Questions D1 (932 questions)

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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]
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)