Questions — Edexcel D1 (505 questions)

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Edexcel D1 2001 January Q4
8 marks Moderate -0.8
This question should be answered on the sheet provided in the answer booklet. A manager has five workers, Mr. Ahmed, Miss Brown, Ms. Clough, Mr. Dingle and Mrs. Evans. To finish an urgent order he needs each of them to work overtime, one on each evening, in the next week. The workers are only available on the following evenings: Mr. Ahmed (A) -- Monday and Wednesday; Miss Brown (B) -- Monday, Wednesday and Friday; Ms. Clough (C) -- Monday; Mr. Dingle (D) -- Tuesday, Wednesday and Thursday; Mrs. Evans (E) -- Wednesday and Thursday. The manager initially suggests that A might work on Monday, B on Wednesday and D on Thursday.
  1. Using the nodes printed on the answer sheet, draw a bipartite graph to model the availability of the five workers. Indicate, in a distinctive way, the manager's initial suggestion. [2 marks]
  2. Obtain an alternating path, starting at C, and use this to improve the initial matching. [3 marks]
  3. Find another alternating path and hence obtain a complete matching. [3 marks]
Edexcel D1 2001 January Q5
13 marks Moderate -0.8
This question should be answered on the sheet provided in the answer booklet. \includegraphics{figure_2} Figure 2 shows the activity network used to model a small building project. The activities are represented by the edges and the number in brackets on each edge represents the time, in hours, taken to complete that activity.
  1. Calculate the early time and the late time for each event. Write your answers in the boxes on the answer sheet. [6 marks]
  2. Hence determine the critical activities and the length of the critical path. [2 marks]
Each activity requires one worker. The project is to be completed in the minimum time.
  1. Schedule the activities for the minimum number of workers using the time line on the answer sheet. Ensure that you make clear the order in which each worker undertakes his activities. [5 marks]
Edexcel D1 2001 January Q6
14 marks Moderate -0.3
This question should be answered on the sheet provided in the answer booklet. \includegraphics{figure_3} Figure 3 shows a capacitated, directed network. The number on each arc indicates the capacity of that arc.
  1. State the maximum flow along
    1. SAET,
    2. SBDT,
    3. SCFT.
    [3 marks]
  2. Show these maximum flows on Diagram 1 on the answer sheet. [1 mark]
  3. Taking your answer to part (b) as the initial flow pattern, use the labelling procedure to find a maximum flow from S to T. Your working should be shown on Diagram 2. List each flow augmenting route you find, together with its flow. [6 marks]
  4. Indicate a maximum flow on Diagram 3. [2 marks]
  5. Prove that your flow is maximal. [2 marks]
Edexcel D1 2001 January Q7
20 marks Moderate -0.3
A tailor makes two types of garment, A and B. He has available 70 m² of cotton fabric and 90 m² of woollen fabric. Garment A requires 1 m² of cotton fabric and 3 m² of woollen fabric. Garment B requires 2 m² of each fabric. The tailor makes \(x\) garments of type A and \(y\) garments of type B.
  1. Explain why this can be modelled by the inequalities $$x + 2y \leq 70,$$ $$3x + 2y \leq 90,$$ $$x \geq 0, y \geq 0.$$ [2 marks]
The tailor sells type A for £30 and type B for £40. All garments made are sold. The tailor wishes to maximise his total income.
  1. Set up an initial Simplex tableau for this problem. [3 marks]
  2. Solve the problem using the Simplex algorithm. [8 marks]
Figure 4 shows a graphical representation of the feasible region for this problem. \includegraphics{figure_4}
  1. Obtain the coordinates of the points A, C and D. [4 marks]
  2. Relate each stage of the Simplex algorithm to the corresponding point in Fig. 4. [3 marks]
Edexcel D1 2002 January Q1
6 marks Easy -1.3
Ann, Bryn, Daljit, Gareth and Nickos have all joined a new committee. Each of them is to be allocated to one of five jobs 1, 2, 3, 4 or 5. The table shows each member's preferences for the jobs.
Ann1 or 2
Bryn3 or 1
Daljit2 or 4
Gareth5 or 3
Nickos1 or 2
Initially Ann, Bryn, Daljit and Gareth are allocated the first job in their lists shown in the table.
  1. Draw a bipartite graph to model the preferences shown in the table and indicate, in a distinctive way, the initial allocation of jobs. [2]
  2. Use the matching improvement algorithm to find a complete matching, showing clearly your alternating path. [3]
  3. Find a second alternating path from the initial allocation. [1]
Edexcel D1 2002 January Q2
7 marks Easy -1.8
  1. Use the binary search algorithm to try to locate the name \(SABINE\) in the following alphabetical list. Explain each step of the algorithm. \begin{align} 1. &\quad ABLE
    2. &\quad BROWN
    3. &\quad COOKE
    4. &\quad DANIEL
    5. &\quad DOUBLE
    6. &\quad FEW
    7. &\quad OSBORNE
    8. &\quad PAUL
    9. &\quad SWIFT
    10. &\quad TURNER \end{align} [5]
  2. Find the maximum number of iterations of the binary search algorithm needed to locate a name in a list of 1000 names. [2]
Edexcel D1 2002 January Q3
9 marks Moderate -0.8
  1. \(A\)\(B\)\(C\)\(D\)\(E\)\(F\)
    \(A\)--101213209
    \(B\)10--715117
    \(C\)127--11183
    \(D\)131511--278
    \(E\)20111827--18
    \(F\)973818--
    The table shows the distances, in metres, between six nodes \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\) of a network.
    1. Use Prim's algorithm, starting at \(A\), to solve the minimum connector problem for this table of distances. Explain your method and indicate the order in which you selected the edges. [4]
    2. Draw your minimum spanning tree and find its total length. [2]
    3. State whether your minimum spanning tree is unique. Justify your answer. [1]
  2. A connected network \(N\) has seven vertices.
    1. State the number of edges in a minimum spanning tree for \(N\). [1]
    A minimum spanning tree for a connected network has \(n\) edges.
    1. State the number of vertices in the network. [1]
Edexcel D1 2002 January Q4
8 marks Moderate -0.8
\includegraphics{figure_1} Figure 1 shows a network of roads. Erica wishes to travel from \(A\) to \(L\) as quickly as possible. The number on each edge gives the time, in minutes, to travel along that road.
  1. Use Dijkstra's algorithm to find a quickest route from \(A\) to \(L\). Complete all the boxes on the answer sheet and explain clearly how you determined the quickest route from your labelling. [7]
  2. Show that there is another route which also takes the minimum time [1]
Edexcel D1 2002 January Q5
14 marks Moderate -0.8
Two fertilizers are available, a liquid \(X\) and a powder \(Y\). A bottle of \(X\) contains 5 units of chemical \(A\), 2 units of chemical \(B\) and \(\frac{1}{2}\) unit of chemical \(C\). A packet of \(Y\) contains 1 unit of \(A\), 2 units of \(B\) and 2 units of \(C\). A professional gardener makes her own fertilizer. She requires at least 10 units of \(A\), at least 12 units of \(B\) and at least 6 units of \(C\). She buys \(x\) bottles of \(X\) and \(y\) packets of \(Y\).
  1. Write down the inequalities which model this situation. [4]
  2. On the grid provided construct and label the feasible region. [3]
A bottle of \(X\) costs £2 and a packet of \(Y\) costs £3.
  1. Write down an expression, in terms of \(x\) and \(y\), for the total cost \(£T\). [1]
  2. Using your graph, obtain the values of \(x\) and \(y\) that give the minimum value of \(T\). Make your method clear and calculate the minimum value of \(T\). [4]
  3. Suggest how the situation might be changed so that it could no longer be represented graphically. [2]
Edexcel D1 2002 January Q6
14 marks Standard +0.3
\includegraphics{figure_2} A company has 3 warehouses \(W_1\), \(W_2\), and \(W_3\). It needs to transport the goods stored there to 2 retail outlets \(R_1\) and \(R_2\). The capacities of the possible routes, in van loads per day, are shown in Fig 2. Warehouses \(W_1\), \(W_2\) and \(W_3\) have 14, 12 and 14 van loads respectively available per day and retail outlets \(R_1\) and \(R_2\) can accept 6 and 25 van loads respectively per day.
  1. On Diagram 1 on the answer sheet add a supersource \(W\), a supersink \(R\) and the appropriate directed arcs to obtain a single-source, single-sink capacitated network. State the minimum capacity of each arc you have added. [3]
  2. State the maximum flow along
    1. \(W\) \(W_1\) \(A\) \(R_1\) \(R\),
    2. \(W\) \(W_3\) \(C\) \(R_2\) \(R\). [2]
  3. Taking your answers to part (b) as the initial flow pattern, use the labelling procedure to obtain a maximum flow through the network from \(W\) to \(R\). Show your working on Diagram 2. List each flow-augmenting route you use, together with its flow. [5]
  4. From your final flow pattern, determine the number of van loads passing through \(B\) each day. [1]
The company has the opportunity to increase the number of vans loads from one of the warehouses \(W_1\), \(W_2\), \(W_3\), to \(A\), \(B\) or \(C\).
  1. Determine how the company should use this opportunity so that it achieves a maximal flow. [3]
Edexcel D1 2002 January Q7
17 marks Moderate -0.8
\includegraphics{figure_3} A project is modelled by the activity network shown in Fig 3. The activities are represented by the edges. The number in brackets on each edge gives the time, in days, taken to complete the activity.
  1. Calculate the early time and the late time for each event. Write these in the boxes on the answer sheet. [4]
  2. Hence determine the critical activities and the length of the critical path. [2]
  3. Obtain the total float for each of the non-critical activities. [3]
  4. On the first grid on the answer sheet, draw a cascade (Gantt) chart showing the information obtained in parts (b) and (c). [4]
Each activity requires one worker. Only two workers are available.
  1. On the second grid on the answer sheet, draw up a schedule and find the minimum time in which the 2 workers can complete the project. [4]
Edexcel D1 2003 January Q1
4 marks Standard +0.3
\includegraphics{figure_1} Use the planarity algorithm to show that the graph in Fig. 1 is planar. [4]
Edexcel D1 2003 January Q2
6 marks Easy -1.2
At Tesafe supermarket there are 5 trainee staff, Homan \((H)\), Jenna \((J)\), Mary \((M)\), Tim \((T)\) and Yoshie \((Y)\). They each must spend one week in each of 5 departments, Delicatessen \((D)\), Frozen foods \((F)\), Groceries \((G)\), Pet foods \((P)\), Soft drinks \((S)\). Next week every department requires exactly one trainee. The table below shows the departments in which the trainees have yet to spend time.
TraineeDepartments
\(H\)\(D, F, P\)
\(J\)\(G, D, F\)
\(M\)\(S, P, G\)
\(T\)\(F, S, G\)
\(Y\)\(D\)
Initially \(H\), \(J\), \(M\) and \(T\) are allocated to the first department in their list.
  1. Draw a bipartite graph to model this situation and indicate the initial matching in a distinctive way. [2] Starting from this matching,
  2. use the maximum matching algorithm to find a complete matching. You must make clear your alternating path and your complete matching. [4]
Edexcel D1 2003 January Q3
9 marks Moderate -0.5
A manager wishes to purchase seats for a new cinema. He wishes to buy three types of seat; standard, deluxe and majestic. Let the number of standard, deluxe and majestic seats to be bought be \(x\), \(y\) and \(z\) respectively. He decides that the total number of deluxe and majestic seats should be at most half of the number of standard seats. The number of deluxe seats should be at least 10\% and at most 20\% of the total number of seats. The number of majestic seats should be at least half of the number of deluxe seats. The total number of seats should be at least 250. Standard, deluxe and majestic seats each cost £20, £26 and £36, respectively. The manager wishes to minimize the total cost, £\(C\), of the seats. Formulate this situation as a linear programming problem, simplifying your inequalities so that all the coefficients are integers. [9]
Edexcel D1 2003 January Q4
9 marks Standard +0.3
\includegraphics{figure_2} The arcs in Fig. 2 represent roads in a town. The weight on each arc gives the time, in minutes, taken to drive along that road. The times taken to drive along \(AB\) and \(DE\) vary depending upon the time of day. A police officer wishes to drive along each road at least once, starting and finishing at \(A\). The journey is to be completed in the least time.
  1. Briefly explain how you know that a route between \(B\) and \(E\) will have to be repeated. [1]
  2. List the possible routes between \(B\) and \(E\). State how long each would take, in terms of \(x\) where appropriate. [2]
  3. Find the range of values that \(x\) must satisfy so that \(DE\) would be one of the repeated arcs. [3] Given that \(x = 7\),
  4. find the total time needed for the police officer to carry out this journey. [3]
Edexcel D1 2003 January Q5
10 marks Moderate -0.8
\includegraphics{figure_3} A project is modelled by the activity network in Fig. 3. The activities are represented by the arcs. One worker is required for each activity. The number in brackets on each arc gives the time, in hours, to complete the activity. The earliest event time and the latest event time are given by the numbers in the left box and right box respectively.
  1. State the value of \(x\) and the value of \(y\). [2]
  2. List the critical activities. [2]
  3. Explain why at least 3 workers will be needed to complete this project in 38 hours. [2]
  4. Schedule the activities so that the project is completed in 38 hours using just 3 workers. You must make clear the start time and finish time of each activity. [4]
Edexcel D1 2003 January Q6
9 marks Easy -1.3
25 22 30 18 29 21 27 21 The list of numbers above is to be sorted into descending order.
    1. Perform the first pass of a bubble sort, giving the state of the list after each exchange.
    2. Perform further passes, giving the state of the list after each pass, until the algorithm terminates.
    [5]
The numbers represent the lengths, in cm, of pieces to be cut from rods of length 50 cm.
    1. Show the result of applying the first fit decreasing bin packing algorithm to this situation.
    2. Determine whether your solution to \((b)\) \((i)\) has used the minimum number of 50 cm rods.
    [4]
Edexcel D1 2003 January Q7
14 marks Standard +0.3
\includegraphics{figure_4} Figure 4 shows a capacitated directed network. The number on each arc is its capacity. The numbers in circles show a feasible flow from sources \(A\) and \(B\) to sinks \(I\), \(J\) and \(K\). Take this as the initial flow pattern.
  1. On Diagram 1 in the answer booklet, add a supersource \(S\) and a supersink \(W\) to obtain a capacitated network with a single source and single sink. State the minimum capacities of the arcs you have added. [3]
    1. Use the given initial flow and the labelling procedure on Diagram 2 to find the maximum flow through the network. You must list each flow-augmenting route you use together with its flow.
    2. Verify that your flow is maximal.
    [9]
  3. Show your maximum flow pattern on Diagram 3. [2]
Edexcel D1 2003 January Q8
14 marks Moderate -0.3
The tableau below is the initial tableau for a maximising linear programming problem.
Basic Variable\(x\)\(y\)\(z\)\(r\)\(s\)Value
\(r\)234108
\(s\)3310110
\(P\)-8-9-5000
  1. For this problem \(x \geq 0\), \(y \geq 0\), \(z \geq 0\). Write down the other two inequalities and the objective function. [3]
  2. Solve this linear programming problem. [8]
  3. State the final value of \(P\), the objective function, and of each of the variables. [3]
Edexcel D1 2004 January Q1
6 marks Easy -2.0
Define the terms
  1. bipartite graph, [2]
  2. alternating path, [2]
  3. matching, [1]
  4. complete matching. [1]
Edexcel D1 2004 January Q2
6 marks Moderate -0.3
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 tableau was obtained.
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
\(s\)30201\(-\frac{2}{3}\)\(\frac{2}{3}\)
\(r\)40\(\frac{7}{2}\)108\(\frac{9}{2}\)
\(y\)5170037
P30200863
  1. State, giving your reason, whether this tableau represents the optimal solution. [1]
  2. State the values of every variable. [3]
  3. Calculate the profit made on each unit of \(y\). [2]
Edexcel D1 2004 January Q3
7 marks Moderate -0.5
\includegraphics{figure_1} Figure 1 shows a network of roads represented by arcs. The capacity of the road represented by that arc is shown on each arc. The numbers in circles represent a possible flow of 26 from \(B\) to \(L\). Three cuts \(C_1\), \(C_2\) and \(C_3\) are shown on Fig. 1.
  1. Find the capacity of each of the three cuts. [3]
  2. Verify that the flow of 26 is maximal. [1]
The government aims to maximise the possible flow from \(B\) to \(L\) by using one of two options. Option 1: Build a new road from \(E\) to \(J\) with capacity 5. or Option 2: Build a new road from \(F\) to \(H\) with capacity 3.
  1. By considering both options, explain which one meets the government's aim [3]
Edexcel D1 2004 January Q4
9 marks Standard +0.3
\includegraphics{figure_2} An engineer needs to check the state of a number of roads to see whether they need resurfacing. The roads that need to be checked are represented by the arcs in Fig. 2. The number on each arc represents the length of that road in km. To check all the roads, he needs to travel along each road at least once. He wishes to minimise the total distance travelled. The engineer's office is at \(G\), so he starts and ends his journey at \(G\).
  1. Use an appropriate algorithm to find a route for the engineer to follow. State your route and its length. [6]
The engineer lives at \(D\). He believes he can reduce the distance travelled by starting from home and inspecting all the roads on the way to his office at \(G\).
  1. State whether the engineer is correct in his belief. If so, calculate how much shorter his new route is. If not, explain why not. [3]
Edexcel D1 2004 January Q5
10 marks Easy -1.2
\includegraphics{figure_3} Figure 3 describes an algorithm in the form of a flow chart, where \(a\) is a positive integer. List \(P\), which is referred to in the flow chart, comprises the prime numbers 2, 3, 5, 7, 11, 13, 17, ...
  1. Starting with \(a = 90\), implement this algorithm. Show your working in the table in the answer book. [7]
  2. Explain the significance of the output list. [2]
  3. Write down the final value of \(c\) for any initial value of \(a\). [1]
Edexcel D1 2004 January Q6
10 marks Moderate -0.8
ABCDEF
A\(-\)73\(-\)811
B7\(-\)42\(-\)7
C34\(-\)59\(-\)
D\(-\)25\(-\)63
E8\(-\)96\(-\)\(-\)
F117\(-\)3\(-\)\(-\)
The matrix represents a network of roads between six villages \(A\), \(B\), \(C\), \(D\), \(E\) and \(F\). The value in each cell represents the distance, in km, along these roads.
  1. Show this information on the diagram in the answer book. [2]
  2. Use Kruskal's algorithm to determine the minimum spanning tree. State the order in which you include the arcs and the length of the minimum spanning tree. Draw the minimum spanning tree. [5]
  3. Starting at \(D\), use Prim's algorithm on the matrix given in the answer book to find the minimum spanning tree. State the order in which you include the arcs. [3]