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OCR D2 Q2
9 marks Moderate -0.3
2. The payoff matrix for player \(A\) in a two-person zero-sum game is shown below.
\cline { 3 - 5 } \multicolumn{2}{c|}{}\(B\)
\cline { 2 - 5 } \multicolumn{2}{c|}{}IIIIII
\multirow{3}{*}{\(A\)}I35- 2
\cline { 2 - 5 }II7\({ } ^ { - } 4\)- 1
\cline { 2 - 5 }III9\({ } ^ { - } 4\)8
  1. Applying the dominance rule, explain, with justification, which strategy can be ignored by
    1. player \(A\),
    2. player \(B\).
  2. For the reduced table, find the optimal strategy for
    1. player \(A\),
    2. player \(B\).
OCR D2 Q3
9 marks Standard +0.3
  1. A sheet is provided for use in answering this question.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{df7b056f-1446-43f1-a2fd-c0d56533550e-3_588_1285_287_296} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a capacitated, directed network. The numbers on each arc indicate the minimum and maximum capacity of that arc. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{df7b056f-1446-43f1-a2fd-c0d56533550e-3_648_1288_1155_296} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a feasible flow through the same network.
  1. Using this as your initial flow pattern, use the labelling procedure to find a maximal flow. You should list each flow-augmenting route you use together with its flow and draw the maximal flow pattern.
    (6 marks)
  2. Find a cut of the same value as your maximum flow and explain why this proves it gives the maximim possible flow.
OCR D2 Q4
11 marks Moderate -0.3
4.
ActivityTimePrecedence
A12
B5
C10
D8A
E5A, B , C
F9C
G11D, E
H6G, F
I6H
J2H
K3I
Construct an activity network to show the tasks involved in widening a bridge over the B451.
  1. Find those tasks which lie on the critical path and list them in order.
  2. State the minimum length of time needed to widen the bridge.
  3. Represent the tasks on a Gantt diagram. Tasks \(F\) and \(J\) each require 3 workers, tasks \(B\), \(D\) and \(I\) each require 2 workers and the remaining tasks each require one worker.
  4. Draw a resource histogram showing how it is possible for a team of 4 workers to complete the project in the minimum possible time.
OCR D2 Q5
11 marks Standard +0.8
  1. A company wishes to plan its production of a particular item over the coming four months based on its current orders. In each month the company can manufacture up to three of the item with the costs according to how many it makes being as follows:
No. of Items Produced0123
Cost in Pounds05500970013100
There are no items in stock at the start of the period and the company wishes to meet all its orders on time and have no stock left at the end of the 4-month period. If any items are not to be supplied in the month they are made there is also a storage cost incurred of \(\pounds 400\) per item per month. The orders for each of the four months being considered are as follows:
MonthMarchAprilMayJune
No. of Orders1241
Use dynamic programming to find how many of the item the company should make in each of these four months in order to minimise the total cost for this period. \section*{Please hand this sheet in for marking} \includegraphics[max width=\textwidth, alt={}, center]{df7b056f-1446-43f1-a2fd-c0d56533550e-6_588_1285_504_276} \includegraphics[max width=\textwidth, alt={}, center]{df7b056f-1446-43f1-a2fd-c0d56533550e-6_588_1280_1361_276}
OCR D2 Q1
4 marks Moderate -0.5
  1. The payoff matrix for player \(A\) in a two-person zero-sum game is shown below.
\cline { 3 - 5 } \multicolumn{2}{c|}{}\(B\)
\cline { 2 - 5 } \multicolumn{2}{c|}{}IIIIII
\multirow{3}{*}{\(A\)}I- 340
\cline { 2 - 5 }II221
\cline { 2 - 5 }III3- 2- 1
Find the optimal strategy for each player and the value of the game. \section*{2. \$ FMMUMLTIP HIR3 UFHGHQFH} \includegraphics[max width=\textwidth, alt={}, center]{06378fdc-2d77-4bdc-810a-1ce9de180c3d-2_49_232_264_310}
ஏ\% 11) \includegraphics[max width=\textwidth, alt={}, center]{06378fdc-2d77-4bdc-810a-1ce9de180c3d-2_49_428_374_310}
ه' ாணம் \includegraphics[max width=\textwidth, alt={}, center]{06378fdc-2d77-4bdc-810a-1ce9de180c3d-2_56_451_484_310} \includegraphics[max width=\textwidth, alt={}, center]{06378fdc-2d77-4bdc-810a-1ce9de180c3d-2_56_442_541_310} \includegraphics[max width=\textwidth, alt={}, center]{06378fdc-2d77-4bdc-810a-1ce9de180c3d-2_49_435_598_310} \includegraphics[max width=\textwidth, alt={}, center]{06378fdc-2d77-4bdc-810a-1ce9de180c3d-2_51_426_651_310} \includegraphics[max width=\textwidth, alt={}, center]{06378fdc-2d77-4bdc-810a-1ce9de180c3d-2_56_485_705_310} \includegraphics[max width=\textwidth, alt={}, center]{06378fdc-2d77-4bdc-810a-1ce9de180c3d-2_54_485_762_310} \includegraphics[max width=\textwidth, alt={}, center]{06378fdc-2d77-4bdc-810a-1ce9de180c3d-2_58_479_817_310} \includegraphics[max width=\textwidth, alt={}, center]{06378fdc-2d77-4bdc-810a-1ce9de180c3d-2_54_460_872_310} \includegraphics[max width=\textwidth, alt={}, center]{06378fdc-2d77-4bdc-810a-1ce9de180c3d-2_56_501_927_310} \section*{Fig. 1} Construct an activity network
Use appropriate forward and backward scanning to find
  1. the minimum number of days needed to complete the entire project,
  2. the activities which lie on the critical path.
OCR D2 Q3
8 marks Moderate -0.8
  1. Arthur is planning a bus journey from town \(A\) to town \(L\). There are various routes he can take but he will have to change buses three times - at \(B , C\) or \(D\), at \(E , F , G\) or \(H\) and at \(I , J\) or \(K\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{06378fdc-2d77-4bdc-810a-1ce9de180c3d-3_760_1410_351_315} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows the bus routes that Arthur can use. The number on each arc shows the average waiting time, in minutes, for a bus to come on that route. As the forecast is for rain, Arthur wishes to plan his journey so that the total waiting time is as small as possible. Use dynamic programming to find the route that Arthur should use.
OCR D2 Q4
10 marks Standard +0.3
4. A construction company has three teams of workers available, each of which is to be assigned to one of four jobs at a site. The following table shows the estimated cost, in tens of pounds, of each team doing each job:
WindowsConservatoryDoorsGreenhouse
Team A2780881
Team B2860571
Team C3090773
Use the Hungarian algorithm to find an allocation of jobs which will minimise the total cost. Show the state of the table after each stage in the algorithm and state the cost of the final assignment.
OCR D2 Q5
11 marks Moderate -0.3
5. The following matrix gives the capacities of the pipes in a system.
To From\(S\)\(T\)\(A\)\(B\)\(C\)D
S--1626--
T------
A----135
B-16---11
C-11----
D-11----
  1. Represent this information as a digraph.
  2. Find the minimum cut, expressing it in the form \(\{ \} \mid \{ \}\), and state its value.
  3. Starting from having no flow in the system, use the labelling procedure to find a maximal flow through the system. You should list each flow-augmenting route you use, together with its flow.
  4. Explain how you know that this flow is maximal.
OCR D2 Q6
21 marks Challenging +1.2
6. The payoff matrix for player \(A\) in a two-person zero-sum game is shown below.
\cline { 3 - 5 } \multicolumn{2}{c|}{}\(B\)
\cline { 3 - 5 } \multicolumn{2}{c|}{}IIIIII
\multirow{2}{*}{\(A\)}I- 23- 1
\cline { 2 - 5 }II4- 52
  1. Formulate this information as a linear programming problem, the solution to which will give the optimal strategy for player \(B\).
    (7 marks)
  2. By solving this linear programming problem, find the optimal strategy for player \(B\) and the value of the game.
OCR D2 Q1
8 marks Moderate -0.8
  1. A linear programming problem is defined as follows:
$$\begin{array} { l l } \text { Maximise } & P = 3 x + 3 y + 4 z \\ \text { subject to } & x + 2 y + z \leq 30 \\ & 5 x + y + 3 z \leq 60 \\ \text { and } & x \geq 0 , y \geq 0 , z \geq 0 . \end{array}$$
  1. Display the problem in a Simplex Tableau.
  2. Starting with a pivot chosen from the \(z\)-column, perform one iteration of your tableau.
  3. Write down the resulting values of \(x , y , z\) and \(P\) and state with a reason whether or not these values give an optimal solution.
OCR D2 Q2
8 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d88fdf1e-7547-434d-ba87-7f816e4386ba-1_627_1116_1190_388} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a capacitated, directed network. The numbers on each arc indicate the maximum capacity of that arc. In addition to the restrictions on flow through the arcs a maximum flow of 6 units is allowed to pass through vertex \(C\).
  1. Redraw the network to take into account this restriction.
  2. Starting with an initial flow of 6 units along SADT and 6 units along SBT use the labelling procedure to find a maximal flow. You should list each flow-augmenting route you use together with its flow and draw the maximal flow pattern.
OCR D2 Q3
10 marks Standard +0.3
3. A travel company offers a touring holiday which stops at four locations, \(A , B , C\) and \(D\). The tour may be taken with the locations appearing in any order, but the number of days spent in each location is dependent on its position in the tour, as shown in the table below.
\multirow{2}{*}{}Stage
1234
A7856
\(B\)6965
C9857
\(D\)7766
Showing the state of the table at each stage, use the Hungarian algorithm to find the order in which to complete the tour so as to maximise the total number of days. State the maximum total number of days that can be spent in the four locations.
OCR D2 Q4
10 marks Moderate -0.8
4. A rally consisting of four stages is being planned. The first stage will begin at \(A\) and the last stage will end at \(L\). Various routes are being considered, with the end of one stage being the start of the next. The organisers want the total length of the route chosen to be as small as possible. The table below shows the length, in miles, of each of the possible stages.
\multirow{2}{*}{}Finishing point
BCDE\(F\)G\(H\)\(I\)\(J\)\(K\)\(L\)
\multirow{11}{*}{Starting point}A54.51310
B8114
C510.5
D96
E12715
\(F\)522
G893
\(H\)1029
I5
J6
K10
Use dynamic programming to find the route which satisfies the wish of the organisers.
OCR D2 Q5
12 marks Moderate -0.8
  1. A project involves six tasks, some of which cannot be started until others have been completed. This is shown in the table below.
TaskDuration (minutes)Immediate predecessors
A18-
B23-
C13\(A , B\)
D9A
E28\(B , D\)
\(F\)23C
  1. Draw an activity network for this project.
  2. By labelling your network, find the critical path and the minimum duration of the project. An extra condition is now imposed. Task \(A\) may not begin until task \(B\) has been underway for at least 10 minutes.
  3. Draw a new network taking into account this restriction.
  4. Find a revised value for the minimum duration of the project and state the new critical path.
OCR D2 Q6
12 marks Standard +0.8
6. The payoff matrix for player \(A\) in a two-person zero-sum game is shown below.
\cline { 3 - 4 } \multicolumn{2}{c|}{}\(B\)
\cline { 3 - 4 }III
\multirow{2}{*}{\(A\)}I4\({ } ^ { - } 8\)
\cline { 2 - 4 }II2\({ } ^ { - } 4\)
\cline { 2 - 4 }III\({ } ^ { - } 8\)2
  1. Explain why the game does not have a saddle point.
  2. Using a graphical method, find the optimal strategy for player \(B\).
  3. Find the optimal strategy for player \(A\).
  4. Find the value of the game.
OCR D2 Q1
7 marks Moderate -0.5
  1. A couple are making the arrangements for their wedding. They are deciding whether to have the ceremony at their church, a local castle or a nearby registry office. The reception will then be held in a marquee, at the castle or at a local hotel. Both the castle and hotel offer catering services but the couple are also considering using Deluxe Catering or Cuisine, who can both provide the food at any venue.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b8eb80d5-5af5-4a8b-8335-6fae95f3aa73-1_938_1514_520_248} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The network in Figure 1 shows the costs incurred (including transport), in hundreds of pounds, according to the choice the couple make for each stage of the day. Use dynamic programming to find how the couple can minimise the total cost of their wedding and state the total cost of this arrangement.
OCR D2 Q2
8 marks Standard +0.3
2. A two-person zero-sum game is represented by the payoff matrix for player \(A\) shown below.
\cline { 3 - 5 } \multicolumn{2}{c|}{}\(B\)
\cline { 3 - 5 } \multicolumn{2}{c|}{}IIIIII
\multirow{2}{*}{\(A\)}I1- 12
\cline { 2 - 5 }II35- 1
  1. Represent the expected payoffs to \(A\) against \(B\) 's strategies graphically and hence determine which strategy is not worth considering for player \(B\).
  2. Find the best strategy for player \(A\) and the value of the game.
OCR D2 Q3
9 marks Easy -1.2
3. Four people are contributing to the entertainment section of an email magazine. For one issue reviews are required for a film, a musical, a ballet and a concert such that each person reviews one show. The people in charge of the magazine will pay each person's expenses and the cost, in pounds, for each reviewer to attend each show are given below.
FilmMusicalBalletConcert
Andrew5201218
Betty6181516
Carlos421915
Davina5161113
Use the Hungarian algorithm to find an optimal assignment which minimises the total cost. State the total cost of this allocation.
OCR D2 Q4
9 marks Standard +0.3
  1. A sheet is provided for use in answering this question.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b8eb80d5-5af5-4a8b-8335-6fae95f3aa73-3_881_1310_319_315} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a capacitated, directed network.
The numbers in bold denote the capacities of each arc.
The numbers in circles show a feasible flow of 48 through the network.
  1. Find the values of \(x\) and \(y\).
    1. Use the labelling procedure to find the maximum flow through this network, listing each flow-augmenting route you use together with its flow.
    2. Show your maximum flow pattern and state its value.
    1. Find a minimum cut, listing the arcs through which it passes.
    2. Explain why this proves that the flow found in part (b) is a maximum.
OCR D2 Q5
12 marks Moderate -0.3
5. A leisure company owns boats of each of the following types: 2-person boats which are 4 metres long and weigh 50 kg .
4-person boats which are 3 metres long and weigh 20 kg .
8-person boats which are 14 metres long and weigh 100 kg .
The leisure company is willing to donate boats to a local sports club to accommodate up to 40 people at any one time. However, storage facilities mean that the combined length of the boats must not be more than 75 metres. Also, it must be possible to transport all the boats on a single trailer which has a maximum load capacity of 600 kg . The club intends to hire the boats out to help with the cost of maintaining them. It plans to charge \(\pounds 10 , \pounds 12\) and \(\pounds 8\) per day, for the 2 -, 4 - and 8 -person boats respectively and wishes to maximise its daily revenue ( \(\pounds R\) ). Let \(x , y\) and \(z\) represent the number of 2-, 4- and 8-person boats respectively given to the club.
  1. Model this as a linear programming problem. Using the Simplex algorithm the following initial tableau is obtained:
    \(R\)\(x\)\(y\)\(z\)\(s\)\(t\)\(u\)
    1\({ } ^ { - } 10\)\({ } ^ { - } 12\)\({ } ^ { - } 8\)0000
    012410020
    0431401075
    0521000160
  2. Explain the purpose of the variables \(s , t\) and \(u\).
  3. By increasing the value of \(y\) first, work out the next two complete tableaus.
  4. Explain how you know that your final tableau gives an optimal solution and state this solution in practical terms.
OCR D2 Q6
15 marks Moderate -0.3
  1. A project consists of the activities listed in the table below. For each activity the table shows how long it will take, which other activites must be completed before it can be done and the number of workers needed to complete it.
ActivityDuration (hours)Immediate Predecessor(s)No. of Workers
A3-9
B2A5
C5\(A\)6
D3C5
E6\(B , D\)2
\(F\)13D5
\(G\)4E6
\(H\)12E4
I3\(F\)4
J5H, I3
K7\(G , J\)8
  1. Draw an activity network for the project.
  2. Find the critical path and the minimum time in which the project can be completed.
  3. Represent all of the activities on a Gantt diagram.
  4. By drawing a resource histogram, find out the maximum number of workers required at any one time if each activity is begun as soon as possible.
  5. Draw another resource histogram to show how the project can be completed in the minimum time possible using a maximum of 10 workers at any one time. Sheet for answering question 4 \section*{Please hand this sheet in for marking} \includegraphics[max width=\textwidth, alt={}, center]{b8eb80d5-5af5-4a8b-8335-6fae95f3aa73-6_729_1227_482_338} \includegraphics[max width=\textwidth, alt={}, center]{b8eb80d5-5af5-4a8b-8335-6fae95f3aa73-6_723_1223_1466_338}
Edexcel AEA 2024 June Q1
7 marks Challenging +1.2
1．In the binomial expansion of $$( 1 - 8 x ) ^ { p } \quad | x | < \frac { 1 } { 8 }$$ where \(p\) is a positive constant，
－the sum of the coefficient of \(x\) and the coefficient of \(x ^ { 2 }\) is equal to the coefficient of \(x ^ { 3 }\) －the coefficient of \(x ^ { 2 }\) is positive
Determine the value of \(p\) ． \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-02_2264_56_315_1977}
Edexcel AEA 2024 June Q2
6 marks Challenging +1.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-04_904_826_255_623} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve defined by the equation $$y ^ { 2 } + 3 y - 6 \sin y = 4 - x ^ { 2 }$$ The point \(P ( x , y )\) lies on the curve．
The distance from the origin，\(O\) ，to \(P\) is \(D\) ．
（a）Write down an equation for \(D ^ { 2 }\) in terms of \(y\) only．
（b）Hence determine the minimum value of \(D\) giving your answer in simplest form． \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-04_2266_53_312_1977}
Edexcel AEA 2024 June Q3
14 marks Challenging +1.8
3．（i）Determine the value of \(k\) such that $$\arctan \frac { 1 } { 2 } - \arctan \frac { 1 } { 3 } = \arctan k$$ （ii）（a）Prove that $$\cos 3 A \equiv 4 \cos ^ { 3 } A - 3 \cos A$$ Given that \(a = \cos 20 ^ { \circ }\) （b）write down，in terms of \(a\) ，an expression for \(\cos 40 ^ { \circ }\) （c）determine，in terms of \(a\) ，a simplified expression for \(\cos 80 ^ { \circ }\) （d）Use part（a）to show that $$4 a ^ { 3 } - 3 a = \frac { 1 } { 2 }$$ （e）Hence，or otherwise，show that $$\cos 20 ^ { \circ } \cos 40 ^ { \circ } \cos 80 ^ { \circ } = \frac { 1 } { 8 }$$
Edexcel AEA 2024 June Q4
16 marks Challenging +1.8
4．（a）Use the substitution \(x = \sqrt { 3 } \tan u\) to show that $$\int \frac { 1 } { 3 + x ^ { 2 } } \mathrm {~d} x = p \arctan ( p x ) + c$$ where \(p\) is a real constant to be determined and \(c\) is an arbitrary constant．
（b）Use the substitution \(x = \frac { 3 u + 3 } { u - 3 }\) to determine the exact value of \(I\) where $$I = \int _ { - 3 } ^ { 1 } \frac { \ln ( 3 - x ) } { 3 + x ^ { 2 } } \mathrm {~d} x$$ giving your answer in simplest form． \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-10_2264_47_314_1984}