Questions — Edexcel (10514 questions)

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Edexcel Paper 3 Specimen Q7
8 marks Standard +0.3
  1. A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\).
A particle of mass \(m\) is placed on the plane and then projected up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is \(\mu\).
The particle moves up the plane with a constant deceleration of \(\frac { 4 } { 5 } \mathrm {~g}\).
  1. Find the value of \(\mu\). The particle comes to rest at the point \(A\) on the plane.
  2. Determine whether the particle will remain at \(A\), carefully justifying your answer.
Edexcel Paper 3 Specimen Q8
10 marks Moderate -0.3
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively]
A radio controlled model boat is placed on the surface of a large pond.
The boat is modelled as a particle.
At time \(t = 0\), the boat is at the fixed point \(O\) and is moving due north with speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Relative to \(O\), the position vector of the boat at time \(t\) seconds is \(\mathbf { r }\) metres.
At time \(t = 15\), the velocity of the boat is \(( 10.5 \mathbf { i } - 0.9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
The acceleration of the boat is constant.
  1. Show that the acceleration of the boat is \(( 0.7 \mathbf { i } - 0.1 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  2. Find \(\mathbf { r }\) in terms of \(t\).
  3. Find the value of \(t\) when the boat is north-east of \(O\).
  4. Find the value of \(t\) when the boat is moving in a north-east direction.
Edexcel Paper 3 Specimen Q9
13 marks Challenging +1.2
9. Figure 1 A uniform ladder \(A B\), of length \(2 a\) and weight \(W\), has its end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 4 }\).
The end \(B\) of the ladder is resting against a smooth vertical wall, as shown in Figure 1.
A builder of weight \(7 W\) stands at the top of the ladder.
To stop the ladder from slipping, the builder's assistant applies a horizontal force of magnitude \(P\) to the ladder at \(A\), towards the wall.
The force acts in a direction which is perpendicular to the wall.
The ladder rests in equilibrium in a vertical plane perpendicular to the wall and makes an angle \(\alpha\) with the horizontal ground, where \(\tan \alpha = \frac { 5 } { 2 }\).
The builder is modelled as a particle and the ladder is modelled as a uniform rod.
  1. Show that the reaction of the wall on the ladder at \(B\) has magnitude \(3 W\).
  2. Find, in terms of \(W\), the range of possible values of \(P\) for which the ladder remains in equilibrium. Often in practice, the builder's assistant will simply stand on the bottom of the ladder.
  3. Explain briefly how this helps to stop the ladder from slipping.
Edexcel Paper 3 Specimen Q10
13 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e678bf51-6dca-4ad7-808b-dfa31b04dc63-22_719_1333_246_365} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A boy throws a stone with speed \(U \mathrm {~ms} ^ { - 1 }\) from a point \(O\) at the top of a vertical cliff. The point \(O\) is 18 m above sea level.
The stone is thrown at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\).
The stone hits the sea at the point \(S\) which is at a horizontal distance of 36 m from the foot of the cliff, as shown in Figure 2.
The stone is modelled as a particle moving freely under gravity with \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Find
  1. the value of \(U\),
  2. the speed of the stone when it is 10.8 m above sea level, giving your answer to 2 significant figures.
  3. Suggest two improvements that could be made to the model.
Edexcel Paper 3 Specimen Q1
14 marks Standard +0.3
  1. Kaff coffee is sold in packets. A seller measures the masses of the contents of a random sample of 90 packets of Kaff coffee from her stock. The results are shown in the table below.
Mass \(w ( \mathrm {~g} )\)Midpoint \(y ( \mathrm {~g} )\)Frequency f
\(240 \leq w < 245\)242.58
\(245 \leq w < 248\)246.515
\(248 \leq w < 252\)250.035
\(252 \leq w < 255\)253.523
\(255 \leq w < 260\)257.59
$$\text { (You may use } \sum \mathrm { fy } ^ { 2 } = 5644 \text { 171.75) }$$ A histogram is drawn and the class \(245 \leq w < 248\) is represented by a rectangle of width 1.2 cm and height 10 cm .
  1. Calculate the width and the height of the rectangle representing the class \(255 \leq w < 260\).
  2. Use linear interpolation to estimate the median mass of the contents of a packet of Kaff coffee to 1 decimal place.
  3. Estimate the mean and the standard deviation of the mass of the contents of a packet of Kaff coffee to 1 decimal place. The seller claims that the mean mass of the contents of the packets is more than the stated mass. Given that the stated mass of the contents of a packet of Kaff coffee is 250 g and the actual standard deviation of the contents of a packet of Kaff coffee is 4 g ,
  4. test, using a 5\% level of significance, whether or not the seller's claim is justified. State your hypotheses clearly.
    (You may assume that the mass of the contents of a packet is normally distributed.)
  5. Using your answers to parts (b) and (c), comment on the assumption that the mass of the contents of a packet is normally distributed.
    (Total 14 marks)
Edexcel Paper 3 Specimen Q2
7 marks Moderate -0.3
2. A researcher believes that there is a linear relationship between daily mean temperature and daily total rainfall. The 7 places in the northern hemisphere from the large data set are used. The mean of the daily mean temperatures, \(t ^ { \circ } \mathrm { C }\), and the mean of the daily total rainfall, \(s \mathrm {~mm}\), for the month of July in 2015 are shown on the scatter diagram below. \includegraphics[max width=\textwidth, alt={}, center]{565bfa73-8095-4242-80b6-cd47aaff6a31-03_844_1339_497_372}
  1. With reference to the scatter diagram, explain why a linear regression model may not be suitable for the relationship between \(t\) and s .
    (1) The researcher calculated the product moment correlation coefficient for the 7 places and obtained \(r = 0.658\).
  2. Stating your hypotheses clearly, test at the \(10 \%\) level of significance, whether or not the product moment correlation coefficient for the population is greater than zero.
    (3)
  3. Using your knowledge of the large data set, suggest the names of the 2 places labelled \(G\) and \(H\).
    (1)
  4. Using your knowledge from the large data set, and with reference to the locations of the two places labelled \(G\) and \(H\), give a reason why these places have the highest temperatures in July.
    (2)
  5. Suggest how you could make better use of the large data set to investigate the relationship between daily mean temperature and daily total rainfall.
    (1)
    (Total 7 marks)
Edexcel Paper 3 Specimen Q3
10 marks Standard +0.8
3. For a particular type of bulb, \(36 \%\) grow into plants with blue flowers and the remainder grow into plants with white flowers. Bulbs are sold in mixed bags of 40 Russell selects a random sample of 5 bags of bulbs.
  1. Find the probability that fewer than 2 of these bags will contain more bulbs that grow into plants with blue flowers than grow into plants with white flowers.
    (4) Maggie takes a random sample of \(n\) bulbs.
    Using a normal approximation, the probability that more than 244 of these \(n\) bulbs will grow into blue flowers is 0.0521 to 4 decimal places.
  2. Find the value of \(n\).
    (6)
    (Total 10 marks)
Edexcel Paper 3 Specimen Q4
11 marks Standard +0.3
4. The Venn diagram shows the probabilities of students' lunch boxes containing a drink, sandwiches and a chocolate bar. \includegraphics[max width=\textwidth, alt={}, center]{565bfa73-8095-4242-80b6-cd47aaff6a31-05_655_899_392_484} \(D\) is the event that a lunch box contains a drink, \(S\) is the event that a lunch box contains sandwiches, \(C\) is the event that a lunch box contains a chocolate bar, \(u , v\) and \(w\) are probabilities.
  1. Write down \(\mathrm { P } \left( S \cap D ^ { \prime } \right)\). One day, 80 students each bring in a lunch box.
    Given that all 80 lunch boxes contain sandwiches and a drink,
  2. estimate how many of these 80 lunch boxes will contain a chocolate bar. Given that the events \(S\) and \(C\) are independent and that \(\mathrm { P } ( D \mid C ) = \frac { 14 } { 15 }\),
  3. calculate the value of \(u\), the value of \(v\) and the value of \(w\).
    (7)
    (Total 11 marks)
Edexcel Paper 3 Specimen Q5
8 marks Moderate -0.8
5. The lifetimes of batteries sold by company \(X\) are normally distributed, with mean 150 hours and standard deviation 25 hours. A box contains 12 batteries from company \(X\).
  1. Find the expected number of these batteries that have a lifetime of more than 160 hours. The lifetimes of batteries sold by company \(Y\) are normally distributed, with mean 160 hours and \(80 \%\) of these batteries have a lifetime of less than 180 hours.
  2. Find the standard deviation of the lifetimes of batteries from company \(Y\). Both companies sell their batteries for the same price.
  3. State which company you would recommend. Give reasons for your answer.
Edexcel FD1 AS 2020 June Q1
6 marks Moderate -0.8
1. \(3.7 \quad 2.5\) \(5.4 \quad 1.9\) 2.7
3.2
3.1
2.7
4.2
2.0
  1. Use the first-fit bin packing algorithm to determine how the numbers listed above can be packed into bins of size 8.5 The first-fit bin packing algorithm is to be used to pack \(n\) numbers into bins. The number of comparisons is used to measure the order of the first-fit bin packing algorithm.
  2. By considering the worst case, determine the order of the first-fit bin packing algorithm in terms of \(n\). You must make your method and working clear.
Edexcel FD1 AS 2020 June Q2
14 marks Moderate -0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-03_693_1379_233_342} \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. 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.
    1. State the minimum project completion time.
    2. List the critical activities.
  4. Calculate the maximum number of hours by which activity H could be delayed without affecting the shortest possible completion time of the project. You must make the numbers used in your calculation clear.
  5. Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.
  6. Draw a cascade chart for this project on Grid 1 in the answer book.
  7. Using the answer to (f), explain why it is not possible to complete the project in the shortest possible time using the number of workers found in (e).
Edexcel FD1 AS 2020 June Q3
11 marks Challenging +1.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-04_720_1470_233_296} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} [The weight of the network is \(5 x + 246\) ]
  1. Explain why it is not possible to draw a graph with an odd number of vertices of odd valency. Figure 2 represents a network of 14 roads in a town. The expression on each arc gives the time, in minutes, to travel along the corresponding road. Prim's algorithm, starting at A, is applied to the network. The order in which the arcs are selected is \(\mathrm { AD } , \mathrm { DH } , \mathrm { DG } , \mathrm { FG } , \mathrm { EF } , \mathrm { CG } , \mathrm { BD }\). It is given that the order in which the arcs are selected is unique.
  2. Using this information, find the smallest possible range of values for \(x\), showing your working clearly. A route that minimises the total time taken to traverse each road at least once is required. The route must start and finish at the same vertex. Given that the time taken to traverse this route is 318 minutes,
  3. use an appropriate algorithm to determine the value of \(x\), showing your working clearly.
Edexcel FD1 AS 2020 June Q4
9 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-05_1472_1320_233_376} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region. Figure 3 also shows an objective line for the problem and the optimal vertex, which is labelled as \(V\). The value of the objective at \(V\) is 556
Express the linear programming problem in algebraic form. List the constraints as simplified inequalities with integer coefficients and determine the objective. Please check the examination details below before entering your candidate information
Candidate surname
Other names Pearson Edexcel
Centre Number
Candidate Number Level 3 GCE \includegraphics[max width=\textwidth, alt={}, center]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-09_122_433_356_991}
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□ \section*{Thursday 14 May 2020} Afternoon
Paper Reference 8FMO/27 \section*{Further Mathematics} Advanced Subsidiary
Further Mathematics options
27: Decision Mathematics 1
(Part of options D, F, H and K) \section*{Answer Book} Do not return the question paper with the answer book.
1. \(\begin{array} { l l l l l l l l l l } 3.7 & 2.5 & 5.4 & 1.9 & 2.7 & 3.2 & 3.1 & 2.7 & 4.2 & 2.0 \end{array}\)
  1. (a)
Activity
Immediately
preceding
activities
A
B
C
D
Activity
Immediately
preceding
activities
E
F
G
H
Activity
Immediately
preceding
activities
I
J
K
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-12_734_1646_925_196} \captionsetup{labelformat=empty} \caption{Diagram 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-13_1116_1475_979_296} \captionsetup{labelformat=empty} \caption{Grid 1}
\end{figure} 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-14_716_1467_255_299} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} [The weight of the network is \(5 x + 246\) ] 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-18_1470_1319_255_388} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}
Edexcel FD2 2020 June Q1
8 marks Standard +0.3
  1. Four workers, A, B, C and D, are to be assigned to four tasks, 1, 2, 3 and 4 . Each worker must be assigned to exactly one task and each task must be done by exactly one worker.
Worker A cannot do task 3 and worker B cannot do task 4 The table below shows the profit, in pounds, that each worker would earn if assigned to each of the tasks.
1234
A2920-23
B323028-
C35323425
D29312730
  1. Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total profit. You must make your method clear and show the table after each stage.
  2. Determine the resulting total profit.
Edexcel FD2 2020 June Q2
7 marks Standard +0.3
2. Jenny can choose one of three options, A, B or C, when playing a game. The profit, in pounds, associated with each outcome and their corresponding probabilities are shown on the decision tree in Figure 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0fc09f9-06ea-4528-a2de-f409112d85cc-03_947_1319_349_374} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Calculate the optimal EMV to determine Jenny's best course of action. You must make your working clear. For a profit of \(\pounds x\), Jenny's utility is given by \(1 - \mathrm { e } ^ { - \frac { x } { 400 } }\)
  2. Using expected utility as the criterion for the best course of action, determine what Jenny should do now to maximise her profit. You must make your working clear.
Edexcel FD2 2020 June Q3
16 marks Challenging +1.2
3. Table 1 shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , to three sales points, \(\mathrm { P } , \mathrm { Q }\) and R . It also shows the number of units held at each supply point and the number of units required at each sales point. A minimum cost solution is required. \begin{table}[h]
PQRSupply
A25241742
B7121468
C13112025
D16151340
Demand597244
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} Table 2 shows an initial solution given by the north-west corner method. \begin{table}[h]
PQR
A42
B1751
C214
D40
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
  1. Taking AR as the entering cell, use the stepping-stone method to find an improved solution. Make your method clear.
  2. Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by stating
Edexcel FD2 2020 June Q4
8 marks Challenging +1.2
  1. The complementary function for the second order recurrence relation
$$u _ { n + 2 } + \alpha u _ { n + 1 } + \beta u _ { n } = 20 ( - 3 ) ^ { n } \quad n \geqslant 0$$ is given by $$u _ { n } = A ( 2 ) ^ { n } + B ( - 1 ) ^ { n }$$ where \(A\) and \(B\) are arbitrary non-zero constants.
  1. Find the value of \(\alpha\) and the value of \(\beta\). Given that \(2 u _ { 0 } = u _ { 1 }\) and \(u _ { 4 } = 164\)
  2. find the solution of this second order recurrence relation to obtain an expression for \(u _ { n }\) in terms of \(n\).
    (6)
Edexcel FD2 2020 June Q5
10 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0fc09f9-06ea-4528-a2de-f409112d85cc-06_830_1397_205_333} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a capacitated, directed network. The network represents a system of pipes through which fluid can flow. The weights on the arcs show the lower capacities and upper capacities for the corresponding pipes, in litres per second.
  1. State the source node.
  2. Explain why the sink node must be G.
  3. Calculate the capacity of the cut \(C _ { 1 }\)
  4. Assuming that a feasible flow exists,
    1. explain why arc JH must be at its upper capacity,
    2. explain why arcs AD and CD must be at their lower capacities.
  5. Use Diagram 1 in the answer book to show a flow of 18 litres per second through the system.
  6. Prove that the answer to (e) is the maximum flow through the system.
Edexcel FD2 2020 June Q6
14 marks Challenging +1.8
6.
\multirow{6}{*}{Player A}Player B
\multirow[b]{2}{*}{Option Q}Option XOption YOption Z
153
Option R4-31
Option S2-4-2
Option T3-20
A two person zero-sum game is represented by the pay-off matrix for player A, shown above.
  1. Explain, with justification, why this matrix may be reduced to a \(3 \times 3\) matrix by removing option S from player A's choices.
  2. Verify that there is no stable solution to the reduced game. Player A intends to make a random choice between options \(\mathrm { Q } , \mathrm { R }\) and T , choosing option Q with probability \(p _ { 1 }\), option R with probability \(p _ { 2 }\) and option T with probability \(p _ { 3 }\) Player A wants to find the optimal values of \(p _ { 1 } , p _ { 2 }\) and \(p _ { 3 }\) using the Simplex algorithm. Player A formulates the following linear programme, writing the constraints as inequalities. Maximise \(P = V\), where \(V =\) the value of original game + 3 $$\begin{aligned} \text { subject to } & V \leqslant 4 p _ { 1 } + 7 p _ { 2 } + 6 p _ { 3 } \\ & V \leqslant 8 p _ { 1 } + p _ { 3 } \\ & V \leqslant 6 p _ { 1 } + 4 p _ { 2 } + 3 p _ { 3 } \\ & p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 \\ & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , V \geqslant 0 \end{aligned}$$
  3. Explain why \(V\) cannot exceed any of the following expressions $$4 p _ { 1 } + 7 p _ { 2 } + 6 p _ { 3 } \quad 8 p _ { 1 } + p _ { 3 } \quad 6 p _ { 1 } + 4 p _ { 2 } + 3 p _ { 3 }$$
  4. Explain why it is necessary to use the constraint \(p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1\) The Simplex algorithm is used to solve the linear programming problem.
    Given that the optimal value of \(p _ { 1 } = \frac { 7 } { 11 }\) and the optimal value of \(p _ { 3 } = 0\)
  5. calculate the value of the game to player A .
    (3) Player B intends to make a random choice between options \(\mathrm { X } , \mathrm { Y }\) and Z , choosing option X with probability \(q _ { 1 }\), option Y with probability \(q _ { 2 }\) and option Z with probability \(q _ { 3 }\)
  6. Determine the optimal strategy for player B, making your working clear.
Edexcel FD2 2020 June Q7
12 marks Standard +0.3
7. A manufacturer can export five batches of footwear each year. Each exported batch contains just one type of footwear. The types of footwear are trainers, sandals or high heels. The table below shows the profit, in \(\pounds 1000\) s, for the number of batches of each type of footwear.
1234
A
B
C
D
1234
A
B
C
D
1234
A
B
C
D
1234
A
B
C
D
1234
A
B
C
D
1234
A
B
C
D
1234
A
B
C
D
1234
A
B
C
D
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0fc09f9-06ea-4528-a2de-f409112d85cc-12_956_1333_258_283} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0fc09f9-06ea-4528-a2de-f409112d85cc-13_954_1322_260_278} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} 3. \begin{table}[h]
PQRSupply
A25241742
B7121468
C13112025
D16151340
Demand597244
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
PQR
A
B
C
D
PQR
A
B
C
D
PQR
A
B
C
D
PQR
A
B
C
D
PQR
A
B
C
D
PQR
A
B
C
D
PQR
A
B
C
D
PQR
A
B
C
D
PQR
A
B
C
D
4. .
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0fc09f9-06ea-4528-a2de-f409112d85cc-21_666_1239_1155_413} \captionsetup{labelformat=empty} \caption{Diagram 1}
\end{figure} 6. Player A \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Player B}
\cline { 2 - 4 } \multicolumn{1}{c|}{}Option XOption YOption Z
Option Q153
Option R4- 31
Option S2- 4- 2
Option T3- 20
\end{table} 7.
StageStateActionDestinationValue
Trainers0000
StageStateActionDestinationValue
Edexcel FD2 2022 June Q1
6 marks Moderate -0.8
  1. Four workers, A, B, C and D, are to be assigned to four tasks, 1, 2, 3 and 4. Each task must be assigned to just one worker and each worker must do only one task.
The cost of assigning each worker to each task is shown in the table below.
The total cost is to be minimised.
1234
A32453448
B37395046
C46444042
D43454852
  1. Reducing rows first, use the Hungarian algorithm to obtain an allocation that minimises the total cost. You must make your method clear and show the table after each stage.
  2. State the minimum total cost.
Edexcel FD2 2022 June Q2
4 marks Standard +0.8
2. The general solution of the second order recurrence relation $$u _ { n + 2 } + k _ { 1 } u _ { n + 1 } + k _ { 2 } u _ { n } = 0 \quad n \geqslant 0$$ is given by $$u _ { n } = ( A + B n ) ( - 3 ) ^ { n }$$ where \(A\) and \(B\) are arbitrary non-zero constants.
  1. Find the value of \(k _ { 1 }\) and the value of \(k _ { 2 }\) Given that \(u _ { 0 } = u _ { 1 } = 1\)
  2. find the value of \(A\) and the value of \(B\).
Edexcel FD2 2022 June Q3
7 marks Moderate -0.8
3. The table below shows the transport options, usual travel times, possible delay times and corresponding probabilities of delay for a journey. All times are in minutes.
Transport optionUsual travel timePossible delay timeProbability of delay
\multirow{2}{*}{Car}\multirow{2}{*}{52}100.10
250.02
\multirow{2}{*}{Train}\multirow{2}{*}{45}150.05
250.03
\multirow{2}{*}{Coach}\multirow{2}{*}{55}50.05
150.01
  1. Draw a decision tree to model the transport options and the possible outcomes.
  2. State the minimum expected travel time and the corresponding transport option indicated by the decision tree.
Edexcel FD2 2022 June Q4
9 marks Moderate -0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cea07472-f93b-4a7b-b362-89fb8c0af4a9-04_931_1312_219_379} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a capacitated, directed network of pipes. The uncircled number on each arc represents the capacity of the corresponding pipe. The numbers in circles represent an initial flow.
  1. List the saturated arcs.
  2. State the value of the initial flow.
  3. Explain why arc FT cannot be full to capacity.
  4. State the capacity of cut \(C _ { 1 }\) and the capacity of cut \(C _ { 2 }\)
  5. By inspection find one flow-augmenting route to increase the flow by three units. You must state your route.
  6. Prove that, once the flow-augmenting route found in part (e) has been applied, the flow is maximal.
Edexcel FD2 2022 June Q5
9 marks Standard +0.8
5. A standard transportation problem is described in the linear programming formulation below. Let \(X _ { i j }\) be the number of units transported from \(i\) to \(j\) where \(i \in \{ \mathrm {~A} , \mathrm {~B} , \mathrm { C } , \mathrm { D } \}\) $$j \in \{ \mathrm { R } , \mathrm {~S} , \mathrm {~T} \} \text { and } x _ { i j } \geqslant 0$$ Minimise \(P = 23 x _ { \mathrm { AR } } + 17 x _ { \mathrm { AS } } + 24 x _ { \mathrm { AT } } + 15 x _ { \mathrm { BR } } + 29 x _ { \mathrm { BS } } + 32 x _ { \mathrm { BT } }\) $$+ 25 x _ { \mathrm { CR } } + 25 x _ { \mathrm { CS } } + 27 x _ { \mathrm { CT } } + 19 x _ { \mathrm { DR } } + 20 x _ { \mathrm { DS } } + 25 x _ { \mathrm { DT } }$$ subject to $$\begin{aligned} & \sum x _ { \mathrm { A } j } \leqslant 34 \\ & \sum x _ { \mathrm { B } j } \leqslant 27 \\ & \sum x _ { \mathrm { C } j } \leqslant 41 \\ & \sum x _ { \mathrm { D } j } \leqslant 18 \\ & \sum x _ { i \mathrm { R } } \geqslant 44 \\ & \sum x _ { i \mathrm {~S} } \geqslant 37 \\ & \sum x _ { i \mathrm {~T} } \geqslant k \end{aligned}$$ Given that the problem is balanced,
  1. state the value of \(k\).
  2. Explain precisely what the constraint \(\sum x _ { i \mathrm { R } } \geqslant 44\) means in the transportation problem.
  3. Use the north-west corner method to obtain the cost of an initial solution to this transportation problem.
  4. Perform one iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by showing the route and the