Questions - Edexcel FD2 - A-Level Maths

	Cost of travel option	Journey time (in hours) without delays	Probability of a 1-hour delay	Probability of a 2-hour delay	Probability of a 3-hour delay	Probability of a 24-hour delay
Plane	£200	3	0.09	0.05	0	0.03
Train	£130	5	0.07	0.03	0	0
Coach	£70	6	0.15	0.1	0.05	0

By drawing a decision tree, evaluate the EMV of the total cost of Sebastien's journey for each node of your tree.
Hence state the travel option that minimises the EMV of the total cost of Sebastien's journey.
A cube root utility function is applied to the total costs of each option. Determine the travel option with the best expected utility and state the corresponding value.

Edexcel FD2 2024 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{931ccf1d-4b02-448c-b492-846b0f42c057-07_709_1507_214_280} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The staged, directed network in Figure 2 represents the roads that connect 12 towns, S, A, B, C, D, E, F, G, H, I, J and T. The number on each arc shows the time, in hours, it takes to drive between these towns. Elena plans to drive from S to T . She must arrive at T by 9 pm .

By completing the table in the answer book, use dynamic programming to find the latest time that Elena can start her journey from S to arrive at T by 9 pm .
Hence write down the route that Elena should take.

Edexcel FD2 2024 June Q7

\multirow{2}{*}{}		Player B
		Option X	Option Y	Option Z
\multirow{3}{*}{Player A}	Option R	3	2	-3
	Option S	4	-2	1
	Option T	-1	3	6

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

Verify that there is no stable solution to this game. Player A intends to make a random choice between options $\mathrm { R } , \mathrm { S }$ and T , choosing option R with probability $p _ { 1 }$, option S 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 objective function for the corresponding linear programme. $$\text { Maximise } P = V \quad \text { where } V = \text { the value of the game } + 3$$

Determine an initial Simplex tableau, making your variables and working clear. After several iterations of the Simplex algorithm, a possible final tableau is

b.v.	$V$	$p _ { 1 }$	$p _ { 2 }$	$p _ { 3 }$	r	$s$	$t$	$u$	Value
$p _ { 3 }$	0	0	0	1	$\frac { 1 } { 10 }$	$- \frac { 3 } { 80 }$	$- \frac { 1 } { 16 }$	$\frac { 33 } { 80 }$	$\frac { 33 } { 80 }$
$p _ { 2 }$	0	0	1	0	$- \frac { 1 } { 10 }$	$\frac { 13 } { 80 }$	$- \frac { 1 } { 16 }$	$\frac { 17 } { 80 }$	$\frac { 17 } { 80 }$
V	1	0	0	0	$\frac { 1 } { 2 }$	$\frac { 5 } { 16 }$	$\frac { 3 } { 16 }$	$\frac { 73 } { 16 }$	$\frac { 73 } { 16 }$
$p _ { 1 }$	0	1	0	0	0	$- \frac { 1 } { 8 }$	$\frac { 1 } { 8 }$	$\frac { 3 } { 8 }$	$\frac { 3 } { 8 }$
$P$	0	0	0	0	$\frac { 1 } { 2 }$	$\frac { 5 } { 16 }$	$\frac { 3 } { 16 }$	$\frac { 73 } { 16 }$	$\frac { 73 } { 16 }$

1. State the best strategy for player A.
2. Calculate the value of the game for player B. Player B intends to make a random choice between options $\mathrm { X } , \mathrm { Y }$ and Z .
Determine the best strategy for player B, making your method and working clear.
(3)

Edexcel FD2 2024 June Q8

8. A sequence $\left\{ u _ { n } \right\}$, where $n \geqslant 0$, satisfies the recurrence relation $$2 u _ { n + 2 } + 5 u _ { n + 1 } = 3 u _ { n } + 8 n + 2$$

Find the general solution of this recurrence relation.
(5) A particular solution of this recurrence relation has $u _ { 0 } = 1$ and $u _ { 1 } = k$, where $k$ is a positive constant. All terms of the sequence are positive.
Determine the value of $k$.
(3)

Edexcel FD2 Specimen Q1

(a) Find the general solution of the recurrence relation

$$u _ { n + 2 } = u _ { n + 1 } + u _ { n } , \quad n \geqslant 1$$ Given that $u _ { 1 } = 1$ and $u _ { 2 } = 1$
(b) find the particular solution of the recurrence relation.

Edexcel FD2 Specimen Q2

	D	E	F	Available
A	15	19	9	25
B	11	18	10	55
C	11	12	18	20
Required	38	24	38

A company has three factories, $\mathrm { A } , \mathrm { B }$ and C . It supplies mattresses to three shops, $\mathrm { D } , \mathrm { E }$ and F . The table shows the transportation cost, in pounds, of moving one mattress from each factory to each shop. It also shows the number of mattresses available at each factory and the number of mattresses required at each shop. A minimum cost solution is required.

Use the north-west corner method to obtain an initial solution.
Show how the transportation algorithm is used to solve this problem. You must state, at each appropriate step, the
- shadow costs,
- improvement indices,
- route,
- entering cell and exiting cell,
  and explain clearly how you know that your final solution is optimal.

Edexcel FD2 Specimen Q3

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 at most one task and each task must be done by just one worker. The amount, in pounds, that each worker would earn while assigned to each task is shown in the table below.

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	P	Q	R	S
A	32	32	33	35
B	28	35	31	37
C	35	29	33	36
D	36	30	36	33

The Hungarian algorithm is to be used to find the maximum total amount which may be earned by the four workers.

Explain how the table should be modified.
Reducing rows first, use the Hungarian algorithm to obtain an allocation which maximises the total earnings, stating how each table was formed.
Formulate the problem as a linear programming problem. You must define your decision variables and make your objective function and constraints clear.

Edexcel FD2 Specimen Q4

4. A game uses a standard pack of 52 playing cards. A player gives 5 tokens to play and then picks a card. If they pick a $2,3,4,5$ or 6 then they gain 15 tokens. If any other card is picked they lose. If they lose, the card is replaced and they can choose to pick again for another 5 tokens. This time if they pick either an ace or a king they gain 40 tokens. If any other card is picked they lose. Daniel is deciding whether to play this game.

Draw a decision tree to model Daniel's possible decisions and the possible outcomes.
Calculate Daniel's optimal EMV and state the optimal strategy indicated by the decision tree.

Edexcel FD2 Specimen Q5

	B plays 1	B plays 2	B plays 3	B plays 4
A plays 1	4	-2	3	2
A plays 2	3	-1	2	0
A plays 3	-1	2	0	3

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

Explain, with justification, how this matrix may be reduced to a $3 \times 3$ matrix.
Find the play-safe strategy for each player and verify that there is no stable solution to this game. The game is formulated as a linear programming problem for player A .
The objective is to maximise $P = V$, where $V$ is the value of the game to player A.
One of the constraints is that $p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1$, where $p _ { 1 } , p _ { 2 } , p _ { 3 }$ are the probabilities that player A plays 1, 2, 3 respectively.
Formulate the remaining constraints for this problem. Write these constraints as inequalities. The Simplex algorithm is used to solve the linear programming problem.
The solution obtained is $p _ { 1 } = 0 , p _ { 2 } = \frac { 3 } { 7 } , p _ { 3 } = \frac { 4 } { 7 }$
Calculate the value of the game to player A.

Edexcel FD2 Specimen Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a2bc4f5d-f7db-4ce7-860b-f53a743c7e2c-7_821_1433_205_317} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a capacitated, directed network. The number on each arc $( x , y )$ represents the lower $( x )$ capacity and upper $( y )$ capacity of that arc.

Calculate the value of the cut $C _ { 1 }$ and cut $C _ { 2 }$
Explain why the flow through the network must be at least 12 and at most 16
Explain why arcs DG, AG, EG and FG must all be at their lower capacities.
Determine a maximum flow pattern for this network and draw it on Diagram 1 in the answer book. You do not need to use the labelling procedure.
1. State the value of the maximum flow through the network.
2. Explain why the value of the maximum flow is equal to the value of the minimum flow through the network. Node E becomes blocked and no flow can pass through it. To maintain the maximum flow through the network the upper capacity of exactly one arc is increased.
Explain how it is possible to maintain the maximum flow found in (d).

Edexcel FD2 Specimen Q7

7. A company assembles boats. They can assemble up to five boats in any one month, but if they assemble more than three they will have to hire additional space at a cost of $\pounds 800$ per month. The company can store up to two boats at a cost of $\pounds 350$ each per month.
The overhead costs are $\pounds 1500$ in any month in which work is done.
Boats are delivered at the end of each month. There are no boats in stock at the beginning of January and there must be none in stock at the end of May. The order book for boats is

Month	January	February	March	April	May
Number ordered	3	2	6	3	4

Use dynamic programming to determine the production schedule which minimises the costs to the company. Show your working in the table provided in the answer book and state the minimum production cost.

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

\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.

Edexcel FD2 2022 June Q4

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.

List the saturated arcs.
State the value of the initial flow.
Explain why arc FT cannot be full to capacity.
State the capacity of cut $C _ { 1 }$ and the capacity of cut $C _ { 2 }$
By inspection find one flow-augmenting route to increase the flow by three units. You must state your route.
Prove that, once the flow-augmenting route found in part (e) has been applied, the flow is maximal.

Edexcel FD2 2022 June Q5

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,

state the value of $k$.
Explain precisely what the constraint $\sum x _ { i \mathrm { R } } \geqslant 44$ means in the transportation problem.
Use the north-west corner method to obtain the cost of an initial solution to this transportation problem.
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
- shadow costs
- improvement indices
- entering cell and exiting cell.

Edexcel FD2 2022 June Q6

Bernie makes garden sheds. He can build up to four sheds each month.

If he builds more than two sheds in any one month, he must hire an additional worker at a cost of $\pounds 250$ for that month. In any month in which sheds are made, the overhead costs are $\pounds 35$ for each shed made that month. A maximum of three sheds can be held in storage at the end of any one month, at a cost of $\pounds 80$ per shed per month. Sheds must be delivered at the end of the month.
The order schedule for sheds is

Month	January	February	March	April	May
Number ordered	1	3	3	5	2

There are no sheds in storage at the beginning of January and Bernie plans to have no sheds left in storage after the May delivery. Use dynamic programming to determine the production schedule that minimises the costs given above. Complete the working in the table provided in the answer book and state the minimum cost.

Edexcel FD2 2022 June Q7

\multirow{2}{*}{}		Player B
		Option W	Option X	Option Y	Option Z
\multirow{3}{*}{Player A}	Option Q	4	3	-1	-2
	Option R	-3	5	-4	$k$
	Option S	-1	6	3	-3

A two person zero-sum game is represented by the pay-off matrix for player A shown above. It is given that $k$ is an integer.

Show that Q is the play-safe option for player A regardless of the value of $k$. Given that Z is the play-safe option for player B ,
determine the range of possible values of $k$. You must make your working clear.
Explain why player B should never play option X. You must make your reasoning clear. Player A intends to make a random choice between options $\mathrm { Q } , \mathrm { R }$ and S , choosing option Q with probability $p _ { 1 }$, option R with probability $p _ { 2 }$ and option S with probability $p _ { 3 }$ Player A wants to find the optimal values of $p _ { 1 } , p _ { 2 }$ and $p _ { 3 }$ using the Simplex algorithm.
Given that $k > - 4$, player A formulates the following objective function for the corresponding linear program. $$\text { Maximise } P = V \text {, where } V = \text { the value of the original game } + 4$$
1. Formulate the constraints of the linear programming problem for player A. You should write the constraints as equations.
2. Write down an initial Simplex tableau, making your variables clear. The Simplex algorithm is used to solve the linear programming problem. It is given that in the final Simplex tableau the optimal value of $p _ { 1 } = \frac { 7 } { 37 }$, the optimal value of $p _ { 2 } = \frac { 17 } { 37 }$ and all the slack variables are zero.
Determine the value of $k$, making your method clear.

b.v.	\(V\)	\(p _ { 1 }\)	\(p _ { 2 }\)	\(p _ { 3 }\)	r	\(s\)	\(t\)	\(u\)	Value
\(p _ { 3 }\)	0	0	0	1	\(\frac { 1 } { 10 }\)	\(- \frac { 3 } { 80 }\)	\(- \frac { 1 } { 16 }\)	\(\frac { 33 } { 80 }\)	\(\frac { 33 } { 80 }\)
\(p _ { 2 }\)	0	0	1	0	\(- \frac { 1 } { 10 }\)	\(\frac { 13 } { 80 }\)	\(- \frac { 1 } { 16 }\)	\(\frac { 17 } { 80 }\)	\(\frac { 17 } { 80 }\)
V	1	0	0	0	\(\frac { 1 } { 2 }\)	\(\frac { 5 } { 16 }\)	\(\frac { 3 } { 16 }\)	\(\frac { 73 } { 16 }\)	\(\frac { 73 } { 16 }\)
\(p _ { 1 }\)	0	1	0	0	0	\(- \frac { 1 } { 8 }\)	\(\frac { 1 } { 8 }\)	\(\frac { 3 } { 8 }\)	\(\frac { 3 } { 8 }\)
\(P\)	0	0	0	0	\(\frac { 1 } { 2 }\)	\(\frac { 5 } { 16 }\)	\(\frac { 3 } { 16 }\)	\(\frac { 73 } { 16 }\)	\(\frac { 73 } { 16 }\)

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