Questions — Edexcel D1 (505 questions)

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Edexcel D1 2004 January Q7
13 marks Moderate -0.3
Becky's bird food company makes two types of bird food. One type is for bird feeders and the other for bird tables. Let \(x\) represent the quantity of food made for bird feeders and \(y\) represent the quantity of food made for bird tables. Due to restrictions in the production process, and known demand, the following constraints apply. $$x + y \leq 12,$$ $$y < 2x,$$ $$2y \geq 7,$$ $$y + 3x \geq 15.$$
  1. On the axes provided, show these constraints and label the feasible region \(R\). [5]
The objective is to minimise \(C = 2x + 5y\).
  1. Solve this problem, making your method clear. Give, as fractions, the value of \(C\) and the amount of each type of food that should be produced. [4]
Another objective (for the same constraints given above) is to maximise \(P = 3x + 2y\), where the variables must take integer values.
  1. Solve this problem, making your method clear. State the value of \(P\) and the amount of each type of food that should be produced. [4]
Edexcel D1 2004 January Q8
14 marks Moderate -0.8
\includegraphics{figure_4} A trainee at a building company is using critical path analysis to help plan a project. Figure 4 shows the trainee's activity network. Each activity is represented by an arc and the number in brackets on each arc is the duration of the activity, in hours.
  1. Find the values of \(x\), \(y\) and \(z\). [3]
  2. State the total length of the project and list the critical activities. [2]
  3. Calculate the total float time on
    1. activity \(N\),
    2. activity \(H\). [3]
The trainee's activity network is checked by the supervisor who finds a number of errors and omissions in the diagram. The project should be represented by the following precedence table.
ActivityMust be preceded by:Duration
\(A\)\(-\)4
\(B\)\(-\)3
\(C\)\(-\)5
\(D\)\(B\)2
\(E\)\(A, D\)8
\(F\)\(B\)2
\(G\)\(C\)2
\(H\)\(C\)3
\(I\)\(F, G\)4
\(J\)\(F, G\)2
\(K\)\(F, G\)7
\(L\)\(E, I\)9
\(M\)\(H, J\)3
\(N\)\(E, I, K, M\)3
\(P\)\(E, I\)6
\(Q\)\(H, J\)5
\(R\)\(Q\)7
  1. By adding activities and dummies amend the diagram in the answer book so that it represents the precedence table. (The durations of activities \(A\), \(B\), ..., \(N\) are all correctly given on the diagram in the answer book.) [4]
  2. Find the total time needed to complete this project. [2]
Edexcel D1 2005 January Q1
6 marks Moderate -0.8
\includegraphics{figure_1} The bipartite graph in Figure 1 shows a mapping between six people, Andy (\(A\)), David (\(D\)), Joan (\(J\)), Preety (\(P\)), Sally (\(S\)) and Trevor (\(T\)), and six tasks 1, 2, 3, 4, 5 and 6. The initial matching is \(A\) to 2, \(D\) to 1, \(J\) to 3 and \(P\) to 4.
  1. Indicate this initial matching in a distinctive way on the bipartite graph drawn in the answer book. [1]
  2. Starting from this initial matching, use the maximum matching algorithm to find a complete matching. List clearly the alternating paths you use. [5]
Edexcel D1 2005 January Q2
7 marks Moderate -0.3
The precedence table for activities involved in producing a computer game is shown below.
ActivityMust be preceded by
\(A\)\(-\)
\(B\)\(A\)
\(C\)\(B\)
\(D\)\(A, C\)
\(E\)\(A\)
\(F\)\(E\)
\(G\)\(E\)
\(H\)\(G\)
\(I\)\(D, F\)
\(J\)\(G, I\)
\(K\)\(G, I\)
\(L\)\(H, K\)
An activity on arc network is to be drawn to model this production process.
  1. Explain why it is necessary to use at least two dummies when drawing the activity network. [2]
  2. Draw the activity network using exactly two dummies. [5]
Edexcel D1 2005 January Q3
8 marks Easy -1.2
\includegraphics{figure_2} The network in Figure 2 shows the distances, in metres, between 10 wildlife observation points. The observation points are to be linked by footpaths, to form a network along the arcs indicated, using the least possible total length.
  1. Find a minimum spanning tree for the network in Figure 2, showing clearly the order in which you selected the arcs for your tree, using
    1. Kruskal's algorithm, [3]
    2. Prim's algorithm, starting from \(A\). [3]
    Given that footpaths are already in place along \(AB\) and \(FI\) and so should be included in the spanning tree,
  2. explain which algorithm you would choose to complete the tree, and how it should be adapted. (You do not need to find the tree.) [2]
Edexcel D1 2005 January Q4
11 marks Easy -1.3
\(650 \quad 431 \quad 245 \quad 643 \quad 455 \quad 134 \quad 710 \quad 234 \quad 162 \quad 452\)
  1. The list of numbers above is to be sorted into descending order. Perform a Quick Sort to obtain the sorted list, giving the state of the list after each pass, indicating the pivot elements. [5]
The numbers in the list represent the lengths, in mm, of some pieces of wood. The wood is sold in one metre lengths.
  1. Use the first-fit decreasing bin packing algorithm to determine how these pieces could be cut from the minimum number of one metre lengths. (You should ignore wastage due to cutting.) [4]
  2. Determine whether your solution to part (b) is optimal. Give a reason for your answer. [2]
Edexcel D1 2005 January Q5
11 marks Moderate -0.5
\includegraphics{figure_3} Figure 3 shows a network of paths. The number on each arc gives the distance, in metres, of that path.
  1. Use Dijkstra's algorithm to find the shortest distance from \(A\) to \(H\). [5]
  2. Solve the route inspection problem for the network shown in Figure 3. You should make your method and working clear. State a shortest route, starting at \(A\), and find its length. [The total weight of the network is 1241] [6]
Edexcel D1 2005 January Q6
14 marks Moderate -0.8
\includegraphics{figure_4} Figure 4 shows a capacitated directed network. The number on each arc is its capacity.
  1. State the maximum flow along
    1. \(SADT\),
    2. \(SCET\),
    3. \(SBFT\). [2]
  2. Show these maximum flows on Diagram 1 in the answer book. [1]
Take your answer to part (b) as the initial flow pattern.
    1. Use the labelling procedure to find a maximum flow from \(S\) to \(T\). Your working should be shown on Diagram 2 in the answer book. List each flow-augmenting route you use, together with its flow. [5]
    2. Draw your final flow pattern on Diagram 3 in the answer book. [2]
    3. Prove that your flow is maximal. [3]
  2. Give an example of a practical situation that could have been modelled by the original network. [1]
Edexcel D1 2005 January Q7
18 marks Standard +0.3
Flatland UK Ltd makes three types of carpet, the Lincoln, the Norfolk and the Suffolk. The carpets all require units of black, green and red wool. For each roll of carpet, the Lincoln requires 1 unit of black, 1 of green and 3 of red, the Norfolk requires 1 unit of black, 2 of green and 2 of red, and the Suffolk requires 2 units of black, 1 of green and 1 of red. There are up to 30 units of black, 40 units of green and 50 units of red available each day. Profits of £50, £80 and £60 are made on each roll of Lincoln, Norfolk and Suffolk respectively. Flatland UK Ltd wishes to maximise its profit. Let the number of rolls of the Lincoln, Norfolk and Suffolk made daily be \(x\), \(y\) and \(z\) respectively.
  1. Formulate the above situation as a linear programming problem, listing clearly the constraints as inequalities in their simplest form, and stating the objective function. [4]
This problem is to be solved using the Simplex algorithm. The most negative number in the profit row is taken to indicate the pivot column at each stage.
  1. Stating your row operations, show that after one complete iteration the tableau becomes
    Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
    \(r\)\(\frac{1}{2}\)0\(1\frac{1}{2}\)1\(-\frac{1}{2}\)010
    \(y\)\(\frac{1}{2}\)1\(\frac{1}{2}\)0\(\frac{1}{2}\)020
    \(t\)2000\(-1\)110
    \(P\)\(-10\)0\(-20\)04001600
    [4]
  2. Explain the practical meaning of the value 10 in the top row. [2]
    1. Perform one further complete iteration of the Simplex algorithm.
    2. State whether your current answer to part (d)(i) is optimal. Give a reason for your answer.
    3. Interpret your current tableau, giving the value of each variable. [8]
Edexcel D1 2006 January Q1
7 marks Moderate -0.8
\includegraphics{figure_1} A taxi firm has six taxis \(A\), \(B\), \(C\), \(D\), \(E\) and \(F\), available for six journeys, 1, 2, 3, 4, 5 and 6, which are booked for 9 a.m. tomorrow. The bipartite graph shown in Figure 1 shows the possible matchings. Initially \(A\), \(B\), \(C\) and \(D\) are matched to 5, 2, 3 and 6 respectively, as indicated in Figure 2.
  1. Explain why it is necessary to perform the maximum matching algorithm twice in order to try to obtain a complete matching. [1]
  2. Use the maximum matching algorithm twice to obtain a complete matching. List clearly the alternating paths you use. [6]
Edexcel D1 2006 January Q2
15 marks Easy -1.3
\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
\(A\)\(-\)4811792\(-\)\(-\)\(-\)
\(B\)48\(-\)\(-\)\(-\)\(-\)6355
\(C\)117\(-\)\(-\)28\(-\)\(-\)85
\(D\)92\(-\)28\(-\)58132\(-\)
\(E\)\(-\)\(-\)\(-\)58\(-\)124\(-\)
\(F\)\(-\)63\(-\)132124\(-\)\(-\)
\(G\)\(-\)5585\(-\)\(-\)\(-\)\(-\)
The table shows the lengths, in metres, of the paths between seven vertices \(A\), \(B\), \(C\), \(D\), \(E\), \(F\) and \(G\) in a network N.
  1. Use Prim's algorithm, starting at \(A\), to solve the minimum connector problem for this table of distances. You must clearly state the order in which you selected the edges of your tree, and the weight of your final tree. Draw your tree using the vertices given in Diagram 1 in the answer book. [5]
  2. Draw N using the vertices given in Diagram 2 in the answer book. [3]
  3. Solve the Route Inspection problem for N. You must make your method of working clear. State a shortest route and find its length. (The weight of N is 802.) [7]
Edexcel D1 2006 January Q3
9 marks Easy -1.2
\includegraphics{figure_3} An algorithm is described by the flow chart shown in Figure 3.
  1. Complete the table in the answer book recording the results of each step as the algorithm is applied. (Notice that values of \(A\), \(B\), \(C\) and \(D\) are to be given to 3 decimal places, and the values of \(E\) to 1 significant figure.) [8]
  2. Write down the output from the algorithm. [1]
Edexcel D1 2006 January Q4
11 marks Moderate -0.8
  1. Define the terms
    1. cut,
    2. minimum cut,
    as applied to a directed network flow. [2]
\includegraphics{figure_4} Figure 4 shows a capacitated directed network and two cuts \(C_1\) and \(C_2\). The number on each arc is its capacity.
  1. State the values of the cuts \(C_1\) and \(C_2\). [3]
Given that one of these two cuts is a minimum cut,
  1. Find a maximum flow pattern by inspection, and show it on the diagram in the answer book. [3]
  2. Find a second minimum cut for this network. [1]
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 \(G\).
  1. State, with a reason, which of these arcs should be added, and the value of the increased flow. [2]
Edexcel D1 2006 January Q5
15 marks Moderate -0.8
\includegraphics{figure_5} The network in Figure 5 shows the activities involved in a process. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, taken to complete the activity.
  1. Calculate the early time and late time for each event, showing them on the diagram in the answer book. [4]
  2. Determine the critical activities and the length of the critical path. [2]
  3. On the grid in the answer book, draw a cascade (Gantt) chart for the process. [4]
Each activity requires only one worker, and workers may not share an activity.
  1. Use your cascade chart to determine the minimum numbers of workers required to complete the process in the minimum time. Explain your reasoning clearly. [2]
  2. Schedule the activities, using the number of workers you found in part \((d)\), so that the process is completed in the shortest time. [3]
Edexcel D1 2006 January Q6
18 marks Moderate -0.8
A company produces two types of party bag, Infant and Junior. Both types of bag contain a balloon, a toy and a whistle. In addition the Infant bag contains 3 sweets and 3 stickers and the Junior bag contains 10 sweets and 2 stickers. The sweets and stickers are produced in the company's factory. The factory can produce up to 3000 sweets per hour and 1200 stickers per hour. The company buys a large supply of balloons, toys and whistles. Market research indicates that at least twice as many Infant bags as Junior bags should be produced. Both types of party bag are sold at a profit of 15p per bag. All the bags are sold. The company wishes to maximise its profit. Let \(x\) be the number of Infant bags produced and \(y\) be the number of Junior bags produced per hour.
  1. Formulate the above situation as a linear programming problem. [5]
  2. Represent your inequalities graphically, indicating clearly the feasible region. [6]
  3. Find the number of Infant bags and Junior bags that should be produced each hour and the maximum hourly profit. Make your method clear. [3]
In order to increase the profit further, the company decides to buy additional equipment. It can buy equipment to increase the production of either sweets or stickers, but not both.
  1. Using your graph, explain which equipment should be bought, giving your reasoning. [2]
The manager of the company does not understand why the balloons, toys and whistles have not been considered in the above calculations.
  1. Explain briefly why they do not need to be considered. [2]
Edexcel D1 2007 January Q1
Easy -1.8
Use the binary search algorithm to try to locate the name NIGEL in the following alphabetical list. Clearly indicate how you chose your pivots and which part of the list is being rejected at each stage. 1. Bhavika 2. Clive 3. Elizabeth 4. John 5. Mark 6. Nicky 7. Preety 8. Steve 9. Trevor 10. Verity (Total 4 marks)
Edexcel D1 2007 January Q2
Moderate -0.8
\includegraphics{figure_1} \includegraphics{figure_2} Figure 1 shows the possible allocations of five people, Ellen, George, Jo, Lydia and Yi Wen to five tasks, 1, 2, 3, 4 and 5. Figure 2 shows an initial matching.
  1. Find an alternating path linking George with 5. List the resulting improved matching this gives. (3)
  2. Explain why it is not possible to find a complete matching. (1)
George now has task 2 added to his possible allocation.
  1. Using the improved matching found in part (a) as the new initial matching, find an alternating path linking Yi Wen with task 1 to find a complete matching. List the complete matching. (3)
(Total 7 marks)
Edexcel D1 2007 January Q3
Moderate -0.8
\includegraphics{figure_3}
  1. Write down the name given to the type of graph drawn in Figure 3. (1)
A Hamiltonian cycle for the graph in Figure 3 begins A, 3, B, ....
  1. Complete this Hamiltonian cycle. (2)
  2. Starting with the Hamiltonian cycle found in (b), use the planarity algorithm to determine if the graph is planar. (3)
(Total 6 marks)
Edexcel D1 2007 January Q4
Moderate -0.8
A three-variable linear programming problem in \(x\), \(y\) and \(z\) is to be solved. The objective is to maximise the profit \(P\). The following initial tableau was obtained.
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)Value
\(r\)2041080
\(s\)14201160
\(P\)\(-2\)\(-8\)\(-20\)000
  1. Taking the most negative number in the profit row to indicate the pivot column, perform one complete iteration of the simplex algorithm, to obtain tableau \(T\). State the row operations that you use. (5)
  2. Write down the profit equation shown in tableau \(T\). (1)
  3. State whether tableau \(T\) is optimal. Give a reason for your answer. (1)
(Total 7 marks)
Edexcel D1 2007 January Q5
Moderate -0.3
  1. Explain why a network cannot have an odd number of vertices of odd degree. (2)
\includegraphics{figure_4} Figure 4 shows a network of paths in a public park. The number on each arc represents the length of that path in metres. Hamish needs to walk along each path at least once to check the paths for frost damage starting and finishing at \(A\). He wishes to minimise the total distance he walks.
  1. Use the route inspection algorithm to find which paths, if any, need to be traversed twice. (4)
  2. Find the length of Hamish's route. [The total weight of the network in Figure 4 is 4180 m.] (1)
(Total 7 marks)
Edexcel D1 2007 January Q6
Moderate -0.8
\includegraphics{figure_5} A project is modelled by the activity network shown in Figure 5. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, to complete the activity. The numbers in circles are the event numbers. Each activity requires one worker.
  1. Explain the purpose of the dotted line from event 6 to event 8. (1)
  2. Calculate the early time and late time for each event. Write these in the boxes in the answer book. (4)
  3. Calculate the total float on activities \(D\), \(E\) and \(F\). (3)
  4. Determine the critical activities. (2)
  5. Given that the sum of all the times of the activities is 95 hours, calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working. (2)
  6. Given that workers may not share an activity, schedule the activities so that the process is completed in the shortest time using the minimum number of workers. (4)
(Total 16 marks)
Edexcel D1 2007 January Q7
Easy -1.3
\includegraphics{figure_6} The captain of the Malde Mare takes passengers on trips across the lake in her boat. The number of children is represented by \(x\) and the number of adults by \(y\). Two of the constraints limiting the number of people she can take on each trip are $$x < 10$$ and $$2 \leq y \leq 10$$ These are shown on the graph in Figure 6, where the rejected regions are shaded out.
  1. Explain why the line \(x = 10\) is shown as a dotted line. (1)
  2. Use the constraints to write down statements that describe the number of children and the number of adults that can be taken on each trip. (3)
For each trip she charges £2 per child and £3 per adult. She must take at least £24 per trip to cover costs. The number of children must not exceed twice the number of adults.
  1. Use this information to write down two inequalities. (2)
  2. Add two lines and shading to Diagram 1 in your answer book to represent these inequalities. Hence determine the feasible region and label it R. (4)
  3. Use your graph to determine how many children and adults would be on the trip if the captain takes:
    1. the minimum number of passengers,
    2. the maximum number of passengers.
    (4)
(Total 14 marks)
Edexcel D1 2007 January Q8
Standard +0.3
\includegraphics{figure_7} In solving a network flow problem using the labelling procedure, the diagram in Figure 7 was created. The arrow on each arc indicates the direction of the flow along that arc. The arrows above and below each arc show the direction and value of the flow as indicated by the labelling procedure.
  1. Add a supersource S, a supersink T and appropriate arcs to Diagram 2 in the answer book, and complete the labelling procedure for these arcs. (3)
  2. Write down the value of the initial flow shown in Figure 7. (1)
  3. Use Diagram 2, the initial flow and the labelling procedure to find the maximal flow of 124 through this network. List each flow-augmenting path you use, together with its flow. (5)
  4. Show your flow on Diagram 3 and state its value. (3)
  5. Prove that your flow is maximal. (2)
(Total 14 marks)
Edexcel D1 2003 June Q1
5 marks Moderate -0.8
Six workers \(A\), \(B\), \(C\), \(D\), \(E\) and \(F\) are to be matched to six tasks 1, 2, 3, 4, 5 and 6. The table below shows the tasks that each worker is able to do.
WorkerTasks
A2, 3, 5
B1, 3, 4, 5
C2
D3, 6
E2, 4, 5
F1
A bipartite graph showing this information is drawn in the answer booklet. Initially, \(A\), \(B\), \(D\) and \(E\) are allocated to tasks 2, 1, 3 and 5 respectively. Starting from the given initial matching, use the matching improvement algorithm to find a complete matching, showing your alternating paths clearly. [5]
Edexcel D1 2003 June Q2
6 marks Moderate -0.3
  1. Explain why it is impossible to draw a network with exactly three odd vertices. [2]
\includegraphics{figure_1} The Route Inspection problem is solved for the network in Fig. 1 and the length of the route is found to be 100.
  1. Determine the value of \(x\), showing your working clearly. [4]