Questions - Edexcel D1 - A-Level Maths

Starting from this initial matching, use the maximum matching algorithm to find an improved matching. You must list the alternating path used, and your improved matching.
Explain why it is not possible to find a complete matching. D now has task 2 added to their possible allocation.
Using the improved matching found in part (a) as the new initial matching, use the maximum matching algorithm to find a complete matching. You must list the alternating path used and your complete matching.
(3)

Edexcel D1 2009 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ef029462-ffed-4cdf-87bc-56c8a13d671f-5_806_1211_264_427} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} (The total weight of the network in Figure 3 is 543 km .)
Figure 3 models a network of railway tracks that have to be inspected. The number on each arc is the length, in km , of that section of railway track.
Each track must be traversed at least once and the length of the inspection route must be minimised.
The inspection route must start and finish at the same vertex.

Use an appropriate algorithm to find the length of the shortest inspection route. You should make your method and working clear. It is now permitted to start and finish the inspection at two distinct vertices.
State which two vertices should be chosen to minimise the length of the new route. Give a reason for your answer.

Edexcel D1 2009 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ef029462-ffed-4cdf-87bc-56c8a13d671f-6_609_1283_260_392} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a network of roads through eight villages, $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G }$ and H . The number on each arc is the length of that road in km .

Use Dijkstra's algorithm to find the shortest route from A to H. State your shortest route and its length.
(5) There is a fair in village C and you cannot drive through the village. A shortest route from A to H which avoids C needs to be found.
State this new minimal route and its length.
(2)

Edexcel D1 2009 January Q7

7. A linear programming problem is modelled by the following constraints $$\begin{aligned} 8 x + 3 y & \leqslant 480
8 x + 7 y & \geqslant 560
y & \geqslant 4 x
x , y & \geqslant 0 \end{aligned}$$

Use the grid provided in your answer book to represent these inequalities graphically. Hence determine the feasible region and label it R . The objective function, $F$, is given by $$F = 3 x + y$$
Making your method clear, determine
1. the minimum value of the function $F$ and the coordinates of the optimal point,
2. the maximum value of the function $F$ and the coordinates of the optimal point.

Edexcel D1 2009 January Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ef029462-ffed-4cdf-87bc-56c8a13d671f-8_574_1362_242_349} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} The network in Figure 5 shows the activities involved in a process. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, taken to complete the activity.

Calculate the early time and the late time for each event, showing them on the diagram in the answer book.
Determine the critical activities and the length of the critical path.
Calculate the total float on activities F and G . You must make the numbers you used in your calculation clear.
On the grid in the answer book, draw a cascade (Gantt) chart for the process. Given that each task requires just one worker,
use your cascade chart to determine the minimum number of workers required to complete the process in the minimum time. Explain your reasoning clearly.

Edexcel D1 2010 January Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{17bc9fb2-13bf-4ffa-93ac-bef170467570-2_611_629_360_717} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the possible allocation of six people, Alice (A), Brian (B), Christine (C), David (D), Elizabeth (E) and Freddy (F), to six tasks, 1, 2, 3, 4, 5 and 6. An initial matching is Alice to task 1, Christine to task 3, David to task 4 and Elizabeth to task 5.

Show this initial matching on Diagram 1 in the answer book.
(1)
Starting from this initial matching, use the maximum matching algorithm to find a complete matching. List clearly the alternating paths that you use, and give your final matching.
(5)

Edexcel D1 2010 January Q2

2. Prim's algorithm finds a minimum spanning tree for a connected graph.

Explain the terms
1. connected graph,
2. tree,
3. spanning tree.

Name an alternative algorithm for finding a minimum spanning tree. \begin{table}[h]

	Cambridge	London	Norwich	Oxford	Portsmouth	Salisbury	York
Cambridge (C)	-	60	62	81	132	139	156
London (L)	60	-	116	56	74	88	211
Norwich (N)	62	116	-	144	204	201	181
Oxford (O)	81	56	144	-	84	63	184
Portsmouth (P)	132	74	204	84	-	43	269
Salisbury (S)	139	88	201	63	43	-	248
York (Y)	156	211	181	184	269	248	-

\captionsetup{labelformat=empty} \caption{Figure 2}

\end{table} Figure 2 shows the distances by road, in miles, between seven cities.

1. Use Prim's algorithm, starting at London, to find the minimum spanning tree for these cities. You must clearly state the order in which you selected the edges of your tree, and the weight of the final tree.
2. Draw your tree using the vertices given in Diagram 2 in the answer book.
  (5)

Edexcel D1 2010 January Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{17bc9fb2-13bf-4ffa-93ac-bef170467570-4_579_1273_239_397} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} [The total weight of the network is 167]
Figure 3 represents a network of paths. The number on each arc gives the time, in minutes, to travel along that path.

Use Dijkstra's algorithm to find the quickest route from A to H. State your quickest route and the time taken.
(5) Kevin must walk along each path at least once and return to his starting point.
Use an appropriate algorithm to find the time of Kevin's quickest possible route, starting and finishing at A. You should make your method and working clear.

Edexcel D1 2010 January Q4

4. A builder is asked to replace the guttering on a house. The lengths needed, in metres, are $$0.6,4.0,2.5,3.2,0.5,2.6,0.4,0.3,4.0 \text { and } 1.0$$ Guttering is sold in 4 m lengths.

Carry out a quick sort to produce a list of the lengths needed in descending order. You should show the result of each pass and identify your pivots clearly.
Apply the first-fit decreasing bin-packing algorithm to your ordered list to determine the total number of 4 m lengths needed.
Does the answer to part (b) use the minimum number of 4 m lengths? You must justify your answer.

Edexcel D1 2010 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{17bc9fb2-13bf-4ffa-93ac-bef170467570-6_2228_613_269_861} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} An algorithm is described by the flowchart shown in Figure 4.

Given that $\mathrm { S } = 25000$, complete the table in the answer book to show the results obtained at each step when the algorithm is applied. This algorithm is designed to model a possible system of income tax, T, on an annual salary, £S.
Write down the amount of income tax paid by a person with an annual salary of $\pounds 25000$.
Find the maximum annual salary of a person who pays no tax.

Edexcel D1 2010 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{17bc9fb2-13bf-4ffa-93ac-bef170467570-7_614_1315_1027_374} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 is the activity network relating to a building project. The number in brackets on each arc gives the time taken, in days, to complete the activity.

Explain the significance of the dotted line from event (2) to event (3).
Complete the precedence table in the answer booklet.
Calculate the early time and the late time for each event, showing them on the diagram in the answer booklet.
Determine the critical activities and the length of the critical path.
On the grid in the answer booklet, draw a cascade (Gantt) chart for the project.

Edexcel D1 2010 January Q7

7. You are in charge of buying new cupboards for a school laboratory. The cupboards are available in two different sizes, standard and large.
The maximum budget available is $\pounds 1800$. Standard cupboards cost $\pounds 150$ and large cupboards cost $\pounds 300$.
Let $x$ be the number of standard cupboards and $y$ be the number of large cupboards.

Write down an inequality, in terms of $x$ and $y$, to model this constraint.
(2) The cupboards will be fitted along a wall 9 m long. Standard cupboards are 90 cm long and large cupboards are 120 cm long.
Show that this constraint can be modelled by $$3 x + 4 y \leqslant 30$$ You must make your reasoning clear. Given also that $y \geqslant 2$,
explain what this constraint means in the context of the question. The capacity of a large cupboard is $40 \%$ greater than the capacity of a standard cupboard. You wish to maximise the total capacity.
Show that your objective can be expressed as $$\text { maximise } 5 x + 7 y$$
Represent your inequalities graphically, on the axes in your answer booklet, indicating clearly the feasible region, R.
Find the number of standard cupboards and large cupboards that need to be purchased. Make your method clear.

Edexcel D1 2011 January Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0360f78d-e18c-4c47-a2ec-ddd705a4175f-2_750_1285_388_390} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a network of roads between eight villages, $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G }$ and H . The number on each arc gives the length, in miles, of the corresponding road.

Use Dijkstra's algorithm to find the shortest distance from A to H .
(5)
State your shortest route.
(1)
Write down the shortest route from H to C and state its length.
(2)

Edexcel D1 2011 January Q2

2. $$\begin{array} { l l l l l l l l } 23 & 29 & 11 & 34 & 10 & 14 & 35 & 17 \end{array}$$ The numbers represent the sizes, in megabytes (MB), of eight files.
The files are to be stored on 50 MB discs.

Calculate a lower bound for the number of discs needed to store all eight files.
Use the first-fit bin packing algorithm to fit the files onto the discs.
Perform a bubble sort on the numbers in the list to sort them into descending order. You need only write down the final result of each pass.
Use the first-fit decreasing bin packing algorithm to fit the files onto the discs.

Edexcel D1 2011 January Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0360f78d-e18c-4c47-a2ec-ddd705a4175f-4_986_1255_269_402} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure}

Use Kruskal's algorithm to find a minimum spanning tree for the network shown in Figure 2. 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 spanning tree.
(3)
Starting at A, use Prim's algorithm to find a minimum spanning tree for the network in Figure 2. You must clearly state the order in which you include the arcs in your tree.
(3)
Draw a minimum spanning tree for the network in Figure 2 using the vertices given in Diagram 1 of the answer book. State the weight of the minimum spanning tree.
(2) A new spanning tree is required which includes the arcs DI and HG, and which has the lowest possible total weight.
Explain which algorithm you would choose to complete the tree, and how the algorithm should be adapted. (You do not need to find the tree.)

Edexcel D1 2011 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0360f78d-e18c-4c47-a2ec-ddd705a4175f-5_736_602_276_301} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0360f78d-e18c-4c47-a2ec-ddd705a4175f-5_730_588_278_1171} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Six workers, Anthony, Beth, David, Jacob, Kantola and Miri, are to be allocated to six tasks, 1, 2, $3,4,5$ and 6 . Figure 3 shows the possible allocations of the workers, and an initial matching is shown in Figure 4.

Write down the technical name given to the type of diagram shown in Figure 3.
Use the maximum matching algorithm once to find an improved matching. You must state the alternating path you use and your improved matching. Anthony now agrees to add task 6 to his possible allocations.
Starting with your improved matching, use the maximum matching algorithm to obtain a complete matching. You must state the alternating path you use and your complete matching.

Edexcel D1 2011 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0360f78d-e18c-4c47-a2ec-ddd705a4175f-6_867_1381_260_342} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} [The total weight of the network is 31.6 km ]
Figure 5 models a network of roads. The road markings on these roads are to be renewed. The number on each arc represents the length, in km , of that road. In order to renew the road markings, each road must be traversed at least once.

Use the route inspection algorithm, starting and finishing at A , to find a suitable route, which should be stated. You must make your method and working clear.
State the roads that must be traversed twice and the length of the route.
(3) The machine that will be used to renew the road markings can only be delivered to D . It will start at D, but it may finish at any vertex.
Each road must still be traversed at least once.
Given that the route is to be minimised, determine where the machine should finish. Give reasons to justify your answer.
(3)

Edexcel D1 2011 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0360f78d-e18c-4c47-a2ec-ddd705a4175f-7_1214_1581_251_242} \captionsetup{labelformat=empty} \caption{Figure 6}

\end{figure} The graph in Figure 6 is being used to solve a linear programming problem. Two of the constraints have been drawn on the graph and the rejected regions shaded out.

Write down the constraints shown on the graph. Two further constraints are $$\begin{aligned} x + y & \geqslant 30
\text { and } \quad 5 x + 8 y & \leqslant 400 \end{aligned}$$
Add two lines and shading to Graph 1 in your answer book to represent these constraints. Hence determine the feasible region and label it R . The objective is to $$\text { minimise } 15 x + 10 y$$
Draw a profit line on Graph 1 and use it to find the optimal solution. You must label your profit line clearly.
(3)

Edexcel D1 2011 January Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0360f78d-e18c-4c47-a2ec-ddd705a4175f-8_888_1701_198_180} \captionsetup{labelformat=empty} \caption{Figure 7}

\end{figure} The network in Figure 7 shows the activities that need to be undertaken to complete a maintenance project. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the activity. The numbers in circles are the events. Each activity requires one worker. The project is to be completed in the shortest possible time.

Complete the precedence table for this network in the answer book.
Explain why each of the following is necessary.
1. The dummy from event 6 to event 7 .
2. The dummy from event 8 to event 9 .
Complete Diagram 2 in the answer book to show the early and the late event times.
State the critical activities.
Calculate the total float on activity K . You must make the numbers used in your calculation clear.
Calculate a lower bound for the number of workers needed to complete the project in the minimum time.

Edexcel D1 2012 January Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e02c4a9a-d2ab-489f-b838-9b4d902c4457-2_782_974_379_539} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 represents the distances, in km, between eight vertices, $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G }$ and H in a network.

Use Kruskal's algorithm to find the minimum spanning tree for the network. 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 spanning tree.
(3)
Starting at A, use Prim's algorithm to find the minimum spanning tree. You must clearly state the order in which you selected the arcs of your tree.
(3)
Draw the minimum spanning tree using the vertices given in Diagram 1 in the answer book.
State the weight of the tree.
(1)

Edexcel D1 2012 January Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e02c4a9a-d2ab-489f-b838-9b4d902c4457-3_650_1357_260_354} \captionsetup{labelformat=empty} \caption{Figure 2
[0pt] [The weight of the network is 129 miles]}

\end{figure} Figure 2 models a network of canals. The number on each arc gives the length, in miles, of that canal. Brett needs to travel along each canal to check that it is in good repair. He wishes to minimise the length of his route.

Use the route inspection algorithm to find the length of his route. State the arcs that should be repeated. You should make your method and working clear. A canal between B and F , of length 12 miles, is to be opened and needs to be included in Brett's inspection route.
Determine if the addition of this canal will increase or decrease the length of Brett's minimum route. You must make your reasoning clear.

Edexcel D1 2012 January Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e02c4a9a-d2ab-489f-b838-9b4d902c4457-4_743_457_237_434} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e02c4a9a-d2ab-489f-b838-9b4d902c4457-4_743_455_237_1165} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Define the terms

bipartite graph,
matching. Figure 3 shows the possible allocations of six people, Charles (C), Emily (E), George (G), Harriet (H), Jack (J) and Shen (S), to six tasks, 1, 2, 3, 4, 5 and 6. Figure 4 shows an initial matching.
Starting from this initial matching, use the maximum matching algorithm to find an improved matching. You should list the alternating path you use and state your improved matching. Emily has task 5 added to her possible allocations and Harriet has task 3 added to her possible allocations.
Starting from the improved matching found in part (c), use the maximum matching algorithm to find a complete matching. You should list the alternating path you use and state your improved matching.
(3)
(Total 10 marks)

Edexcel D1 2012 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e02c4a9a-d2ab-489f-b838-9b4d902c4457-5_679_1420_228_312} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 models a network of roads. The number on each edge gives the time, in minutes, taken to travel along that road. Olivia wishes to travel from A to J as quickly as possible.

Use Dijkstra's algorithm to find the shortest time needed to travel from A to J. State the shortest route. On a particular day Olivia must include G in her route.
Find a route of minimal time from A to J that includes G , and state its length

Edexcel D1 2012 January Q5

Use the first-fit bin packing algorithm to determine how the numbers listed above can be packed into bins of size 20.
The list of numbers is to be sorted into descending order. Use a bubble sort to obtain the sorted list, giving the state of the list after each complete pass.
Apply the first-fit decreasing bin packing algorithm to your ordered list to pack the numbers into bins of size 20.
Determine whether your answer to (c) uses the minimum number of bins. You must justify your answer.

Edexcel D1 2012 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e02c4a9a-d2ab-489f-b838-9b4d902c4457-7_2226_1628_299_221} \captionsetup{labelformat=empty} \caption{Figure 6}

\end{figure} Edgar has recently bought a field in which he intends to plant apple trees and plum trees. He can use linear programming to determine the number of each type of tree he should plant. Let $x$ be the number of apple trees he plants and $y$ be the number of plum trees he plants. Two of the constraints are $$\begin{aligned} & x \geqslant 40
& y \leqslant 50 \end{aligned}$$ These are shown on the graph in Figure 6, where the rejected region is shaded out.

Use these two constraints to write down two statements that describe the number of apple trees and plum trees Edgar can plant. Two further constraints are $$\begin{aligned} 3 x + 4 y & \leqslant 360
x & \leqslant 2 y \end{aligned}$$
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 . Edgar will make a profit of $\pounds 60$ from each apple tree and $\pounds 20$ from each plum tree. He wishes to maximise his profit, P.
Write down the objective function.
Use an objective line to determine the optimal point of the feasible region, R . You must make your method clear.
Find Edgar's maximum profit.

Questions — Edexcel D1 (480 questions)

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