Questions - Edexcel - A-Level Maths

Calculate the capacity of
1. cut $C _ { 1 }$
2. cut $C _ { 2 }$
Using only the capacities of cuts $C _ { 1 }$ and $C _ { 2 }$, state what can be deduced about the maximum flow through the system. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{262aa0e6-479f-447a-94db-aeb901b3c6fe-6_775_1516_169_278} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows an initial flow through the same network.
State the value of the initial flow.
By entering values along $B C , C F$ and $D T$, complete the labelling procedure on Diagram 1 in the answer book.
Use the labelling procedure to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow.
Use your answer to (e) to find a maximum flow pattern for this system of pipes and draw it on Diagram 2 in the answer book.
Prove that the answer to (f) is optimal. A vertex restriction is now applied to $B$ so that no more than 16 litres per second can flow through it.
1. Complete Diagram 3 in the answer book to show this restriction.
2. State the value of the maximum flow with this restriction.

Edexcel FD2 2021 June Q6

Standard +0.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{262aa0e6-479f-447a-94db-aeb901b3c6fe-7_782_1426_219_322} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The staged, directed network in Figure 3 represents a series of roads connecting 12 towns, $S , A , B , C , D , E , F , G , H , I , J$ and $T$. The number on each arc shows the distance between these towns, in miles. Bradley is planning a four-day cycle ride from $S$ to $T$.
He plans to leave his home at $S$. On the first night he will stay at $A , B$ or $C$, on the second night he will stay at $D , E , F$ or $G$, on the third night he will stay at $H , I$ or $J$, and he will arrive at his friend's house at $T$ on the fourth day. Bradley decides that the maximum distance he will cycle on any one day should be as small as possible.

Write down the type of dynamic programming problem that Bradley needs to solve.
Use dynamic programming to complete the table in the answer book.
Hence write down the possible routes that Bradley could take.

Edexcel FD2 2021 June Q7

Challenging +1.2

7. Alexis and Becky are playing a zero-sum game. Alexis has two options, Q and R . Becky has three options, $\mathrm { X } , \mathrm { Y }$ and Z .
Alexis intends to make a random choice between options Q and R , choosing option Q with probability $p _ { 1 }$ and option R with probability $p _ { 2 }$ Alexis wants to find the optimal values of $p _ { 1 }$ and $p _ { 2 }$ and formulates the following linear programme, writing the constraints as inequalities. $$\begin{aligned} & \text { Maximise } P = V \\ & \text { where } V = 3 + \text { the value of the gan } \\ & \text { subject to } V \leqslant 6 p _ { 1 } + p _ { 2 } \\ & \qquad \begin{aligned} & V \leqslant 8 p _ { 2 } \\ & V \leqslant 4 p _ { 1 } + 2 p _ { 2 } \\ & p _ { 1 } + p _ { 2 } \leqslant 1 \\ & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , V \geqslant 0 \end{aligned} \end{aligned}$$

Complete the pay-off matrix for Alexis in the answer book.
Use a graphical method to find the best strategy for Alexis.
Calculate the value of the game to Alexis. Becky 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 best strategy for Becky, making your method and working clear.

Edexcel FD2 2023 June Q1

Moderate -0.5

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ae354c58-6de8-4f94-b404-2f0feecb5bf3-02_953_1687_251_191} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a capacitated, directed network of pipes. The number on each arc represents the capacity of that pipe. The numbers in circles represent a feasible flow from S to T .

State the value of the feasible flow.
(1)
Find the capacity of cut $\mathrm { C } _ { 1 }$ and the capacity of cut $\mathrm { C } _ { 2 }$
(2)
By inspection, find a flow-augmenting route to increase the flow by two units. You must state your route.
(1)
Prove that, once the flow-augmenting route found in (c) has been applied, the flow is now maximal.
(3)

Edexcel FD2 2023 June Q2

Moderate -0.8

2. An outdoor theatre is holding a summer gala performance. The theatre owner must decide whether to take out insurance against rain for this performance. The theatre owner estimates that

on a fine day, the total profit will be $\pounds 15000$
on a wet day, the total loss will be $\pounds 20000$

Insurance against rain costs $\pounds 2000$. If the performance must be cancelled due to rain, then the theatre owner will receive $\pounds 16000$ from the insurer. If the performance is not cancelled due to rain, then the theatre owner will receive nothing from the insurer. The probability of rain on the day of the gala performance is 0.2
Draw a decision tree and hence determine whether the theatre owner should take out the insurance against rain for this performance.

Edexcel FD2 2023 June Q3

Standard +0.3

3. The table below shows the stock held at each supply point and the stock required at each demand point in a standard transportation problem. The table also shows the cost, in pounds, of transporting the stock from each supply point to each demand point.

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	Q	R	S	Supply
A	23	18	12	45
B	8	10	14	27
C	11	14	21	34
D	19	15	11	50
Demand	75	37	44

The problem is partially described by the linear programming formulation below.
Let $x _ { i j }$ be the number of units transported from i to j $$\begin{aligned} & \text { where } \quad i \in \{ A , B , C , D \} \\ & \quad j \in \{ Q , R , S \} \text { and } x _ { i j } \geqslant 0 \\ & \text { Minimise } P = 23 x _ { A Q } + 18 x _ { A R } + 12 x _ { A S } + 8 x _ { B Q } + 10 x _ { B R } + 14 x _ { B S } \\ & \quad + 11 x _ { C Q } + 14 x _ { C R } + 21 x _ { C S } + 19 x _ { D Q } + 15 x _ { D R } + 11 x _ { D S } \end{aligned}$$

Write down, as inequalities, the constraints of the linear program.
Use the north-west corner method to obtain an initial solution to this transportation problem.
Taking AS as the entering cell, use the stepping-stone method to find an improved solution. Make your route clear.
Perform one further 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 2023 June Q4

Standard +0.8

Four students, $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D , are to be allocated to four rounds, $1,2,3$ and 4 , in a competition. Each student is to take part in exactly one round and no two students may play in the same round.

Each student has been given an estimated score for each round. The estimated scores for each student are shown in the table below.

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	1	2	3	4
A	34	20	18	15
B	49	31	12	34
C	48	27	23	26
D	52	45	42	42

Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total estimated score. You must make your method clear and show the table after each stage.
Find this total estimated score.

Edexcel FD2 2023 June Q5

Challenging +1.2

5. A sequence $\left\{ u _ { n } \right\}$, where $\mathrm { n } \geqslant 0$, satisfies the second order recurrence relation $$u _ { n + 2 } = \frac { 1 } { 2 } \left( u _ { n + 1 } + u _ { n } \right) + 3 \text { where } u _ { 0 } = 15 \quad u _ { 1 } = 20$$

By considering the sequence $\left\{ v _ { n } \right\}$, where $u _ { n } = v _ { n } + 2 n$ for $\mathrm { n } \geqslant 0$, determine an expression for $u _ { n }$ as a function of n .
Describe the long-term behaviour of $u _ { n }$

Edexcel FD2 2023 June Q6

Standard +0.3

6. Polly is a motivational speaker who is planning her engagements for the next four weeks. Polly will

visit four different countries in these four weeks
visit just one country each week
leave from her home, S , and return there only after visiting the four countries
travel directly from one country to the next

Polly wishes to determine a schedule of four countries to visit.
Table 1 shows the countries Polly could visit each week. \begin{table}[h]

Week	1	2	3	4
Possible countries to visit	A or B	C, D or E	F or G	H, I or J

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

\end{table} Table 2 shows the speaker fee, in $\pounds 100$ s, Polly would expect to earn in each country. \begin{table}[h]

Country	A	B	C	D	E	F	G	H	I	J
Earnings in $\boldsymbol { \pounds } \mathbf { 1 0 0 s }$	47	45	48	47	49	44	45	47	49	48

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

\end{table} Table 3 shows the cost, in $\pounds 100$ s, of travelling between the countries. \begin{table}[h]

	A	B	C	D	E	F	G	H	I	J
S	5	2						7	8	8
A			3	4	5
B			5	4	6
C						7	5
D						6	7
E						7	6
F								6	7	8
G								7	8	6

\captionsetup{labelformat=empty} \caption{Table 3}

\end{table} Polly's expected income is the value of the speaker fee minus the cost of travel.
She wants to find a schedule that maximises her total expected income for the four weeks. Use dynamic programming to determine the optimal schedule. Complete the table provided in the answer book and state the maximum expected income.
(13)

Edexcel FD2 2023 June Q7

Standard +0.3

7. Martina decides to open a bank account to help her to save for a holiday. Each month she puts $\pounds \mathrm { k }$ into the account and allows herself to spend one quarter of what was in the account at the end of the previous month. Let $u _ { n }$ (where $\mathrm { n } \geqslant 1$ ) represent the amount in the account at the end of month n .
Martina has $\pounds \mathrm {~K}$ in the account at the end of the first month.

By setting up a first order recurrence relation for $u _ { n + 1 }$ in terms of $u _ { n }$, determine an expression for $u _ { n }$ in terms of n and k At the end of the 8th month, Martina needs to have at least $\pounds 1750$ in the account to pay for her holiday.
Determine, to the nearest penny, the minimum amount of money that Martina should put into the account each month.

Edexcel FD2 2023 June Q8

Challenging +1.8

8. A two-person zero-sum game is represented by the pay-off matrix for player A shown below. \section*{Player B} Player A

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Option X	Option Y	Option Z
Option Q	- 3	2	5
Option R	2	- 1	0
Option S	4	- 2	- 1
Option T	- 4	0	2

Verify that there is no stable solution to this game.
Explain why player A should never play option T. 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 calculate the optimal values of $p _ { 1 } , p _ { 2 }$ and $p _ { 3 }$ using the Simplex algorithm.
1. Formulate the game as a linear programming problem for player A. You should write the constraints as equations.
2. Write down an initial Simplex tableau for this linear programming problem, making your variables clear. The linear programming problem is solved using the Simplex algorithm. The optimal value of $p _ { 1 }$ is $\frac { 6 } { 11 }$ and the optimal value of $p _ { 2 }$ is 0
Find the best strategy for player B, defining any variables you use.

Edexcel FD2 2024 June Q1

Standard +0.3

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{931ccf1d-4b02-448c-b492-846b0f42c057-02_696_1347_214_367} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a capacitated, directed network of pipes. The numbers in circles represent an initial flow from S to T . The other number on each arc represents the capacity, in litres per second, of the corresponding pipe.

1. State the value of $x$
2. State the value of $y$
State the value of the initial flow.
State the capacity of cut $C _ { 1 }$
Find, by inspection, a flow-augmenting route to increase the flow by four units. You must state your route. The flow-augmenting route from (d) is used to increase the flow from S to T .
Prove that the flow is now maximal. A vertex restriction is now applied so that no more than 12 litres per second can flow through E.
1. Complete Diagram 1 in the answer book to show this restriction.
2. State the value of the maximum flow through the network with this restriction.

Edexcel FD2 2024 June Q2

Standard +0.8

2. The general solution of the first order recurrence relation $$u _ { n + 1 } + a u _ { n } = b n ^ { 2 } + c n + d \quad n \geqslant 0$$ is given by $$u _ { n } = A ( 3 ) ^ { n } + 5 n ^ { 2 } + 1$$ where $A$ is an arbitrary non-zero constant.
By considering expressions for $u _ { n + 1 }$ and $u _ { n }$, find the values of the constants $a , b , c$ and $d$.

Edexcel FD2 2024 June Q3

Challenging +1.2

3. The table below shows the cost, in pounds, of transporting one unit of stock from each of four supply points, $\mathrm { E } , \mathrm { F } , \mathrm { G }$ and H , to three sales points, $\mathrm { A } , \mathrm { B }$ and C . It also shows the stock held at each supply point and the amount required at each sales point.
A minimum cost solution is required.

	A	B	C	Supply
E	23	28	22	21
F	26	19	29	32
G	29	24	20	29
H	24	26	19	23
Demand	45	19	23

Explain why it is necessary to add a dummy demand point.
On Table 1 in the answer book, insert appropriate values in the dummy demand column, D. After finding an initial feasible solution and applying one iteration of the stepping-stone method, the table becomes
$A$ $B$ $C$ $D$
$E$ 21
$F$ 19 13
$G$ 6 23
$H$ 5 18
Starting with GD as the next entering cell, perform two further iterations of the stepping-stone method to obtain an improved solution. You must make your method clear by showing your routes and stating the
- shadow costs
- improvement indices
- entering and exiting cells
- State the cost of the solution found in (c).
- Determine whether the solution obtained in (c) is optimal, giving a reason for your answer.

Edexcel FD2 2024 June Q4

Standard +0.8

Four workers, A, B, C and D, are to be assigned to four tasks, P, Q, R and S.

Each task must be assigned to just one worker and each worker can do only one task.
Worker B cannot be assigned to task Q and worker D cannot be assigned to task R.
The amount, in pounds, that each worker would earn when assigned to each task is shown in the table below.

	P	Q	R	S
A	65	72	69	75
B	71	-	68	65
C	70	69	73	77
D	73	70	-	71

The Hungarian algorithm can be used to find the maximum total amount that would be earned by the four workers.

1. Explain how to modify the table so that the Hungarian algorithm could be applied.
2. Modify the table as described in (a)(i).
Formulate the above situation as a linear programming problem. You must define the decision variables and make the objective function and constraints clear.

Edexcel FD2 2024 June Q5

Standard +0.3

5. Sebastien needs to make a journey. He can choose between travelling by plane, by train or by coach. Sebastien knows the exact costs of all three travel options, but he also wants to account for his travel time, including any possible delays. The cost of Sebastien's time is $\pounds 50$ per hour.
The table below shows the costs, the journey times (without delays), and the corresponding probabilities of delays, for each travel option.

	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

Standard +0.3

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

Challenging +1.2

\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

Challenging +1.2

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

Standard +0.8

(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

Challenging +1.2

	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

Moderate -0.5

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

Moderate -0.3

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

Challenging +1.2

	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

Challenging +1.8

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).

	\(A\)	\(B\)	\(C\)	\(D\)
\(E\)	21
\(F\)	19	13
\(G\)		6	23
\(H\)	5			18

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	\(\frac { 1 } { 2 }\)	\(\frac { 5 } { 16 }\)	\(\frac { 3 } { 16 }\)	\(\frac { 73 } { 16 }\)	\(\frac { 73 } { 16 }\)

Country	A	B	C	D	E	F	G	H	I	J
Earnings in \(\boldsymbol { \pounds } \mathbf { 1 0 0 s }\)	47	45	48	47	49	44	45	47	49	48

Questions — Edexcel (9670 questions)

Browse by module

Browse by board

Edexcel FD2 2021 June Q5

Edexcel FD2 2021 June Q6

Edexcel FD2 2021 June Q7

Edexcel FD2 2023 June Q1

Edexcel FD2 2023 June Q2

Edexcel FD2 2023 June Q3

Edexcel FD2 2023 June Q4

Edexcel FD2 2023 June Q5

Edexcel FD2 2023 June Q6

Edexcel FD2 2023 June Q7

Edexcel FD2 2023 June Q8

Edexcel FD2 2024 June Q1

Edexcel FD2 2024 June Q2

Edexcel FD2 2024 June Q3

Edexcel FD2 2024 June Q4

Edexcel FD2 2024 June Q5

Edexcel FD2 2024 June Q6

Edexcel FD2 2024 June Q7

Edexcel FD2 2024 June Q8

Edexcel FD2 Specimen Q1

Edexcel FD2 Specimen Q2

Edexcel FD2 Specimen Q3

Edexcel FD2 Specimen Q4

Edexcel FD2 Specimen Q5

Edexcel FD2 Specimen Q6