (e) is optimal.
Edexcel FD2 AS 2019 June Q4
4. The table below gives the pay-off matrix for a zero-sum game between two players, Aljaz and Brendan. The values in the table show the pay-offs for Aljaz. You may not need to use all of these tables
You may not need to use all the rows and columns
\includegraphics[max width=\textwidth, alt={}, center]{bbdfa492-6578-484a-a0b5-fcdb78020b83-06_437_832_1201_139}
Edexcel FD2 AS 2020 June Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9f21789-1c5b-42f5-9c5a-3b29d9346c46-02_751_1557_214_255} \captionsetup{labelformat=empty} \caption{Figure 1}
\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. 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.
Edexcel FD2 AS 2020 June Q2
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.
\cline { 2 - 5 } \multicolumn{1}{c|}{}PQRS
A72985984
B67876886
C70-6279
D78936481
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.
Edexcel FD2 AS 2020 June Q3
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]
\cline { 3 - 5 } \multicolumn{2}{c|}{}Team B
\cline { 3 - 5 } \multicolumn{2}{c|}{}PaulQaasimRashid
\multirow{3}{*}{Team A}Mischa4- 62
\cline { 2 - 5 }Noel0- 26
\cline { 2 - 5 }Olive- 620
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
    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.
    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.
Edexcel FD2 AS 2020 June Q4
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
Candidate surname
Other names Pearson Edexcel
Centre Number
Candidate Number Level 3 GCE
\includegraphics[max width=\textwidth, alt={}, center]{a9f21789-1c5b-42f5-9c5a-3b29d9346c46-05_122_433_356_991}
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□ \section*{Thursday 14 May 2020} You may not need to use all of these tables.
\cline { 2 - 5 } \multicolumn{1}{c|}{}\(P\)\(Q\)\(R\)\(S\)
\(A\)
\(B\)
\(C\)
\(D\)
\cline { 2 - 5 } \multicolumn{1}{c|}{}\(P\)\(Q\)\(R\)\(S\)
\(A\)
\(B\)
\(C\)
\(D\)
\cline { 2 - 5 } \multicolumn{1}{c|}{}PQRS
A
B
C
D
\cline { 2 - 5 } \multicolumn{1}{c|}{}\(P\)\(Q\)\(R\)\(S\)
\(A\)
\(B\)
\(C\)
\(D\)
\cline { 2 - 5 } \multicolumn{1}{c|}{}PQRS
A
B
C
D
\cline { 2 - 5 } \multicolumn{1}{c|}{}PQRS
A
B
C
D
\cline { 2 - 5 } \multicolumn{1}{c|}{}\(P\)\(Q\)\(R\)\(S\)
\(A\)
\(B\)
\(C\)
\(D\)
\cline { 2 - 5 } \multicolumn{1}{c|}{}\(P\)\(Q\)\(R\)\(S\)
\(A\)
\(B\)
\(C\)
\(D\)
3. \begin{table}[h]
\cline { 3 - 5 } \multicolumn{2}{c|}{}Team B
\cline { 3 - 5 } \multicolumn{2}{c|}{}PaulQaasimRashid
\multirow{3}{*}{Team A}Mischa4- 62
\cline { 2 - 5 }Noel0- 26
\cline { 2 - 5 }Olive- 620
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} 4. .
Edexcel FD2 AS 2021 June Q1
  1. Five workers, A, B, C, D and E, are available to complete four tasks, P, Q, R and S.
Each task must be assigned to exactly one worker and each worker can do at most one task.
Worker B cannot be assigned to task R.
The amount, in pounds, that each worker will earn if they are assigned to each task is shown in the table below.
\cline { 2 - 5 } \multicolumn{1}{c|}{}\(\mathbf { P }\)\(\mathbf { Q }\)\(\mathbf { R }\)\(\mathbf { S }\)
\(\mathbf { A }\)55565857
\(\mathbf { B }\)6061-64
\(\mathbf { C }\)59606263
\(\mathbf { D }\)64667169
\(\mathbf { E }\)65687266
The Hungarian algorithm is to be used to find the maximum total amount that can be earned by the five workers.
  1. Explain how the table should be modified to allow the Hungarian algorithm to be used, giving reasons for your answer.
  2. Reducing rows first, use the Hungarian algorithm to obtain the maximum possible total earnings. You should explain how any initial row and column reductions were made and how you determined if the table was optimal at each stage.
Edexcel FD2 AS 2021 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{261e22b8-0063-419c-a388-6831a427fb65-03_860_1705_276_182} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a capacitated, directed network. The number on each arc represents the capacity of that arc. The numbers in circles represent an initial flow.
  1. State the value of the initial flow.
    (1)
  2. Obtain the capacity of the cut that passes through the arcs \(\mathrm { AG } , \mathrm { CG } , \mathrm { GF } , \mathrm { FT } , \mathrm { FH }\) and EH .
    (1)
  3. Complete the initialisation of the labelling procedure on Diagram 1 in the answer book by entering values along \(\mathrm { SD } , \mathrm { BD } , \mathrm { BE }\) and GF .
    (2)
  4. Use the labelling procedure to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow.
    (3)
  5. Use the answer to part (d) to add a maximum flow pattern to Diagram 2 in the answer book.
    (1)
  6. Prove that your answer to part (e) is optimal.
    (3)
Edexcel FD2 AS 2021 June Q3
3. In your answer to this question you must show detailed reasoning. A two-person zero-sum game is represented by the following pay-off matrix for player A.
\cline { 2 - 3 } \multicolumn{1}{c|}{}B plays \(X\)B plays \(Y\)
A plays \(Q\)4- 3
A plays \(R\)2- 1
A plays \(S\)- 35
A plays \(T\)- 13
  1. Verify that there is no stable solution to this game. Player B plays their option X with probability \(p\).
  2. Use a graphical method to find the optimal value of \(p\) and hence find the best strategy for player B.
  3. Find the value of the game to player A .
  4. Hence find the best strategy for player A .
Edexcel FD2 AS 2021 June Q4
4. Sarah takes out a mortgage of \(\pounds 155000\) to buy a house. Interest is added each month on the outstanding balance at a constant rate of \(r\) \% each month. Sarah makes fixed monthly repayments to reduce the amount owed. Each month, interest is added, and then her monthly repayment is used to reduce the outstanding amount owed. The recurrence relationship for the amount of the mortgage outstanding after \(n + 1\) months is modelled by $$u _ { n + 1 } = 1.0025 u _ { n } - x \quad n \geqslant 0$$ where \(\pounds u _ { n }\) is the amount of the mortgage outstanding after \(n\) months and \(\pounds x\) is the monthly repayment.
  1. State the value of \(r\).
  2. Solve the recurrence relation to find an expression for \(u _ { n }\) in terms of \(x\) and \(n\). Given that the mortgage will be paid off in exactly 30 years,
  3. determine, to 2 decimal places, the least possible value of \(x\). \section*{(Total for Question 4 is 9 marks)} TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS
    END
Edexcel FD2 AS 2022 June Q1
  1. 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 and worker D cannot be assigned to task S.
The time, in minutes, that each worker takes to complete each task is shown in the table below.
\cline { 2 - 5 } \multicolumn{1}{c|}{}PQRS
A54485152
B55515358
C52-5354
D676368-
The Hungarian algorithm is to be used to find the minimum total time for the four workers to complete the tasks.
  1. Modify the table so that the Hungarian algorithm may be used.
  2. Reducing rows first, use the Hungarian algorithm to obtain an allocation that minimises the total time. You should explain how any initial row and column reductions are made and also how you determine if the table is optimal at each stage.
Edexcel FD2 AS 2022 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d7e250dc-9e38-4f65-a51a-c6a08082f310-03_1120_1757_212_153} \captionsetup{labelformat=empty} \caption{Figure 1}
\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.
  2. List the eight saturated arcs.
  3. Explain why arc EH can never be full to capacity.
  4. Find the capacity of
    1. cut \(C _ { 1 }\)
    2. cut \(C _ { 2 }\)
  5. Write down a flow-augmenting route that increases the flow by three units. Given that the flow through the network is increased by three units,
  6. prove that this new flow is maximal.
Edexcel FD2 AS 2022 June Q3
3. Terry and June play a zero-sum game. The pay-off matrix shows the number of points that Terry scores for each combination of strategies.
\cline { 2 - 4 } \multicolumn{2}{c|}{}June
\cline { 3 - 4 } \multicolumn{2}{c|}{}Option XOption Y
\multirow{4}{*}{Terry}Option A14
\cline { 2 - 4 }Option B- 26
\cline { 2 - 4 }Option C- 15
\cline { 2 - 4 }Option D8- 4
  1. Explain the meaning of 'zero-sum' game.
  2. Verify that there is no stable solution to the game.
  3. Write down the pay-off matrix for June.
    1. Find the best strategy for June, defining any variables you use.
    2. State the value of the game to Terry. Let Terry play option A with probability \(t\).
  5. By writing down a linear equation in \(t\), find the best strategy for Terry.
Edexcel FD2 AS 2022 June Q4
4. A sequence \(\left\{ u _ { n } \right\}\), where \(n \geqslant 0\), satisfies the recurrence relation $$u _ { n + 1 } + 3 u _ { n } = n + k$$ where \(k\) is a non-zero constant.
Given that \(u _ { 0 } = 1\)
  1. solve the recurrence relation, giving \(u _ { n }\) in terms of \(k\) and \(n\). Given that \(u _ { n }\) is a linear function of \(n\),
  2. use your answer to part (a) to find the value of \(u _ { 100 }\) TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS END
Edexcel FD2 AS 2023 June Q1
  1. Five workers, A, B, C, D and E, are available to complete four tasks, P, Q, R and S.
Each worker can only be assigned to at most one task, and each task must be done by at most one worker. Worker B cannot be assigned to task Q and worker E cannot be assigned to task S.
The time, in minutes, that each worker takes to complete each task is shown in the table below.
PQRS
A38393737
B39-3940
C41444042
D40413938
E363941-
The Hungarian algorithm is to be used to find the least total time to complete all four tasks.
  1. Explain how the table should be modified so that the Hungarian algorithm can be applied.
    1. Use the Hungarian algorithm to obtain an allocation that minimises the total time.
    2. Explain how you determined if the table was optimal at each stage.
  3. Calculate the least total time to complete all four tasks.
Edexcel FD2 AS 2023 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ddebe845-4280-471b-8da0-cb7211cea756-03_855_1820_210_127} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} An engineer monitors a system of pipes through which a fluid flows from the source, S , to the sink, T . The engineer initialises the labelling procedure for this system, and the excess capacities and potential backflows are shown on the arrows either side of each arc, as shown in Figure 1.
  1. State the value of the initial flow.
  2. Obtain the capacity of the cut that passes through the arcs \(\mathrm { SA } , \mathrm { SB } , \mathrm { CE } , \mathrm { FE }\) and FJ .
  3. Use the labelling procedure to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow.
  4. Use your answer to (c) to draw a maximum flow pattern on Diagram 1 in the answer book.
  5. Prove that the answer to (d) is optimal.
Edexcel FD2 AS 2023 June Q3
3. A two-person zero-sum game is represented by the following pay-off matrix for player \(A\).
\(B\) plays X\(B\) plays Y
\(A\) plays Q2-2
\(A\) plays R-15
A plays S34
\(A\) plays T02
    1. Show that this game is stable.
    2. State the value of this game to player \(B\). Option S is removed from player A's choices and the reduced game, with option S removed, is no longer stable.
  2. Write down the reduced pay-off matrix for player \(B\). Let \(B\) play option X with probability \(p\) and option Y with probability \(1 - p\).
  3. Use a graphical method to find the optimal value of \(p\) and hence find the best strategy for player \(B\) in the reduced game.
    1. Find the value of the reduced game to player \(A\).
    2. State which option player \(A\) should never play in the reduced game.
    3. Hence find the best strategy for player \(A\) in the reduced game.
Edexcel FD2 AS 2023 June Q4
4. A sequence \(\left\{ u _ { n } \right\}\), where \(n \geqslant 0\), satisfies the recurrence relation $$u _ { n + 1 } = \frac { 3 } { 2 } u _ { n } - 2 n ^ { 2 } - 4 \quad u _ { 0 } = k$$ where \(k\) is an integer.
  1. Determine an expression for \(u _ { n }\) in terms of \(n\) and \(k\).
    (6) Given that \(u _ { 10 } > 5000\)
  2. determine the minimum possible value of \(k\).
    (2)
Edexcel FD2 AS 2024 June Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{40023f8e-6874-400e-84b5-60d98b648afc-02_1010_1467_353_399} \captionsetup{labelformat=empty} \caption{Figure 1}
\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.