Questions — Edexcel FD2 (51 questions)

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Edexcel FD2 2019 June Q1
  1. Table 1 shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , to each of four demand points, \(\mathrm { P } , \mathrm { Q } , \mathrm { R }\) and S . It also shows the stock held at each supply point and the stock required at each demand point. A minimum cost solution is required.
\begin{table}[h]
PQRSSupply
A1514171123
B109161242
C111381018
D1513161719
Demand25451220
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} Table 2 shows an initial solution given by the north-west corner method. \begin{table}[h]
PQRS
A23
B240
C5121
D19
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
  1. Taking DQ as the entering cell, use the stepping-stone method to find an improved solution. Make your method clear.
  2. Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by stating the
    • shadow costs
    • improvement indices
    • route
    • entering cell and exiting cell.
    • Determine whether the solution obtained from this second iteration is optimal, giving a reason for your answer.
    • State the cost of the solution found in (b).
Edexcel FD2 2019 June Q2
  1. Four workers, Ted (T), Harold (H), James (J) and Margaret (M), are to be assigned to four tasks, 1, 2, 3 and 4. Each worker must be assigned to just one task and each task must be done by just one worker.
The profit, in pounds, resulting from allocating each worker to each task, is shown in the table below. The profit is to be maximised.
1234
T103977480
H201155145155
J111807792
M203188137184
  1. Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total profit. You must make your method clear and show the table after each stage.
  2. Determine the resulting total profit.
Edexcel FD2 2019 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5274a614-7862-49f0-ad1d-b59b73aa51ad-04_1047_1691_210_187} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} In Figure 1 the weight of \(\operatorname { arc } \mathrm { SB }\) is denoted by \(x\) where \(x \geqslant 0\)
  1. Explain why Dijkstra's algorithm cannot be used on the directed network in Figure 1.
    (1) It is given that the minimum weight route from S to T passes through B .
  2. Use dynamic programming to find
    1. the range of possible values of \(x\)
    2. the minimum weight route from S to T .
      (12)
Edexcel FD2 2019 June Q4
4.
\multirow{2}{*}{}Player B
Option XOption YOption Z
\multirow{3}{*}{Player A}Option P3-20
Option Q-44-2
Option R12-1
A two person zero-sum game is represented by the pay-off matrix for player A shown above.
  1. Verify that there is no stable solution to this game. Player A intends to make a random choice between options \(\mathrm { P } , \mathrm { Q }\) and R , choosing option P with probability \(p _ { 1 }\), option Q with probability \(p _ { 2 }\) and option R with probability \(p _ { 3 }\) Player A wants to find the optimal values of \(p _ { 1 } , p _ { 2 }\) and \(p _ { 3 }\) using the Simplex algorithm. Player A formulates the following linear programming problem for the game, writing the constraints as inequalities. $$\begin{aligned} & \text { Maximise } P = V
    & \text { subject to } V \geqslant 3 p _ { 1 } - 4 p _ { 2 } + p _ { 3 }
    &
    & V \geqslant - 2 p _ { 1 } + 4 p _ { 2 } + 2 p _ { 3 }
    & V \geqslant - 2 p _ { 2 } - p _ { 3 }
    & p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1
    & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , V \geqslant 0 \end{aligned}$$
  2. Correct the errors made by player A in the linear programming formulation of the game, giving reasons for your answer.
  3. Write down an initial Simplex tableau for the corrected linear programming problem. The Simplex algorithm is used to solve the corrected linear programming problem. The optimal values are \(p _ { 1 } = 0.6 , p _ { 2 } = 0\) and \(p _ { 3 } = 0.4\)
  4. Calculate the value of the game to player A.
  5. Determine the optimal strategy for player B, making your reasoning clear.
Edexcel FD2 2019 June Q5
5. An increasing sequence \(\left\{ u _ { n } \right\}\) for \(n \in \mathbb { N }\) is such that the difference between the \(n\)th term of \(\left\{ u _ { n } \right\}\) and the mean of the previous two terms of \(\left\{ u _ { n } \right\}\) is always 6
  1. Show that, for \(n \geqslant 3\) $$2 u _ { n } - u _ { n - 1 } - u _ { n - 2 } = 12$$ Given that \(u _ { 1 } = 2\) and \(u _ { 2 } = 8\)
  2. find the solution of this second order recurrence relation to obtain an expression for \(u _ { n }\) in terms of \(n\).
  3. Show that as \(n \rightarrow \infty , u _ { n } \rightarrow k n\) where \(k\) is a constant to be determined. You must give reasons for your answer.
Edexcel FD2 2019 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5274a614-7862-49f0-ad1d-b59b73aa51ad-07_983_1513_210_283} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a capacitated, directed network. The network represents a system of pipes through which fluid flows from a source, S , to a sink, T . The numbers ( \(l , u\) ) on each arc represent, in litres per second, the lower capacity, \(l\), and the upper capacity, \(u\), of the corresponding pipe. Two cuts \(C _ { 1 }\) and \(C _ { 2 }\) are shown.
  1. Find the capacity of
    1. \(\operatorname { cut } C _ { 1 }\)
    2. \(\operatorname { cut } C _ { 2 }\)
  2. Explain why the arcs AE and CE cannot be at their upper capacities simultaneously.
  3. Explain why a flow of 31 litres per second through the system is not possible.
  4. Hence determine a minimum feasible flow and a maximum feasible flow through the system. You must draw these feasible flows on the diagrams in the answer book and give reasons to justify your answer. You should not apply the labelling procedure to find these flows.
Edexcel FD2 2021 June Q1
  1. Four workers, A, B, C and D, are to be assigned to three tasks, 1, 2 and 3 . Each task must be assigned to just one worker and each worker can do one task only.
Worker A cannot do task 2 and worker D cannot do task 3
The cost of assigning each worker to each task is shown in the table below.
The total cost is to be minimised.
123
A53-62
B485759
C556358
D6949-
Formulate the above situation as a linear programming problem. You must define your decision variables and make the objective function and constraints clear.
(6) \section*{(Total for Question 1 is 6 marks)}
Edexcel FD2 2021 June Q2
  1. Alka is considering paying \(\pounds 5\) to play a game. The game involves rolling two fair six-sided dice. If the sum of the numbers on the two dice is at least 8 , she receives \(\pounds 10\), otherwise she loses and receives nothing.
If Alka loses, she can pay a further \(\pounds 5\) to roll the dice again. If both dice show the same number then she receives \(\pounds 35\), otherwise she loses and receives nothing.
  1. Draw a decision tree to model Alka’s possible decisions and the possible outcomes.
  2. Determine Alka’s optimal EMV and state the optimal strategy indicated by the decision tree.
Edexcel FD2 2021 June Q3
3. The table below shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , to four sales points, \(\mathrm { P } , \mathrm { Q } , \mathrm { R }\) and S . It also shows the number of units held at each supply point and the number of units required at each sales point. A minimum cost solution is required.
PQRSSupply
A1819171328
B1615141943
C2117222329
D1620192136
Demand25414030
  1. Use the north-west corner method to obtain an initial solution.
  2. Taking AS as the entering cell, use the stepping-stone method to find an improved solution. Make your method clear.
  3. Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by showing the route and stating the
    • shadow costs
    • improvement indices
    • entering cell and exiting cell
    • State the cost of the solution found in (c).
    • Determine whether the solution obtained in (c) is optimal, giving a reason for your answer.
Edexcel FD2 2021 June Q4
  1. Sequences \(\left\{ x _ { n } \right\}\) and \(\left\{ y _ { n } \right\}\) for \(n \in \mathbb { N }\), are defined by
$$\begin{gathered} x _ { n + 1 } = 2 y _ { n } + 3 \quad \text { and } \quad y _ { n + 1 } = 3 x _ { n + 1 } - 4 x _ { n }
x _ { 1 } = 1 \quad \text { and } \quad y _ { 1 } = a \end{gathered}$$ where \(a\) is a constant.
  1. Show that \(x _ { n + 2 } - 6 x _ { n + 1 } + 8 x _ { n } = 3\)
  2. Solve the second-order recurrence relation given in (a) to obtain an expression for \(x _ { n }\) in terms of \(a\) and \(n\). Given that \(x _ { 7 } = 28225\)
  3. find the value of \(a\).
Edexcel FD2 2021 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{262aa0e6-479f-447a-94db-aeb901b3c6fe-5_1095_1666_212_203} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a capacitated, directed network. The network represents a system of pipes through which fluid can flow. The weights on the arcs show the lower and upper capacities for the corresponding pipes, in litres per second.
  1. Calculate 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 flow through the system. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{262aa0e6-479f-447a-94db-aeb901b3c6fe-6_775_1516_169_278} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows an initial flow through the same network.
  3. State the value of the initial flow.
  4. By entering values along \(B C , C F\) and \(D T\), complete the labelling procedure on Diagram 1 in the answer book.
  5. 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.
  6. Use your answer to (e) to find a maximum flow pattern for this system of pipes and draw it on Diagram 2 in the answer book.
  7. Prove that the answer to (f) is optimal. A vertex restriction is now applied to \(B\) so that no more than 16 litres per second can flow through it.
    1. Complete Diagram 3 in the answer book to show this restriction.
    2. State the value of the maximum flow with this restriction.
Edexcel FD2 2021 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{262aa0e6-479f-447a-94db-aeb901b3c6fe-7_782_1426_219_322} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The staged, directed network in Figure 3 represents a series of roads connecting 12 towns, \(S , A , B , C , D , E , F , G , H , I , J\) and \(T\). The number on each arc shows the distance between these towns, in miles. Bradley is planning a four-day cycle ride from \(S\) to \(T\).
He plans to leave his home at \(S\). On the first night he will stay at \(A , B\) or \(C\), on the second night he will stay at \(D , E , F\) or \(G\), on the third night he will stay at \(H , I\) or \(J\), and he will arrive at his friend's house at \(T\) on the fourth day. Bradley decides that the maximum distance he will cycle on any one day should be as small as possible.
  1. Write down the type of dynamic programming problem that Bradley needs to solve.
  2. Use dynamic programming to complete the table in the answer book.
  3. Hence write down the possible routes that Bradley could take.
Edexcel FD2 2021 June Q7
7. Alexis and Becky are playing a zero-sum game. Alexis has two options, Q and R . Becky has three options, \(\mathrm { X } , \mathrm { Y }\) and Z .
Alexis intends to make a random choice between options Q and R , choosing option Q with probability \(p _ { 1 }\) and option R with probability \(p _ { 2 }\) Alexis wants to find the optimal values of \(p _ { 1 }\) and \(p _ { 2 }\) and formulates the following linear programme, writing the constraints as inequalities. $$\begin{aligned} & \text { Maximise } P = V
& \text { where } V = 3 + \text { the value of the gan }
& \text { subject to } V \leqslant 6 p _ { 1 } + p _ { 2 }
& \qquad \begin{aligned} & V \leqslant 8 p _ { 2 }
& V \leqslant 4 p _ { 1 } + 2 p _ { 2 }
& p _ { 1 } + p _ { 2 } \leqslant 1
& p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , V \geqslant 0 \end{aligned} \end{aligned}$$
  1. Complete the pay-off matrix for Alexis in the answer book.
  2. Use a graphical method to find the best strategy for Alexis.
  3. Calculate the value of the game to Alexis. Becky intends to make a random choice between options \(\mathrm { X } , \mathrm { Y }\) and Z , choosing option X with probability \(q _ { 1 }\), option Y with probability \(q _ { 2 }\) and option Z with probability \(q _ { 3 }\)
  4. Determine the best strategy for Becky, making your method and working clear.
Edexcel FD2 2023 June Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae354c58-6de8-4f94-b404-2f0feecb5bf3-02_953_1687_251_191} \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 that pipe. The numbers in circles represent a feasible flow from S to T .
  1. State the value of the feasible flow.
    (1)
  2. Find the capacity of cut \(\mathrm { C } _ { 1 }\) and the capacity of cut \(\mathrm { C } _ { 2 }\)
    (2)
  3. By inspection, find a flow-augmenting route to increase the flow by two units. You must state your route.
    (1)
  4. Prove that, once the flow-augmenting route found in (c) has been applied, the flow is now maximal.
    (3)
Edexcel FD2 2023 June Q2
2. An outdoor theatre is holding a summer gala performance. The theatre owner must decide whether to take out insurance against rain for this performance. The theatre owner estimates that
  • on a fine day, the total profit will be \(\pounds 15000\)
  • on a wet day, the total loss will be \(\pounds 20000\)
Insurance against rain costs \(\pounds 2000\). If the performance must be cancelled due to rain, then the theatre owner will receive \(\pounds 16000\) from the insurer. If the performance is not cancelled due to rain, then the theatre owner will receive nothing from the insurer. The probability of rain on the day of the gala performance is 0.2
Draw a decision tree and hence determine whether the theatre owner should take out the insurance against rain for this performance.
Edexcel FD2 2023 June Q3
3. The table below shows the stock held at each supply point and the stock required at each demand point in a standard transportation problem. The table also shows the cost, in pounds, of transporting the stock from each supply point to each demand point.
\cline { 2 - 5 } \multicolumn{1}{c|}{}QRSSupply
A23181245
B8101427
C11142134
D19151150
Demand753744
The problem is partially described by the linear programming formulation below.
Let \(x _ { i j }\) be the number of units transported from i to j $$\begin{aligned} & \text { where } \quad i \in \{ A , B , C , D \}
& \quad j \in \{ Q , R , S \} \text { and } x _ { i j } \geqslant 0
& \text { Minimise } P = 23 x _ { A Q } + 18 x _ { A R } + 12 x _ { A S } + 8 x _ { B Q } + 10 x _ { B R } + 14 x _ { B S }
& \quad + 11 x _ { C Q } + 14 x _ { C R } + 21 x _ { C S } + 19 x _ { D Q } + 15 x _ { D R } + 11 x _ { D S } \end{aligned}$$
  1. Write down, as inequalities, the constraints of the linear program.
  2. Use the north-west corner method to obtain an initial solution to this transportation problem.
  3. Taking AS as the entering cell, use the stepping-stone method to find an improved solution. Make your route clear.
  4. Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by showing the route and the
    • shadow costs
    • improvement indices
    • entering cell and exiting cell
Edexcel FD2 2023 June Q4
  1. Four students, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , are to be allocated to four rounds, \(1,2,3\) and 4 , in a competition. Each student is to take part in exactly one round and no two students may play in the same round.
Each student has been given an estimated score for each round. The estimated scores for each student are shown in the table below.
\cline { 2 - 5 } \multicolumn{1}{c|}{}1234
A34201815
B49311234
C48272326
D52454242
  1. Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total estimated score. You must make your method clear and show the table after each stage.
  2. Find this total estimated score.
Edexcel FD2 2023 June Q5
5. A sequence \(\left\{ u _ { n } \right\}\), where \(\mathrm { n } \geqslant 0\), satisfies the second order recurrence relation $$u _ { n + 2 } = \frac { 1 } { 2 } \left( u _ { n + 1 } + u _ { n } \right) + 3 \text { where } u _ { 0 } = 15 \quad u _ { 1 } = 20$$
  1. By considering the sequence \(\left\{ v _ { n } \right\}\), where \(u _ { n } = v _ { n } + 2 n\) for \(\mathrm { n } \geqslant 0\), determine an expression for \(u _ { n }\) as a function of n .
  2. Describe the long-term behaviour of \(u _ { n }\)
Edexcel FD2 2023 June Q6
6. Polly is a motivational speaker who is planning her engagements for the next four weeks. Polly will
  • visit four different countries in these four weeks
  • visit just one country each week
  • leave from her home, S , and return there only after visiting the four countries
  • travel directly from one country to the next
Polly wishes to determine a schedule of four countries to visit.
Table 1 shows the countries Polly could visit each week. \begin{table}[h]
Week1234
Possible countries to visitA or BC, D or EF or GH, I or J
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} Table 2 shows the speaker fee, in \(\pounds 100\) s, Polly would expect to earn in each country. \begin{table}[h]
CountryABCDEFGHIJ
Earnings in \(\boldsymbol { \pounds } \mathbf { 1 0 0 s }\)47454847494445474948
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table} Table 3 shows the cost, in \(\pounds 100\) s, of travelling between the countries. \begin{table}[h]
ABCDEFGHIJ
S52788
A345
B546
C75
D67
E76
F678
G786
\captionsetup{labelformat=empty} \caption{Table 3}
\end{table} Polly's expected income is the value of the speaker fee minus the cost of travel.
She wants to find a schedule that maximises her total expected income for the four weeks. Use dynamic programming to determine the optimal schedule. Complete the table provided in the answer book and state the maximum expected income.
(13)
Edexcel FD2 2023 June Q7
7. Martina decides to open a bank account to help her to save for a holiday. Each month she puts \(\pounds \mathrm { k }\) into the account and allows herself to spend one quarter of what was in the account at the end of the previous month. Let \(u _ { n }\) (where \(\mathrm { n } \geqslant 1\) ) represent the amount in the account at the end of month n .
Martina has \(\pounds \mathrm {~K}\) in the account at the end of the first month.
  1. By setting up a first order recurrence relation for \(u _ { n + 1 }\) in terms of \(u _ { n }\), determine an expression for \(u _ { n }\) in terms of n and k At the end of the 8th month, Martina needs to have at least \(\pounds 1750\) in the account to pay for her holiday.
  2. Determine, to the nearest penny, the minimum amount of money that Martina should put into the account each month.
Edexcel FD2 2023 June Q8
8. A two-person zero-sum game is represented by the pay-off matrix for player A shown below. \section*{Player B} Player A
\cline { 2 - 4 } \multicolumn{1}{c|}{}Option XOption YOption Z
Option Q- 325
Option R2- 10
Option S4- 2- 1
Option T- 402
  1. Verify that there is no stable solution to this game.
  2. Explain why player A should never play option T. You must make your reasoning clear. Player A intends to make a random choice between options \(\mathrm { Q } , \mathrm { R }\) and S , choosing option \(Q\) with probability \(p _ { 1 }\), option \(R\) with probability \(p _ { 2 }\) and option \(S\) with probability \(p _ { 3 }\) Player A wants to calculate the optimal values of \(p _ { 1 } , p _ { 2 }\) and \(p _ { 3 }\) using the Simplex algorithm.
    1. Formulate the game as a linear programming problem for player A. You should write the constraints as equations.
    2. Write down an initial Simplex tableau for this linear programming problem, making your variables clear. The linear programming problem is solved using the Simplex algorithm. The optimal value of \(p _ { 1 }\) is \(\frac { 6 } { 11 }\) and the optimal value of \(p _ { 2 }\) is 0
  4. Find the best strategy for player B, defining any variables you use.
Edexcel FD2 2024 June Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{931ccf1d-4b02-448c-b492-846b0f42c057-02_696_1347_214_367} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a capacitated, directed network of pipes. The numbers in circles represent an initial flow from S to T . The other number on each arc represents the capacity, in litres per second, of the corresponding pipe.
    1. State the value of \(x\)
    2. State the value of \(y\)
  2. State the value of the initial flow.
  3. State the capacity of cut \(C _ { 1 }\)
  4. Find, by inspection, a flow-augmenting route to increase the flow by four units. You must state your route. The flow-augmenting route from (d) is used to increase the flow from S to T .
  5. Prove that the flow is now maximal. A vertex restriction is now applied so that no more than 12 litres per second can flow through E.
    1. Complete Diagram 1 in the answer book to show this restriction.
    2. State the value of the maximum flow through the network with this restriction.
Edexcel FD2 2024 June Q2
2. The general solution of the first order recurrence relation $$u _ { n + 1 } + a u _ { n } = b n ^ { 2 } + c n + d \quad n \geqslant 0$$ is given by $$u _ { n } = A ( 3 ) ^ { n } + 5 n ^ { 2 } + 1$$ where \(A\) is an arbitrary non-zero constant.
By considering expressions for \(u _ { n + 1 }\) and \(u _ { n }\), find the values of the constants \(a , b , c\) and \(d\).
Edexcel FD2 2024 June Q3
3. The table below shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { E } , \mathrm { F } , \mathrm { G }\) and H , to three sales points, \(\mathrm { A } , \mathrm { B }\) and C . It also shows the stock held at each supply point and the amount required at each sales point.
A minimum cost solution is required.
ABCSupply
E23282221
F26192932
G29242029
H24261923
Demand451923
  1. Explain why it is necessary to add a dummy demand point.
  2. On Table 1 in the answer book, insert appropriate values in the dummy demand column, D. After finding an initial feasible solution and applying one iteration of the stepping-stone method, the table becomes
    \(A\)\(B\)\(C\)\(D\)
    \(E\)21
    \(F\)1913
    \(G\)623
    \(H\)518
  3. Starting with GD as the next entering cell, perform two further iterations of the stepping-stone method to obtain an improved solution. You must make your method clear by showing your routes and stating the
    • shadow costs
    • improvement indices
    • entering and exiting cells
    • State the cost of the solution found in (c).
    • Determine whether the solution obtained in (c) is optimal, giving a reason for your answer.
Edexcel FD2 2024 June Q4
  1. Four workers, A, B, C and D, are to be assigned to four tasks, P, Q, R and S.
Each task must be assigned to just one worker and each worker can do only one task.
Worker B cannot be assigned to task Q and worker D cannot be assigned to task R.
The amount, in pounds, that each worker would earn when assigned to each task is shown in the table below.
PQRS
A65726975
B71-6865
C70697377
D7370-71
The Hungarian algorithm can be used to find the maximum total amount that would be earned by the four workers.
    1. Explain how to modify the table so that the Hungarian algorithm could be applied.
    2. Modify the table as described in (a)(i).
  2. Formulate the above situation as a linear programming problem. You must define the decision variables and make the objective function and constraints clear.