Questions - Edexcel - 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) $A$
- Monday (M) $B$ ◯
- Tuesday (Tu) $C \odot$
- Wednesday (W) $D$ ◯
- Thursday (Th) $E$ -
- Friday (F)
  (b)
  (c)
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)
1. SAET $\_\_\_\_$
2. SBDT $\_\_\_\_$
3. 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 M4 Q1

13 marks Challenging +1.2

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-03_457_638_233_598} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A fixed smooth plane is inclined to the horizontal at an angle of $45 ^ { \circ }$. A particle $P$ is moving horizontally and strikes the plane. Immediately before the impact, $P$ is moving in a vertical plane containing a line of greatest slope of the inclined plane. Immediately after the impact, $P$ is moving in a direction which makes an angle of $30 ^ { \circ }$ with the inclined plane, as shown in Figure 1. Find the fraction of the kinetic energy of $P$ which is lost in the impact.

Edexcel M4 Q2

8 marks Challenging +1.8

2. At time $t = 0$, a particle $P$ of mass $m$ is projected vertically upwards with speed $\sqrt { \frac { g } { k } }$, where $k$ is a constant. At time $t$ the speed of $P$ is $v$. The particle $P$ moves against air resistance whose magnitude is modelled as being $m k v ^ { 2 }$ when the speed of $P$ is $v$. Find, in terms of $k$, the distance travelled by $P$ until its speed first becomes half of its initial speed.

Edexcel M4 Q3

10 marks Challenging +1.2

At noon a motorboat $P$ is 2 km north-west of another motorboat $Q$. The motorboat $P$ is moving due south at $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The motorboat $Q$ is pursuing motorboat $P$ at a speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and sets a course in order to get as close to motorboat $P$ as possible.

Find the course set by $Q$, giving your answer as a bearing to the nearest degree.
Find the shortest distance between $P$ and $Q$.
Find the distance travelled by $Q$ from its position at noon to the point of closest approach.

Edexcel M4 Q4

10 marks Challenging +1.2

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-08_479_807_246_571} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A light inextensible string of length $2 a$ has one end attached to a fixed point $A$. The other end of the string is attached to a particle $P$ of mass $m$. A second light inextensible string of length $L$, where $L > \frac { 12 a } { 5 }$, has one of its ends attached to $P$ and passes over a small smooth peg fixed at a point $B$. The line $A B$ is horizontal and $A B = 2 a$. The other end of the second string is attached to a particle of mass $\frac { 7 } { 20 } m$, which hangs vertically below $B$, as shown in Figure 2.

Show that the potential energy of the system, when the angle $P A B = 2 \theta$, is $$\frac { 1 } { 5 } m g a ( 7 \sin \theta - 10 \sin 2 \theta ) + \text { constant. }$$
Show that there is only one value of $\cos \theta$ for which the system is in equilibrium and find this value.
Determine the stability of the position of equilibrium.

Edexcel M4 Q5

8 marks Standard +0.8

Two small smooth spheres $A$ and $B$, of mass 2 kg and 1 kg respectively, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of $A$ is $( \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $B$ is $- 2 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Immediately after the collision the velocity of $A$ is $\mathbf { j } \mathrm { m } \mathrm { s } ^ { - 1 }$.

Show that the velocity of $B$ immediately after the collision is $2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the impulse of $B$ on $A$ in the collision, giving your answer as a vector, and hence show that the line of centres is parallel to $\mathbf { i } + \mathbf { j }$.
Find the coefficient of restitution between $A$ and $B$.

Edexcel M4 Q6

14 marks Challenging +1.2

6. A light elastic spring $A B$ has natural length $2 a$ and modulus of elasticity $2 m n ^ { 2 } a$, where $n$ is a constant. A particle $P$ of mass $m$ is attached to the end $A$ of the spring. At time $t = 0$, the spring, with $P$ attached, lies at rest and unstretched on a smooth horizontal plane. The other end $B$ of the spring is then pulled along the plane in the direction $A B$ with constant acceleration $f$. At time $t$ the extension of the spring is $x$.

Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + n ^ { 2 } x = f .$$
Find $x$ in terms of $n , f$ and $t$. Hence find
the maximum extension of the spring,
the speed of $P$ when the spring first reaches its maximum extension.
1. \hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are unit vectors due east and due north respectively]
A man cycles at a constant speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on level ground and finds that when his velocity is $u \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ the velocity of the wind appears to be $v ( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, where $v$ is a positive constant. When the man cycles with velocity $\frac { 1 } { 5 } u ( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, the velocity of the wind appears to be $w \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $w$ is a positive constant. Find, in terms of $u$, the true velocity of the wind.

Edexcel M4 Q7

12 marks Challenging +1.8

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-38_451_1077_315_370} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a framework $A B C$, consisting of two uniform rods rigidly joined together at $B$ so that $\angle A B C = 90 ^ { \circ }$. The $\operatorname { rod } A B$ has length $2 a$ and mass $4 m$, and the $\operatorname { rod } B C$ has length $a$ and mass $2 m$. The framework is smoothly hinged at $A$ to a fixed point, so that the framework can rotate in a fixed vertical plane. One end of a light elastic string, of natural length $2 a$ and modulus of elasticity $3 m g$, is attached to $A$. The string passes through a small smooth ring $R$ fixed at a distance $2 a$ from $A$, on the same horizontal level as $A$ and in the same vertical plane as the framework. The other end of the string is attached to $B$. The angle $A R B$ is $\theta$, where $0 < \theta < \frac { \pi } { 2 }$.

Show that the potential energy $V$ of the system is given by $$V = 8 a m g \sin 2 \theta + 5 a m g \cos 2 \theta + \text { constant }$$
Find the value of $\theta$ for which the system is in equilibrium.
Determine the stability of this position of equilibrium. A smooth uniform sphere $S$, of mass $m$, is moving on a smooth horizontal plane when it collides obliquely with another smooth uniform sphere $T$, of the same radius as $S$ but of mass $2 m$, which is at rest on the plane. Immediately before the collision the velocity of $S$ makes an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$, with the line joining the centres of the spheres. Immediately after the collision the speed of $T$ is $V$. The coefficient of restitution between the spheres is $\frac { 3 } { 4 }$.

Find, in terms of $V$, the speed of $S$
1. immediately before the collision,
2. immediately after the collision.
Find the angle through which the direction of motion of $S$ is deflected as a result of the collision.

Edexcel D2 Q1

Easy -1.2

1. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{195b1c1f-5ce3-4762-80c3-34c26382b88b-002_675_1052_378_485}

\end{figure} Figure 1 shows a network of roads connecting six villages $A , B , C , D , E$ and $F$. The lengths of the roads are given in km .

Complete the table in the answer booklet, in which the entries are the shortest distances between pairs of villages. You should do this by inspection. The table can now be taken to represent a complete network.
Use the nearest-neighbour algorithm, starting at $A$, on your completed table in part (a). Obtain an upper bound to the length of a tour in this complete network, which starts and finishes at $A$ and visits every village exactly once.
(3)
Interpret your answer in part (b) in terms of the original network of roads connecting the six villages.
(1)
By choosing a different vertex as your starting point, use the nearest-neighbour algorithm to obtain a shorter tour than that found in part (b). State the tour and its length.
(2)

Edexcel D2 Q3

Standard +0.3

3. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{195b1c1f-5ce3-4762-80c3-34c26382b88b-004_764_1514_283_141}

\end{figure} The network in Fig. 2 shows possible routes that an aircraft can take from $S$ to $T$. The numbers on the directed arcs give the amount of fuel used on that part of the route, in appropriate units. The airline wishes to choose the route for which the maximum amount of fuel used on any part of the route is as small as possible. This is the rninimax route.

Complete the table in the answer booklet.
(8)
Hence obtain the minimax route from $S$ to $T$ and state the maximum amount of fuel used on any part of this route.
(2)

Questions — Edexcel (10514 questions)

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Edexcel D2 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	-