Questions — Edexcel (9685 questions)

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Edexcel D1 2013 June Q4
8 marks Easy -1.8
4.
  1. Sam (S)
  2. Janelle (J)
  3. Haoyu (H)
  4. Alfie (A)
  5. Cyrus (C)
  6. Komal (K)
  7. Polly (P)
  8. David (D)
  9. Tom (T)
  10. Lydia (L)
A binary search is to be performed on the names in the list above to locate the name Lydia.
  1. Using an appropriate algorithm, rearrange the list so that a binary search can be performed, showing the state of the list after each complete iteration. State the name of the algorithm you have used.
  2. Use the binary search algorithm to locate the name Lydia in the list you obtained in (a). You must make your method clear.
Edexcel D1 2013 June Q5
10 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1493d74b-e9ef-4c9a-91f6-877c1eaa74e2-06_712_1520_246_267} \captionsetup{labelformat=empty} \caption{Figure 4
[0pt] [The total weight of the network is 181 miles]}
\end{figure} Figure 4 represents a network of power cables that have to be inspected. The number on each arc represents the length, in km , of that cable. A route of minimum length that traverses each cable at least once and starts and finishes at A needs to be found.
  1. Use the route inspection algorithm to find the arcs that will need to be traversed twice. You must make your method and working clear.
  2. Write down a possible shortest inspection route, giving its length. It is now decided to start and finish the inspection route at two distinct vertices. The route must still traverse each cable at least once.
  3. Determine possible starting and finishing points so that the length of the route is minimised. You must give reasons for your answer.
Edexcel D1 2013 June Q6
7 marks Moderate -0.3
6.
ActivityImmediately preceding activities
A-
B-
CA
DA
EB
FC D
GD
HF G
IH
JH
KI J
  1. Draw the activity network described in the precedence table, using activity on arc and exactly two dummies.
    (5)
  2. Explain why each of the two dummies is necessary.
    (2)
Edexcel D1 2013 June Q7
7 marks Moderate -0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1493d74b-e9ef-4c9a-91f6-877c1eaa74e2-08_748_1563_251_242} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 represents a network of roads. The number on each arc represents the length, in miles, of the corresponding road. A large crane is required at J and it may be transported from either \(\mathrm { C } _ { 1 }\) or \(\mathrm { C } _ { 2 }\). A route of minimum length is required. It is decided to use Dijkstra's algorithm to find the shortest routes between \(\mathrm { C } _ { 1 }\) and J and between \(\mathrm { C } _ { 2 }\) and J .
  1. Explain why J , rather than \(\mathrm { C } _ { 1 }\) or \(\mathrm { C } _ { 2 }\), should be chosen as the starting vertex.
    (1)
  2. Use Dijkstra's algorithm to find the shortest route needed to transport the crane. State your route and its length.
    (6)
Edexcel D1 2013 June Q8
16 marks Moderate -0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1493d74b-e9ef-4c9a-91f6-877c1eaa74e2-09_1118_1134_214_486} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} A company makes two types of garden bench, the 'Rustic' and the 'Contemporary'. The company wishes to maximise its profit and decides to use linear programming. Let \(x\) be the number of 'Rustic' benches made each week and \(y\) be the number of 'Contemporary' benches made each week. The graph in Figure 6 is being used to solve this linear programming problem.
Two of the constraints have been drawn on the graph and the rejected region shaded out.
  1. Write down the constraints shown on the graph giving your answers as inequalities in terms of \(x\) and \(y\). It takes 4 working hours to make one 'Rustic' bench and 3 working hours to make one 'Contemporary' bench. There are 120 working hours available in each week.
  2. Write down an inequality to represent this information. Market research shows that 'Rustic' benches should be at most \(\frac { 3 } { 4 }\) of the total benches made each week.
  3. Write down, and simplify, an inequality to represent this information. Your inequality must have integer coefficients.
  4. Add two lines and shading to Diagram 1 in your answer book to represent the inequalities of (b) and (c). Hence determine and label the feasible region, R. The profit on each 'Rustic' bench and each 'Contemporary' bench is \(\pounds 45\) and \(\pounds 30\) respectively.
  5. Write down the objective function, P , in terms of \(x\) and \(y\).
  6. Determine the coordinates of each of the vertices of the feasible region and hence use the vertex method to determine the optimal point.
  7. State the maximum weekly profit the company could make.
    (Total 16 marks)
Edexcel D1 2013 June Q1
7 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b32eb57-c9cd-46ec-a328-12050148bdf7-2_766_570_324_342} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b32eb57-c9cd-46ec-a328-12050148bdf7-2_755_570_331_1146} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows the possible allocations of six people, Alex (A), Ben (B), Harriet (H), Izzy (I), Leo (L) and Rowan (R), to six tasks, \(1,2,3,4,5\) and 6.
  1. Write down the technical name given to the type of diagram shown in Figure 1. Figure 2 shows an initial matching.
  2. Starting from the given initial matching, use the maximum matching algorithm to find a complete matching. You should list the alternating paths you use, and state your improved matching after each iteration.
    (6)
Edexcel D1 2013 June Q2
11 marks Moderate -0.8
2.
0.6
0.2
0.4
0.5
0.7
0.1
0.9
0.3
1.5
1.6
  1. Use the first-fit bin packing algorithm to determine how the numbers listed above can be packed into bins of size 2.
    (3)
  2. The list of numbers is to be sorted into descending order. Use a quick sort to obtain the sorted list. You must make your pivots clear.
  3. Apply the first-fit decreasing bin packing algorithm to your ordered list to pack the numbers into bins of size 2 .
  4. Determine whether your answer to (c) uses the minimum number of bins. You must justify your answer.
Edexcel D1 2013 June Q3
10 marks Easy -1.3
3.
ABCDEF
A-1569--
B15-12-14-
C612-710-
D9-7-1117
E-141011-5
F---175-
The table shows the times, in days, needed to repair the network of roads between six towns, A, B, C, D, E and F, following a flood.
  1. Use Prim's algorithm, starting at A , to find the minimum connector for this network. You must list the arcs that form your tree in the order that you selected them.
  2. Draw your minimum connector using the vertices given in Diagram 1 in the answer book.
  3. Add arcs from D, E and F to Diagram 2 in the answer book, so that it shows the network of roads shown by the table.
  4. Use Kruskal's algorithm to find the minimum connector. You should list the arcs in the order in which you consider them. In each case, state whether you are adding the arc to your minimum connector.
  5. State the minimum time needed, in days, to reconnect the six towns.
Edexcel D1 2013 June Q4
8 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b32eb57-c9cd-46ec-a328-12050148bdf7-5_780_1717_276_175} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 represents a network of roads. The number on each arc represents the length, in miles, of the corresponding road. Liz wishes to travel from S to T .
  1. Use Dijkstra's algorithm to find the shortest path from S to T . State your path and its length. On a particular day, Liz must include F in her route.
  2. Find the shortest path from S to T that includes F , and state its length.
Edexcel D1 2013 June Q5
10 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b32eb57-c9cd-46ec-a328-12050148bdf7-6_829_1547_257_259} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} \section*{[The total weight of the network is 344 miles]} Figure 4 represents a railway network. The number on each arc represents the length, in miles, of that section of the railway. Sophie needs to travel along each section to check that it is in good condition.
She must travel along each arc of the network at least once, and wants to find a route of minimum length. She will start and finish at A.
  1. Use the route inspection algorithm to find the arcs that will need to be traversed twice. You must make your method and working clear.
  2. Write down a possible shortest inspection route, giving its length. Sophie now decides to start the inspection route at E. The route must still traverse each arc at least once but may finish at any vertex.
  3. Determine the finishing point so that the length of the route is minimised. You must give reasons for your answer and state the length of your route.
Edexcel D1 2013 June Q6
12 marks Easy -1.2
6. Harry wants to rent out boats at his local park. He can use linear programming to determine the number of each type of boat he should buy. Let \(x\) be the number of 2 -seater boats and \(y\) be the number of 4 -seater boats.
One of the constraints is $$x + y \geqslant 90$$
  1. Explain what this constraint means in the context of the question. Another constraint is $$2 x \leqslant 3 y$$
  2. Explain what this constraint means in the context of the question. A third constraint is $$y \leqslant x + 30$$
  3. Represent these three constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region R . Each 2 -seater boat costs \(\pounds 100\) and each 4 -seater boat costs \(\pounds 300\) to buy. Harry wishes to minimise the total cost of buying the boats.
  4. Write down the objective function, C , in terms of \(x\) and \(y\).
  5. Determine the number of each type of boat that Harry should buy. You must make your method clear and state the minimum cost.
Edexcel D1 2013 June Q7
17 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b32eb57-c9cd-46ec-a328-12050148bdf7-8_724_1730_241_167} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} \section*{[The sum of the duration of all activities is 172 days]} 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 days, to complete the activity. Each activity requires one worker. The project is to be completed in the shortest possible time.
  1. Complete Diagram 1 in the answer book to show the early event times and late event times.
  2. Calculate the total float for activity M. You must make the numbers you use in your calculation clear.
  3. For each of the situations below, explain the effect that the delay would have on the project completion date.
    1. A 2 day delay on the early start of activity P.
    2. A 2 day delay on the early start of activity Q .
  4. Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time. Diagram 2 in the answer book shows a partly completed cascade chart for this project.
  5. Complete the cascade chart.
  6. Use your cascade chart to determine a second lower bound on the number of workers needed to complete the project in the shortest possible time. You must make specific reference to times and activities.
  7. State which of the two lower bounds found in (d) and (f) is better. Give a reason for your answer.
    (Total 17 marks)
Edexcel D1 2014 June Q1
11 marks Easy -1.2
1. $$\begin{array} { l l l l l l l l l } 31 & 10 & 38 & 45 & 19 & 47 & 35 & 28 & 12 \end{array}$$
  1. Use the first-fit bin packing algorithm to determine how the numbers listed above can be packed into bins of size 60
  2. Carry out a quick sort to produce a list of the numbers in descending order. You should show the result of each pass and identify your pivots clearly.
  3. Use the first-fit decreasing bin packing algorithm to determine how the numbers listed can be packed into bins of size 60
  4. Determine whether the number of bins used in (c) is optimal. Give a reason for your answer.
Edexcel D1 2014 June Q2
7 marks Moderate -0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{23cc3c59-35d8-4120-9965-952c0ced5b3d-3_437_399_251_511} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{23cc3c59-35d8-4120-9965-952c0ced5b3d-3_437_401_251_1151} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows the possible allocations of five employees, Ali (A), Campbell (C), Hugo (H), Janelle (J) and Polly (P), to five tasks 1, 2, 3, 4 and 5.
  1. Explain why it is not possible to find a complete matching. It is decided that one of the employees should be trained so that a complete matching becomes possible. There are only enough funds for one employee to be trained. Two employees volunteer to undergo training. Janelle can be trained to do task 1 or Hugo can be trained to do task 5.
  2. Decide which employee, Janelle or Hugo, should undergo training. Give a reason for your answer. You may now assume that the employee you identified in (b) has successfully undergone training. Figure 2 shows an initial matching.
  3. Starting from the given initial matching, use the maximum matching algorithm to find a complete matching. You should list the alternating path that you use, and state the complete matching.
Edexcel D1 2014 June Q3
9 marks Moderate -0.5
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{23cc3c59-35d8-4120-9965-952c0ced5b3d-4_605_1378_248_370} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 represents a network of roads. The number on each arc represents the time taken, in minutes, to traverse each road.
  1. Use Dijkstra's algorithm to find the quickest route from S to T. State your quickest route and the time taken.
    (6) It is now necessary to include E in the route.
  2. Determine the effect that this will have on the time taken for the journey. You must state your new quickest route and the time it takes.
    (3)
Edexcel D1 2014 June Q4
13 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{23cc3c59-35d8-4120-9965-952c0ced5b3d-5_846_798_205_639} \captionsetup{labelformat=empty} \caption{Figure 4
[0pt] [The total weight of the network is 359 cm]}
\end{figure} Figure 4 represents the network of sensor wires used in a medical scanner. The number on each arc represents the length, in cm, of that section of wire. After production, each scanner is tested.
A machine will be programmed to inspect each section of wire.
It will travel along each arc of the network at least once, starting and finishing at A. Its route must be of minimum length.
  1. Use the route inspection algorithm to find the length of a shortest inspection route. You must make your method and working clear. The machine will inspect 15 cm of wire per second.
  2. Calculate the total time taken, in seconds, to test 120 scanners. It is now possible for the machine to start at one vertex and finish at a different vertex. An inspection route of minimum length is still required.
  3. Explain why the machine should be programmed to start at a vertex with odd degree. Due to constraints at the factory, only B or D can be chosen as the starting point and there will also be a 2 second pause between tests.
  4. Determine the new minimum total time now taken to test 120 scanners. You must state which vertex you are starting from and make your calculations clear.
    (4)
Edexcel D1 2014 June Q5
11 marks Moderate -0.3
5. A linear programming problem in \(x\) and \(y\) is described as follows. Maximise \(\quad P = 2 x + 3 y\) subject to $$\begin{aligned} x & \geqslant 25 \\ y & \geqslant 25 \\ 7 x + 8 y & \leqslant 840 \\ 4 y & \leqslant 5 x \\ 5 y & \geqslant 3 x \\ x , y & \geqslant 0 \end{aligned}$$
  1. Add lines and shading to Diagram 1 in the answer book to represent these constraints. Hence determine the feasible region and label it R .
  2. Use the objective line method to find the optimal vertex, V, of the feasible region. You must clearly draw and label your objective line and the vertex V.
  3. Calculate the exact coordinates of vertex V. Given that an integer solution is required,
  4. determine the optimal solution with integer coordinates. You must make your method clear.
Edexcel D1 2014 June Q6
7 marks Moderate -0.5
6. (i) Draw the activity network described in the precedence table below, using activity on arc and the minimum number of dummies.
ActivityImmediately preceding activities
\(A\)-
B-
C-
DA, \(C\)
EB
\(F\)E
GA
\(H\)\(D , F\)
I\(D , F\)
JH, I
(ii) Explain why each of your dummies is necessary.
Edexcel D1 2014 June Q7
11 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{23cc3c59-35d8-4120-9965-952c0ced5b3d-8_620_1221_251_427} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} 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 days, to complete the activity. Each activity requires one worker. The project is to be completed in the shortest possible time.
  1. Complete Diagram 1 in the answer book to show the early event times and late event times.
  2. Calculate the total float for activity D. You must make the numbers you use in your calculation clear.
  3. Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working. The project is to be completed in the minimum time using as few workers as possible.
  4. Schedule the activities using Grid 1 in the answer book.
Edexcel D1 2014 June Q8
6 marks Moderate -0.8
8. A manufacturer of frozen yoghurt is going to exhibit at a trade fair. He will take two types of frozen yoghurt, Banana Blast and Strawberry Scream. He will take a total of at least 1000 litres of yoghurt.
He wants at least \(25 \%\) of the yoghurt to be Banana Blast. He also wants there to be at most half as much Banana Blast as Strawberry Scream. Each litre of Banana Blast costs \(\pounds 3\) to produce and each litre of Strawberry Scream costs \(\pounds 2\) to produce. The manufacturer wants to minimise his costs. Let \(x\) represent the number of litres of Banana Blast and \(y\) represent the number of litres of Strawberry Scream. Formulate this as a linear programming problem, stating the objective and listing the constraints as simplified inequalities with integer coefficients. You should not attempt to solve the problem.
(Total 6 marks)
Edexcel D1 2014 June Q1
5 marks Easy -1.2
1.
ArtBiologyChemistryDramaEnglishFrenchGraphics
Art (A)-619373504842
Biology (B)61-11482836358
Chemistry (C)93114-59947788
Drama (D)738259-8910441
English (E)50839489-9175
French (F)48637710491-68
Graphics (G)425888417568-
The table shows the travelling times, in seconds, to walk between seven departments in a college.
  1. Use Prim's algorithm, starting at Art, to find the minimum spanning tree for the network represented by the table. You must clearly state the order in which you select the edges of your tree.
    (3)
  2. Draw the minimum spanning tree using the vertices given in Diagram 1 in the answer book.
  3. State the weight of the tree.
Edexcel D1 2014 June Q2
7 marks Moderate -0.8
2. (a) Draw the activity network described in the precedence table below, using activity on arc and exactly two dummies.
ActivityImmediately preceding activities
A-
B-
C-
D\(A , B\)
EC
FA, B
GA, B
HE, F
ID
JD, G
K\(H\)
(b) Explain why each of the two dummies is necessary.
Edexcel D1 2014 June Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{818ba207-5839-4698-aacb-75dab88b218f-04_821_1374_210_374} \captionsetup{labelformat=empty} \caption{Figure 1
[0pt] [The total weight of the network is 451]}
\end{figure} Figure 1 models a network of tracks in a forest that need to be inspected by a park ranger. The number on each arc is the length, in km, of that section of the forest track. Each track must be traversed at least once and the length of the inspection route must be minimised. The inspection route taken by the ranger must start and end at vertex A.
  1. Use the route inspection algorithm to find the length of a shortest inspection route. State the arcs that should be repeated. You should make your method and working clear.
  2. State the number of times that vertex J would appear in the inspection route. The landowner decides to build two huts, one hut at vertex K and the other hut at a different vertex. In future, the ranger will be able to start his inspection route at one hut and finish at the other. The inspection route must still traverse each track at least once.
  3. Determine where the other hut should be built so that the length of the route is minimised. You must give reasons for your answer and state a possible route and its length.
Edexcel D1 2014 June Q4
9 marks Moderate -0.8
4. Six pupils, Ashley (A), Fran (F), Jas (J), Ned (N), Peter (P) and Richard (R), each wish to learn a musical instrument. The school they attend has six spare instruments; a clarinet (C), a trumpet (T), a violin (V), a keyboard (K), a set of drums (D) and a guitar (G). The pupils are asked which instruments they would prefer and their preferences are given in the table above. It is decided that each pupil must learn a different instrument and each pupil needs to be allocated to exactly one of their preferred instruments.
  1. Using Diagram 1 in the answer book, draw a bipartite graph to show the possible allocations of pupils to instruments. Initially Ashley, Fran, Jas and Richard are each allocated to their first preference.
  2. Show this initial matching on Diagram 2 in the answer book.
  3. Starting with the initial matching from (b), apply the maximum matching algorithm once to find an improved matching. You must state the alternating path you use and give your improved matching.
  4. Explain why a complete matching is not possible. Fran decides that as a third preference she would like to learn to play the guitar. Peter decides that as a second preference he would like to learn to play the drums.
  5. Starting with the improved matching found in (c), use the maximum matching algorithm to obtain a complete matching. You must state the alternating path you use and your complete matching.
Edexcel D1 2014 June Q5
9 marks Moderate -0.5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{818ba207-5839-4698-aacb-75dab88b218f-06_851_1490_191_317} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Sharon is planning a road trip from Preston to York. Figure 2 shows the network of roads that she could take on her trip. The number on each arc is the length of the corresponding road in miles.
  1. Use Dijkstra's algorithm to find the shortest route from Preston (P) to York (Y). State the shortest route and its length.
    (6) Sharon has a friend, John, who lives in Manchester (M). Sharon decides to travel from Preston to York via Manchester so she can visit John. She wishes to minimise the length of her route.
  2. State the new shortest route. Hence calculate the additional distance she must travel to visit John on this trip. You must make clear the numbers you use in your calculation.
    (3)