Questions — Edexcel (10514 questions)

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Edexcel FD1 AS 2021 June Q2
12 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d3f5dcb4-3e23-4d78-965a-a1acaac13819-03_885_1493_226_287} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A project is modelled by the activity network shown in Figure 1. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, to complete the corresponding activity. The exact duration, \(x\), of activity N is unknown, but it is given that \(5 < x < 10\) Each activity requires one worker. The project is to be completed in the shortest possible time.
  1. Complete the precedence table in the answer book.
  2. Complete Diagram 1 in the answer book to show the early event times and the late event times.
  3. List the critical activities. It is given that activity J can be delayed by up to 4 hours without affecting the shortest possible completion time of the project.
  4. Determine the value of \(x\). You must make the numbers used in your calculation clear.
  5. Draw a cascade chart for this project on Grid 1 in the answer book.
Edexcel FD1 AS 2021 June Q3
9 marks Challenging +1.2
3. Donald plans to bake and sell cakes. The three types of cake that he can bake are brownies, flapjacks and muffins. Donald decides to bake 48 brownies and muffins in total.
Donald decides to bake at least 5 brownies for every 3 flapjacks.
At most \(40 \%\) of the cakes will be muffins.
Donald has enough ingredients to bake 60 brownies or 45 flapjacks or 35 muffins.
Donald plans to sell each brownie for \(\pounds 1.50\), each flapjack for \(\pounds 1\) and each muffin for \(\pounds 1.25\) He wants to maximise the total income from selling the cakes. Let \(x\) represent the number of brownies, let \(y\) represent the number of flapjacks and let \(z\) represent the number of muffins that Donald will bake. Formulate this as a linear programming problem in \(x\) and \(y\) only, stating the objective function and listing the constraints as simplified inequalities with integer coefficients. You should not attempt to solve the problem.
Edexcel FD1 AS 2021 June Q4
8 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d3f5dcb4-3e23-4d78-965a-a1acaac13819-05_712_1433_223_315} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Dijkstra's algorithm has been applied to the network in Figure 2.
A working value has only been replaced at a node if the new working value is smaller.
  1. State the length of the shortest path from A to G .
  2. Complete the table in the answer book giving the weight of each arc listed. (Note that arc CE and arc EF are not in the table.)
  3. State the shortest path from A to G. It is now given that
Edexcel FD1 AS 2022 June Q1
9 marks Easy -1.2
  1. 55534345928373452334247
The list of eleven numbers shown above is to be sorted into ascending order.
  1. Carry out a quick sort to produce the sorted list. You should show the result of each pass and identify your pivots clearly.
    (4) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c8134d3b-71cb-4b92-ac54-81a4ff8f3011-03_814_1545_614_260} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure}
  2. Use Kruskal's algorithm to find the minimum spanning tree for the network in Figure 1. You should list the arcs in the order in which you consider them. For each arc, state whether or not you are adding it to your minimum spanning tree.
    1. Draw the minimum spanning tree on Diagram 1 in the answer book.
    2. State the total weight of the tree.
Edexcel FD1 AS 2022 June Q2
8 marks Moderate -0.3
2.
ActivityImmediately preceding activities
A-
B-
C-
D-
EA
FA, B, C
GC
HC
IE
JE, F, G
KD, H
  1. Draw the activity network described in the precedence table above, using activity on arc. Your activity network must contain the minimum number of dummies only.
  2. Explain why it is necessary to draw a dummy from the end of activity A . Every activity shown in the precedence table has the same duration.
  3. State which activity cannot be critical, justifying your answer.
Edexcel FD1 AS 2022 June Q3
14 marks Moderate -0.5
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c8134d3b-71cb-4b92-ac54-81a4ff8f3011-05_702_1479_201_293} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} [The total weight of the network is 120]
  1. Explain what is meant by the term "path".
  2. State, with a reason, whether the network in Figure 2 is Eulerian, semi-Eulerian or neither. Figure 2 represents a network of cycle tracks between eight villages, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G }\) and H . The number on each arc represents the length, in km , of the corresponding track. Samira lives in village A, and wishes to visit her friend, Daisy, who lives in village H.
  3. Use Dijkstra's algorithm to find the shortest path that Samira can take. An extra cycle track of length 9 km is to be added to the network. It will either go directly between C and D or directly between E and G . Daisy plans to cycle along every track in the new network, starting and finishing at H .
    Given that the addition of either track CD or track EG will not affect the final values obtained in (c),
  4. use a suitable algorithm to find out which of the two possible extra tracks will give Daisy the shortest route, making your method and working clear. You must
Edexcel FD1 AS 2022 June Q4
9 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c8134d3b-71cb-4b92-ac54-81a4ff8f3011-06_1504_1733_210_173} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the constraints of a maximisation linear programming problem in \(x\) and \(y\), where \(x \geqslant 0\) and \(y \geqslant 0\). The unshaded area, including its boundaries, forms the feasible region, \(R\). An objective line has been drawn and labelled on the graph.
  1. List the constraints as simplified inequalities with integer coefficients. The optimal value of the objective function is 216
    1. Calculate the exact coordinates of the optimal vertex.
    2. Hence derive the objective function. Given that \(x\) represents the number of small flower pots and \(y\) represents the number of large flower pots supplied to a customer,
  3. deduce the optimal solution to the problem. TOTAL FOR DECISION MATHEMATICS 1 IS 40 MARKS END
Edexcel FD1 AS 2023 June Q1
4 marks Easy -1.2
1. $$\begin{array} { l l l l l l l l l l l } 67 & 59 & 46 & 71 & 40 & 48 & 53 & 63 & 45 & 54 & 56 \end{array}$$ The list of eleven numbers shown above is to be sorted into descending order.
Carry out a quick sort to produce the sorted list. You should show the result of each pass and identify the pivots clearly.
Edexcel FD1 AS 2023 June Q2
10 marks Moderate -0.5
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9edb5209-4244-4916-b3ee-d77e395e8cab-03_750_1490_262_285} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A project is modelled by the activity network shown in Figure 1. The activities are represented by the arcs. The number in brackets on each arc gives the time required, in hours, to complete the corresponding activity. The numbers in circles are the event numbers. Each activity requires one worker, and the project is to be completed in the shortest possible time.
  1. Explain the significance of the dummy activity from event 3 to event 4
  2. Complete Diagram 1 in the answer book to show the early event times and the late event times.
  3. State the critical activities.
  4. Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.
  5. Draw a Gantt chart for this project on Grid 1 in the answer book.
Edexcel FD1 AS 2023 June Q3
11 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9edb5209-4244-4916-b3ee-d77e395e8cab-04_977_1472_259_294} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 represents a network of train tracks. The number on each edge represents the length, in kilometres, of the corresponding track.
Dyfan wishes to travel from A to J via C. Dyfan wishes to minimise the distance they travel. Given that Dijkstra's algorithm is to be applied only once to find Dyfan's route,
  1. explain why the algorithm should begin at C.
  2. Use Dijkstra's algorithm to find the shortest route from A to J via C. State this route and its length.
  3. Use Prim's algorithm, starting at C , to find a minimum spanning tree for the network. You must clearly state the order in which you select the edges of your tree.
  4. State the total length, in km , of the minimum spanning tree.
Edexcel FD1 AS 2023 June Q4
7 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9edb5209-4244-4916-b3ee-d77e395e8cab-05_997_1379_260_456} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the constraints of a linear programming problem in \(x\) and \(y\). The unshaded area, including its boundaries, forms the feasible region, \(R\). An objective line has been drawn and labelled on the graph.
  1. State the inequalities that define the feasible region. The maximum value of the objective function is \(\frac { 160 } { 3 }\) The minimum value of the objective function is \(\frac { 883 } { 41 }\)
  2. Determine the objective function, showing your working clearly.
Edexcel FD1 AS 2023 June Q5
8 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9edb5209-4244-4916-b3ee-d77e395e8cab-06_873_739_178_664} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} [The weight of the network is \(20 x + 3\) ] Figure 4 shows a graph G that contains 8 arcs and 6 vertices.
  1. State the minimum number of arcs that would need to be added to make G into an Eulerian graph.
  2. Explain whether or not the route \(\mathrm { A } - \mathrm { C } - \mathrm { F } - \mathrm { E } - \mathrm { C } - \mathrm { D } - \mathrm { B }\) is an example of a path on G. Figure 4 represents a network of 8 roads in a city. The expression on each arc gives the time, in minutes, to travel along the corresponding road. You are given that \(x > 1.6\) A route is required that
    The route inspection algorithm is applied to the network in Figure 4 and the time taken for the route is found to be at most 189 minutes. Given that the inspection route contains two roads that need to be traversed twice,
  3. determine the range of possible values of \(x\), making your reasoning clear.
Edexcel FD1 AS 2024 June Q1
9 marks Moderate -0.8
1. $$\begin{array} { l l l l l l l l l l l } 4 & 6.5 & 7 & 1.3 & 2 & 5 & 1.5 & 6 & 4.5 & 6 & 1 \end{array}$$ The list of eleven numbers shown above is to be sorted into descending order.
  1. Carry out a quick sort to produce the sorted list. You should show the result of each pass and identify the pivots clearly.
  2. Use the first-fit decreasing bin packing algorithm to pack the numbers into bins of size 10
  3. Determine whether your answer to part (b) uses the minimum number of bins. You must justify your answer. A different list of eleven numbers is to be sorted into descending order using a bubble sort. The list after the second pass is
    1.6
    1.7
    1.5
    3.8
    3.3
    4.5
    4.8
    5.6
    5.4
    6.7
    9.1
  4. Explain how you know that at least one of the first two passes of the bubble sort was not carried out correctly.
Edexcel FD1 AS 2024 June Q2
8 marks Moderate -0.8
2. A company manages an awards evening. The table below lists the activities required to set up the room for the evening, and their immediately preceding activities. Each activity requires exactly one person.
ActivityImmediately preceding activities
A-
BA
CA
DC
EC
FB, D, E
GE
HB
JH, F, G
Figure 1 shows a partially completed activity network used to model the project. Each activity is represented by an arc. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ca57c64b-0b33-4179-be7f-684bd6ea2162-04_440_813_1689_726} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Add the remaining five activities to Diagram 1 in the answer book to complete the activity network, using exactly two dummies. In addition to setting up the room, the company must prepare the meals for the guests. Figure 2 shows the activity network for preparing the main courses. The numbers in brackets represent the time, in minutes, to complete each task. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ca57c64b-0b33-4179-be7f-684bd6ea2162-05_793_1515_451_373} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure}
  2. Complete Diagram 2 in the answer book to show the early event times and the late event times for the activity network shown in Figure 2.
  3. State the critical activities.
  4. Given that the main courses need to be ready to be served (with all activities completed) at 8 pm , state the latest time that activity \(R\) can start.
Edexcel FD1 AS 2024 June Q3
11 marks Challenging +1.2
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ca57c64b-0b33-4179-be7f-684bd6ea2162-06_764_1547_314_355} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} [The total weight of the network is \(139 + x + y\) ]
  1. Explain what is meant by the term "tree". Figure 3 represents a network of walkways in a warehouse.
    The arcs represent the walkways and the nodes represent junctions between them.
    The number on each arc represents the length, in metres, of the corresponding walkway.
    The values \(x\) and \(y\) are unknown, however it is known that \(x\) and \(y\) are integers and that $$9 < x < y < 14$$
    1. Use Dijkstra's algorithm to find the shortest route from A to M.
    2. State an expression for the length of the shortest route from A to M . The warehouse manager wants to check that all of the walkways are in good condition.
      Their inspection route starts at B and finishes at C .
      The inspection route must traverse each walkway at least once and be as short as possible.
  3. State the arcs that are traversed twice.
  4. State the number of times that H appears in the inspection route. The warehouse manager finds that the total length of the inspection route is 172 metres.
  5. Determine the value of \(x\) and the value of \(y\)
Edexcel FD1 AS 2024 June Q4
12 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ca57c64b-0b33-4179-be7f-684bd6ea2162-07_1105_1249_312_512} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows three of the six constraints for a linear programming problem in \(x\) and \(y\) The unshaded region and its boundaries satisfy these three constraints.
  1. State these three constraints as simplified inequalities with integer coefficients. The variables \(x\) and \(y\) represent the number of orange fish and the number of blue fish, respectively, that are to be kept in an aquarium. The number of fish in the aquarium is subject to these three further constraints
    • there must be at least one blue fish
    • the orange fish must not outnumber the blue fish by more than ten
    • there must be no more than five blue fish for every orange fish
    • Write each of these three constraints as a simplified inequality with integer coefficients.
    • Represent these three constraints by adding lines and shading to Diagram 1 in the answer book, labelling the feasible region, \(R\)
    The total value (in pounds) of the fish in the aquarium is given by the objective function $$\text { Maximise } P = 3 x + 5 y$$
    1. Use the objective line method to determine the optimal point of the feasible region, giving its coordinates as exact fractions.
    2. Hence find the maximum total value of the fish in the aquarium, stating the optimal number of orange fish and the optimal number of blue fish. \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Please check the examination details below before entering your candidate information}
      Candidate surnameOther names
      Centre NumberCandidate Number
      □□
      \end{table} \section*{Pearson Edexcel Level 3 GCE} \section*{Friday 17 May 2024} Afternoon \section*{Further Mathematics} Advanced Subsidiary
      Further Mathematics options
      27: Decision Mathematics 1
      (Part of options D, F, H and K) \section*{D1 Answer Book} Do not return the question paper with the answer book.
      1. \(\begin{array} { l l l l l l l l l l l } 4 & 6.5 & 7 & 1.3 & 2 & 5 & 1.5 & 6 & 4.5 & 6 & 1 \end{array}\) 2.
      \includegraphics[max width=\textwidth, alt={}]{ca57c64b-0b33-4179-be7f-684bd6ea2162-12_435_815_392_463}
      \section*{Diagram 1} Use this diagram only if you need to redraw your activity network. \includegraphics[max width=\textwidth, alt={}, center]{ca57c64b-0b33-4179-be7f-684bd6ea2162-12_442_820_2043_458} Copy of Diagram 1
      VJYV SIHI NI JIIYM ION OCV346 SIHI NI JLIYM ION OCV34V SIHI NI IIIIM ION OC
      Key: \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{ca57c64b-0b33-4179-be7f-684bd6ea2162-13_1217_1783_451_236} \captionsetup{labelformat=empty} \caption{Diagram 2}
      \end{figure} 3. \includegraphics[max width=\textwidth, alt={}, center]{ca57c64b-0b33-4179-be7f-684bd6ea2162-14_2463_1240_339_465}
      Shortest route from A to M:
      Length of shortest route from A to M:
      \includegraphics[max width=\textwidth, alt={}]{ca57c64b-0b33-4179-be7f-684bd6ea2162-16_3038_2264_0_0}
      \includegraphics[max width=\textwidth, alt={}]{ca57c64b-0b33-4179-be7f-684bd6ea2162-17_1103_1247_397_512}
      \section*{Diagram 1} \section*{There is a copy of Diagram 1 on page 11 if you need to redraw your graph.}
      VJYV SIHI NI JIIIM ION OCV341 S1H1 NI JLIYM ION OAV34V SIHI NI IIIVM ION OC
      Use this diagram only if you need to redraw your graph. \includegraphics[max width=\textwidth, alt={}, center]{ca57c64b-0b33-4179-be7f-684bd6ea2162-19_1108_1252_1606_509} Copy of Diagram 1
Edexcel FD1 AS Specimen Q1
12 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e2c1dc4-3724-4bba-961c-1c2ae7e649c4-2_698_1173_447_443} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} [The total weight of the network is 189]
Figure 1 represents a network of pipes in a building. The number on each arc is the length, in metres, of the corresponding pipe.
  1. Use Dijkstra's algorithm to find the shortest path from A to F . State the path and its length. On a particular day, Gabriel needs to check each pipe. A route of minimum length, which traverses each pipe at least once and which starts and finishes at A, needs to be found.
  2. Use an appropriate algorithm to find the pipes that will need to be traversed twice. You must make your method and working clear.
  3. State the minimum length of Gabriel's route. A new pipe, BG, is added to the network. A route of minimum length that traverses each pipe, including BG, needs to be found. The route must start and finish at A. Gabriel works out that the addition of the new pipe increases the length of the route by twice the length of BG .
  4. Calculate the length of BG. You must show your working.
Edexcel FD1 AS Specimen Q2
7 marks Moderate -0.5
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e2c1dc4-3724-4bba-961c-1c2ae7e649c4-3_1463_1194_239_440} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A teacher buys pens and pencils. The number of pens, \(x\), and the number of pencils, \(y\), that he buys can be represented by a linear programming problem as shown in Figure 2, which models the following constraints: $$\begin{aligned} 8 x + 3 y & \leqslant 480 \\ 8 x + 7 y & \geqslant 560 \\ y & \geqslant 4 x \\ x , y & \geqslant 0 \end{aligned}$$ The total cost, in pence, of buying the pens and pencils is given by $$C = 12 x + 15 y$$ Determine the number of pens and the number of pencils which should be bought in order to minimise the total cost. You should make your method and working clear.
Edexcel FD1 AS Specimen Q3
7 marks Moderate -0.3
3.
ActivityTime taken (days)Immediately preceding activities
A5-
B7-
C3-
D4A, B
E4D
F2B
G4B
H5C, G
I10C, G
The table above shows the activities required for the completion of a building project. For each activity, the table shows the time taken in days to complete the activity and the immediately preceding activities. Each activity requires one worker. The project is to be completed in the shortest possible time.
  1. Draw the activity network described in the table, using activity on arc. Your activity network must contain the minimum number of dummies only.
    1. Show that the project can be completed in 21 days, showing your working.
    2. Identify the critical activities.
Edexcel FD1 AS Specimen Q4
9 marks Standard +0.3
4.
  1. Explain why it is not possible to draw a graph with exactly 5 nodes with orders \(1,3,4,4\) and 5 A connected graph has exactly 5 nodes and contains 18 arcs. The orders of the 5 nodes are \(2 ^ { 2 x } - 1,2 ^ { x } , x + 1,2 ^ { x + 1 } - 3\) and \(11 - x\).
    1. Calculate X .
    2. State whether the graph is Eulerian, semi-Eulerian or neither. You must justify your answer.
  3. Draw a graph which satisfies all of the following conditions:
Edexcel FD1 AS Specimen Q5
5 marks Standard +0.8
  1. Jonathan makes two types of information pack for an event, Standard and Value.
Each Standard pack contains 25 posters and 500 flyers.
Each Value pack contains 15 posters and 800 flyers.
He must use at least 150000 flyers.
Between \(35 \%\) and \(65 \%\) of the packs must be Standard packs.
Posters cost 20p each and flyers cost 4p each.
Jonathan wishes to minimise his costs.
Let x and y represent the number of Standard packs and Value packs produced respectively.
Formulate this as a linear programming problem, stating the objective and listing the constraints as simplified inequalities with integer coefficients. You should not attempt to solve the problem. \section*{(Total for Question 5 is 5 marks)} TOTAL IS 40 MARKS
Edexcel FD2 AS 2018 June Q1
5 marks Moderate -0.5
  1. Four workers, A, B, C and D, are to be assigned to four tasks, P, Q, R and S. Each worker must be assigned to exactly one task and each task must be done by only one worker. The time, in hours, that each worker takes to complete each task is shown in the table below.
\cline { 2 - 5 } \multicolumn{1}{c|}{}PQRS
A7.53.589.5
B5277.5
C43.53.58
D653.54
Reducing rows first, use the Hungarian algorithm to obtain an allocation which minimises the total time. You must explain your method and show the table after each stage.
Edexcel FD2 AS 2018 June Q2
15 marks Standard +0.3
2.
  1. Explain what the term 'zero-sum game' means. Two teams, A and B , are to face each other as part of a quiz.
    There will be several rounds to the quiz with 10 points available in each round.
    For each round, the two teams will each choose a team member and these two people will compete against each other until all 10 points have been awarded. The number of points that Team A can expect to gain in each round is shown in the table below.
    \cline { 3 - 5 } \multicolumn{2}{c|}{}Team B
    \cline { 3 - 5 } \multicolumn{2}{c|}{}PaulQaasimRashid
    \multirow{3}{*}{Team A}Mischa563
    \cline { 2 - 5 }Noel417
    \cline { 2 - 5 }Olive458
    The teams are each trying to maximise their number of points.
  2. State the number of points that Team B will expect to gain each round if Team A chooses Noel and Team B chooses Rashid.
  3. Explain why subtracting 5 from each value in the table will model this situation as a zero-sum game.
    1. Find the play-safe strategies for the zero-sum game.
    2. Explain how you know that the game is not stable. At the last minute, Olive becomes unavailable for selection by Team A.
      Team A decides to choose its player for each round so that the probability of choosing Mischa is \(p\) and the probability of choosing Noel is \(1 - p\).
  5. Use a graphical method to find the optimal value of \(p\) for Team A and hence find the best strategy for Team A. For this value of \(p\),
    1. find the expected number of points awarded, per round, to Team A,
    2. find the expected number of points awarded, per round, to Team B.
Edexcel FD2 AS 2018 June Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{905f2578-e4b2-4d4d-8455-298170fd824b-4_781_1159_365_551} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 models the flow of fluid through a system of pipes from a source, S , to a sink, T . The weights on the arcs show the capacities of the corresponding pipes in litres per minute. Two cuts \(C _ { 1 }\) and \(C _ { 2 }\) are shown.
  1. Find the capacity of
    1. cut \(C _ { 1 }\)
    2. cut \(C _ { 2 }\)
  2. Using only the capacities of cuts \(C _ { 1 }\) and \(C _ { 2 }\) state what can be deduced about the maximum possible flow through the system.
  3. On Diagram 1 in the answer book, show how a flow of 120 litres per minute from S to T can be achieved. You do not need to apply the labelling procedure to find this flow.
  4. Prove that 120 litres per minute is the maximum possible flow through the system. A new pipe is planned from S to A . Let the capacity of this pipe be \(x\) litres per minute.
  5. Find, in terms of \(x\) where necessary, the maximum possible flow through the new system.
Edexcel FD2 AS 2018 June Q4
10 marks Moderate -0.8
4. A village has an expected population growth rate (birth rate minus death rate) of \(r \%\) per year. In addition, \(N\) people are expected to move into the village each year. The expected population of the village is modelled by $$u _ { n + 1 } = 1.02 u _ { n } + 50$$ where \(u _ { n }\) is the expected population of the village \(n\) years from now.
  1. State
    1. the value of \(r\),
    2. the value of \(N\). Given that the population 1 year from now is expected to be 560
  2. solve the recurrence relation for \(u _ { n }\)
  3. Hence determine, using algebra, the number of years from now when the model predicts that the population of the village will first be greater than 3000
    (Total for Question 4 is 10 marks)
    TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS END