Questions — Edexcel (10514 questions)

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Edexcel D1 Q5
11 marks Moderate -0.5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63f3ba38-8bca-4684-957f-aca7104f2f3e-05_863_1061_223_360} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The network in Figure 2 represents the streets near Randolph Square in Newtown, Edinburgh. The weightings represent the average time, in seconds, taken to travel along each road in either direction. The large values for the roads \(C D\) and \(J L\) are due to traffic lights. In the course of his work, a van driver must travel along each road at least once each day. He starts and finishes at his depot, \(F\), and wishes to minimise the total time that this takes.
  1. Describe an appropriate algorithm that can be used to find this minimum time.
    (3 marks)
  2. Apply this algorithm to find a route that the driver can take and state the total time he can expect to spend on this journey each day.
    (5 marks)
    The section of road \(A B\) is to be turned into a pedestrian precinct.
  3. Assuming that the driver must still travel along all the other roads at least once each day, find the modified time he can expect to spend on his daily journey Comment on your answer.
    (3 marks)
Edexcel D1 Q6
16 marks Moderate -0.3
6. The table below shows the maximum flows possible within a system.
To From\(S\)\(A\)\(B\)\(C\)D\(T\)
S-353055--
A-----50
B-12-8-20
C----1530
D-----14
T------
For example, the maximum flow from \(B\) to \(A\) is 12 units.
  1. Draw a digraph to represent this information.
  2. Give the capacity of the cut \(\{ S , A , B , C \} \mid \{ D , T \}\).
  3. Find the minimum cut, stating its capacity, and expressing it in the form \(\{ \quad \} \mid \{ \quad \}\).
  4. Use the labelling procedure to find the maximum flow from \(S\) to \(T\). You should list each flow-augmenting route you find together with its flow.
  5. Explain how you know that you have found the maximum possible flow.
  6. Give an example of a practical situation that could be modelled by the original table.
Edexcel D1 Q7
16 marks Standard +0.8
7. A fitness centre runs introductory courses aimed at the following groups of customers: Pensioners, who will be charged \(\pounds 4\) for a 2 -hour session.
Other adults, who will be charged \(\pounds 10\) for a 4 -hour session.
Children, who will be charged \(\pounds 2\) for a 1 -hour session.
Let the number of pensioners, other adults, and children be \(x , y\) and \(z\) respectively.
Regulations state that the number of pensioners, \(x\), must be at most 5 more than the number of adults, \(y\). There must also be at least twice as many adults, \(y\), as there are children, \(z\). The centre is able to supervise up to 40 person-hours each day at the centre and wishes to maximise the revenue \(( \pounds R )\) that can be earned each day from these sessions. You may assume that the places on any courses that the centre runs will be filled.
  1. Modelling this situation as a linear programming problem, write down the constraints and objective function in terms of \(x , y\) and \(z\). Using the Simplex algorithm, the following initial tableau is obtained.
  2. \(\_\_\_\_\)
Edexcel D1 Q1
6 marks Easy -1.2
  1. Make plane drawings of each of the graphs shown in Figure 1. Graph 1 \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e1fd42f7-c97c-4bf2-92d3-69afc8bb6e29-02_1155_664_278_529} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure}
  2. State the name given to Graph 1 and write down the features that identify it.
  3. State, with a reason, whether it is possible to add further arcs to Graph 2 such that it remains a simple connected graph. No further vertices may be added.
    (1 mark)
Edexcel D1 Q2
7 marks Moderate -0.8
2. This question should be answered on the sheet provided. A builder is going to put up houses on a plot of land of area \(12000 \mathrm {~m} ^ { 2 }\).
There are 5 designs to choose from and no more than one of each design can be built.
DesignKendalMilvertonArlingtonElfordGrosvenor
Plot area ('000 \(\mathrm { m } ^ { 2 }\) )3113510
Value ( \(\pounds ^ { \prime } 000 \mathrm {~s}\) )1001904080120
The builder needs to work out which houses he should build in order to maximise the total value of the site. He does this using a tree diagram and each "branch" on the tree is terminated when the total area of land on that branch exceeds \(12000 \mathrm {~m} ^ { 2 }\).
    1. Complete the tree diagram so that each branch is terminated or all choices have been considered.
    2. Hence, determine which designs the builder should use and the total value that the site will have when completed.
  2. Explain how this method could be altered if more than one of each design is allowed.
Edexcel D1 Q3
7 marks Easy -1.2
3. (a) Draw a graph with 6 vertices, each of degree 1 .
(b) Draw two graphs with 6 vertices, each of degree 2 such that:
  1. the graph is connected,
  2. the graph is not connected. A simple connected graph has 5 vertices each of degree \(x\).
    (c) Find the possible values of \(x\) and explain your answer.
    (d) For each value of \(x\) you found in part (c) draw a possible graph.
Edexcel D1 Q4
12 marks Standard +0.3
4. A company produces \(x _ { 1 }\) finished articles at the end of January, \(x _ { 2 }\) finished articles at the end of February, \(x _ { 3 }\) finished articles at the end of March, \(x _ { 4 }\) finished articles at the end of April. Other details for each month are as follows:
MonthJanuaryFebruaryMarchApril
Demand at end of month200350250200
Production costs per article£1000£1800£1600£1900
The cost of storing each finished but unsold article is \(\pounds 500\) per month. Thus, for example, any article unsold at the end of January would incur a \(\pounds 500\) charge if it is stored until the end of February or a \(\pounds 1000\) charge if it is stored until the end of March. There must be no unsold stock at the end of April.
The selling price of each article is \(\pounds 4000\) and the total profit ( \(\pounds P\) ) must be maximised.
  1. Rewrite \(x _ { 4 }\) in terms of the other 3 variables.
  2. Show that the total cost incurred \(( \pounds C )\) is given by: $$C = 600 x _ { 1 } + 900 x _ { 2 } + 200 x _ { 3 } + 1125000$$
  3. Hence, show that \(P = { } ^ { - } 600 x _ { 1 } - 900 x _ { 2 } - 200 x _ { 3 } + 2875000\).
  4. Three of the constraints operating can be expressed as \(x _ { 1 } \geq 200\), \(x _ { 2 } \geq 0\) and \(x _ { 3 } \geq 0\). Write down inequalities representing two further constraints.
    (2 marks)
  5. Explain why it is not appropriate to use a graphical method to solve this problem.
  6. An employee of the company wishes to use the Simplex algorithm to solve the problem. He tries to generate an initial tableau with \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) as the non-basic variables. Explain why this is not appropriate and explain what he should do instead. You are not required to generate an initial tableau or to solve the problem.
    (2 marks)
Edexcel D1 Q5
12 marks Moderate -0.5
5. This question should be answered on the sheet provided. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e1fd42f7-c97c-4bf2-92d3-69afc8bb6e29-05_956_1561_312_242} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a weighted network representing the paths in a certain part of St. Andrews. The numbers on the arcs represent the lengths of the paths in metres.
  1. Use Dijkstra's algorithm to find a route of minimum length from \(P\) to \(F\). You do not need to consider routes via vertex \(Q\). You must show clearly:
    1. the order in which you labelled the vertices,
    2. how you found a route of minimum length from your labelling. Each night a security guard walks along each of the paths in Figure 2 at least once.
  2. The security office is at vertex \(A\), so she must start and finish her inspection at \(A\). Find the minimum distance that she must walk each night.
    (4 marks)
Edexcel D1 Q6
15 marks Standard +0.3
6. This question should be answered on the sheet provided. A town has adopted a one-way system to cope with recent problems associated with congestion in one area. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e1fd42f7-c97c-4bf2-92d3-69afc8bb6e29-06_670_1301_459_331} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 models the one-way system as a capacitated directed network. The numbers on the arcs are proportional to the number of vehicles that can pass along each road in a given period of time.
  1. Find the capacity of the cut which passes through the \(\operatorname { arcs } A E , B F , B G\) and \(C D\).
    (2 marks) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e1fd42f7-c97c-4bf2-92d3-69afc8bb6e29-07_659_1278_196_335} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure} Figure 4 shows a feasible flow of 17 through the same network. For convenience, a supersource, \(S\), and a supersink, \(T\), have been used.
    1. Use the labelling procedure to find the maximum flow through this network. You must list each flow-augmenting route you use together with its flow.
    2. Show your maximum flow pattern and state its value.
  3. Prove that your flow is the maximum possible through the network.
  4. It is suggested that the maximum flow through the network could be increased by making road \(E F\) undirected, so that it has a capacity of 8 in either direction. Using the maximum flow-minimum cut theorem, find the increase in maximum flow this change would allow.
    (2 marks)
  5. An alternative suggestion is to widen a single road in order to increase its capacity. Which road, on its own, could lead to the biggest improvement, and what would be the largest maximum flow this could achieve.
    (2 marks)
Edexcel D1 Q7
16 marks Moderate -0.8
7. A project involves six tasks, some of which cannot be started until others have been completed. This is shown in the table below. \includegraphics[max width=\textwidth, alt={}, center]{e1fd42f7-c97c-4bf2-92d3-69afc8bb6e29-09_2036_1555_349_248}
  1. \(\_\_\_\_\)
  2. \(\_\_\_\_\) \section*{Sheet for answering question 5} NAME \section*{Please hand this sheet in for marking} Sheet for answering question 6
    NAME \section*{Please hand this sheet in for marking}
      1. \includegraphics[max width=\textwidth, alt={}, center]{e1fd42f7-c97c-4bf2-92d3-69afc8bb6e29-11_666_1280_461_374}
      2. \includegraphics[max width=\textwidth, alt={}, center]{e1fd42f7-c97c-4bf2-92d3-69afc8bb6e29-11_657_1276_1356_376} Maximum Flow = \(\_\_\_\_\)
    3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
    4. \(\_\_\_\_\)
    5. \(\_\_\_\_\)
Edexcel D1 Q1
7 marks Easy -1.8
  1. Use the binary search algorithm to locate the name PENICUIK in the following list. \begin{displayquote} ANKERDINE CULROSS DUNOON ELGIN FORFAR FORT WILLIAM HADDINGTON KINCARDINE LARGS MALLAIG MONTROSE PENICUIK ST. ANDREWS THURSO
  2. Use the same algorithm to attempt to locate PENDINE.
  3. Explain the purpose of the mid-point in dividing up the ordered list when using this algorithm. \end{displayquote}
Edexcel D1 Q2
7 marks Moderate -0.8
  1. The following lengths of cloth (in metres) are to be cut from standard 24 metre rolls.
$$\begin{array} { l l l l l l l l l l l l } 6 & 6 & 4 & 6 & 8 & 8 & 4 & 12 & 14 & 6 & 14 & 8 \end{array}$$
  1. Considering only the total amount, what is the least number of rolls that are needed?
  2. Using the first-fit decreasing algorithm, show that the lengths could be cut from 5 rolls.
  3. Using "full-bin" combinations show that it is possible to cut these lengths from the number of rolls found in part (a).
  4. Comment on this result.
Edexcel D1 Q3
11 marks Moderate -0.8
3. This question should be answered on the sheet provided. A firm of auditors is to place one trainee accountant at each of its five offices. These are situated in Dundee, Edinburgh, Glasgow and London. There are two offices in London, one of which is the company's Head Office. The table summarises the trainees' preferences.
TraineePreferences
\(P\)Dundee, London (either)
\(Q\)Dundee, Edinburgh, Glasgow
\(R\)Glasgow, London (Head Office only)
SEdinburgh
\(T\)Edinburgh
  1. Draw a bipartite graph to model this situation.
  2. Explain why no complete matching is possible. Trainee \(T\) is persuaded that either office in London would be just as good for her personal development as the Edinburgh office.
  3. Draw a second bipartite graph to model the changed situation. In an initial matching, trainee \(P\) is placed in the Dundee office, trainee \(R\) in the Glasgow office, and trainee \(S\) in the Edinburgh office.
  4. Draw this initial matching.
  5. Starting from this initial matching, use the maximum matching algorithm to find a complete matching. You must indicate how the algorithm has been applied, stating clearly the alternating paths you use and the final matching.
    (7 marks)
Edexcel D1 Q4
11 marks Standard +0.3
4. The following matrix gives the capacities of the pipes in a system.
To FromS\(T\)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.
Edexcel D1 Q5
11 marks Moderate -0.5
5. This question should be answered on the sheet provided. An algorithm is described by the flow chart shown in Figure 1 below. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e518ab0-9852-4d1d-a4c9-344a5edf9547-05_1337_937_388_404} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Complete the table on the answer sheet recording the results of each instruction as the algorithm is applied and state the final output.
  2. Explain what the algorithm achieves.
  3. Attempt to apply the algorithm again, with the initial value of \(a\), as specified in Box 2, changed to 5 . Explain what happens.
    (2 mark)
  4. Find the set of positive initial values of \(a\) for which the algorithm will work.
    (2 marks)
Edexcel D1 Q6
14 marks Moderate -0.5
6. The manager of a new leisure complex needs to maximise the Revenue \(( \pounds R )\) from providing the following two weekend programmes.
\(\frac { \text { Participants } } { \text { Children } }\)7 hours windsurfing, 2 hours sailing\(\frac { \text { Revenue } } { \pounds 50 }\)
Adults5 hours windsurfing, 6 hours sailing\(\pounds 100\)
The following restrictions apply to each weekend.
No more than 90 participants can be accommodated.
There must be at most 40 adults.
A maximum of 600 person-hours of windsurfing can be offered.
A maximum of 300 person-hours of sailing can be offered.
  1. Formulate the above information as a linear programming problem, listing the constraints as inequalities and stating the objective function \(R\).
  2. On graph paper, illustrate the constraints, indicating clearly the feasible region.
  3. Solve the problem graphically, stating how many adults and how many children should be accepted each weekend and what the revenue will be. The manager is considering buying more windsurfing equipment at a cost of \(\pounds 2000\). This would increase windsurfing provision by \(10 \%\).
  4. State, with a reason, whether such a purchase would be cost effective.
Edexcel D1 Q7
14 marks Standard +0.3
7. This question should be answered on the sheet provided. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e518ab0-9852-4d1d-a4c9-344a5edf9547-07_576_1360_331_278} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows an activity network modelling the tasks involved in widening a bridge over the B451. The arcs represent the tasks and the numbers in brackets gives the time, in days, to complete each task.
  1. Find the early and late times for each event.
  2. Determine those activities which lie on the critical path and list them in order.
  3. State the minimum length of time needed to widen the bridge. Each task needs a single worker.
  4. Show that two men would not be sufficient to widen the bridge in the shortest time.
    (2 marks)
  5. Draw up a schedule showing how 3 men could complete the project in the shortest time. \section*{Please hand this sheet in for marking}
    1. Complete matching:
      \(P\)\(\bullet\)\(\bullet\)\(D\)
      \(Q\)\(\bullet\)\(\bullet\)\(G\)
      \(R\)\(\bullet\)\(\bullet\)\(E\)
      \(S\)\(\bullet\)\(\bullet\)\(L ( H )\)
      \(T\)\(\bullet\)\(\bullet\)\(L\)
      \section*{Please hand this sheet in for marking}
    2. \(x\)\(a\)\(b\)\(( a - b ) < 0.01\) ?
      1005026No
      -2614.923No
      Final output
    3. \(\_\_\_\_\)
    4. \(x\)\(a\)\(b\)\(( a - b ) < 0.01 ?\)
      100
    5. \(\_\_\_\_\) \section*{Please hand this sheet in for marking}
    6. \includegraphics[max width=\textwidth, alt={}, center]{1e518ab0-9852-4d1d-a4c9-344a5edf9547-11_768_1689_427_221}
    7. \(\_\_\_\_\)
    8. \(\_\_\_\_\)
    9. 051015202530354045505560
      Worker 1
      Worker 2
    10. 051015202530354045505560
      Worker 1
      Worker 2
      Worker 3
Edexcel D2 2006 January Q1
13 marks Moderate -0.8
  1. A theme park has four sites, A, B, C and D, on which to put kiosks. Each kiosk will sell a different type of refreshment. The income from each kiosk depends upon what it sells and where it is located. The table below shows the expected daily income, in pounds, from each kiosk at each site.
Hot dogs and beef burgers (H)Ice cream (I)Popcorn, candyfloss and drinks (P)Snacks and hot drinks (S)
Site A267272276261
Site B264271278263
Site C267273275263
Site D261269274257
Reducing rows first, use the Hungarian algorithm to determine a site for each kiosk in order to maximise the total income. State the site for each kiosk and the total expected income. You must make your method clear and show the table after each stage.
(Total 13 marks)
Edexcel D2 2006 January Q2
12 marks Standard +0.8
2. An engineering firm makes motors. They can make up to five in any one month, but if they make more than four they have to hire additional premises at a cost of \(\pounds 500\) per month. They can store up to two motors for \(\pounds 100\) per motor per month. The overhead costs are \(\pounds 200\) in any month in which work is done.
Motors are delivered to buyers at the end of each month. There are no motors in stock at the beginning of May and there should be none in stock after the September delivery. The order book for motors is:
MonthMayJuneJulyAugustSeptember
Number of motors33754
Use dynamic programming to determine the production schedule that minimises the costs, showing your working in the table provided below.
Stage (month)State (Number in store at start of month)Action (Number made in month)Destinatio n (Number in store at end of month)Value (cost)
\section*{Production schedule}
MonthMayJuneJulyAugustSeptember
Number to be
made
Total cost: \(\_\_\_\_\)
Edexcel D2 2006 January Q3
8 marks Moderate -0.5
3. Three depots, F, G and H, supply petrol to three service stations, S, T and U. The table gives the cost, in pounds, of transporting 1000 litres of petrol from each depot to each service station. F, G and H have stocks of 540000,789000 and 673000 litres respectively.
S, T and U require 257000,348000 and 412000 litres respectively. The total cost of transporting the petrol is to be minimised.
STU
F233146
G353851
H415063
Formulate this problem as a linear programming problem. Make clear your decision variables, objective function and constraints.
Edexcel D2 2006 January Q4
16 marks Moderate -0.8
4. The following minimising transportation problem is to be solved.
JKSupply
A12159
B81713
C4912
Demand911
  1. Complete the table below.
    JKLSupply
    A12159
    B81713
    C4912
    Demand91134
  2. Explain why an extra demand column was added to the table above. A possible north-west corner solution is:
    JKL
    A90
    B112
    C12
  3. Explain why it was necessary to place a zero in the first row of the second column. After three iterations of the stepping-stone method the table becomes:
    JKL
    A81
    B13
    C93
  4. Taking the most negative improvement index as the entering square for the stepping stone method, solve the transportation problem. You must make your shadow costs and improvement indices clear and demonstrate that your solution is optimal.
Edexcel D2 2006 January Q5
13 marks Moderate -0.5
5. A two-person zero-sum game is represented by the following pay-off matrix for player A.
B plays 1B plays 2B plays 3B plays 4
A plays 1- 213- 1
A plays 2- 1321
A plays 3- 420- 1
A plays 41- 2- 13
  1. Verify that there is no stable solution to this game.
  2. Explain why the \(4 \times 4\) game above may be reduced to the following \(3 \times 3\) game.
  3. Formulate the \(3 \times 3\) game as a linear programming problem for player A. Write the
    - 213
    - 132
    1- 2- 1
    constraints as inequalities. Define your variables clearly.
Edexcel D2 2006 January Q6
13 marks Moderate -0.5
6. The network in the figure above, shows the distances in km , along the roads between eight towns, A, B, C, D, E, F, G and H. Keith has a shop in each town and needs to visit each one. He wishes to travel a minimum distance and his route should start and finish at A . By deleting D, a lower bound for the length of the route was found to be 586 km .
By deleting F, a lower bound for the length of the route was found to be 590 km .
  1. By deleting C, find another lower bound for the length of the route. State which is the best lower bound of the three, giving a reason for your answer.
  2. By inspection complete the table of least distances. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{(8)
    (8)
    (Total 13 marks)} \includegraphics[alt={},max width=\textwidth]{a5d69a77-c196-483c-a550-1a55363555af-3_780_889_1069_1078}
    \end{figure} (4) The table can now be taken to represent a complete network. The nearest neighbour algorithm was used to obtain upper bounds for the length of the route: Starting at D, an upper bound for the length of the route was found to be 838 km .
    Starting at F, an upper bound for the length of the route was found to be 707 km .
  3. Starting at C , use the nearest neighbour algorithm to obtain another upper bound for the length of the route. State which is the best upper bound of the
    ABCDEFGH
    A-848513817314952
    B84-13077126213222136
    C85130-53888392
    D1387753-49190
    E1731268849-100180215
    F21383100-163115
    G14922292180163-97
    H5213619021511597-
    three, giving a reason for your answer.
    (4) (Total 13 marks)
Edexcel D2 2006 January Q7
11 marks Moderate -0.5
7.
  1. Define the terms
    1. cut,
    2. minimum cut, as applied to a directed network flow. \includegraphics[max width=\textwidth, alt={}, center]{a5d69a77-c196-483c-a550-1a55363555af-4_844_1465_338_299} The figure above shows a capacitated directed network and two cuts \(C _ { 1 }\) and \(C _ { 2 }\). The number on each arc is its capacity.
  2. State the values of the cuts \(C _ { 1 }\) and \(C _ { 2 }\). Given that one of these two cuts is a minimum cut,
  3. find a maximum flow pattern by inspection, and show it on the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{a5d69a77-c196-483c-a550-1a55363555af-4_597_1470_1656_296}
  4. Find a second minimum cut for this network. In order to increase the flow through the network it is decided to add an arc of capacity 100 joining \(D\) either to \(E\) or to \(G\).
  5. State, with a reason, which of these arcs should be added, and the value of the increased flow.
Edexcel D2 2002 June Q1
8 marks Easy -1.2
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{c4c64221-0373-4be9-abe3-5ff281922cdb-01_675_1052_378_485}
\end{figure} Figure 1 shows a network of roads connecting six villages \(A , B , C , D , E\) and \(F\). The lengths of the roads are given in km .
  1. Complete the table in the answer booklet, in which the entries are the shortest distances between pairs of villages. You should do this by inspection. The table can now be taken to represent a complete network.
  2. Use the nearest-neighbour algorithm, starting at \(A\), on your completed table in part (a). Obtain an upper bound to the length of a tour in this complete network, which starts and finishes at \(A\) and visits every village exactly once.
    (3)
  3. Interpret your answer in part (b) in terms of the original network of roads connecting the six villages.
    (1)
  4. By choosing a different vertex as your starting point, use the nearest-neighbour algorithm to obtain a shorter tour than that found in part (b). State the tour and its length.
    (2)