Questions - Edexcel - A-Level Maths

Each paperback takes 4 minutes to print and 1 minute to bind
Each hardcover takes 8 minutes to print and 5 minutes to bind
Each deluxe edition takes 15 minutes to print and 12 minutes to bind

The printing machine is available for at most 150 hours and the binding machine must be used for at least 60 hours. The publisher decides to produce

at least 1600 books in total
at least three times as many paperbacks as hardcovers

The profit on each paperback sold is $\pounds 8$, the profit on each hardcover sold is $\pounds 20$ and the profit on each deluxe edition sold is $\pounds 40$ Let $x , y$ and $z$ represent the number of paperbacks, hardcovers and deluxe editions produced.

Formulate this as a linear programming problem, stating the objective and listing the constraints as simplified inequalities with integer coefficients. The publisher decides to solve this linear programming problem by using the two-stage simplex method.

Set up an initial tableau for solving this problem using the two-stage simplex method. As part of your solution, you must show how

the constraints have been made into equations by using slack variables, exactly two surplus variables and exactly two artificial variables
the rows for the two objective functions are formed

The following tableau is obtained after two iterations of the first stage of the two-stage simplex method.

b.v.	$x$	$y$	$z$	$\mathrm { S } _ { 1 }$	$S _ { 2 }$	$S _ { 3 }$	$s _ { 4 }$	$a _ { 1 }$	$a _ { 2 }$	Value
$\mathrm { S } _ { 1 }$	0	0	0	1	1	3	0	-1	-3	600
$z$	0	$\frac { 4 } { 11 }$	1	0	$- \frac { 1 } { 11 }$	$\frac { 1 } { 11 }$	0	$\frac { 1 } { 11 }$	$- \frac { 1 } { 11 }$	$\frac { 2000 } { 11 }$
$x$	1	$\frac { 7 } { 11 }$	0	0	$\frac { 1 } { 11 }$	$- \frac { 12 } { 11 }$	0	$- \frac { 1 } { 11 }$	$\frac { 12 } { 11 }$	$\frac { 15600 } { 11 }$
$s _ { 4 }$	0	$\frac { 40 } { 11 }$	0	0	$\frac { 1 } { 11 }$	$- \frac { 12 } { 11 }$	1	$- \frac { 1 } { 11 }$	$\frac { 12 } { 11 }$	$\frac { 15600 } { 11 }$
$P$	0	$- \frac { 4 } { 11 }$	0	0	$- \frac { 32 } { 11 }$	$- \frac { 56 } { 11 }$	0	$\frac { 32 } { 11 }$	$\frac { 56 } { 11 }$	$\frac { 204800 } { 11 }$
I	0	0	0	0	0	0	0	1	1	0

Taking the most negative number in the profit row to indicate the pivot column, perform one complete iteration of the second stage of the two-stage simplex method to obtain a new tableau. Make your method clear by stating the row operations you use. After three iterations of the second stage of the two-stage simplex method, the following tableau is obtained.

b.v.	$x$	$y$	$z$	$\mathrm { S } _ { 1 }$	$S _ { 2 }$	$S _ { 3 }$	$s _ { 4 }$	Value
$s _ { 2 }$	0	0	0	1	1	3	0	600
$z$	0	0	1	$\frac { 1 } { 10 }$	0	$\frac { 1 } { 2 }$	$- \frac { 1 } { 10 }$	100
$x$	1	0	0	$- \frac { 3 } { 40 }$	0	$- \frac { 9 } { 8 }$	$- \frac { 7 } { 40 }$	1125
$y$	0	1	0	$- \frac { 1 } { 40 }$	0	$- \frac { 3 } { 8 }$	$\frac { 11 } { 40 }$	375
$P$	0	0	0	$\frac { 29 } { 10 }$	0	$\frac { 7 } { 2 }$	$\frac { 1 } { 10 }$	20500

Given that the publisher produces the optimal number of each version of the book,

1. state the maximum profit the publisher can earn,
2. find the number of hours the binding machine will be used.
Give a reason why the publisher may not earn the profit stated in (d)(i).

Edexcel FD1 2024 June Q1

Moderate -0.8

1. $$\begin{array} { l l l l l l l l l l l } 17 & 8 & 16 & 12 & 24 & 19 & 23 & 11 & 20 & 13 & 4 \end{array}$$ The eleven numbers listed above are to be packed into bins of size $n$ where $n$ is a positive integer. When the first-fit bin packing algorithm is applied to the eleven numbers, the bins are packed as shown below. Bin 1: 17812
Bin 2: 1624
Bin 3: 19114
Bin 4: 2313
Bin 5: 20

Explain why this packing means that the value of $n$ must be 40 The original list of eleven numbers is to be sorted into descending order.
Use a quick sort to obtain the fully sorted list. You must make your pivots clear.
Apply the first-fit decreasing bin packing algorithm to the fully sorted list to pack the numbers into bins of size 40

Edexcel FD1 2024 June Q2

Moderate -0.5

2. The table below represents a network of shortest distances, in miles, to travel between nine castles, A, B, C, D, E, F, G, H and J.

	A	B	C	D	E	F	G	H	J
A	-	50	59	26	50	40	87	63	59
B	50	-	28	61	79	63	45	64	48
C	59	28	-	33	57	35	70	36	45
D	26	61	33	-	24	64	71	37	33
E	50	79	57	24	-	40	64	30	31
F	40	63	35	64	40	-	47	70	71
G	87	45	70	71	64	47	-	34	67
H	63	64	36	37	30	70	34	-	33
J	59	48	45	33	31	71	67	33	-

Use Prim's algorithm, starting at D , to find the minimum spanning tree for this network. You must clearly state the order in which you select the arcs of your tree.
State the weight of the minimum spanning tree found in part (a).
Draw the minimum spanning tree using the vertices given in Diagram 1 in the answer book. A historian needs to visit all of the castles, starting and finishing at the same castle, and wishes to minimise the total distance travelled.
Use your answer to part (b) to calculate an initial upper bound for the length of the historian's route.
1. Use the nearest neighbour algorithm, starting at D , to find an upper bound for the length of the historian's route.
2. Write down the route which gives this upper bound. Using the nearest neighbour algorithm, starting at F , an upper bound of length 352 miles was found.
State the best upper bound that can be obtained by using this information and your answers from parts (d) and (e). Give the reason for your answer.
By deleting A and all of its arcs, find a lower bound for the length of the historian's route.
(2) By deleting J and all of its arcs, a lower bound of length 274 miles was found.
State the best lower bound that can be obtained by using this information and your answer to part (g). Give the reason for your answer.

Edexcel FD1 2024 June Q3

Standard +0.8

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7f7546eb-0c1a-40da-bdf0-31e0574a9867-06_764_1136_258_466} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} [The total weight of the network is 413] Figure 1 represents a network of cycle tracks between ten towns, A, B, C, D, E, F, G, H, J and K. The number on each arc represents the length, in kilometres, of the corresponding track.

Use Dijkstra's algorithm to find the shortest path from A to J. Abi needs to travel along every track shown in Figure 1 to check that they are all in good repair. She needs to start her inspection route at town G and finish her route at either town J or town K. Abi wishes to minimise the total distance required to traverse every track.
By considering all relevant pairings of vertices, determine whether Abi should finish her inspection route at town J or town K. You must
- state which tracks she will repeat in her route
- state the total length of her route
The direct track between town B and town C and the direct track between town H and town K are now closed to all users. A second person, Tarig, is asked to check all the remaining tracks starting at G and finishing at H. Tarig wishes to minimise the total length of his inspection route.
Determine which route, Abi's or Tarig's, is shorter. You must make your working clear.

Edexcel FD1 2024 June Q4

Standard +0.3

4. (a) Explain why it is not possible to draw a graph with exactly six nodes with degrees 1, 2, 3, 4, 5 and 6 A tree, T , has exactly six nodes. The degrees of the six nodes of T are
1
2
$( 4 - x )$
$( 2 x - 5 )$
$( 4 x - 11 )$
$( 3 x - 5 )$
where $x$ is an integer.
(b) Explain how you know that T cannot be Eulerian.
(c) (i) Determine the value of $x$
(ii) Hence state whether T is semi-Eulerian or not. You must justify your answer.
(5)
\includegraphics[max width=\textwidth, alt={}, center]{7f7546eb-0c1a-40da-bdf0-31e0574a9867-07_588_579_977_744} \section*{Figure 2} Figure 2 shows a graph, $G$, with six nodes with degrees $1,2,3,3,3$ and 4
(d) Using the vertices in Diagram 1 in the answer book, draw a graph with exactly six nodes with degrees $1,2,3,3,3$ and 4 that is not isomorphic to G .

Edexcel FD1 2024 June Q5

Standard +0.3

5. Two friends, Anaira and Tommi, play a game involving two positive numbers $x$ and $y$ Anaira gives Tommi the following clues to see if he can correctly determine the value of $x$ and the value of $y$

$x$ is greater than $y$ and the difference between the two is at least 100
$x$ is at most 5 times as large as $y$
the sum of $2 x$ and $3 y$ is at least 350
the sum of $x$ and $y$ is as small as possible

Tommi decides to solve this problem by using the big-M method.

Set up an initial tableau for solving this problem using the big-M method. As part of your solution, you must show
- how the constraints were made into equations using one slack variable, exactly two surplus variables and exactly two artificial variables
- how the objective function was formed
The big-M method is applied until the tableau containing the optimal solution to the problem is found. One row of this final tableau is as follows.
b.v. $x$ $y$ $s _ { 1 }$ $S _ { 2 }$ $\mathrm { S } _ { 3 }$ $a _ { 1 }$ $a _ { 2 }$ Value
$x$ 1 0 $- \frac { 3 } { 5 }$ 0 $- \frac { 1 } { 5 }$ $\frac { 3 } { 5 }$ $\frac { 1 } { 5 }$ 130
1. State the value of $x$
2. Hence deduce the value of $y$, making your reasoning clear.

Edexcel FD1 2024 June Q6

Standard +0.8

6. The precedence table below shows the 12 activities required to complete a project.

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

Draw the activity network described in the precedence table, using activity on arc. Your activity network must contain the minimum number of dummies only.
(5) Each of the activities shown in the precedence table requires one worker. The project is to be completed in the minimum possible time. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7f7546eb-0c1a-40da-bdf0-31e0574a9867-11_303_1547_296_260} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a schedule for the project using three workers.
1. State the critical path for the network.
2. State the minimum completion time for the project.
3. Calculate the total float on activity B.
4. Calculate the total float on activity G. Immediately after the start of the project, it is found that the duration of activity I, as shown in Figure 3, is incorrect. In fact, activity I will take 8 hours.
  The durations of all the other activities remain as shown in Figure 3.
Determine whether the project can still be completed in the minimum completion time using only three workers when the duration of activity I is 8 hours. Your answer must make specific reference to workers, times and activities.

Edexcel FD1 2024 June Q7

Standard +0.3

7. A maximisation linear programming problem in $x , y$ and $z$ is to be solved using the Simplex method. The tableau after the 1st iteration is shown below.

b.v.	$x$	$y$	$z$	$s _ { 1 }$	$S _ { 2 }$	$S _ { 3 }$	Value
$s _ { 1 }$	0	$- \frac { 1 } { 2 }$	$\frac { 3 } { 2 }$	1	$- \frac { 1 } { 2 }$	0	30
$x$	1	$\frac { 1 } { 4 }$	$- \frac { 1 } { 4 }$	0	$\frac { 1 } { 4 }$	0	10
$S _ { 3 }$	0	1	1	0	0	1	26
$P$	0	$- \frac { 1 } { 4 }$	$- \frac { 11 } { 4 }$	0	$\frac { 3 } { 4 }$	0	30

State the column that contains the pivot value for the 1st iteration. You must give a reason for your answer.
By considering the equations represented in the above tableau, formulate the linear programming problem in $x , y$ and $z$ only. State the objective and list the constraints as inequalities with integer coefficients.
Taking the most negative number in the profit row to indicate the pivot column, perform the 2nd iteration of the Simplex algorithm, to obtain a new tableau, T . Make your method clear by stating the row operations you use.

Explain, using T, how you know that an optimal solution to the original linear programming problem has not been found after the 2nd iteration.

State the values of the basic variables after the 2nd iteration. A student attempts the 3rd iteration of the Simplex algorithm and obtains the tableau below.

b.v.	$x$	$y$	$z$	$s _ { 1 }$	$S _ { 2 }$	$\mathrm { S } _ { 3 }$	Value
z	0	0	1	$\frac { 1 } { 2 }$	$- \frac { 1 } { 4 }$	$\frac { 1 } { 4 }$	$\frac { 43 } { 2 }$
$x$	1	0	0	$\frac { 1 } { 4 }$	$\frac { 1 } { 8 }$	$- \frac { 1 } { 8 }$	$\frac { 57 } { 4 }$
$y$	0	1	0	$- \frac { 1 } { 2 }$	$\frac { 1 } { 4 }$	$\frac { 3 } { 4 }$	$\frac { 9 } { 2 }$
$P$	0	1	0	$\frac { 5 } { 4 }$	$\frac { 1 } { 8 }$	$\frac { 7 } { 8 }$	$\frac { 361 } { 4 }$

Explain how you know that the student's attempt at the 3rd iteration is not correct.

Edexcel FD1 Specimen Q1

Easy -1.2

A list of $n$ numbers needs to be sorted into descending order starting at the left-hand end of the list.
1. Describe how to carry out the first pass of a bubble sort on the numbers in the list.
Bubble sort is a quadratic order algorithm.
A computer takes approximately 0.021 seconds to apply a bubble sort to a list of 2000 numbers.
Estimate the time it would take the computer to apply a bubble sort to a list of 50000 numbers. Make your method clear.

Edexcel FD1 Specimen Q2

Standard +0.8

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{37435cc9-1e38-4c55-bd72-e2a1ec415ba7-03_570_663_175_701} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure}

Define what is meant by a planar graph.
Starting at A, find a Hamiltonian cycle for the graph in Figure 1. Arc AG is added to Figure 1 to create the graph shown in Figure 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{37435cc9-1e38-4c55-bd72-e2a1ec415ba7-03_568_666_1226_701} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Taking ABCDEFGA as the Hamiltonian cycle,
use the planarity algorithm to determine whether the graph shown in Figure 2 is planar. You must make your working clear and justify your answer.

Edexcel FD1 Specimen Q3

Standard +0.3

3. (a) Explain clearly the difference between the classical travelling salesperson problem and the practical travelling salesperson problem.

	A	B	C	D	E	F	G
A	-	17	24	16	21	18	41
B	17	-	35	25	30	31	$x$
C	24	35	-	28	20	35	32
D	16	25	28	-	29	19	45
E	21	30	20	29	-	22	35
F	18	31	35	19	22	-	37
G	41	$x$	32	45	35	37	-

The table shows the least distances, in km, by road between seven towns, A, B, C, D, E, F and G . The least distance between B and G is $x \mathrm {~km}$, where $x > 25$ Preety needs to visit each town at least once, starting and finishing at A. She wishes to minimise the total distance she travels.
(b) Starting by deleting B and all of its arcs, find a lower bound for Preety's route. Preety found the nearest neighbour routes from each of A and C . Given that the sum of the lengths of these routes is 331 km ,
(c) find $x$, making your method clear.
(d) Write down the smallest interval that you can be confident contains the optimal length of Preety's route. Give your answer as an inequality.

Edexcel FD1 Specimen Q4

Moderate -0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{37435cc9-1e38-4c55-bd72-e2a1ec415ba7-05_572_799_228_632} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The network in Figure 3 shows the roads linking a depot, D, and three collection points $\mathrm { A } , \mathrm { B }$ and C . The number on each arc represents the length, in miles, of the corresponding road. The road from B to D is a one-way road, as indicated by the arrow.

Explain clearly if Dijkstra's algorithm can be used to find a route from D to A . The initial distance and route tables for the network are given in the answer book.
Use Floyd's algorithm to find a table of least distances. You should show both the distance table and the route table after each iteration.
Explain how the final route table can be used to find the shortest route from D to B . State this route. There are items to collect at $\mathrm { A } , \mathrm { B }$ and C . A van will leave D to make these collections in any order and then return to D. A minimum route is required. Using the final distance table and the Nearest Neighbour algorithm starting at D,
find a minimum route and state its length. Floyd's algorithm and Dijkstra's algorithm are applied to a network. Each will find the shortest distance between vertices of the network.
Describe how the results of these algorithms differ.

Edexcel FD1 Specimen Q5

Standard +0.3

5. A garden centre makes hanging baskets to sell to its customers. Three types of hanging basket are made, Sunshine, Drama and Peaceful. The plants used are categorised as Impact, Flowering or Trailing. Each Sunshine basket contains 2 Impact plants, 4 Flowering plants and 3 Trailing plants. Each Drama basket contains 3 Impact plants, 2 Flowering plants and 4 Trailing plants. Each Peaceful basket contains 1 Impact plant, 3 Flowering plants and 2 Trailing plants. The garden centre can use at most 80 Impact plants, at most 140 Flowering plants and at most 96 Trailing plants each day. The profit on Sunshine, Drama and Peaceful baskets are $\pounds 12 , \pounds 20$ and $\pounds 16$ respectively. The garden centre wishes to maximise its profit. Let $x , y$ and $z$ be the number of Sunshine, Drama and Peaceful baskets respectively, produced each day.

Formulate this situation as a linear programming problem, giving your constraints as inequalities.

State the further restriction that applies to the values of $x , y$ and $z$ in this context. The Simplex algorithm is used to solve this problem. After one iteration, the tableau is

b.v.	$x$	$y$	$z$	$r$	$s$	$t$	Value
$r$	$- \frac { 1 } { 4 }$	0	$- \frac { 1 } { 2 }$	1	0	$- \frac { 3 } { 4 }$	8
$s$	$\frac { 5 } { 2 }$	0	2	0	1	$- \frac { 1 } { 2 }$	92
$y$	$\frac { 3 } { 4 }$	1	$\frac { 1 } { 2 }$	0	0	$\frac { 1 } { 4 }$	24
$P$	3	0	-6	0	0	5	480

State the variable that was increased in the first iteration. Justify your answer.
Determine how many plants in total are being used after only one iteration of the Simplex algorithm.

Explain why for a second iteration of the Simplex algorithm the 2 in the $z$ column is the pivot value. After a second iteration, the tableau is

b.v.	$x$	$y$	$z$	$r$	$s$	$t$	Value
$r$	$\frac { 3 } { 8 }$	0	0	1	$\frac { 1 } { 4 }$	$- \frac { 7 } { 8 }$	31
$s$	$\frac { 5 } { 4 }$	0	1	0	$\frac { 1 } { 2 }$	$- \frac { 1 } { 4 }$	46
$y$	$\frac { 1 } { 8 }$	1	0	0	$- \frac { 1 } { 4 }$	$\frac { 3 } { 8 }$	1
$P$	$\frac { 21 } { 2 }$	0	0	0	3	$\frac { 7 } { 2 }$	756

Use algebra to explain why this tableau is optimal.
State the optimal number of each type of basket that should be made. The manager of the garden centre is able to increase the number of Impact plants available each day from 80 to 100 . She wants to know if this would increase her profit.
Use your final tableau to determine the effect of this increase. (You should not carry out any further calculations.)

Edexcel FD1 Specimen Q6

Moderate -0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{37435cc9-1e38-4c55-bd72-e2a1ec415ba7-08_1113_1319_169_374} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} A project is modelled by the activity network shown in Figure 4. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete that activity. Each activity requires one worker. The project is to be completed in the shortest possible time.

Calculate the early time and the late time for each event, using Diagram 1 in the answer book.
On Grid 1 in the answer book, complete the cascade (Gantt) chart for this project.
On Grid 2 in the answer book, draw a resource histogram to show the number of workers required each day when each activity begins at its earliest time. The supervisor of the project states that only three workers are required to complete the project in the minimum time.
Use Grid 2 to determine if the project can be completed in the minimum time by only three workers. Give reasons for your answer.

Edexcel FD1 Specimen Q7

Standard +0.8

7. A linear programming problem in $x , y$ and $z$ is described as follows. Maximise $P = 3 x + 2 y + 2 z$
subject to $$\begin{aligned} & 2 x + 2 y + z \leqslant 25 \\ & x + 4 y \leqslant 15 \\ & x \geqslant 3 \end{aligned}$$

Explain why the Simplex algorithm cannot be used to solve this linear programming problem. The big-M method is to be used to solve this linear programming problem.
Define, in this context, what M represents. You must use correct mathematical language in your answer. The initial tableau for a big-M solution to the problem is shown below.
b.v. $x$ $y$ $z$ $s _ { 1 }$ $s _ { 2 }$ $s _ { 3 }$ $t _ { 1 }$ Value
$s _ { 1 }$ 2 2 1 1 0 0 0 25
$s _ { 2 }$ 1 4 0 0 1 0 0 15
$t _ { 1 }$ 1 0 0 0 0 -1 1 3
$P$ $- ( 3 + M )$ -2 -2 0 0 M 0 $- 3 M$
Explain clearly how the equation represented in the b.v. $t _ { 1 }$ row was derived.
Show how the equation represented in the b.v. $P$ row was derived. The tableau obtained from the first iteration of the big-M method is shown below.
b.v. $x$ $y$ $z$ $s _ { 1 }$ $s _ { 2 }$ $s _ { 3 }$ $t _ { 1 }$ Value
$s _ { 1 }$ 0 2 1 1 0 2 -2 19
$s _ { 2 }$ 0 4 0 0 1 1 -1 12
$x$ 1 0 0 0 0 -1 1 3
P 0 -2 -2 0 0 -3 $3 +$ M 9
Solve the linear programming problem, starting from this second tableau. You must
- give a detailed explanation of your method by clearly stating the row operations you use and
- state the solution by deducing the final values of $P , x , y$ and $z$.

Edexcel FD2 2019 June Q1

Challenging +1.2

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 each of four demand points, $\mathrm { P } , \mathrm { Q } , \mathrm { R }$ and S . It also shows the stock held at each supply point and the stock required at each demand point. A minimum cost solution is required.

\begin{table}[h]

	P	Q	R	S	Supply
A	15	14	17	11	23
B	10	9	16	12	42
C	11	13	8	10	18
D	15	13	16	17	19
Demand	25	45	12	20

\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	S
A	23
B	2	40
C		5	12	1
D				19

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

\end{table}

Taking DQ 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 the
- shadow costs
- improvement indices
- route
- entering cell and exiting cell.
- Determine whether the solution obtained from this second iteration is optimal, giving a reason for your answer.
- State the cost of the solution found in (b).

Edexcel FD2 2019 June Q2

Standard +0.8

Four workers, Ted (T), Harold (H), James (J) and Margaret (M), are to be assigned to four tasks, 1, 2, 3 and 4. Each worker must be assigned to just one task and each task must be done by just one worker.

The profit, in pounds, resulting from allocating each worker to each task, is shown in the table below. The profit is to be maximised.

	1	2	3	4
T	103	97	74	80
H	201	155	145	155
J	111	80	77	92
M	203	188	137	184

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 2019 June Q3

Challenging +1.2

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5274a614-7862-49f0-ad1d-b59b73aa51ad-04_1047_1691_210_187} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} In Figure 1 the weight of $\operatorname { arc } \mathrm { SB }$ is denoted by $x$ where $x \geqslant 0$

Explain why Dijkstra's algorithm cannot be used on the directed network in Figure 1.
(1) It is given that the minimum weight route from S to T passes through B .
Use dynamic programming to find
1. the range of possible values of $x$
2. the minimum weight route from S to T .
  (12)

Edexcel FD2 2019 June Q4

Challenging +1.2

\multirow{2}{*}{}		Player B
		Option X	Option Y	Option Z
\multirow{3}{*}{Player A}	Option P	3	-2	0
	Option Q	-4	4	-2
	Option R	1	2	-1

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 $\mathrm { P } , \mathrm { Q }$ and R , choosing option P with probability $p _ { 1 }$, option Q with probability $p _ { 2 }$ and option R 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 programming problem for the game, writing the constraints as inequalities. $$\begin{aligned} & \text { Maximise } P = V \\ & \text { subject to } V \geqslant 3 p _ { 1 } - 4 p _ { 2 } + p _ { 3 } \\ & \\ & V \geqslant - 2 p _ { 1 } + 4 p _ { 2 } + 2 p _ { 3 } \\ & V \geqslant - 2 p _ { 2 } - p _ { 3 } \\ & p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 \\ & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , V \geqslant 0 \end{aligned}$$
Correct the errors made by player A in the linear programming formulation of the game, giving reasons for your answer.
Write down an initial Simplex tableau for the corrected linear programming problem. The Simplex algorithm is used to solve the corrected linear programming problem. The optimal values are $p _ { 1 } = 0.6 , p _ { 2 } = 0$ and $p _ { 3 } = 0.4$
Calculate the value of the game to player A.
Determine the optimal strategy for player B, making your reasoning clear.

Edexcel FD2 2019 June Q5

Challenging +1.2

5. An increasing sequence $\left\{ u _ { n } \right\}$ for $n \in \mathbb { N }$ is such that the difference between the $n$th term of $\left\{ u _ { n } \right\}$ and the mean of the previous two terms of $\left\{ u _ { n } \right\}$ is always 6

Show that, for $n \geqslant 3$ $$2 u _ { n } - u _ { n - 1 } - u _ { n - 2 } = 12$$ Given that $u _ { 1 } = 2$ and $u _ { 2 } = 8$
find the solution of this second order recurrence relation to obtain an expression for $u _ { n }$ in terms of $n$.
Show that as $n \rightarrow \infty , u _ { n } \rightarrow k n$ where $k$ is a constant to be determined. You must give reasons for your answer.

Edexcel FD2 2019 June Q6

Challenging +1.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5274a614-7862-49f0-ad1d-b59b73aa51ad-07_983_1513_210_283} \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 flows from a source, S , to a sink, T . The numbers ( $l , u$ ) on each arc represent, in litres per second, the lower capacity, $l$, and the upper capacity, $u$, of the corresponding pipe. Two cuts $C _ { 1 }$ and $C _ { 2 }$ are shown.

Find the capacity of
1. $\operatorname { cut } C _ { 1 }$
2. $\operatorname { cut } C _ { 2 }$
Explain why the arcs AE and CE cannot be at their upper capacities simultaneously.
Explain why a flow of 31 litres per second through the system is not possible.
Hence determine a minimum feasible flow and a maximum feasible flow through the system. You must draw these feasible flows on the diagrams in the answer book and give reasons to justify your answer. You should not apply the labelling procedure to find these flows.

Edexcel FD2 2021 June Q1

Moderate -0.3

Four workers, A, B, C and D, are to be assigned to three tasks, 1, 2 and 3 . Each task must be assigned to just one worker and each worker can do one task only.

Worker A cannot do task 2 and worker D cannot do task 3
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
A	53	-	62
B	48	57	59
C	55	63	58
D	69	49	-

Formulate the above situation as a linear programming problem. You must define your decision variables and make the objective function and constraints clear.
(6) \section*{(Total for Question 1 is 6 marks)}

Edexcel FD2 2021 June Q2

Moderate -0.3

Alka is considering paying $\pounds 5$ to play a game. The game involves rolling two fair six-sided dice. If the sum of the numbers on the two dice is at least 8 , she receives $\pounds 10$, otherwise she loses and receives nothing.

If Alka loses, she can pay a further $\pounds 5$ to roll the dice again. If both dice show the same number then she receives $\pounds 35$, otherwise she loses and receives nothing.

Draw a decision tree to model Alka’s possible decisions and the possible outcomes.
Determine Alka’s optimal EMV and state the optimal strategy indicated by the decision tree.

Edexcel FD2 2021 June Q3

Standard +0.8

3. The table below 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 four sales points, $\mathrm { P } , \mathrm { Q } , \mathrm { R }$ and S . 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.

	P	Q	R	S	Supply
A	18	19	17	13	28
B	16	15	14	19	43
C	21	17	22	23	29
D	16	20	19	21	36
Demand	25	41	40	30

Use the north-west corner method to obtain an initial solution.
Taking AS 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 showing the route and stating the
- shadow costs
- improvement indices
- entering cell and exiting cell
- 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 2021 June Q4

Challenging +1.2

Sequences $\left\{ x _ { n } \right\}$ and $\left\{ y _ { n } \right\}$ for $n \in \mathbb { N }$, are defined by

$$\begin{gathered} x _ { n + 1 } = 2 y _ { n } + 3 \quad \text { and } \quad y _ { n + 1 } = 3 x _ { n + 1 } - 4 x _ { n } \\ x _ { 1 } = 1 \quad \text { and } \quad y _ { 1 } = a \end{gathered}$$ where $a$ is a constant.

Show that $x _ { n + 2 } - 6 x _ { n + 1 } + 8 x _ { n } = 3$
Solve the second-order recurrence relation given in (a) to obtain an expression for $x _ { n }$ in terms of $a$ and $n$. Given that $x _ { 7 } = 28225$
find the value of $a$.

b.v.	\(x\)	\(y\)	\(z\)	\(\mathrm { S } _ { 1 }\)	\(S _ { 2 }\)	\(S _ { 3 }\)	\(s _ { 4 }\)	\(a _ { 1 }\)	\(a _ { 2 }\)	Value
\(\mathrm { S } _ { 1 }\)	0	0	0	1	1	3	0	-1	-3	600
\(z\)	0	\(\frac { 4 } { 11 }\)	1	0	\(- \frac { 1 } { 11 }\)	\(\frac { 1 } { 11 }\)	0	\(\frac { 1 } { 11 }\)	\(- \frac { 1 } { 11 }\)	\(\frac { 2000 } { 11 }\)
\(x\)	1	\(\frac { 7 } { 11 }\)	0	0	\(\frac { 1 } { 11 }\)	\(- \frac { 12 } { 11 }\)	0	\(- \frac { 1 } { 11 }\)	\(\frac { 12 } { 11 }\)	\(\frac { 15600 } { 11 }\)
\(s _ { 4 }\)	0	\(\frac { 40 } { 11 }\)	0	0	\(\frac { 1 } { 11 }\)	\(- \frac { 12 } { 11 }\)	1	\(- \frac { 1 } { 11 }\)	\(\frac { 12 } { 11 }\)	\(\frac { 15600 } { 11 }\)
\(P\)	0	\(- \frac { 4 } { 11 }\)	0	0	\(- \frac { 32 } { 11 }\)	\(- \frac { 56 } { 11 }\)	0	\(\frac { 32 } { 11 }\)	\(\frac { 56 } { 11 }\)	\(\frac { 204800 } { 11 }\)
I	0	0	0	0	0	0	0	1	1	0

b.v.	\(x\)	\(y\)	\(z\)	\(\mathrm { S } _ { 1 }\)	\(S _ { 2 }\)	\(S _ { 3 }\)	\(s _ { 4 }\)	Value
\(s _ { 2 }\)	0	0	0	1	1	3	0	600
\(z\)	0	0	1	\(\frac { 1 } { 10 }\)	0	\(\frac { 1 } { 2 }\)	\(- \frac { 1 } { 10 }\)	100
\(x\)	1	0	0	\(- \frac { 3 } { 40 }\)	0	\(- \frac { 9 } { 8 }\)	\(- \frac { 7 } { 40 }\)	1125
\(y\)	0	1	0	\(- \frac { 1 } { 40 }\)	0	\(- \frac { 3 } { 8 }\)	\(\frac { 11 } { 40 }\)	375
\(P\)	0	0	0	\(\frac { 29 } { 10 }\)	0	\(\frac { 7 } { 2 }\)	\(\frac { 1 } { 10 }\)	20500

b.v.	\(x\)	\(y\)	\(s _ { 1 }\)	\(S _ { 2 }\)	\(\mathrm { S } _ { 3 }\)	\(a _ { 1 }\)	\(a _ { 2 }\)	Value
\(x\)	1	0	\(- \frac { 3 } { 5 }\)	0	\(- \frac { 1 } { 5 }\)	\(\frac { 3 } { 5 }\)	\(\frac { 1 } { 5 }\)	130

b.v.	\(x\)	\(y\)	\(z\)	\(s _ { 1 }\)	\(S _ { 2 }\)	\(S _ { 3 }\)	Value
\(s _ { 1 }\)	0	\(- \frac { 1 } { 2 }\)	\(\frac { 3 } { 2 }\)	1	\(- \frac { 1 } { 2 }\)	0	30
\(x\)	1	\(\frac { 1 } { 4 }\)	\(- \frac { 1 } { 4 }\)	0	\(\frac { 1 } { 4 }\)	0	10
\(S _ { 3 }\)	0	1	1	0	0	1	26
\(P\)	0	\(- \frac { 1 } { 4 }\)	\(- \frac { 11 } { 4 }\)	0	\(\frac { 3 } { 4 }\)	0	30

b.v.	\(x\)	\(y\)	\(z\)	\(s _ { 1 }\)	\(S _ { 2 }\)	\(\mathrm { S } _ { 3 }\)	Value
z	0	0	1	\(\frac { 1 } { 2 }\)	\(- \frac { 1 } { 4 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 43 } { 2 }\)
\(x\)	1	0	0	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 8 }\)	\(- \frac { 1 } { 8 }\)	\(\frac { 57 } { 4 }\)
\(y\)	0	1	0	\(- \frac { 1 } { 2 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 3 } { 4 }\)	\(\frac { 9 } { 2 }\)
\(P\)	0	1	0	\(\frac { 5 } { 4 }\)	\(\frac { 1 } { 8 }\)	\(\frac { 7 } { 8 }\)	\(\frac { 361 } { 4 }\)

	A	B	C	D	E	F	G	H	J
A	-	50	59	26	50	40	87	63	59
B	50	-	28	61	79	63	45	64	48
C	59	28	-	33	57	35	70	36	45
D	26	61	33	-	24	64	71	37	33
E	50	79	57	24	-	40	64	30	31
F	40	63	35	64	40	-	47	70	71
G	87	45	70	71	64	47	-	34	67
H	63	64	36	37	30	70	34	-	33
J	59	48	45	33	31	71	67	33	-

	A	B	C	D	E	F	G	H	J
A	-	50	59	26	50	40	87	63	59
B	50	-	28	61	79	63	45	64	48
C	59	28	-	33	57	35	70	36	45
D	26	61	33	-	24	64	71	37	33
E	50	79	57	24	-	40	64	30	31
F	40	63	35	64	40	-	47	70	71
G	87	45	70	71	64	47	-	34	67
H	63	64	36	37	30	70	34	-	33
J	59	48	45	33	31	71	67	33	-

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	A	B	C	D	E	F	G	H	J
A	-	50	59	26	50	40	87	63	59
B	50	-	28	61	79	63	45	64	48
C	59	28	-	33	57	35	70	36	45
D	26	61	33	-	24	64	71	37	33
E	50	79	57	24	-	40	64	30	31
F	40	63	35	64	40	-	47	70	71
G	87	45	70	71	64	47	-	34	67
H	63	64	36	37	30	70	34	-	33
J	59	48	45	33	31	71	67	33	-