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Edexcel D1 2018 June Q1
8 marks Easy -1.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6b51f3a0-0945-4254-8c63-20e1371e9e3a-02_1189_1531_360_267} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Define the terms
    1. tree,
    2. minimum spanning tree.
  2. Use Prim's algorithm, starting at A , to find a minimum spanning tree for the network shown in Figure 1. You must clearly state the order in which you select the arcs of the tree.
  3. Draw the minimum spanning tree using the vertices given in Diagram 1 in the answer book and state the weight of the tree.
Edexcel D1 2018 June Q2
12 marks Easy -1.2
2. A list of nine numbers needs to be sorted into descending order.
  1. Describe how to carry out the first pass of a bubble sort on the numbers in the list. Mayleen used a sorting algorithm to sort a list of nine numbers into descending order. Mayleen's list after the first pass through the algorithm is given below. $$\begin{array} { l l l l l l l l l } 30 & 33 & 35 & 27 & 20 & 24 & 21 & 15 & 19 \end{array}$$
  2. Explain how you know that Mayleen did not use the bubble sort algorithm. Given that Mayleen used the quick sort algorithm,
  3. write down the number that was used as a pivot for the first pass,
  4. complete the quick sort to obtain a fully sorted list in descending order. You must make your pivots clear.
  5. Use the first-fit decreasing bin packing algorithm to determine how the numbers listed can be packed into bins of size 60 A tenth number, 18, is added to the list of nine numbers.
  6. Determine whether it is possible to pack the ten numbers into 4 bins of size 60 . You must justify your answer.
Edexcel D1 2018 June Q3
7 marks Moderate -0.8
3. (a) Draw the activity network described in the precedence table below, using activity on arc and exactly four dummies.
(5)
ActivityImmediately preceding activities
A-
B-
C-
DA
ED
FA, B
GA, B, C
HA, B, C
IE, F, G
JE, F, G
KE, F, G, H
Given that D is a critical activity,
(b) state which other activities must also be critical.
(2)
Edexcel D1 2018 June Q4
14 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6b51f3a0-0945-4254-8c63-20e1371e9e3a-05_812_1655_269_205} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} [The total weight of the network is 293] Figure 2 models a network of roads. The number on each edge gives the time, in minutes, taken to travel along the corresponding road.
  1. Use Dijkstra's algorithm to find the shortest time needed to travel from A to J . State the quickest route.
    (6) The road represented by edge GJ is closed due to essential maintenance. Sahil needs to travel along all the other roads to check that they are in good repair. Sahil wishes to complete his route as quickly as possible and will start and finish at the same vertex.
  2. Use the route inspection algorithm to find the duration of Sahil's quickest route. State the edges that will need to be traversed twice. You should make your method and working clear.
    (6) Given that Sahil will start and finish at vertex C,
  3. state the number of times that Sahil will visit vertex E and vertex F in his inspection route.
    (2)
Edexcel D1 2018 June Q5
6 marks Moderate -0.5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6b51f3a0-0945-4254-8c63-20e1371e9e3a-06_648_567_273_750} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the possible allocations of five workers, Cole (C), Dorothy (D), Harold (H), Richard (R) and Stephen (S), to five tasks, 1, 2, 3, 4 and 5. In an initial matching, each of three workers is allocated to a different task. For this initial matching, there are three possible alternating paths that start at C . One alternating path is $$C - 3 = S - 4 = D - 5$$ A second alternating path is $$\mathrm { C } - 1 = \mathrm { H } - 2$$
  1. Use this information to deduce the initial matching.
  2. Find the third alternating path that starts at C .
  3. List the improved matching generated by using the alternating path \(\mathrm { C } - 3 = \mathrm { S } - 4 = \mathrm { D } - 5\)
  4. Starting from the improved matching found in (c), use the maximum matching algorithm to obtain a complete matching. You must list the alternating path you use and the final matching.
    (3)
Edexcel D1 2018 June Q6
11 marks Moderate -0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6b51f3a0-0945-4254-8c63-20e1371e9e3a-07_748_1419_269_324} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A 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 corresponding 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 the late event times.
  2. State the critical activities.
  3. Draw a cascade (Gantt) chart for this project on the grid in the answer book.
  4. Use your cascade chart to determine the minimum number of workers needed to complete the project in the shortest possible time. You must make specific reference to times and activities. (You do not need to provide a schedule of the activities.)
Edexcel D1 2018 June Q20
Easy -1.8
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\hline & 1 & I & I & I & 1 & 1 & & I & I & I & I & I & 1 & I & I & I & I & 1 & । &
\hline \multirow{7}{*}{\includegraphics[max width=\textwidth, alt={}]{6b51f3a0-0945-4254-8c63-20e1371e9e3a-23_1401_67_1168_25} } & & & & & & & & & & & & & & & & & & & I &
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\hline & & & & & & & & & & & & & & & & & & & &
\hline & & & & & & & & & & & & & & & & & & & &
\hline & \multicolumn{19}{|r|}{\multirow[b]{2}{*}{(Total 11 marks)}} &
\hline & & & & & & & & & & & & & & & & & & & &
\hline & & & & & & & & & & & & & & & & & & & &
\hline \end{tabular} \end{center} 7.
\includegraphics[max width=\textwidth, alt={}]{6b51f3a0-0945-4254-8c63-20e1371e9e3a-25_1241_1590_299_185}
\section*{Diagram 1}
\includegraphics[max width=\textwidth, alt={}]{6b51f3a0-0945-4254-8c63-20e1371e9e3a-28_2639_1830_121_121}
Edexcel D1 2019 June Q1
7 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{87f0e571-e708-4ca9-adc7-4ed18e144d32-02_474_501_374_429} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{87f0e571-e708-4ca9-adc7-4ed18e144d32-02_474_501_374_1133} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows the possible allocations of six people, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F , 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 2019 June Q2
11 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{87f0e571-e708-4ca9-adc7-4ed18e144d32-03_716_1491_239_294} \captionsetup{labelformat=empty} \caption{Figure 3
[0pt] [The total weight of the network is 48.2]}
\end{figure} A surveyor 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 Figure 3. The number on each arc represents the length of that road in miles. To check all the roads, she needs to travel along each road at least once. She wishes to minimise the total distance travelled. The surveyor's office is at F , so she starts and ends her journey at F .
  1. Find a route for the surveyor to follow. State your route and its length. You must make your method and reasoning clear. The surveyor lives at D and wonders if she can reduce the distance travelled by starting from home and inspecting all the roads on the way to her office at F .
  2. By considering the pairings of all relevant nodes, find the arcs that will need to be traversed twice in the inspection route from D to F. You must make your method and working clear.
  3. Determine which of the two routes, the one starting at F and ending at F , or the one starting at D and ending at F , is longer. You must show your working.
Edexcel D1 2019 June Q3
11 marks Moderate -0.8
3.
ABCDEFGHJ
A-385-----
B3-4------
C84--94---
D5----749-
E--9--4--7
F--474--813
G---4---4-
H---9-84-7
J----713-7-
The table above shows the lengths, in metres, of the paths between nine vertices, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F }\), G, H and J.
  1. Use Prim's algorithm, starting at A , to find a minimum spanning tree for this table of distances. You must clearly state the order in which you select the edges and state its weight. Draw your minimum spanning tree using the vertices in the answer book.
  2. State whether your minimum spanning tree is unique. Justify your answer.
  3. Use Dijkstra's algorithm to find the length of the shortest path from A to J.
Edexcel D1 2019 June Q4
15 marks Moderate -0.8
4. $$\begin{array} { l l l l l l l l l l l } 25 & 9 & 32 & 16 & 17 & 23 & 18 & 12 & 4 & 8 & 40 \end{array}$$ The numbers in the list represent the weights, in kilograms, of eleven suitcases. The suitcases are to be transported in containers that will each hold a maximum weight of 50 kg .
  1. Calculate a lower bound for the number of containers needed. You must make your method clear.
  2. Use the first-fit bin packing algorithm to allocate the suitcases to the containers.
  3. Carry out a quick sort to produce a list of the weights in descending order. You should show the result of each pass and identify your pivots clearly.
  4. Use the first-fit decreasing bin packing algorithm to allocate the suitcases to the containers. The two heaviest suitcases are replaced with two suitcases both of which weigh \(x \mathrm {~kg}\). It is given that the lower bound for the number of containers needed is now one less than the number found in (a).
  5. Determine the range of values for \(x\). You should make your working clear.
Edexcel D1 2019 June Q5
7 marks Moderate -0.8
5.
ActivityImmediately preceding activities
A-
B-
CA
DA
EA, B
FC, D
GD
HD, E
IF, G
JF, G, H
  1. Draw the activity network described in the precedence table above, using activity on arc and exactly 4 dummies.
  2. Explain why one of the activities I or J must be critical. It is given that activity C is a critical activity.
  3. State the activities that are therefore guaranteed to be critical.
Edexcel D1 2019 June Q6
10 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{87f0e571-e708-4ca9-adc7-4ed18e144d32-07_1502_1659_230_210} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region. The vertices of the feasible region are \(A ( 4,7 ) , B ( 5,3 ) , C ( - 1,5 )\) and \(D ( - 2,1 )\).
  1. Determine the inequality that defines the boundary of \(R\) that passes through vertices \(A\) and \(C\), leaving your answer with integer coefficients only. The objective is to maximise \(P = 5 x + y\)
  2. Find the coordinates of the optimal vertex and the corresponding value of \(P\). The objective is changed to maximise \(Q = k x + y\)
  3. If \(k\) can take any value, find the range of values of \(k\) for which \(A\) is the only optimal vertex.
Edexcel D1 Q2
4 marks Easy -1.8
2. Use the binary search algorithm to try to locate the name Hannah in the following alphabetical list. Clearly indicate how you selected your pivots and which part of the list is being rejected at each stage. \begin{displayquote} Adam
Ben
Charlie
Eleanor
Freya
Greg
Jenny
Richard
Toby \end{displayquote}
Edexcel D1 Q3
8 marks Moderate -0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{552f3296-ad61-448b-8168-6709fb359fa2-3_780_1353_248_356} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 represents the distance, in metres, between eight data collection points, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F }\), G and H . The data collection points are to be linked by cables.
  1. Listing the arcs in the order that you select them, find a minimum spanning tree for the network using
    1. Kruskal's algorithm, stating in addition any arcs you reject,
    2. Prim's algorithm, starting from A .
  2. State the minimum amount of cable needed.
  3. Draw your minimum spanning tree using the vertices given in Figure 1 in your answer book.
Edexcel D1 Q4
8 marks Moderate -0.5
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{552f3296-ad61-448b-8168-6709fb359fa2-4_549_586_285_356} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{552f3296-ad61-448b-8168-6709fb359fa2-4_545_583_287_1128} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Six airline pilots, Alice, Dan, Miya, Phil, Sophie and Tom, are to be assigned to six flights, 1, 2, 3, 4, 5 and 6. A bipartite graph showing the possible allocations is shown in Figure 2, and an initial matching is given in Figure 3. The maximum matching algorithm will be used to obtain a complete matching.
  1. Starting from A, find an alternating path that leads to an improved matching and list the improved matching that it gives.
    (3)
  2. Using the improved matching found in part (a) as the new initial matching, find a complete matching. You must state any alternating paths you use and list your final complete matching.
Edexcel D1 Q5
7 marks Moderate -0.5
5.
ActivityImmediately preceding activity
A-
B\(\boldsymbol { A }\)
\(\boldsymbol { C }\)\(\boldsymbol { A }\)
DA
E\(\boldsymbol { B } \boldsymbol { C }\)
FB C
G\(\boldsymbol { D }\)
\(\boldsymbol { H }\)D
IE
\(J\)E F \(G\)
\(K\)E F \(G\)
\(\boldsymbol { L }\)I J
The precedence table shows the activities involved in planning an opening ceremony. An activity on arc network is to be drawn to model this planning process.
  1. Draw the activity network using exactly two dummies.
  2. Explain why each of the two dummies is necessary.
Edexcel D1 Q6
7 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{552f3296-ad61-448b-8168-6709fb359fa2-6_757_1253_262_406} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 models a network of water pipes that need to be inspected. The number on each arc represents the length, in km , of that pipe. A machine is to be used to inspect for leaks. The machine must travel along each pipe at least once, starting and finishing at the same point, and the length of the inspection route is to be minimised.
[0pt] [The total weight of the network is 185 km ]
  1. Starting at A, use an appropriate algorithm to find the length of the shortest inspection route. You should make your method and working clear. Given that it is now permitted to start and finish the inspection at two distinct vertices,
  2. state which two vertices should be chosen to minimise the length of the new route. Give a reason for your answer.
Edexcel D1 Q7
10 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{552f3296-ad61-448b-8168-6709fb359fa2-7_915_1509_267_278} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows the possible bus journeys linking towns, S, A, B, C, D, E, F, G, H and T. Each arc represents a bus journey. The number on each arc represents the cost, in pounds, of travelling along that route.
  1. Use Dijkstra's algorithm, on the diagram in the answer book to find the cheapest route from S to T. State your cheapest route and its cost.
    (6)
  2. Explain how you determined your cheapest route from your labelled diagram. The bus journey from S to B is cancelled due to a driver's illness.
  3. Find the cheapest route from S to T that does not include SB , and state its cost.
Edexcel D1 Q8
10 marks Moderate -0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{552f3296-ad61-448b-8168-6709fb359fa2-8_1051_1385_194_365} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} A company produces two products, X and Y .
Let \(x\) and \(y\) be the hourly production, in kgs, of X and Y respectively.
In addition to \(x \geqslant 0\) and \(y \geqslant 0\), two of the constraints governing the production are $$\begin{gathered} 12 x + 7 y \geqslant 840 \\ 4 x + 9 y \geqslant 720 \end{gathered}$$ These constraints are shown on the graph in Figure 6, where the rejected regions are shaded out. Two further constraints are $$\begin{gathered} x \geqslant 20 \\ 3 x + 2 y \leqslant 360 \end{gathered}$$
  1. Add two lines and shading to Figure 6 in your answer book to represent these inequalities.
  2. Hence determine and label the feasible region, R. The company makes a profit of 70 p and 20 p per kilogram of X and Y respectively.
  3. Write down an expression, in terms of \(x\) and \(y\), for the hourly profit, £P.
  4. Mark points A and B on your graph where A and B represent the maximum and minimum values of P respectively. Make your method clear.
    (4)
Edexcel D1 Q9
12 marks Moderate -0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{552f3296-ad61-448b-8168-6709fb359fa2-9_784_1531_242_267} \captionsetup{labelformat=empty} \caption{Figure 7}
\end{figure} Figure 7 shows an activity network. Each activity is represented by an arc and the number in brackets on each arc is the duration of the activity in days.
  1. Complete Figure 7 in the answer book showing the early and late event times.
  2. List the critical path for this network. The sum of all the activity times is 95 days and each activity requires just one worker. The project must be completed in the minimum time.
  3. Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must make your method clear.
  4. On the grid in your answer book, draw a cascade (Gantt) chart for this network.
Edexcel D1 2002 November Q1
4 marks Standard +0.3
  1. Figure 1 \includegraphics[max width=\textwidth, alt={}, center]{438a62e6-113c-428e-85bf-4b1cbecee0de-2_473_682_348_614}
A Hamilton cycle for the graph in Fig. 1 begins \(A , X , D , V , \ldots\).
  1. Complete this Hamiltonian cycle.
  2. Hence use the planarity algorithm to determine if the graph is planar.
Edexcel D1 2002 November Q2
6 marks Moderate -0.8
2. The precedence table for activities involved in manufacturing a toy is shown below.
ActivityPreceding activity
\(A\)-
\(B\)-
\(C\)-
\(D\)\(A\)
\(E\)\(A\)
\(F\)\(B\)
\(G\)\(B\)
\(H\)\(C , D , E , F\)
\(I\)\(E\)
\(J\)\(E\)
\(K\)\(I\)
\(L\)\(I\)
\(M\)\(G , H , K\)
  1. Draw an activity network, using activity on arc, and exactly one dummy, to model the manufacturing process.
  2. Explain briefly why it is necessary to use a dummy in this case.
Edexcel D1 2002 November Q3
7 marks Moderate -0.5
3. At a water sports centre there are five new instructors. Ali (A), George ( \(G\) ), Jo ( \(J\) ), Lydia ( \(L\) ) and Nadia \(( N )\). They are to be matched to five sports, canoeing \(( C )\), scuba diving \(( D )\), surfing \(( F )\), sailing ( \(S\) ) and water skiing ( \(W\) ). The table indicates the sports each new instructor is qualified to teach.
InstructorSport
\(A\)\(C , F , W\)
\(G\)\(F\)
\(J\)\(D , C , S\)
\(L\)\(S , W\)
\(N\)\(D , F\)
Initially, \(A , G , J\) and \(L\) are each matched to the first sport in their individual list.
  1. Draw a bipartite graph to model this situation and indicate the initial matching in a distinctive way.
  2. Starting from this initial matching, use the maximum matching algorithm to find a complete matching. You must clearly list any alternating paths used. Given that on a particular day \(J\) must be matched to \(D\),
  3. explain why it is no longer possible to find a complete matching. \includegraphics[max width=\textwidth, alt={}, center]{438a62e6-113c-428e-85bf-4b1cbecee0de-4_720_1305_391_236} Figure 2 models an underground network of pipes that must be inspected for leaks. The nodes \(A\), \(B , C , D , E , F , G\) and \(H\) represent entry points to the network. The number on each arc gives the length, in metres, of the corresponding pipe. Each pipe must be traversed at least once and the length of the inspection route must be minimised.
Edexcel D1 2002 November Q5
10 marks Moderate -0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{438a62e6-113c-428e-85bf-4b1cbecee0de-5_763_1490_348_242}
\end{figure}
  1. Use Dijkstra's algorithm to find the shortest route from \(S\) to \(T\) in Fig. 3. Show all necessary working in the boxes in the answer booklet. State your shortest route and its length.
  2. Explain how you determined the shortest route from your labelling.
  3. It is now necessary to go from \(S\) to \(T\) via \(H\). Obtain the shortest route and its length.