Questions — Edexcel D1 (505 questions)

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Edexcel D1 2003 June Q3
4 marks Easy -1.2
  1. Describe the differences between Prim's algorithm and Kruskal's algorithm for finding a minimum connector of a network.
\includegraphics{figure_2}
  1. Listing the arcs in the order that you select them, find a minimum connector for the network in Fig. 2, using
    1. Prim's algorithm,
    2. Kruskal's algorithm.
    [4]
Edexcel D1 2003 June Q4
9 marks Easy -1.8
The following list gives the names of some students who have represented Britain in the International Mathematics Olympiad. Roper \((R)\), Palmer \((P)\), Boase \((B)\), Young \((Y)\), Thomas \((T)\), Kenney \((K)\), Morris \((M)\), Halliwell \((H)\), Wicker \((W)\), Garesalingam \((G)\).
  1. Use the quick sort algorithm to sort the names above into alphabetical order. [5]
  2. Use the binary search algorithm to locate the name Kenney. [4]
Edexcel D1 2003 June Q5
15 marks Moderate -0.3
\includegraphics{figure_3} The network in Fig. 3 shows the activities involved in the process of producing a perfume. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, taken to complete the activity.
  1. Calculate the early time and the late time for each event, showing them on Diagram 1 in the answer booklet. [4]
  2. Hence determine the critical activities. [2]
  3. Calculate the total float time for \(D\). [2]
Each activity requires only one person.
  1. Find a lower bound for the number of workers needed to complete the process in the minimum time. [2]
Given that there are only three workers available, and that workers may not share an activity,
  1. schedule the activities so that the process is completed in the shortest time. Use the time line in the answer booklet. State the new shortest time. [5]
Edexcel D1 2003 June Q6
15 marks Easy -1.3
A company produces two types of self-assembly wooden bedroom suites, the 'Oxford' and the 'York'. After the pieces of wood have been cut and finished, all the materials have to be packaged. The table below shows the time, in hours, needed to complete each stage of the process and the profit made, in pounds, on each type of suite.
OxfordYork
Cutting46
Finishing3.54
Packaging24
Profit (£)300500
The times available each week for cutting, finishing and packaging are 66, 56 and 40 hours respectively. The company wishes to maximise its profit. Let \(x\) be the number of Oxford, and \(y\) be the number of York suites made each week.
  1. Write down the objective function. [1]
  2. In addition to $$2x + 3y \leq 33,$$ $$x \geq 0,$$ $$y \geq 0,$$ find two further inequalities to model the company's situation. [2]
  3. On the grid in the answer booklet, illustrate all the inequalities, indicating clearly the feasible region. [4]
  4. Explain how you would locate the optimal point. [2]
  5. Determine the number of Oxford and York suites that should be made each week and the maximum profit gained. [3]
It is noticed that when the optimal solution is adopted, the time needed for one of the three stages of the process is less than that available.
  1. Identify this stage and state by how many hours the time may be reduced. [3]
Edexcel D1 2003 June Q7
18 marks Standard +0.3
\includegraphics{figure_4} Figure 4 shows a capacitated, directed network. The unbracketed number on each arc indicates the capacity of that arc, and the numbers in brackets show a feasible flow of value 68 through the network.
  1. Add a supersource and a supersink, and arcs of appropriate capacity, to Diagram 2 in the answer booklet. [2]
  2. Find the values of \(x\) and \(y\), explaining your method briefly. [2]
  3. Find the value of cuts \(C_1\) and \(C_2\). [3]
Starting with the given feasible flow of 68,
  1. use the labelling procedure on Diagram 3 to find a maximal flow through this network. List each flow-augmenting route you use, together with its flow. [6]
  2. Show your maximal flow on Diagram 4 and state its value. [3]
  3. Prove that your flow is maximal. [2]
Edexcel D1 2004 June Q1
7 marks Easy -1.2
The organiser of a sponsored walk wishes to allocate each of six volunteers, Alan, Geoff, Laura, Nicola, Philip and Sam to one of the checkpoints along the route. Two volunteers are needed at checkpoint 1 (the start) and one volunteer at each of checkpoint 2, 3, 4 and 5 (the finish). Each volunteer will be assigned to just one checkpoint. The table shows the checkpoints each volunteer is prepared to supervise.
NameCheckpoints
Alan1 or 3
Geoff1 or 5
Laura2, 1 or 4
Nicola5
Philip2 or 5
Sam2
Initially Alan, Geoff, Laura and Nicola are assigned to the first checkpoint in their individual list.
  1. Draw a bipartite graph to model this situation and indicate the initial matching in a distinctive way. [2]
  2. Starting from this initial matching, use the maximum matching algorithm to find an improved matching. Clearly list any alternating paths you use. [3]
  3. Explain why it is not possible to find a complete matching. [2]
Edexcel D1 2004 June Q2
8 marks Easy -1.2
\includegraphics{figure_1} Figure 1 shows a network of roads. The number on each edge gives the time, in minutes, to travel along that road. Avinash wishes to travel from \(S\) to \(T\) as quickly as possible.
  1. Use Dijkstra's algorithm to find the shortest time to travel from \(S\) to \(T\). [4]
  2. Find a route for Avinash to travel from \(S\) to \(T\) in the shortest time. State, with a reason, whether this route is a unique solution. [2]
On a particular day Avinash must include \(C\) in his route.
  1. Find a route of minimal time from \(S\) to \(T\) that includes \(C\), and state its time. [2]
Edexcel D1 2004 June Q3
9 marks Moderate -0.8
\includegraphics{figure_2}
  1. Describe a practical problem that could be modelled using the network in Fig. 2 and solved using the route inspection algorithm. [1]
  2. Use the route inspection algorithm to find which paths, if any, need to be traversed twice. [4]
  3. State whether your answer to part (b) is unique. Give a reason for your answer. [1]
  4. Find the length of the shortest inspection route that traverses each arc at least once and starts and finishes at the same vertex. [1]
Given that it is permitted to start and finish the inspection at two distinct vertices,
  1. find which two vertices should be chosen to minimise the length of the route. Give a reason for your answer. [2]
Edexcel D1 2004 June Q4
9 marks Easy -1.8
  1. Glasgow
  2. Newcastle
  3. Manchester
  4. York
  5. Leicester
  6. Birmingham
  7. Cardiff
  8. Exeter
  9. Southampton
  10. Plymouth
A binary search is to be performed on the names in the list above to locate the name Newcastle.
  1. Explain why a binary search cannot be performed with the list in its present form. [1]
  2. Using an appropriate algorithm, alter the list so that a binary search can be performed. State the name of the algorithm you use. [4]
  3. Use the binary search algorithm on your new list to locate the name Newcastle. [4]
Edexcel D1 2004 June Q5
13 marks Moderate -0.8
\includegraphics{figure_3} Figure 3 shows a capacitated directed network. The number on each arc is its capacity. \includegraphics{figure_4} Figure 4 shows a feasible initial flow through the same network.
  1. Write down the values of the flow \(x\) and the flow \(y\). [2]
  2. Obtain the value of the initial flow through the network, and explain how you know it is not maximal. [2]
  3. Use this initial flow and the labelling procedure on Diagram 1 in this answer book to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow. [5]
  4. Show your maximal flow pattern on Diagram 2. [2]
  5. Prove that your flow is maximal. [2]
Edexcel D1 2004 June Q6
14 marks Moderate -0.8
The Young Enterprise Company "Decide", is going to produce badges to sell to decision maths students. It will produce two types of badges. Badge 1 reads "I made the decision to do maths" and Badge 2 reads "Maths is the right decision". "Decide" must produce at least 200 badges and has enough material for 500 badges. Market research suggests that the number produced of Badge 1 should be between 20% and 40% of the total number of badges made. The company makes a profit of 30p on each Badge 1 sold and 40p on each Badge 2. It will sell all that it produced, and wishes to maximise its profit. Let \(x\) be the number produced of Badge 1 and \(y\) be the number of Badge 2.
  1. Formulate this situation as a linear programming problem, simplifying your inequalities so that all the coefficients are integers. [6]
  2. On the grid provided in the answer book, construct and clearly label the feasible region. [5]
  3. Using your graph, advise the company on the number of each badge it should produce. State the maximum profit "Decide" will make. [3]
Edexcel D1 2004 June Q7
15 marks Moderate -0.8
\includegraphics{figure_5} A project is modelled by the activity network shown in Fig. 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 give the event numbers. Each activity requires one worker.
  1. Explain the purpose of the dotted line from event 4 to event 5. [1]
  2. Calculate the early time and the late time for each event. Write these in the boxes in the answer book. [4]
  3. Determine the critical activities. [1]
  4. Obtain the total float for each of the non-critical activities. [3]
  5. On the grid in the answer book, draw a cascade (Gantt) chart, showing the answers to parts (c) and (d). [4]
  6. Determine the minimum number of workers needed to complete the project in the minimum time. Make your reasoning clear. [2]
Edexcel D1 2005 June Q1
Easy -1.8
The table shows the marks obtained by students in a test. The students are listed in alphabetical order. Carry out a quick sort to produce a list of students in descending order of marks. You should show the result of each pass and identify your pivots clearly.
Ali74
Bobby28
Eun-Jung63
Katie54
Marciana54
Peter49
Rory37
Sophie68
(Total 5 marks)
Edexcel D1 2005 June Q2
7 marks Moderate -0.8
\includegraphics{figure_1}
  1. Starting from \(A\); write down a Hamiltonian cycle for the graph in Figure 1. [2]
  2. Use the planarity algorithm to show that the graph in Figure 1 is planar. [3]
Arcs \(AF\) and \(EF\) are now added to the graph.
  1. Explain why the new graph is not planar. [2]
(Total 7 marks)
Edexcel D1 2005 June Q3
7 marks Moderate -0.3
\includegraphics{figure_2} Figure 2 models a network of roads which need to be inspected to assess if they need to be resurfaced. The number on each arc represents the length, in km, of that road. Each road must be traversed at least once and the length of the inspection route must be minimised.
  1. Starting and finishing at \(A\), solve this route inspection problem. You should make your method and working clear. State the length of the shortest route. (The weight of the network is 77 km.) [5]
Given that it is now permitted to start and finish the inspection at two distinct vertices,
  1. state which two vertices you should choose to minimise the length of the route. Give a reason for your answer. [2]
(Total 7 marks)
Edexcel D1 2005 June Q4
7 marks Moderate -0.8
The precedence table shows the activities involved in a project.
ActivityImmediately preceding activities
\(A\)--
\(B\)--
\(C\)--
\(D\)\(A\)
\(E\)\(A\)
\(F\)\(B\)
\(G\)\(B\)
\(H\)\(C, D\)
\(I\)\(E\)
\(J\)\(F, H\)
\(K\)\(G, J\)
\(L\)\(G\)
\(M\)\(L\)
\(N\)\(L\)
  1. Draw the activity network for this project, using activity on arc and using two dummies. [4]
  2. Explain why each of the two dummies is necessary. [3]
(Total 7 marks)
Edexcel D1 2005 June Q5
8 marks Moderate -0.8
\includegraphics{figure_3} \includegraphics{figure_4} A film critic, Verity, must see five films A, B, C, D and E over two days. The films are being shown at five special critics' preview times: \begin{align} 1 &\text{ (Monday 4 pm),}
2 &\text{ (Monday 7 pm),}
3 &\text{ (Tuesday 1 pm),}
4 &\text{ (Tuesday 4 pm),}
5 &\text{ (Tuesday 7 pm).} \end{align} The bipartite graph in Figure 3 shows the times at which each film is showing. Initially Verity intends to see \begin{align} &\text{Film A on Monday at 4 pm,}
&\text{Film B on Tuesday at 4 pm,}
&\text{Film C on Tuesday at 1 pm,}
&\text{Film D on Monday at 7 pm.} \end{align} This initial matching is shown in Figure 4. Using the maximum matching algorithm and the given initial matching,
  1. find two distinct alternating paths and complete the matchings they give. [6]
Verity's son is very keen to see film D, but he can only go with his mother to the showing on Monday at 7 pm.
  1. Explain why it will not be possible for Verity to take her son to this showing and still see all five films herself. [2]
(Total 8 marks)
Edexcel D1 2005 June Q6
10 marks Easy -1.2
\includegraphics{figure_5} Figure 5 shows a network of roads. The number on each arc represents the length of that road in km.
  1. Use Dijkstra's algorithm to find the shortest route from \(A\) to \(J\). State your shortest route and its length. [5]
  2. Explain how you determined the shortest route from your labelled diagram. [2]
The road from \(C\) to \(F\) will be closed next week for repairs.
  1. Find the shortest route from \(A\) to \(J\) that does not include \(CF\) and state its length. [3]
(Total 10 marks)
Edexcel D1 2005 June Q7
15 marks Moderate -0.3
Polly has a bird food stall at the local market. Each week she makes and sells three types of packs \(A\), \(B\) and \(C\). Pack \(A\) contains 4 kg of bird seed, 2 suet blocks and 1 kg of peanuts. Pack \(B\) contains 5 kg of bird seed, 1 suet block and 2 kg of peanuts. Pack \(C\) contains 10 kg of bird seed, 4 suet blocks and 3 kg of peanuts. Each week Polly has 140 kg of bird seed, 60 suet blocks and 60 kg of peanuts available for the packs. The profit made on each pack of \(A\), \(B\) and \(C\) sold is £3.50, £3.50 and £6.50 respectively. Polly sells every pack on her stall and wishes to maximise her profit, \(P\) pence. Let \(x\), \(y\) and \(z\) be the numbers of packs \(A\), \(B\) and \(C\) sold each week. An initial Simplex tableau for the above situation is
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
\(r\)4510100140
\(s\)21401060
\(t\)12300160
\(P\)\(-350\)\(-350\)\(-650\)0000
  1. Explain the meaning of the variables \(r\), \(s\) and \(t\) in the context of this question. [2]
  2. Perform one complete iteration of the Simplex algorithm to form a new tableau \(T\). Take the most negative number in the profit row to indicate the pivotal column. [5]
  3. State the value of every variable as given by tableau \(T\). [3]
  4. Write down the profit equation given by tableau \(T\). [2]
  5. Use your profit equation to explain why tableau \(T\) is not optimal. [1]
Taking the most negative number in the profit row to indicate the pivotal column,
  1. identify clearly the location of the next pivotal element. [2]
(Total 15 marks)
Edexcel D1 2005 June Q8
16 marks Standard +0.3
\includegraphics{figure_6} Figure 6 shows a capacitated directed network. The number on each arc is its capacity. The numbers in circles show a feasible flow through the network. Take this as the initial flow.
  1. On Diagram 1 and Diagram 2 in the answer book, add a supersource \(S\) and a supersink \(T\). On Diagram 1 show the minimum capacities of the arcs you have added. [2]
Diagram 2 in the answer book shows the first stage of the labelling procedure for the given initial flow.
  1. Complete the initial labelling procedure in Diagram 2. [2]
  2. Find the maximum flow through the network. You must list each flow-augmenting route you use, together with its flow, and state the maximal flow. [6]
  3. Show a maximal flow pattern on Diagram 3. [2]
  4. Prove that your flow is maximal. [2]
  5. Describe briefly a situation for which this network could be a suitable model. [2]
(Total 16 marks)
Edexcel D1 2006 June Q1
4 marks Easy -1.8
52 48 50 45 64 47 53 The list of numbers above is to be sorted into descending order. Perform a bubble sort to obtain the sorted list, giving the state of the list after each completed pass. [4]
Edexcel D1 2006 June Q2
7 marks Easy -1.2
  1. Define the term 'alternating path'. [2]
  2. \includegraphics{figure_1} The bipartite graph in Figure 1 shows the films that six customers wish to hire this Saturday evening. The shop has only one copy of each film. The bold lines indicate an initial matching. Starting from this initial matching use the maximum matching algorithm twice to obtain a complete matching. You should clearly state the alternating paths you use. [5]
Edexcel D1 2006 June Q3
7 marks Moderate -0.8
\includegraphics{figure_2} Figure 2 shows the network of pipes represented by arcs. The length of each pipe, in kilometres, is shown by the number on each arc. The network is to be inspected for leakages, using the shortest route and starting and finishing at A.
  1. Use the route inspection algorithm to find which arcs, if any, need to be traversed twice. [4]
  2. State the length of the minimum route. [The total weight of the network is 394 km.] [1]
It is now permitted to start and finish the inspection at two distinct vertices.
  1. State, with a reason, which two vertices should be chosen to minimise the length of the new route. [2]
Edexcel D1 2006 June Q4
12 marks Easy -1.3
  1. Explain what is meant by the term 'path'. [2]
\includegraphics{figure_3} Figure 3 shows a network of cycle tracks. The number on each edge represents the length, in miles, of that track. Mary wishes to cycle from A to I as part of a cycling holiday. She wishes to minimise the distance she travels.
  1. Use Dijkstra's algorithm to find the shortest path from A to I. Show all necessary working in the boxes in Diagram 1 in the answer book. State your shortest path and its length. [6]
  2. Explain how you determined the shortest path from your labelling. [2]
Mary wants to visit a theme park at E.
  1. Find a path of minimal length that goes from A to I via E and state its length. [2]
Edexcel D1 2006 June Q5
15 marks Moderate -0.8
\includegraphics{figure_4} An engineering project is modelled by the activity network shown in Figure 4. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the activity. Each activity requires one worker. The project is to be completed in the shortest time.
  1. Calculate the early time and late time for each event. Write these in boxes in Diagram 1 in the answer book. [4]
  2. State the critical activities. [1]
  3. Find the total float on activities D and F. You must show your working. [3]
  4. On the grid in the answer book, draw a cascade (Gantt) chart for this project. [4]
The chief engineer visits the project on day 15 and day 25 to check the progress of the work. Given that the project is on schedule,
  1. which activities must be happening on each of these two days? [3]