Questions - A-Level Maths

AQA Further Paper 3 Discrete 2024 June Q4

4 Daniel and Jackson play a zero-sum game. The game is represented by the following pay-off matrix for Daniel.

\multirow{6}{*}{Daniel}	Jackson
	Strategy	W	X	Y	Z
	A	3	-2	1	4
	B	5	1	-4	1
	C	2	-1	1	2
	D	-3	0	2	-1

Neither player has any strategies which can be ignored due to dominance. 4

Prove that the game does not have a stable solution.
Fully justify your answer.
4
Determine the play-safe strategy for each player. Play-safe strategy for Daniel $\_\_\_\_$ Play-safe strategy for Jackson $\_\_\_\_$

AQA Further Paper 3 Discrete 2024 June Q5

5

1. Determine the electrical connections that should be installed.
  5
(ii) Find the minimum possible total time needed to install the required electrical connections.
5
Following the installation of the electrical connections, some of the car parks have an indirect connection to the stadium's main electricity power supply. Give one limitation of this installation.

AQA Further Paper 3 Discrete 2024 June Q6

6
A company delivers parcels to houses in a village, using a van. The network below shows the roads in the village. Each node represents a road junction and the weight of each arc represents the length, in miles, of the road between the junctions.
\includegraphics[max width=\textwidth, alt={}, center]{8d4db82a-0daf-487a-a6eb-be3ce8e59141-08_1208_1193_502_407} The total length of all of the roads in the village is 31.4 miles. On one particular day, the driver is due to make deliveries to at least one house on each road, so the van must travel along each road at least once. However, the driver has forgotten to add fuel to the van and it only has 4.5 litres of fuel to use to make its deliveries. The van uses, on average, 1 litre of fuel to travel 7.8 miles along the roads of this village. Whilst making each delivery, the driver turns off the van's engine so it does not use any fuel. Determine whether the van has enough fuel for the driver to make all of the deliveries to houses on each road of the village, starting and finishing at the same junction. Fully justify your answer.
\includegraphics[max width=\textwidth, alt={}, center]{8d4db82a-0daf-487a-a6eb-be3ce8e59141-09_2489_1778_175_107}

AQA Further Paper 3 Discrete 2024 June Q7

7

By considering associativity, show that the set of integers does not form a group under the binary operation of subtraction. Fully justify your answer.
7
The group G is formed by the set $$\{ 1,7,8,11,12,18 \}$$ under the operation of multiplication modulo 19 7
1. Complete the Cayley table for $G$
  ${ } ^ { \times } 19$ 1 7 8 11 12 18
  1 1 7 8 11 12 18
  7 7 11
  8 8 7
  11 11 7
  12 12 11
  18 18 1
  7
(ii) State the inverse of 11 in $G$
7
1. State, with a reason, the possible orders of the proper subgroups of $G$ 7
(ii) Find all the proper subgroups of $G$
Give your answers in the form $\left( \langle g \rangle , \mathrm { x } _ { 19 } \right)$ where $g \in G$
7
(iii) The group $H$ is such that $G \cong H$ State a possible name for $H$

AQA Further Paper 3 Discrete 2024 June Q8

8
Figure 1 shows a network of water pipes. The number on each arc represents the upper capacity for each pipe in litres per second. The numbers in the circles represent an initial feasible flow of 103 litres per second. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{8d4db82a-0daf-487a-a6eb-be3ce8e59141-12_979_1074_589_466}

\end{figure} 8

On Figure 1 above, add a supersource $S$ and a supersink $T$ to the network. 8
Using flow augmentation, find the maximum flow through the network. You must indicate any flow augmenting paths clearly in the table below. You may use Figure 2, on the opposite page, in your solution.
Augmenting Path Extra Flow
Maximum Flow $\_\_\_\_$ litres per second \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{8d4db82a-0daf-487a-a6eb-be3ce8e59141-13_960_1074_315_466}
\end{figure} 8
While the flow through the network is at its maximum value, the pipe EG develops a leak. To repair the leak, an engineer turns off the flow of water through EG
The engineer claims that the maximum flow of water through the network will reduce by 31 litres per second. Comment on the validity of the engineer's claim.

AQA Further Paper 3 Discrete 2024 June Q9

9 Janet and Samantha play a zero-sum game. The game is represented by the following pay-off matrix for Janet. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Samantha}

\multirow{5}{*}{Janet}	Strategy	$\mathbf { S } _ { \mathbf { 1 } }$	$\mathbf { S } _ { \mathbf { 2 } }$	$\mathbf { S } _ { \mathbf { 3 } }$
	$\mathbf { J } _ { \mathbf { 1 } }$	2	7	6
	$\mathbf { J } _ { \mathbf { 2 } }$	5	5	1
	$\mathbf { J } _ { \mathbf { 3 } }$	4	3	8
	$\mathbf { J } _ { \mathbf { 4 } }$	1	6	4

\end{table} $\mathbf { 9 }$ (a) Explain why Janet should never play strategy $\mathbf { J } _ { \mathbf { 4 } }$ 9 (b) Janet wants to maximise her winnings from the game.
She defines the following variables.
$p _ { 1 } =$ the probability of Janet playing strategy $\mathbf { J } _ { \mathbf { 1 } }$
$p _ { 2 } =$ the probability of Janet playing strategy $\mathbf { J } _ { 2 }$
$p _ { 3 } =$ the probability of Janet playing strategy $\mathbf { J } _ { \mathbf { 3 } }$
$v =$ the value of the game for Janet
Janet then formulates her situation as the following linear programming problem. $$\begin{array} { l l } \text { Maximise } & P = v
\text { subject to } & 2 p _ { 1 } + 5 p _ { 2 } + 4 p _ { 3 } \geq v
& 7 p _ { 1 } + 5 p _ { 2 } + 3 p _ { 3 } \geq v
& 6 p _ { 1 } + p _ { 2 } + 8 p _ { 3 } \geq v
\text { and } & p _ { 1 } + p _ { 2 } + p _ { 3 } \leq 1
& p _ { 1 } , p _ { 2 } , p _ { 3 } \geq 0 \end{array}$$ 9 (b) (i) Complete the initial Simplex tableau for Janet's situation in the grid below. Find the probability of Janet playing strategy $\mathbf { J } _ { \mathbf { 3 } }$ when she is playing to maximise her winnings from the game.
\includegraphics[max width=\textwidth, alt={}, center]{8d4db82a-0daf-487a-a6eb-be3ce8e59141-17_2491_1755_173_123} A project is undertaken by Higton Engineering Ltd. The project is broken down into 11 separate activities $A , B , \ldots , K$
Figure 3 below shows a completed activity network for the project, along with the earliest start time, duration, latest finish time and the number of workers required for each activity. All times and durations are given in days. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{8d4db82a-0daf-487a-a6eb-be3ce8e59141-18_930_1714_724_148}

\end{figure}

AQA Further Paper 3 Discrete 2024 June Q10

10

Write down the critical path. 10
Using Figure 4 below, draw a resource histogram for the project to show how the project can be completed in the minimum possible time. Assume that each activity is to start as early as possible. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{8d4db82a-0daf-487a-a6eb-be3ce8e59141-19_760_1707_568_153}
\end{figure} 10
Higton Engineering Ltd only has four workers available to work on the project. Find the minimum completion time for the project. Use Figure 5 below in your answer. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{8d4db82a-0daf-487a-a6eb-be3ce8e59141-19_510_1703_1786_155}
\end{figure} Minimum completion time $\_\_\_\_$
\includegraphics[max width=\textwidth, alt={}, center]{8d4db82a-0daf-487a-a6eb-be3ce8e59141-20_2491_1755_173_123} Additional page, if required.
Write the question numbers in the left-hand margin. number
.....
\section*{Additional page, if required. Write the question numbers in the left-hand margin.
\includegraphics[max width=\textwidth, alt={}]{8d4db82a-0daf-487a-a6eb-be3ce8e59141-22_67_195_233_644}} □ - box box \section*{Additional page, if required. Write the question numbers in the left-hand margin.}
\includegraphics[max width=\textwidth, alt={}]{8d4db82a-0daf-487a-a6eb-be3ce8e59141-24_2484_1748_178_130}

Edexcel FD1 AS 2019 June Q1

(a) Draw the graph $\mathrm { K } _ { 5 }$
(b) (i) In the context of graph theory explain what is meant by 'semi-Eulerian'.
(ii) Draw two semi-Eulerian subgraphs of $\mathrm { K } _ { 5 }$, each having five vertices but with a different number of edges.
(c) Explain why a graph with exactly five vertices with vertex orders 1, 2, 2, 3 and 4 cannot be a tree.
The following algorithm produces a numerical approximation for the integral

$$I = \int _ { \mathrm { A } } ^ { \mathrm { B } } x ^ { 4 } \mathrm {~d} x$$ Step 1 Start
Step 2 Input the values of A, B and N
Step 3 Let $\mathrm { H } = ( \mathrm { B } - \mathrm { A } ) / \mathrm { N }$
Step 4 Let $\mathrm { C } = \mathrm { H } / 2$
Step 5 Let $\mathrm { D } = 0$
Step 6 Let $\mathrm { D } = \mathrm { D } + \mathrm { A } ^ { 4 } + \mathrm { B } ^ { 4 }$
Step $7 \quad$ Let $\mathrm { E } = \mathrm { A }$
Step 8 Let $\mathrm { E } = \mathrm { E } + \mathrm { H }$
Step 9 If $\mathrm { E } = \mathrm { B }$ go to Step 12
Step $10 \quad$ Let $\mathrm { D } = \mathrm { D } + 2 \times \mathrm { E } ^ { 4 }$
Step 11 Go to Step 8
Step 12 Let $\mathrm { F } = \mathrm { C } \times \mathrm { D }$
Step 13 Output F
Step 14 Stop
For the case when $\mathrm { A } = 1 , \mathrm {~B} = 3$ and $\mathrm { N } = 4$,
(a) (i) complete the table in the answer book to show the results obtained at each step of the algorithm.
(ii) State the final output.
(b) Calculate, to 3 significant figures, the percentage error between the exact value of $I$ and the value obtained from using the approximation to $I$ in this case.
3.

Activity	Immediately preceding activities
A	-
B	-
C	A
D	A
E	A
F	B, C
G	B, C
H	D
I	D, E, F, G
J	D, E, F, G
K	G

(a) Draw the activity network described in the precedence table above, using activity on arc. Your activity network must contain the minimum number of dummies. Every activity shown in the precedence table has the same duration.
(b) Explain why activity B cannot be critical.
(c) State which other activities are not critical.
4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{103a0bcf-3adf-407c-aa98-a784b0b39bf5-04_577_1357_230_354} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} [The total weight of the network is $135 + 4 x + 2 y$ ] The weights on the arcs in Figure 1 represent distances. The weights on the arcs CE and GH are given in terms of $x$ and $y$, where $x$ and $y$ are positive constants and $7 < x + y < 20$ There are three paths from A to H that have the same minimum length.
(a) Use Dijkstra's algorithm to find $x$ and $y$. An inspection route starting at A and finishing at H is found. The route traverses each arc at least once and is of minimum length.
(b) State the arcs that are traversed twice.
(c) State the number of times that vertex C appears in the inspection route.
(d) Determine the length of the inspection route.
5. Ben is a wedding planner. He needs to order flowers for the weddings that are taking place next month. The three types of flower he needs to order are roses, hydrangeas and peonies. Based on his experience, Ben forms the following constraints on the number of each type of flower he will need to order.

At least three-fifths of all the flowers must be roses.
For every 2 hydrangeas there must be at most 3 peonies.
The total number of flowers must be exactly 1000

The cost of each rose is $\pounds 1$, the cost of each hydrangea is $\pounds 5$ and the cost of each peony is $\pounds 4$ Ben wants to minimise the cost of the flowers. Let $x$ represent the number of roses, let $y$ represent the number of hydrangeas and let $z$ represent the number of peonies that he will order.
(a) 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. Ben decides to order the minimum number of roses that satisfy his constraints.
(b) (i) Calculate the number of each type of flower that he will order to minimise the cost of the flowers.
(ii) Calculate the corresponding total cost of this order. Please check the examination details below before entering your candidate information
Candidate surname
Other names Pearson Edexcel Level 3 GCE Centre Number
\includegraphics[max width=\textwidth, alt={}, center]{103a0bcf-3adf-407c-aa98-a784b0b39bf5-09_122_433_356_991} Candidate Number
□
□
□ \section*{Thursoay 16 May 2019} Afternoon
Paper Reference 8FMO-27 \section*{Further Mathematics} \section*{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.

Edexcel FD1 AS 2019 June Q3

3.

Activity	Immediately preceding activities
A	-
B	-
C	A
D	A
E	A
F	B, C
G	B, C
H	D
I	D, E, F, G
J	D, E, F, G
K	G

Draw the activity network described in the precedence table above, using activity on arc. Your activity network must contain the minimum number of dummies. Every activity shown in the precedence table has the same duration.
Explain why activity B cannot be critical.
State which other activities are not critical.

Edexcel FD1 AS 2019 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{103a0bcf-3adf-407c-aa98-a784b0b39bf5-04_577_1357_230_354} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} [The total weight of the network is $135 + 4 x + 2 y$ ] The weights on the arcs in Figure 1 represent distances. The weights on the arcs CE and GH are given in terms of $x$ and $y$, where $x$ and $y$ are positive constants and $7 < x + y < 20$ There are three paths from A to H that have the same minimum length.

Use Dijkstra's algorithm to find $x$ and $y$. An inspection route starting at A and finishing at H is found. The route traverses each arc at least once and is of minimum length.
State the arcs that are traversed twice.
State the number of times that vertex C appears in the inspection route.
Determine the length of the inspection route.

Edexcel FD1 AS 2019 June Q5

5. Ben is a wedding planner. He needs to order flowers for the weddings that are taking place next month. The three types of flower he needs to order are roses, hydrangeas and peonies. Based on his experience, Ben forms the following constraints on the number of each type of flower he will need to order.

At least three-fifths of all the flowers must be roses.
For every 2 hydrangeas there must be at most 3 peonies.
The total number of flowers must be exactly 1000

The cost of each rose is $\pounds 1$, the cost of each hydrangea is $\pounds 5$ and the cost of each peony is $\pounds 4$ Ben wants to minimise the cost of the flowers. Let $x$ represent the number of roses, let $y$ represent the number of hydrangeas and let $z$ represent the number of peonies that he will order.

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. Ben decides to order the minimum number of roses that satisfy his constraints.
1. Calculate the number of each type of flower that he will order to minimise the cost of the flowers.
2. Calculate the corresponding total cost of this order. Please check the examination details below before entering your candidate information
  Candidate surname
  Other names Pearson Edexcel Level 3 GCE Centre Number
  \includegraphics[max width=\textwidth, alt={}, center]{103a0bcf-3adf-407c-aa98-a784b0b39bf5-09_122_433_356_991} Candidate Number
  □
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  □ \section*{Thursoay 16 May 2019} Afternoon
  Paper Reference 8FMO-27 \section*{Further Mathematics} \section*{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. 2. You may not need to use all the rows in this table. It may not be necessary to complete all boxes in each row.
  A B N H C D E F
  3.
  \includegraphics[max width=\textwidth, alt={}]{103a0bcf-3adf-407c-aa98-a784b0b39bf5-16_2530_1776_207_148}
  5.

Edexcel FD1 AS 2020 June Q1

1. $3.7 \quad 2.5$
$5.4 \quad 1.9$
2.7
3.2
3.1
2.7
4.2
2.0

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.
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

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.

Complete the precedence table in the answer book.
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.
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.
Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.
Draw a cascade chart for this project on Grid 1 in the answer book.
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

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 $5 x + 246$ ]

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.
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,
use an appropriate algorithm to determine the value of $x$, showing your working clearly.

Edexcel FD1 AS 2020 June Q4

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}
□
□
□
□ \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.
$\begin{array} { l l l l l l l l l l } 3.7 & 2.5 & 5.4 & 1.9 & 2.7 & 3.2 & 3.1 & 2.7 & 4.2 & 2.0 \end{array}$

(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

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.

	1	2	3	4
A	29	20	-	23
B	32	30	28	-
C	35	32	34	25
D	29	31	27	30

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.
Determine the resulting total profit.

Edexcel FD2 2020 June Q2

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}

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 $1 - \mathrm { e } ^ { - \frac { x } { 400 } }$
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

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]

	P	Q	R	Supply
A	25	24	17	42
B	7	12	14	68
C	13	11	20	25
D	16	15	13	40
Demand	59	72	44

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table} Table 2 shows an initial solution given by the north-west corner method. \begin{table}[h]

	P	Q	R
A	42
B	17	51
C		21	4
D			40

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

\end{table}

Taking AR as the entering cell, use the stepping-stone method to find an improved solution. Make your method clear.
Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by stating
- shadow costs
- improvement indices
- route
- entering cell and exiting cell.
- Determine whether the solution obtained from this second iteration is optimal, giving the reason for your answer.
- Formulate this situation as a linear programming problem. You must define your decision variables and make the objective function and constraints clear.
- Explain why the Simplex algorithm cannot be used to solve transportation linear programming problems such as that formulated in (d).

Edexcel FD2 2020 June Q4

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.

Find the value of $\alpha$ and the value of $\beta$. Given that $2 u _ { 0 } = u _ { 1 }$ and $u _ { 4 } = 164$
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

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.

State the source node.
Explain why the sink node must be G.
Calculate the capacity of the cut $C _ { 1 }$
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.
Use Diagram 1 in the answer book to show a flow of 18 litres per second through the system.
Prove that the answer to (e) is the maximum flow through the system.

Edexcel FD2 2020 June Q6

6.

\multirow{6}{*}{Player A}	Player B
	\multirow[b]{2}{*}{Option Q}	Option X	Option Y	Option Z
		1	5	3
	Option R	4	-3	1
	Option S	2	-4	-2
	Option T	3	-2	0

A two person zero-sum game is represented by the pay-off matrix for player A, shown above.

Explain, with justification, why this matrix may be reduced to a $3 \times 3$ matrix by removing option S from player A's choices.
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 $$\begin{aligned} \text { subject to } & V \leqslant 4 p _ { 1 } + 7 p _ { 2 } + 6 p _ { 3 }
& V \leqslant 8 p _ { 1 } + p _ { 3 }
& V \leqslant 6 p _ { 1 } + 4 p _ { 2 } + 3 p _ { 3 }
& p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1
& p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , V \geqslant 0 \end{aligned}$$
Explain why $V$ cannot exceed any of the following expressions $$4 p _ { 1 } + 7 p _ { 2 } + 6 p _ { 3 } \quad 8 p _ { 1 } + p _ { 3 } \quad 6 p _ { 1 } + 4 p _ { 2 } + 3 p _ { 3 }$$
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 $p _ { 3 } = 0$
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 $q _ { 3 }$
Determine the optimal strategy for player B, making your working clear.

Edexcel FD2 2020 June Q7

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.

	1	2	3	4
A
B
C
D

	1	2	3	4
A
B
C
D

	1	2	3	4
A
B
C
D

	1	2	3	4
A
B
C
D

	1	2	3	4
A
B
C
D

	1	2	3	4
A
B
C
D

	1	2	3	4
A
B
C
D

	1	2	3	4
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]

	P	Q	R	Supply
A	25	24	17	42
B	7	12	14	68
C	13	11	20	25
D	16	15	13	40
Demand	59	72	44

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table}

	P	Q	R
A
B
C
D

	P	Q	R
A
B
C
D

	P	Q	R
A
B
C
D

	P	Q	R
A
B
C
D

	P	Q	R
A
B
C
D

	P	Q	R
A
B
C
D

	P	Q	R
A
B
C
D

	P	Q	R
A
B
C
D

	P	Q	R
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 X	Option Y	Option Z
Option Q	1	5	3
Option R	4	- 3	1
Option S	2	- 4	- 2
Option T	3	- 2	0

\end{table} 7.

Stage	State	Action	Destination	Value
Trainers	0	0	0	0

Stage	State	Action	Destination	Value

Edexcel FD2 2022 June Q1

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.

	1	2	3	4
A	32	45	34	48
B	37	39	50	46
C	46	44	40	42
D	43	45	48	52

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.
State the minimum total cost.

Edexcel FD2 2022 June Q2

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.

Find the value of $k _ { 1 }$ and the value of $k _ { 2 }$ Given that $u _ { 0 } = u _ { 1 } = 1$
find the value of $A$ and the value of $B$.

Edexcel FD2 2022 June Q3

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 option	Usual travel time	Possible delay time	Probability of delay
\multirow{2}{*}{Car}	\multirow{2}{*}{52}	10	0.10
		25	0.02
\multirow{2}{*}{Train}	\multirow{2}{*}{45}	15	0.05
		25	0.03
\multirow{2}{*}{Coach}	\multirow{2}{*}{55}	5	0.05
		15	0.01

Draw a decision tree to model the transport options and the possible outcomes.
State the minimum expected travel time and the corresponding transport option indicated by the decision tree.

\multirow{5}{*}{Janet}	Strategy	\(\mathbf { S } _ { \mathbf { 1 } }\)	\(\mathbf { S } _ { \mathbf { 2 } }\)	\(\mathbf { S } _ { \mathbf { 3 } }\)
	\(\mathbf { J } _ { \mathbf { 1 } }\)	2	7	6
	\(\mathbf { J } _ { \mathbf { 2 } }\)	5	5	1
	\(\mathbf { J } _ { \mathbf { 3 } }\)	4	3	8
	\(\mathbf { J } _ { \mathbf { 4 } }\)	1	6	4

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\({ } ^ { \times } 19\)	1	7	8	11	12	18
1	1	7	8	11	12	18
7	7	11
8	8		7
11	11			7
12	12				11
18	18					1

Augmenting Path	Extra Flow