Questions - Edexcel D1 - A-Level Maths

Edexcel D1 Q7

Moderate -0.5

7. A tailor makes two types of garment, $A$ and $B$. He has available $70 \mathrm {~m} ^ { 2 }$ of cotton fabric and $90 \mathrm {~m} ^ { 2 }$ of woollen fabric. Garment $A$ requires $1 \mathrm {~m} ^ { 2 }$ of cotton fabric and $3 \mathrm {~m} ^ { 2 }$ of woollen fabric. Garment $B$ requires $2 \mathrm {~m} ^ { 2 }$ of each fabric. The tailor makes $x$ garments of type $A$ and $y$ garments of type $B$.

Explain why this can be modelled by the inequalities $$\begin{aligned} & x + 2 y \leq 70 \\ & 3 x + 2 y \leq 90 \\ & x \geq 0 , y \geq 0 \end{aligned}$$ The tailor sells type $A$ for $\pounds 30$ and type $B$ for $\pounds 40$. All garments made are sold. The tailor wishes to maximise his total income.
Set up an initial Simplex tableau for this problem.
Solve the problem using the Simplex algorithm. Figure 4 shows a graphical representation of the feasible region for this problem. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-008_686_1277_1319_453} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
Obtain the coordinates of the points A, $C$ and $D$.
Relate each stage of the Simplex algorithm to the corresponding point in Fig. 4. 6689 Decision Mathematics 1 (New Syllabus) Order of selecting edges
Final tree
(b) Minimum total length of cable
Question 4 to be answered on this page
(a) $A$
Question 5 to be answered on this page
Key
(a) Early
Time
Late
Time \includegraphics[max width=\textwidth, alt={}, center]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-011_433_227_534_201} $F ( 3 )$ \includegraphics[max width=\textwidth, alt={}, center]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-011_117_222_1016_992}
H(4) K(6)
(b) Critical activities
Length of critical path $\_\_\_\_$ (c) \includegraphics[max width=\textwidth, alt={}, center]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-011_492_1604_1925_266} Question 6 to be answered on pages 4 and 5
(a) (i) SAET $\_\_\_\_$ (ii) SBDT $\_\_\_\_$ (iii) SCFT $\_\_\_\_$ (b) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-012_691_1307_893_384} \captionsetup{labelformat=empty} \caption{Diagram 1}
\end{figure} (c) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-013_699_1314_167_382} \captionsetup{labelformat=empty} \caption{Diagram 2}
\end{figure} Flow augmenting routes
(d) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-013_693_1314_1368_382} \captionsetup{labelformat=empty} \caption{Diagram 3}
\end{figure} (e) $\_\_\_\_$

Edexcel D1 Q10

Easy -1.2

10 The numbers in the list above represent the lengths, in metres, of ten lengths of fabric. They are to be cut from rolls of fabric of length 60 m .

Calculate a lower bound for the number of rolls needed.
(2)
Use the first-fit bin packing algorithm to determine how these ten lengths can be cut from rolls of length 60 m .
Use full bins to find an optimal solution that uses the minimum number of rolls.

Edexcel D1 Q12

Easy -1.2

12
10 The numbers in the list above represent the lengths, in metres, of ten lengths of fabric. They are to be cut from rolls of fabric of length 60 m .

Calculate a lower bound for the number of rolls needed.
(2)
Use the first-fit bin packing algorithm to determine how these ten lengths can be cut from rolls of length 60 m .
Use full bins to find an optimal solution that uses the minimum number of rolls.

Edexcel D1 Q13

Moderate -1.0

13
16
5
8
2
15
12
10 6.

\includegraphics[max width=\textwidth, alt={}]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-558_2226_1632_322_157}

\section*{Diagram 1}

Edexcel D1 2009 June Q1

5 marks Easy -1.3

1.

	$\mathbf { A }$	$\mathbf { B }$	$\mathbf { C }$	$\mathbf { D }$	$\mathbf { E }$	$\mathbf { F }$
$\mathbf { A }$	-	135	180	70	95	225
$\mathbf { B }$	135	-	215	125	205	240
$\mathbf { C }$	180	215	-	150	165	155
$\mathbf { D }$	70	125	150	-	100	195
$\mathbf { E }$	95	205	165	100	-	215
$\mathbf { F }$	225	240	155	195	215	-

The table shows the lengths, in km, of potential rail routes between six towns, $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }$ and F .

Use Prim's algorithm, starting from A , to find a minimum spanning tree for this table. You must list the arcs that form your tree in the order that they are selected.
Draw your tree using the vertices given in Diagram 1 in the answer book.
State the total weight of your tree.

Edexcel D1 2009 June Q2

9 marks Moderate -0.8

2.
32
45
17
23
38
28
16
9
12
10 The numbers in the list above represent the lengths, in metres, of ten lengths of fabric. They are to be cut from rolls of fabric of length 60 m .

Calculate a lower bound for the number of rolls needed.
Use the first-fit bin packing algorithm to determine how these ten lengths can be cut from rolls of length 60 m .
Use full bins to find an optimal solution that uses the minimum number of rolls.

Edexcel D1 2009 June Q3

7 marks Moderate -0.8

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c1482d20-7bce-46cb-9ac8-c659ecad30de-3_755_624_283_283} \captionsetup{labelformat=empty} \caption{Figure 1}

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

\includegraphics[alt={},max width=\textwidth]{c1482d20-7bce-46cb-9ac8-c659ecad30de-3_750_620_285_1146} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 1 shows the possible allocations of six workers, Charlotte (C), Eleanor (E), Harry (H), Matt (M), Rachel (R) and Simon (S) to six tasks, 1, 2, 3, 4, 5 and 6. Figure 2 shows an initial matching.

List an alternating path, starting at H and ending at 4 . Use your path to find an improved matching. List your improved matching.
Explain why it is not possible to find a complete matching. Simon (S) now has task 3 added to his possible allocation.
Taking the improved matching found in (a) as the new initial matching, use the maximum matching algorithm to find a complete matching. List clearly the alternating path you use and your complete matching.
(3)

Edexcel D1 2009 June Q4

9 marks Easy -1.8

4. Miri
Jessie
Edward
Katie
Hegg
Beth
Louis
Philip
Natsuko
Dylan

Use the quick sort algorithm to sort the above list into alphabetical order.
(5)
Use the binary search algorithm to locate the name Louis.

Edexcel D1 2009 June Q5

9 marks Standard +0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c1482d20-7bce-46cb-9ac8-c659ecad30de-5_940_1419_262_322} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} [The total weight of the network is 625 m ]
Figure 3 models a network of paths in a park. The number on each arc represents the length, in m , of that path.
Rob needs to travel along each path to inspect the surface. He wants 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.
(6) The surface on each path is to be renewed. A machine will be hired to do this task and driven along each path.
The machine will be delivered to point G and will start from there, but it may be collected from any point once the task is complete.
Given that each path must be traversed at least once, determine the finishing point so that the length of the route is minimised. Give a reason for your answer and state the length of your route.
(3)

Edexcel D1 2009 June Q6

7 marks Moderate -0.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c1482d20-7bce-46cb-9ac8-c659ecad30de-6_899_1493_262_285} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 represents a network of roads. The number on each arc gives the length, in km , of that road.

Use Dijkstra's algorithm to find the shortest distance from A to I. State your shortest route.
(6)
State the shortest distance from A to G .
(1)

Edexcel D1 2009 June Q7

14 marks Easy -1.3

7. Rose makes hanging baskets which she sells at her local market. She makes two types, large and small. Rose makes $x$ large baskets and $y$ small baskets. Each large basket costs $\pounds 7$ to make and each small basket costs $\pounds 5$ to make. Rose has $\pounds 350$ she can spend on making the baskets.

Write down an inequality, in terms of $x$ and $y$, to model this constraint.
(2) Two further constraints are $$\begin{aligned} & y \leqslant 20 \text { and } \\ & y \leqslant 4 x \end{aligned}$$
Use these two constraints to write down statements that describe the numbers of large and small baskets that Rose can make.
On the grid provided, show these three constraints and $x \geqslant 0 , y \geqslant 0$. Hence label the feasible region, R. Rose makes a profit of $\pounds 2$ on each large basket and $\pounds 3$ on each small basket. Rose wishes to maximise her profit, £P.
Write down the objective function.
Use your graph to determine the optimal numbers of large and small baskets Rose should make, and state the optimal profit.

Edexcel D1 2009 June Q17

Easy -1.2

17
23
38
28
16
9
12
10 The numbers in the list above represent the lengths, in metres, of ten lengths of fabric. They are to be cut from rolls of fabric of length 60 m .

Calculate a lower bound for the number of rolls needed.
Use the first-fit bin packing algorithm to determine how these ten lengths can be cut from rolls of length 60 m .
Use full bins to find an optimal solution that uses the minimum number of rolls.
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c1482d20-7bce-46cb-9ac8-c659ecad30de-3_755_624_283_283} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c1482d20-7bce-46cb-9ac8-c659ecad30de-3_750_620_285_1146} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows the possible allocations of six workers, Charlotte (C), Eleanor (E), Harry (H), Matt (M), Rachel (R) and Simon (S) to six tasks, 1, 2, 3, 4, 5 and 6. Figure 2 shows an initial matching.
1. List an alternating path, starting at H and ending at 4 . Use your path to find an improved matching. List your improved matching.
2. Explain why it is not possible to find a complete matching. Simon (S) now has task 3 added to his possible allocation.
3. Taking the improved matching found in (a) as the new initial matching, use the maximum matching algorithm to find a complete matching. List clearly the alternating path you use and your complete matching.
  (3)
  4. Miri
  Jessie
  Edward
  Katie
  Hegg
  Beth
  Louis
  Philip
  Natsuko
  Dylan
4. Use the quick sort algorithm to sort the above list into alphabetical order.
  (5)
5. Use the binary search algorithm to locate the name Louis.
  5. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{c1482d20-7bce-46cb-9ac8-c659ecad30de-5_940_1419_262_322} \captionsetup{labelformat=empty} \caption{Figure 3}
  \end{figure} [The total weight of the network is 625 m ]
  Figure 3 models a network of paths in a park. The number on each arc represents the length, in m , of that path.
  Rob needs to travel along each path to inspect the surface. He wants to minimise the length of his route.
6. 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.
  (6) The surface on each path is to be renewed. A machine will be hired to do this task and driven along each path.
  The machine will be delivered to point G and will start from there, but it may be collected from any point once the task is complete.
7. Given that each path must be traversed at least once, determine the finishing point so that the length of the route is minimised. Give a reason for your answer and state the length of your route.
  (3)
  6. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{c1482d20-7bce-46cb-9ac8-c659ecad30de-6_899_1493_262_285} \captionsetup{labelformat=empty} \caption{Figure 4}
  \end{figure} Figure 4 represents a network of roads. The number on each arc gives the length, in km , of that road.
8. Use Dijkstra's algorithm to find the shortest distance from A to I. State your shortest route.
  (6)
9. State the shortest distance from A to G .
  (1)
  7. Rose makes hanging baskets which she sells at her local market. She makes two types, large and small. Rose makes $x$ large baskets and $y$ small baskets. Each large basket costs $\pounds 7$ to make and each small basket costs $\pounds 5$ to make. Rose has $\pounds 350$ she can spend on making the baskets.
10. Write down an inequality, in terms of $x$ and $y$, to model this constraint.
  (2) Two further constraints are $$\begin{aligned} & y \leqslant 20 \text { and } \\ & y \leqslant 4 x \end{aligned}$$
11. Use these two constraints to write down statements that describe the numbers of large and small baskets that Rose can make.
12. On the grid provided, show these three constraints and $x \geqslant 0 , y \geqslant 0$. Hence label the feasible region, R. Rose makes a profit of $\pounds 2$ on each large basket and $\pounds 3$ on each small basket. Rose wishes to maximise her profit, £P.
13. Write down the objective function.
14. Use your graph to determine the optimal numbers of large and small baskets Rose should make, and state the optimal profit.
  8. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{c1482d20-7bce-46cb-9ac8-c659ecad30de-8_809_1541_283_262} \captionsetup{labelformat=empty} \caption{Figure 5}
  \end{figure} A construction 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.
15. Complete Diagram 2 in the answer book, showing the early and late event times.
16. State the critical activities.
17. Find the total float for activities M and H . You must make the numbers you use in your calculations clear.
18. On the grid provided, draw a cascade (Gantt) chart for this project. An inspector visits the project at 1 pm on days 16 and 31 to check the progress of the work.
19. Given that the project is on schedule, which activities must be happening on each of these days?

Edexcel D1 2011 June Q1

9 marks Moderate -0.5

1.

Jenny

Edexcel D1 2011 June Q10

Easy -1.8

10. & Freya
A binary search is to be performed on the names in the list above to locate the name Kim.

Explain why a binary search cannot be performed with the list in its present form.
Using an appropriate algorithm, alter 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.
Use the binary search algorithm to locate the name Kim in the list you obtained in (b). You must make your method clear.
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7a968b0-18dd-44cf-b934-4beb8f2290ac-3_858_1169_244_447} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
1. Define the terms
  1. tree,
  2. minimum spanning tree.
    (3)
  3. Use Kruskal's algorithm to find a minimum spanning tree for the network shown in Figure 1. 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)
  4. Draw your minimum spanning tree using the vertices given in Diagram 1 in the answer book.
2. State whether your minimum spanning tree is unique. Justify your answer.
  (1)
  3. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{f7a968b0-18dd-44cf-b934-4beb8f2290ac-4_1492_1298_210_379} \captionsetup{labelformat=empty} \caption{Figure 2}
  \end{figure} Figure 2 shows the constraints of a linear programming problem in $x$ and $y$, where $R$ is the feasible region.
3. Write down the inequalities that form region $R$. The objective is to maximise $3 x + y$.
4. Find the optimal values of $x$ and $y$. You must make your method clear.
5. Obtain the optimal value of the objective function. Given that integer values of $x$ and $y$ are now required,
6. write down the optimal values of $x$ and $y$.
  4. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{f7a968b0-18dd-44cf-b934-4beb8f2290ac-5_623_577_287_383} \captionsetup{labelformat=empty} \caption{Figure 3}
  \end{figure} \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{f7a968b0-18dd-44cf-b934-4beb8f2290ac-5_620_582_287_1098} \captionsetup{labelformat=empty} \caption{Figure 4}
  \end{figure} Figure 3 shows the possible allocations of five workers, Adam (A), Catherine (C), Harriet (H), Josh (J) and Richard (R) to five tasks, 1, 2, 3, 4 and 5. Figure 4 shows an initial matching.
  There are three possible alternating paths that start at A .
  One of them is $$A - 3 = R - 4 = C - 5$$
7. Find the other two alternating paths that start at A .
8. List the improved matching generated by using the alternating path $\mathrm { A } - 3 = \mathrm { R } - 4 = \mathrm { C } - 5$.
9. Starting from the improved matching found in (b), use the maximum matching algorithm to obtain a complete matching. You must list the alternating path used and your final matching.
  5. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{f7a968b0-18dd-44cf-b934-4beb8f2290ac-6_835_913_219_575} \captionsetup{labelformat=empty} \caption{Figure 5
  [0pt] [The total weight of the network is 98 km ]}
  \end{figure} Figure 5 models a network of gas pipes that have to be inspected. The number on each arc represents the length, in km, of that pipe. A route of minimum length that traverses each pipe at least once and starts and finishes at A needs to be found.
10. Use the route inspection algorithm to find the pipes that will need to be traversed twice. You must make your method and working clear.
11. Write down a possible shortest inspection route, giving its length. It is now decided to start the inspection route at D . The route must still traverse each pipe at least once but may finish at any node.
12. 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.
  6. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{f7a968b0-18dd-44cf-b934-4beb8f2290ac-7_823_1374_226_347} \captionsetup{labelformat=empty} \caption{Figure 6}
  \end{figure} Figure 6 shows a network of cycle tracks. The number on each arc gives the length, in km, of that track.
13. Use Dijkstra's algorithm to find the shortest route from A to H. State your shortest route and its length.
14. Explain how you determined your shortest route from your labelled diagram. The track between E and F is now closed for resurfacing and cannot be used.
15. Find the shortest route from A to H and state its length.
  (2)
  7. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{f7a968b0-18dd-44cf-b934-4beb8f2290ac-8_798_1497_258_283} \captionsetup{labelformat=empty} \caption{Figure 7}
  \end{figure} A project is modelled by the activity network shown in Figure 7. 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.
16. Complete the precedence table in the answer book.
  (3)
17. Complete Diagram 1 in the answer book, to show the early event times and late event times.
18. State the critical activities.
19. On the grid in your answer book, draw a cascade (Gantt) chart for this project.
20. By considering the activities that must take place between time 7 and time 16, explain why it is not possible to complete this project with just 3 workers in the minimum time.
  8. A firm is planning to produce two types of radio, type A and type B. Market research suggests that, each week:
  Each type A radio requires 3 switches and each type B radio requires 2 switches. The firm can only buy 200 switches each week. The profit on each type A radio is $\pounds 15$.
  The profit on each type B radio is $\pounds 12$.
  The firm wishes to maximise its weekly profit.
  Formulate this situation as a linear programming problem, defining your variables.
  (Total 7 marks)

Edexcel D1 Specimen Q1

4 marks 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

Edexcel D1 Specimen Q10

Easy -1.8

10. & Verity
2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0f22a56f-ed07-4a9d-bc07-2fa5d69b3d51-03_549_526_194_413} \captionsetup{labelformat=empty} \caption{Figure 1}

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

\includegraphics[alt={},max width=\textwidth]{0f22a56f-ed07-4a9d-bc07-2fa5d69b3d51-03_547_524_196_1110} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} 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.

Find an alternating path linking George with 5. List the resulting improved matching this gives.
Explain why it is not possible to find a complete matching. George now has task 2 added to his possible allocation.
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. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0f22a56f-ed07-4a9d-bc07-2fa5d69b3d51-04_586_1417_205_317} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The network in Figure 3 shows the distances, in metres, between 10 wildlife observation points. The observation points are to be linked by footpaths, to form a network along the arcs indicated, using the least possible total length.
1. Find a minimum spanning tree for the network in Figure 3, showing clearly the order in which you selected the arcs for your tree, using
  1. Kruskal's algorithm,
  2. Prim's algorithm, starting from $A$. Given that footpaths are already in place along $A B$ and $F I$ and so should be included in the spanning tree,
  3. explain which algorithm you would choose to complete the tree, and how it should be adapted. (You do not need to find the tree.)
    4. $\quad \begin{array} { l l l l l l l l l l } 650 & 431 & 245 & 643 & 455 & 134 & 710 & 234 & 162 & 452 \end{array}$
  4. The list of numbers above is to be sorted into descending order. Perform a Quick Sort to obtain the sorted list, giving the state of the list after each pass, indicating the pivot elements. The numbers in the list represent the lengths, in mm, of some pieces of wood. The wood is sold in one metre lengths.
  5. Use the first-fit decreasing bin packing algorithm to determine how these pieces could be cut from the minimum number of one metre lengths. (You should ignore wastage due to cutting.)
  6. Determine whether your solution to part (b) is optimal. Give a reason for your answer.
    5. (a) Explain why a network cannot have an odd number of vertices of odd degree. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0f22a56f-ed07-4a9d-bc07-2fa5d69b3d51-06_615_1143_338_461} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} 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.
  7. Use the route inspection algorithm to find which paths, if any, need to be traversed twice.
  8. Find the length of Hamish's route.
    [0pt] [The total weight of the network in Figure 4 is 4180 m .]
    (1)
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0f22a56f-ed07-4a9d-bc07-2fa5d69b3d51-07_627_1408_223_331} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} Figure 5 shows a network of roads. The number on each arc represents the length of that road in km .
  9. Use Dijkstra's algorithm to find the shortest route from $A$ to $J$. State your shortest route and its length.
  10. Explain how you determined the shortest route from your labelled diagram. The road from $C$ to $F$ will be closed next week for repairs.
  11. Find a shortest route from $A$ to $J$ that does not include $C F$ and state its length.
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0f22a56f-ed07-4a9d-bc07-2fa5d69b3d51-08_1501_1650_201_210} \captionsetup{labelformat=empty} \caption{Figure 6}
    \end{figure} 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 $$2 \leqslant y \leqslant 10$$ These are shown on the graph in Figure 6, where the rejected regions are shaded out.
  12. Explain why the line $x = 10$ is shown as a dotted line.
  13. 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 $\pounds 2$ per child and $\pounds 3$ per adult. She must take at least $\pounds 24$ per trip to cover costs. The number of children must not exceed twice the number of adults.
  14. 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.
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0f22a56f-ed07-4a9d-bc07-2fa5d69b3d51-10_622_1441_194_312} \captionsetup{labelformat=empty} \caption{Figure 7}
    \end{figure} An engineering project is modelled by the activity network shown in Figure 7. 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.
  3. Calculate the early time and late time for each event. Write these in the boxes in Diagram 1 in the answer book.
  4. State the critical activities.
  5. Find the total float on activities $D$ and $F$. You must show your working.
  6. On the grid in the answer book, draw a cascade (Gantt) chart for this project. 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,
  7. which activities must be happening on each of these two days?

Edexcel D1 2018 Specimen Q1

8 marks Moderate -0.8

The table shows the least distances, in km, between six towns, A, B, C, D, E and F.

	A	B	C	D	E	F
A	--	122	217	137	109	82
B	122	--	110	130	128	204
C	217	110	--	204	238	135
D	137	130	204	--	98	211
E	109	128	238	98	--	113
F	82	204	135	211	113	--

Liz must visit each town at least once. She will start and finish at A and wishes to minimise the total distance she will travel.

Starting with the minimum spanning tree given in your answer book, use the shortcut method to find an upper bound below 810 km for Liz's route. You must state the shortcut(s) you use and the length of your upper bound. \hfill [2]
Use the nearest neighbour algorithm, starting at A, to find another upper bound for the length of Liz's route. \hfill [2]
Starting by deleting F, and all of its arcs, find a lower bound for the length of Liz's route. \hfill [3]
Use your results to write down the smallest interval which you are confident contains the optimal length of the route. \hfill [1]

Edexcel D1 2018 Specimen Q2

10 marks Easy -1.3

Kruskal's algorithm finds a minimum spanning tree for a connected graph with $n$ vertices.

Explain the terms
1. connected graph,
2. tree,
3. spanning tree.
\hfill [3]
Write down, in terms of $n$, the number of arcs in the minimum spanning tree. \hfill [1]

The table below shows the lengths, in km, of a network of roads between seven villages, A, B, C, D, E, F and G.

	A	B	C	D	E	F	G
A	--	17	--	19	30	--	--
B	17	--	21	23	--	--	--
C	--	21	--	27	29	31	22
D	19	23	27	--	--	40	--
E	30	--	29	--	--	33	25
F	--	--	31	40	33	--	39
G	--	--	22	--	25	39	--

Complete the drawing of the network on Diagram 1 in the answer book by adding the necessary arcs from vertex C together with their weights. \hfill [2]
Use Kruskal's algorithm to find a minimum spanning tree for the network. You should list the arcs in the order that you consider them. In each case, state whether you are adding the arc to your minimum spanning tree. \hfill [3]
State the weight of the minimum spanning tree. \hfill [1]

Edexcel D1 2018 Specimen Q3

15 marks Easy -1.2

12.1 \quad 9.3 \quad 15.7 \quad 10.9 \quad 17.4 \quad 6.4 \quad 20.1 \quad 7.9 \quad 8.1 \quad 14.0

Use the first-fit bin packing algorithm to determine how the numbers listed above can be packed into bins of size 33 \hfill [3]

The list is to be sorted into descending order.

1. Starting at the left-hand end of the list, perform two passes through the list using a bubble sort. Write down the state of the list that results at the end of each pass.
2. Write down the total number of comparisons and the total number of swaps performed during your two passes.
\hfill [4]
Use a quick sort on the original list to obtain a fully sorted list in descending order. You must make your pivots clear. \hfill [4]
Use the first-fit decreasing bin packing algorithm to determine how the numbers listed can be packed into bins of size 33 \hfill [3]
Determine whether your answer to (d) uses the minimum number of bins. You must justify your answer. \hfill [1]

Edexcel D1 2018 Specimen Q4

15 marks Moderate -0.5

\includegraphics{figure_1} Figure 1 models a network of roads. The number on each edge gives the time, in minutes, taken to travel along that road. Oliver wishes to travel by road from A to K as quickly as possible.

Use Dijkstra's algorithm to find the shortest time needed to travel from A to K. State the quickest route. \hfill [6]

On a particular day Oliver must travel from B to K via A.

Find a route of minimal time from B to K that includes A, and state its length. \hfill [2]

Oliver needs to travel along each road to check that it is in good repair. He wishes to minimise the total time required to traverse the network.

Use the route inspection algorithm to find the shortest time needed. You must state all combinations of edges that Oliver could repeat, making your method and working clear. \hfill [7]

Edexcel D1 2018 Specimen Q5

11 marks Moderate -0.8

A linear programming problem in $x$ and $y$ is described as follows. Maximise P = $5x + 3y$ subject to: $x \geqslant 3$ $$x + y \leqslant 9$$ $$15x + 22y \leqslant 165$$ $$26x - 50y \leqslant 325$$

Add lines and shading to Diagram 2 in the answer book to represent these constraints. Hence determine the feasible region and label it R. \hfill [4]
Use the objective line method to find the optimal vertex, V, of the feasible region. You must draw and label your objective line and label vertex V clearly. \hfill [2]
Calculate the exact coordinates of vertex V and hence calculate the corresponding value of P at V. \hfill [3]

The objective is now to minimise $5x + 3y$, where $x$ and $y$ are integers.

Write down the minimum value of $5x + 3y$ and the corresponding value of $x$ and corresponding value of $y$. \hfill [2]

Edexcel D1 2018 Specimen Q6

16 marks Moderate -0.8

\includegraphics{figure_2} A project is modelled by the activity network shown in Figure 2. The activities are represented by the arcs. The number in brackets on each arc gives the time required, in hours, to complete the activity. The numbers in circles are the event numbers. Each activity requires one worker.

Explain the significance of the dummy activity
1. from event 5 to event 6
2. from event 7 to event 9.
\hfill [2]
Complete Diagram 3 in the answer book to show the early event times and the late event times. \hfill [4]
State the minimum project completion time. \hfill [1]
Calculate a lower bound for the minimum number of workers required to complete the project in the minimum time. You must show your working. \hfill [2]
On Grid 1 in your answer book, draw a cascade (Gantt) chart for this project. \hfill [4]
On Grid 2 in your answer book, construct a scheduling diagram to show that this project can be completed with three workers in just one more hour than the minimum project completion time. \hfill [3]

Edexcel D1 2001 January Q1

6 marks Easy -1.2

This question should be answered on the sheet provided in the answer booklet. A school wishes to link 6 computers. One is in the school office and one in each of rooms $A$, $B$, $C$, $D$ and $E$. Cables need to be laid to connect the computers. The school wishes to use a minimum total length of cable. The table shows the shortest distances, in metres, between the various sites.

	Office	Room A	Room B	Room C	Room D	Room E
Office	--	8	16	12	10	14
Room A	8	--	14	13	11	9
Room B	16	14	--	12	15	11
Room C	12	13	12	--	11	8
Room D	10	11	15	11	--	10
Room E	14	9	11	8	10	--

Starting at the school office, use Prim's algorithm to find a minimum spanning tree. Indicate the order in which you select the edges and draw your final tree. [5 marks]
Using your answer to part (a), calculate the minimum total length of cable required. [1 mark]

Edexcel D1 2001 January Q2

7 marks Easy -1.3

Use the binary search algorithm to locate the name HUSSAIN in the following alphabetical list. Explain each step of the algorithm. 1. ALLEN 2. BALL 3. COOPER 4. EVANS 5. HUSSAIN 6. JONES 7. MICHAEL 8. PATEL 9. RICHARDS 10. TINDALL 11. WU [6 marks]
State the maximum number of comparisons that need to be made to locate a name in an alphabetical list of 11 names. [1 mark]

Edexcel D1 2001 January Q3

7 marks Moderate -0.3

\includegraphics{figure_1}

Using an appropriate algorithm, obtain a suitable route starting and finishing at A. [5 marks]
Calculate the total length of this route. [2 marks]

Questions — Edexcel D1 (505 questions)

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Edexcel D1 2001 January Q1

Edexcel D1 2001 January Q2

Edexcel D1 2001 January Q3

	\(\mathbf { A }\)	\(\mathbf { B }\)	\(\mathbf { C }\)	\(\mathbf { D }\)	\(\mathbf { E }\)	\(\mathbf { F }\)
\(\mathbf { A }\)	-	135	180	70	95	225
\(\mathbf { B }\)	135	-	215	125	205	240
\(\mathbf { C }\)	180	215	-	150	165	155
\(\mathbf { D }\)	70	125	150	-	100	195
\(\mathbf { E }\)	95	205	165	100	-	215
\(\mathbf { F }\)	225	240	155	195	215	-

	A	B	C	D	E	F
A	--	122	217	137	109	82
B	122	--	110	130	128	204
C	217	110	--	204	238	135
D	137	130	204	--	98	211
E	109	128	238	98	--	113
F	82	204	135	211	113	--

	A	B	C	D	E	F	G
A	--	17	--	19	30	--	--
B	17	--	21	23	--	--	--
C	--	21	--	27	29	31	22
D	19	23	27	--	--	40	--
E	30	--	29	--	--	33	25
F	--	--	31	40	33	--	39
G	--	--	22	--	25	39	--

	A	B	C	D	E	F
A	--	122	217	137	109	82
B	122	--	110	130	128	204
C	217	110	--	204	238	135
D	137	130	204	--	98	211
E	109	128	238	98	--	113
F	82	204	135	211	113	--

	A	B	C	D	E	F	G
A	--	17	--	19	30	--	--
B	17	--	21	23	--	--	--
C	--	21	--	27	29	31	22
D	19	23	27	--	--	40	--
E	30	--	29	--	--	33	25
F	--	--	31	40	33	--	39
G	--	--	22	--	25	39	--

	A	B	C	D	E	F
A	--	122	217	137	109	82
B	122	--	110	130	128	204
C	217	110	--	204	238	135
D	137	130	204	--	98	211
E	109	128	238	98	--	113
F	82	204	135	211	113	--

	A	B	C	D	E	F	G
A	--	17	--	19	30	--	--
B	17	--	21	23	--	--	--
C	--	21	--	27	29	31	22
D	19	23	27	--	--	40	--
E	30	--	29	--	--	33	25
F	--	--	31	40	33	--	39
G	--	--	22	--	25	39	--