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Teaching guidanceAS and A-level Further Maths(7366, 7367)Discrete

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Version 1.0, August 2017

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AS and A-LEVEL FURTHER MATHEMATICS TEACHING GUIDANCE

Contents

General information - disclaimer 5

Subject content 5

Graphs

DA 6 1

Networks

DB 21 1

Network flows

DC 36 1

Linear programming

DD 67 1

Critical path analysis

DE 77 1

Game theory for zero-sum games

DF 102 1

Binary operations and group theory

DG 114 1

1

1

Appendix 1 Mathematical notation for AS and A-level qualifications in Mathematics and Further Mathematics

A1 133 1

Appendix 2 Mathematical formulae and identities

A2 142 1


General information - disclaimer

This AS and A-level Further Mathematics teaching guidance will help you plan your teaching by further

explaining how we have interpreted content of the specification and providing examples of how the content of the specification may be assessed. The teaching guidance notes do not always cover the whole content statement.

The examples included in this guidance have been chosen to illustrate the level at which this content will be assessed. The wording and format used in this guidance do not always represent how questions would appear in a question paper. Not all questions in this guidance have been through the same rigorous checking process as the ones used in our question papers.

Several questions have been taken from legacy specifications and therefore represent higher levels of AO1 than will be found in a suite of exam papers for these AS and A-level Further Mathematics specifications.

This guidance is not intended to restrict what can be assessed in the question papers based on the specification. Questions will be set in a variety of formats including both familiar and unfamiliar contexts.

All knowledge from the GCSE Mathematics specification is assumed.

Subject content

This Teaching guidance is designed to illustrate the detail within the content of the AS and A-level

Further Mathematics specification.

Half the subject content was set out the Department for Education (DfE). The remaining half was defined by AQA, based on feedback from Higher Education and teachers.

Content in bold type is contained within the AS Further Mathematics qualification as well as the A-level Further Mathematics qualification.

Content in standard type is contained only within the A-level Further Mathematics qualification.

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DA Graphs

DA1 Understand and use the language of graphs, including vertex, edge, trail, cycle, connected, degree, subgraph, subdivision, multiple edge and loop.

Assessed at AS and A-level

Teaching guidance

Students should be able to:

• understand that graphs are composed of vertices and edges, and that each end of an edge must terminate at a vertex

• identify trails and cycles within graphs, where a trail is a walk between vertices that does not repeat any edges, and a cycle is a trail that starts and ends at the same vertex

• recognise when a graph is connected and when a graph is not connected

• identify the degree of a vertex, including multiple edges and loops

• recognise subgraphs of a graph

• recognise subdivisions of a graph.


Examples 1 The graph G is shown in Figure 1.

Figure 1

Which of the following is a subgraph of G?

Circle your answer.

Notes: A subgraph G´ of a graph G is such that the vertex set of G´ is a subset of the vertex set

of G and the edge set of G´ is a subset of the edge set of G.

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2 Figure 2 shows a labelled graph.

Figure 2

For the graph in Figure 2, write down:

(a) a trail of 8 edges starting at A

(b) a cycle starting at C.

3 The graph G is shown below.

Draw a graph with the following properties:

• it is not connected

• it is a subgraph of G

• it has a vertex of degree 4.


4 In the graph

The degree of vertex V is

3 4 5 6

Circle your answer.

5 The graph G is shown in Figure 3.

Figure 3

The graph H is shown in Figure 4.

Figure 4

The graph H is a

cycle of

graph G subdivision of graph G

subgraph of graph G

trail of graph G

Circle your answer.

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DA2 Identify or prove properties of a graph including that a graph is Eulerian, semi-Eulerian or Hamiltonian.


Teaching guidance


• identify graphs as being Eulerian by noting that Eulerian graphs are connected and the degree of each vertex is even

• identify graphs as semi-Eulerian by noting that semi-Eulerian graphs are connected and the graph has exactly two vertices of odd-degree

• identify graphs as Hamiltonian by using a given result (see example 3 below) or by finding a (Hamiltonian) cycle that does not return to any previously visited vertices (except for the starting vertex).

Examples 1 Draw a graph with the following properties:

• it has five vertices

• it contains a vertex that has a degree of four

• it has six edges

• it is Eulerian.

2 The labelled graph G is shown in Figure 1.

Figure 1

(a) Prove that the graph G is Hamiltonian.

(b) Prove that the graph G is semi-Eulerian.

(c) The graph G is a subgraph of H, which is an Eulerian graph.

Draw a possible layout for the graph H.


3 Ore’s theorem states that if “the sum of the degrees of each pair of non-adjacent vertices in the

graph is greater than or equal to the number of vertices in the graph, then the graph is Hamiltonian”.

The graph R is shown in Figure 2. Figure 2

Use Ore’s theorem to prove that the graph R is Hamiltonian.

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DA3 Understand and use Euler’s formula for connected planar graphs.


Teaching guidance

Students should:

• know and be able to use Euler’s formula, v e f− + = 2 where v is the number of vertices in the graph, e is the number of edges in the graph and f is the number of faces in the graph (including the infinite face), for connected planar graphs that are drawn without any edge intersections

• recognise graphs as being planar by redrawing edges so that no edges intersect.

Examples

1 Consider the following graph.

(a) By redrawing the graph, show that the graph is planar.

(b) Using part (a), verify Euler’s formula for connected planar graphs.

2 A connected planar graph has 2x + 1 vertices, 5x – 1 edges and x2 faces, where x is an integer.

(a) Determine the value of x.

(b) Hence, write down the number of vertices, edges and faces of the graph.

3 Euler’s formula is not applicable to which of the following graphs?

Circle your answer.


DA4 Use Kuratowski’s Theorem to determine the planarity of graphs.

Only assessed at A-level

Teaching guidance


• use Kuratowski’s theorem to prove a graph is planar

• use Kuratowski’s theorem to prove a graph is not planar

• identify the complete graph K5

• identify the bipartite graph K3,3

• understand what is meant by subgraph

• understand what is meant by subdivision.

Examples

1 The graph G is shown in Figure 1.

Figure 1

Determine whether or not the graph G is planar.

Fully justify your answer.

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2 The graph D is shown in Figure 2.

Figure 2

Determine whether or not the graph D is planar.


3 The graph B is shown in Figure 3.

Figure 3

(a) Prove that the graph B is planar.

(b) Verify Euler’s formula for the graph B.


DA5 Understand and use complete graphs and bipartite graphs, including adjacency matrices and the complement of a graph.


Teaching guidance


• understand and use the notation Kn for complete graphs with n vertices

• understand and use the notation Km,n for complete bipartite graphs with two sets of vertices, one set having m vertices and the other set having n vertices

• use adjacency matrices, ie draw a graph given its adjacency matrix and construct an adjacency matrix given a graph, including multiple edges and loops

• recognise and draw the complement of a graph for graphs with no loops.

Examples

1 (a) Draw the complete graph K4.

(b) Hence, verify Euler’s formula for connected planar graphs using the complete graph K4.

2 The graph G, which has five labelled vertices A, B, C, D and E, is given by the following

adjacency matrix.

A B C D E

A 0 2 1 0 1

B 2 0 0 1 0

C 1 0 2 0 1

D 0 1 0 0 1

E 1 0 1 1 0

(a) Using the adjacency matrix, determine whether or not the graph G is semi-Eulerian.


(b) Draw the graph of G.

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3 A bipartite graph B has the following adjacency matrix.

V W X Y Z

A 0 1 1 0 1

B 2 0 0 1 0

C 0 1 0 0 1

D 0 1 0 0 1

E 1 0 0 1 0

Draw the bipartite graph B.

4 The graph H is shown in Figure 1.

Figure 1

Draw the complement of the graph H.

5 (a) For the complete bipartite graph Km,n, find, in terms of m and n:

(ii) the number of vertices

(ii) the number of edges.

(b) State the condition(s) for m and n for the complete bipartite graph Km,n to be planar.


DA6 Understand and use simple graphs, simple-connected graphs and trees.


Teaching guidance


• understand that simple graphs are graphs that contain no loops or multiple edges

• use the properties of simple graphs to solve graph theory problems

• understand that simple-connected graphs are graphs that are connected and contain no loops or multiple edges

• use the properties of simple-connected graphs to solve graph theory problems

• understand that trees are graphs that are connected and contain no cycles

• use the properties of trees to solve graph theory problems.

Examples

1 (a) A simple-connected graph X has eight vertices.

(i) State the minimum number of edges of the graph.

(ii) Find the maximum number of edges of the graph.

(b) A simple-connected graph Y has n vertices.

(i) State the minimum number of edges of the graph.

(ii) Find the maximum number of edges of the graph.

(c) A simple graph Z has six vertices and each of the vertices has the same degree d.

(i) State the possible values of d.

(ii) If Z is connected, state the possible values of d.

(iii) If Z is Eulerian, state the possible values of d.

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2 The graph T is shown in Figure 1.

Figure 1

Which row correctly identifies the properties of the graph T?

Tree? Eulerian? Simple?

A Yes Yes No

B Yes No Yes

C No No No

D No Yes Yes

3 Draw a graph with the following properties:

• it has five vertices

• it contains a vertex which has degree four

• it has seven edges

• it is simple.

4 The graph G, which has five labelled vertices A, B, C, D and E is given by the following

adjacency matrix where x and y are non-negative integers.

A B C D E

A 0 x 1 0 1

B x 0 1 0 0

C 1 1 y 0 0

D 0 0 0 0 1

E 1 0 0 1 0

Determine the conditions on x and y for the graph G to be:

(a) a tree

(b) simple

(c) semi-Eulerian.


DA7 Recognise and find isomorphism between graphs.


Teaching guidance


• recognise two graphs as being isomorphic if there is a one-to-one relationship between the vertices of the two graphs based on, for instance, vertex degrees and adjacency matrices

• state the isomorphism between two isomorphic graphs using vertex labels

• identify a graph that is isomorphic to a given graph.

Examples 1 The graph G is shown in Figure 1.

Figure 1

Which of the following graphs is isomorphic to the graph G?

Circle your answer.

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2 The labelled graphs G and H are shown in Figure 1 and Figure 2 respectively.

Figure 1

Figure 2

Determine whether or not the graphs G and H are isomorphic.


3 The graph R is shown in Figure 3.

Figure 3

Which of the following adjacency matrices represents a graph that is isomorphic to the graph R?

Circle your answer.

2 1 0 0 1 0

1 0 1 0 0 0

0 1 0 1 1 0

0 0 1 0 1 1

1 0 1 1 0 1

0 0 0 1 1 0

0 0 1 0 1 0

0 0 1 1 0 0

1 1 0 1 0 1

0 1 1 0 1 0

1 0 0 1 0 0

0 0 1 0 0 0

0 2 1 1 0 1

2 0 1 1 1 1

1 1 0 0 0 0

1 1 0 0 1 1

0 1 0 1 0 0

1 1 0 1 0 0

0 1 0 1 1 0

1 0 1 0 1 1

0 1 0 0 0 1

1 0 0 0 1 0

1 1 0 1 0 1

0 1 1 0 1 0


DB Networks

DB1 Understand and use the language of networks including: node, arc and weight.


Teaching guidance


• understand that networks are composed of nodes and arcs, that each end of an arc must terminate at a node, and that each arc has a weight

• identify and use the degree of a node

• understand that arcs may be directed (shown by an arrow) or may be undirected (shown by the absence of an arrow)

• understand and use the weight of an arc, which is shown by a number or symbol on the arc

• construct networks using nodes, arcs and weights from, for instance, a table of weights.

Notes

We distinguish between a graph and a network: a network is a graph with the addition of a weight assigned to each arc and the terms arc and node are used instead of edge and vertex. Students will not be penalised for interchanging arc/edge or node/vertex in examinations.

Examples 1 Which of the following is a network?

Circle your answer

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2 A network is given by the table

From

To A B C D

A – 4 5 2

B 4 – 6 3

C 3 6 – 7

D 2 1 7 –

where A, B, C and D are nodes.

Draw the network.

3 A network N is given by the table

From

To A B C D

A – 4 5 2

B 4 – 6 3

C 3 6 – 7

D 2 1 7 –

where A, B, C and D are nodes.

Which of the following is the network N?


DB2 Solve network optimisation problems using spanning trees.


Teaching guidance


• construct networks using nodes, arcs and weights in order to solve optimisation problems using spanning trees

• use Prim’s algorithm to find an optimal spanning tree

• use Kruskal’s algorithm to find an optimal spanning tree.

Notes

Students must understand that the use of a known algorithm to solve such problems is essential, because the algorithm is known to produce an optimal result. Trial and improvement methods do not in themselves guarantee the required optimality.

Examples

1 The network below shows 8 towns, A, B, ..., H. The weight of each arc represents the length of the road, in miles, between towns.

During the winter, the council treats some of the roads with salt so that each town can be safely reached on treated roads from any other town. It costs £30 to treat a mile of road.

Find the minimum cost to the council of making it possible for each town to be safely reached on treated roads from any other town.

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2 The following network shows the blueprint for ground that could be excavated between nine villages, A, B, …, I to create ditches.

The weight of each arc represents the capacity of water, in millions of litres, that each ditch can store.

A company that is to excavate the ground wants to ensure that exactly eight ditches are created, that the ditches form a connected network and that the total capacity of water that can be stored by the ditches is maximised.

Find the maximum capacity of water that can be stored by the ditches.


3 A group of eight friends, A, B, …, H, keep in touch by sending text messages. The cost, in

pence, of sending a message between each pair of friends is shown in the following table.

A B C D E F G H

A – 15 10 12 16 11 14 17

B 15 – 15 14 15 16 16 15

C 10 15 – 11 10 12 14 9

D 12 14 11 – 11 12 14 12

E 16 15 10 11 – 13 15 14

F 11 16 12 12 13 – 14 8

G 14 16 14 14 15 14 – 13

H 17 15 9 12 14 8 13 –

One of the group wishes to pass on a piece of news to all the other friends, either by direct text

or by the message being passed on from friend to friend, at the minimum total cost.

(a) Find the minimum total cost.

(b) Person H leaves the group. Find the new minimum total cost.

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DB3 Solve route inspection problems.


Teaching guidance


• identify problems as being route inspection problems

• use the Chinese postman algorithm (to convert a network with at most four odd-degree nodes into an Eulerian network) to determine the optimal weight of an Eulerian cycle

• use the Chinese postman algorithm (to convert a network with at most four odd-degree nodes into a semi-Eulerian network) to determine the optimal weight of an Eulerian trail

• determine how many times a node has been visited during an Eulerian cycle or Eulerian trail.


Examples 1 The following network shows the times, in minutes, taken by a policeman to walk along roads

connecting 12 places, A, B, ..., L, on his beat.

The total of all the times in the network is 224 minutes.

Find the minimum time for the policeman to walk each road at least once, starting and finishing at the police station, which is located at node A.

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2 Sarah sells vegetables from her van. She drives around the streets of a small village. The network shows the streets in the village. The weight of each arc represents the time, in minutes, to drive along that street.

Sarah starts her journey from her house located at node A and drives along all the streets at least once before returning to her house.

(a) Find:

(i) the minimum time for Sarah’s journey

(ii) the number of times Sarah will visit node C during her journey.

(b) Toto is standing for the position of Mayor in the local elections. He intends to travel along all the roads at least once. He can start his journey at any node and can finish his journey at any node.

(i) Find the minimum time for Toto’s journey.

(ii) State the node(s) from which Toto could start his journey in order for him to achieve the minimum time.


3 The diagram shows a network of sixteen roads on a housing estate. The weight of each arc

represents the length, in metres, of the road.

Chris, an ice-cream salesman, travels along each road at least once on his journey around the housing estate.

Find the minimum distance Chris could travel on his journey.

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DB4 Find and interpret upper bounds and lower bounds for the Travelling Salesperson problem.


Teaching guidance


• use the nearest neighbour algorithm to find upper bounds for the total weight of a Hamiltonian cycle within a network or table of weights

• use the lower bound algorithm to find lower bounds for the total weight of a Hamiltonian cycle within a network

• understand that the smallest upper bound and the largest lower bound provide a closed-interval for the optimal length of a Hamiltonian cycle in a network

• interpret upper bounds and lower bounds in the context of the problem.

Examples 1 Amanda, a tourist, wishes to visit five places in Rome: Basilica (B), Coliseum (C), Pantheon (P),

Trevi Fountain (T) and Vatican (V). She is to start her tour at one of the places and visit each of the other places, before returning to her starting place. The table shows the times, in minutes, to travel between these places. Amanda wishes to keep her travelling time to a minimum.

B C P T V

B – 43 57 52 18

C 43 – 18 13 56

P 57 18 – 8 48

T 52 13 8 – 51

V 18 56 48 51 –

(a) Find the total travelling time for the cycle TPVBCT.

(b) Find the total travelling time for another cycle starting and finishing at T.

(c) Compare and interpret your answers for parts (a)(i) and (a)(ii) in the context of the question.


2 Jamie is a mobile hairdresser based at A.

His day’s appointments are at five places: B, C, D, E and F. He can arrange the appointments in any order. He intends to travel from one place to the next until he has visited all of the places, starting and finishing at A. The following table shows the times, in minutes, that it takes to travel between the six places.

A B C D E F

A – 15 11 14 27 12

B 15 – 13 19 24 15

C 11 13 – 10 19 12

D 14 19 10 – 26 15

E 27 24 19 26 – 27

F 12 15 12 15 27 –

Find a lower bound for the time taken for Jamie’s journey. 3 Fred delivers bread to five shops, A, B, C, D and E. Fred starts his journey at shop B, and travels

to each of the other shops once before returning to shop B. Fred wishes to keep his travelling time to a minimum. The table shows the travelling times, in minutes, between the shops.

A B C D E

A – 3 11 15 5

B 3 – 18 12 4

C 11 18 – 5 16

D 15 12 5 – 10

E 5 4 16 10 –

Find a minimum travelling time for Fred’s journey that cannot be improved. Fully justify your answer.

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DB5 Evaluate, modify and refine models that use networks.


Teaching guidance


• understand and identify the limitations of models used with networks in the context of the problem

• discuss the suitability of the model and suggest ways in which a model used with networks may be improved

• refine a model used with networks and evaluate the outcomes of the improved model.


Examples 1 Shruti is travelling by train to a number of cities. She is to start at M and visit each other city at

least once before returning to M. The diagram shows the travelling times, in minutes, between cities. Where no time is shown, there is no direct journey available.

The table below shows the minimum travelling times between all pairs of cities.

B E L M N

B – 230 82 102 192

E 230 – 148 244 258

L 82 148 – 126 110

M 102 244 126 – 236

N 192 258 110 236 –

(a) Find an upper bound for Shruti’s minimum total travelling time.

(b) Suggest a reason, in the context of the question, why Shruti’s total travelling time will be larger than that found in part (a)?

(c) Suggest a way in which the answer found in part (a) could be made more realistic.

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2 The network below shows the times, in minutes, taken by a police car to drive along roads connecting 12 places, A, B,…, L.

Each day, the police car has to drive along each road at least once, starting and finishing at A.

Total of all times = 134 minutes

(a) Find the minimum driving time for the police car.

(b) Explain why the driving time for the police car is likely to differ from the answer found in part (a).


3 Phil, a squash coach, wishes to buy some equipment for his club. In a town centre, there are six

shops, G, I, N, R, S and T, that sell the equipment.

The time, in seconds, to walk between each pair of shops at 9am is shown in the table.

Phil intends to check prices by visiting each of the six shops before returning to his starting point.

G I N R S T

G – 81 82 86 72 76

I 81 – 80 82 68 73

N 82 80 – 84 70 74

R 86 82 84 – 74 70

S 72 68 70 74 – 64

T 76 73 74 70 64 –

(a) Find an upper bound for Phil’s minimum walking time, stating the corresponding cycle.

(b) Each of the six shops that Phil visited in the morning has a sale that starts at 2pm.

Phil again intends to check prices by visiting each of the six shops before returning to his starting point.

Suggest, with a reason, how Phil’s minimum walking time may be different to that found in part (a).

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DC Network flows

DC1 Interpret flow problems represented by a network of directed arcs.


Teaching guidance


• understand that sources are nodes that have all directed arcs pointing outwards from the node

• identify any sources within a network containing directed arcs

• understand that sinks are nodes that have all directed arcs pointing inwards to the node

• identify any sinks within a network containing directed arcs

• understand and interpret flow problems in the context of the problem, including the use of any units that are given

• understand that the flow into any node must equal the flow out of the same node.


Examples 1 A network flow problem is modelled by the network shown in Figure 1.

Figure 1

Which row in the table below correctly identifies a source and a sink?

Circle your answer.

Source Sink

A A G

B D F

C B F

D G D

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2 A network of gas pipes is used to transport gas from a mine to a storage facility, as shown in Figure 1.

The capacity of each pipe, in thousands of litres per second, is given by the number not circled on each.

The number inside the circles on each arc represents the actual flow.

Figure 1

(a) Find the values of:

(i) x

(ii) y

(iii) z

(b) A flow of 58 thousand litres of gas flows through the network of gas pipes per second.

Find the flow along the arc IJ.


3 Figure 1 below shows a network of corridors from the entrance of a college to the main

assembly hall.

The capacity of each corridor, ie the number of people that can pass along the corridor per minute, is given by the number not circled on each arc.

The number inside the circles on each arc represents the actual flow of people along each corridor per minute.

Figure 1

(a) State:

(i) the location of the entrance of the college

(ii) the location of the main assembly hall.

(b) The assembly hall has seating for 750 people and room for a further 500 to stand. The

entrance to the college opens at 8:40 am. Find:

(i) the number of people entering the hall per minute

(ii) the time by which a person wanting a seat must arrive at the assembly hall to guarantee themselves getting a seat.

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DC2 Find the value of a cut and understand its meaning.


Teaching guidance


• determine the value of a cut for a network containing arcs with only upper capacities

• understand and use the value of a cut to put an upper bound on the flow through the network, in the context of the problem

• understand and use the set notation for cuts {source-side nodes}/{sink-side nodes}.

Examples

1 Figure 1 shows a network of pipes. The capacity of each pipe is given by the weight on each arc.

Figure 1

(a) Find the value of:

(i) the cut C1

(ii) the cut C2

(b) Using your answers to part (a), deduce what can be determined about the flow through the network.


2 Figure 1 shows a network of pipelines through which oil can travel from the oil field to the

refinery via intermediate stations.

The weight on each arc represents the capacities in millions of barrels per hour that can flow through each pipeline.

Figure 1

(a) Find the value of the cut marked C on Figure 1.

(b) Find the value of the cut {S, A, B}/{D, E, F, T}.

(c) Interpret your answers to part (a) and (b) in the context of the question.

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3 Figure 1 shows the routes along corridors from two playgrounds to the theatre in a school. The weight of each arc represents the maximum number of students that can travel along the corridor in one minute.

Figure 1

(a) State, with a reason, which node represents the theatre.

(b) (i) Find the value of the cut shown in Figure 1.

(ii) Interpret your answer to part (b)(i) in the context of the question.



Teaching guidance

Students should be able to determine the value of a cut for a network containing arcs that have both upper and lower capacities.

Examples 1 Figure 1 shows a network representing a system of water pipes which have lower and upper

capacities in litres per second.

Figure 1

(a) Find the value of the Cut X on Figure 1.

(b) Hence, state what can be deduced about the flow through the network.

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2 Figure 1 shows a network of pipes that carry oil from one part of an oil refinery to another part of the oil refinery. The weights on each arc represent the lower and upper capacity for the flow of oil through the pipes, in billions of litres per hour.

Figure 1

(a) (i) Find the value of the cut {A, B, C, D}/{E, F, G, H}.

(ii) Find the value of the cut {A, B, C, E}/{D, F, G, H}.

(b) Using your answers to part (a), make a deduction about the flow of oil through the network of pipes.


3 An engineer models a system of water pipes in a water treatment works using a network, as

shown in Figure 1.

The weights on each arc represent the lower and upper capacities for the flow of water through each pipe, in thousands of litres per minute.

Figure 1

(a) Find the value of:

(i) the cut {A, B}/{D, F}

(ii) the cut {A, B, D}/{F}.

(b) Hence, state what can be deduced about the flow through the network.

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DC3 Use and interpret the maximum flow-minimum cut theorem.


Teaching guidance


• understand that there are a finite number of distinct cuts and that, if the value of each distinct cut is known, the value of the maximum flow through the network is equal to the minimum value of the cuts

• use the maximum flow-minimum cut theorem to prove that a flow is a maximum.

Examples 1 When investigating a network flow problem, a student finds a cut with a value of 50 and a flow

through the network with a value of 50.

State and explain what the student can deduce from their findings.

2 When investigating a network flow problem, a student finds a cut with a value of 50 and a flow

through the network with a value of 30.

State and explain what the student can deduce from their findings.


3 An engineer models a system of water pipes in a water treatment works using a network, as

shown in Figure 1. The weights on each arc represent the lower and upper capacities for the flow of water through each pipe, in thousands of litres per minute.

Figure 1

Find the maximum flow through the network.


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DC4 Introduce supersources and supersinks to a network.


Teaching guidance


• identify situations where supersources and supersinks are necessary

• use supersources and supersinks, including using appropriate weights on any arcs that may have been introduced (with both lower and upper capacities at A-level only).


Examples 1 A greengrocer has two suppliers, A and B, and three storage depots, C, D and E. He needs to

transport his stock to three retail outlets X, Y and Z. The capacities of the possible routes, in van loads per week, are shown in Figure 1.

Figure 1

On Figure 2 (below), add a supersource, S, and supersink, T, and all necessary arcs.

Show the capacity on each arc you have added.

Figure 2

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2 Water from two reservoirs, R1 and R2, is used to supply three towns, T1, T2 and T3. In Figure 1, the capacity of each pipe is given by the number not circled on each arc. The numbers in circles represent an initial flow.

Figure 1

Add a supersource and supersink to Figure 2 (below), including any necessary arcs.

Figure 2


3 Water has to be transferred from two mountain lakes P and Q to three urban reservoirs U, V and

W. There are pumping stations at X, Y and Z.

The possible routes with the capacities along each arc, in millions of litres per hour, are shown in Figure 1.

Figure 1

(a) On Figure 2 (below), add a supersource and a supersink and appropriate arcs so as to produce a directed network with a single source and a single sink.

Figure 2

(b) (i) Find the value of the cut C.

(ii) State what can be deduced about the maximum flow of water between the lakes and the reservoirs.

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DC5 Augment flows and determine the maximum flow in a network.


Teaching guidance


• use flow augmentation from an initial flow to find a maximum flow through a network with only upper capacities

• state the maximum flow through a network in the context of the problem

• find a flow pattern corresponding to the maximum flow through a network.


Examples 1 A retail company has warehouses at P, Q and R, and goods are to be transported to retail

outlets at Y and Z, as shown on Figure 1. There are also retaining depots at U, V, W and X. The possible routes with the capacities along each arc, in van loads per week, are shown on Figure 1.

Figure 1

(a) On Figure 2, use flow augmentation starting from a flow of zero to find the maximum number of van loads per week from the warehouses to the retail outlets. List any flow augmenting paths you use on Figure 3.

Figure 2 Figure 3

Route Flow

(b) Prove that the number of van loads per week from the warehouses to the retail outlets you found in part (a) is the maximum.


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2 Figure 1 shows a network of pipelines through which oil can travel. The oil flows from the oil field to the refinery via intermediate stations.

The weights on the arcs show the capacities in millions of barrels per hour that can flow through each pipeline.

Figure 1

Using Figure 2, find the maximum flow of oil through the pipelines in two hours.

Figure 2

Route Flow


3 A network of corridors in a university graduation building is shown in Figure 1. Guests attending

a graduation ceremony move steadily along the corridors from the entrance to the graduation hall.

The weights on the arcs show the maximum number of people that can move along each corridor per minute.

Figure 1

The organist will begin playing music once 740 guests have arrived in the graduation hall. The

entrance to the graduation building opens at 9am.

Using Figure 2, find the time at which the organist begins playing music.

Figure 2

Route Flow

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DC6 Solve problems including arcs with upper and lower capacities.


Teaching guidance

Students should be able to use flow augmentation from an initial flow to find a maximum flow through a network with lower and upper capacities.


Examples 1 Figure 1 shows a network of water pipes in a small factory. The weights of each arc represents

the lower and upper capacities for each pipe in litres per second.

(a) Find the value of the Cut Q.

(b) Figure 2 shows most of the values of a feasible flow of 34 litres per second through the network.

(i) Complete Figure 2 by inserting any missing values.

Figure 2

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(ii) Using this feasible flow as the initial flow, use flow augmentation on Figure 3 (opposite) to find the maximum flow through the network.

Indicate any flow-augmenting paths in the table.

Figure 3

(c) State the maximum flow and indicate the maximum flow on Figure 4.

Figure 4

(d) Prove that the flow you found in parts (b) and (c) is maximum.


Path Extra flow


2 Figure 1 shows a network of pipes with the lower and upper capacities for each pipe in litres per

second.

Figure 1

(a) (i) Find the value of the cut C.

(ii) Find the value of the cut {S, P}/{Q, R, U, V, W, T}.

(ii) Hence, state what can deduced about the maximum flow through the network.

(b) Figure 2 shows a partially completed diagram for a feasible flow through the network. Complete Figure 2 by adding any missing values.

Figure 2

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(c) Using you answer to part (b), use flow augmentation on Figure 3 to find the maximum flow through the network. List any flow augmenting paths that you use in the table.

Figure 3

Path Additional flow

(d) Prove that the flow you found in part (c) is maximum.



3 Figure 1 shows a network of water pipes with the lower and upper capacities for each pipe in

litres per second.

Figure 1

(a) Find a feasible flow of 20 litres per second through the network.

Illustrate your answer on Figure 2.

Figure 2

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(b) Using your answer to part (a), use flow augmentation on Figure 3 (on the next page) to find the maximum amount of water than can flow through the network. List any flow augmenting paths in the table.

Figure 3

Path Flow

(c) Prove that the flow found in part (b) is maximum.


DC7 Refine network flow problems including using nodes of restricted capacity.


Teaching guidance

Students should be able to refine solutions to network flow problems when further restrictions or modifications to a network are introduced.

Examples 1 The network in Figure 1 shows the routes along corridors from two arrival gates to the passport

control area, P, in a small airport. The weight of each arc represents the maximum number of passengers that can travel along a particular corridor in one minute.

Figure 1

(a) State the nodes that represent the arrival gates.

(b) Find the value of the cut shown in Figure 1.

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(c) Using flow augmentation starting from an initial flow of zero on Figure 2 (insert), find the maximum number of passengers that can travel from the arrival gates to the passport control area P in one minute.

You must list any flow augmenting paths that you use in the table.

Route Value of flow

(d) One a particular day, there is an obstruction allowing no more than 50 passengers per minute through node V.

Evaluate how this obstruction changes your answer to part (c).


2 Figure 1 shows the routes along corridors from the playgrounds A and G to the assembly hall in

a school. The weight of each arc represents the maximum number of pupils that can travel along the corridor in one minute.

Figure 1

(a) Find the value of the cut shown in Figure 1.

(b) Use Figure 2 to find the maximum flow through the network.

Figure 2

Route Flow

(c) Prove that the flow you found in part (b) is maximum.


(d) On a particular day, there is an obstruction allowing no more than 15 pupils per minute to pass through node E.

State the maximum number of pupils that can move through the network per minute on this

particular day.

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3 Figure 1 shows the evacuation routes along corridors in a college, from two teaching areas to the exit, in case of a fire alarm sounding.

Figure 1

The two teaching areas are at A and G and the exit is at X.

The weight of each arc represents the maximum number of people that can travel along a particular corridor in one minute.

(a) Find the value of the cut shown on Figure 1.

(b) Use Figure 2 to find the maximum flow through the network. You should indicate any flow augmenting routes in the table.

Figure 2

Route Flow

(c) During one particular fire drill, there is an obstruction allowing no more than 45 people per minute to pass through node B.

State the maximum number of people that can move through the network per minute during this fire drill.


DD Linear programming

DD1 Formulate constrained optimisation problems.


Teaching guidance


• use and introduce (where necessary) variables for quantities within the linear programming problem

• interpret written constraints in linear programming problems and write them as inequalities involving the variables in the problem, including inequalities that ensure non-negative solutions (where necessary)

• write down an objective function for a linear programming problem

• interpret the objective of a linear programming problem and write this as a command to maximise or minimise an objective function subject to inequalities on the variables in the problem.

Examples

1 A bakery makes two sizes of pizza: large and medium.

Every day the bakery must make at least 40 of each size of pizza.

Every day the bakery must make at least 120 pizzas in total, but not more than 400 pizzas in total.

Each large pizza takes 4 minutes to make, and each medium pizza takes 2 minutes to make. There are four workers available, each for five hours a day, to make the pizzas.

The bakery makes a profit of £3 on each large pizza sold, and £1 on each medium pizza sold and would like to maximise its total profit.

Formulate this situation as a linear programming problem.

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2 A company produces two types of gift box: standard and luxury.

Each day, the company must produce at least 20 of each type and at least 70 in total.

The boxes are produced using three different machines.

Machine A must be used for at least 100 minutes each day.

Machine B is available for a maximum of 5 hours each day.

Machine C is available for a maximum of 8 hours each day.

The table shows the time, in minutes, each type of box spends on the three different machines.

Type of box Time on machine A Time on machine B Time on machine C

Standard 1 2 4

Luxury 2 3 4

Each day, the company produces x standard and y luxury gift boxes.

The company wishes to produce the maximum number of boxes each day.

Formulate the company’s situation as a linear programming problem.


3 A football team is performing badly. The manager of the team can buy some new players to try

and gain some extra points. The manager can buy three different types of player: forwards, midfielders and defenders.

The following table shows the information available to the manager.

Type of player Cost per player (£ in millions) Number of points gained per player

Forward 12 3

Midfielder 7 2

Defender 5 1

The manager has £40 million available to spend on new players.

In order to keep the fans of the football team happy, the manager must buy at least one forward player.

The manager wants to gain as many points as possible from buying new players.

Formulate the manager’s situation as a linear programming problem.

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DD2 Solve constrained optimisation problems via graphical methods.


Teaching guidance


• draw axes on a graph paper insert and use a suitable scale in order to solve the linear programming problem

• plot the constraints of the linear programming problem, shading out any regions that are inconsistent with the constraints

• identify the feasible region on a graph

• identify optimal vertices of a feasible region, for instance using an objective line

• understand that linear programming problems requiring integer solutions for the variables may require comparing the local integer coordinates to an optimal vertex

• state the outcomes of a graphical method in terms of the context of the problem.

Examples 1 A linear programming problem is as follows:

Maximise P = 2x + 5y

Subject to x + y ≥ 20

y ≤ 2x

x + 2y ≤ 80

and x ≥ 0, y ≥ 0

Using a graph paper insert (not included here), solve the linear programming problem.


2 Ryan is a florist. Every day he makes two types of bouquet: standard and luxury. To make

bouquets, he uses roses, carnations and lilies, as shown in the table.

Type of bouquet Number of roses Number of carnations Number of lilies

Standard 3 6 4

Luxury 6 3 4

Every day, Ryan has 600 roses, 600 carnations and 480 lilies available.

He makes a profit of £1.50 on each standard bouquet sold and £2.50 on each luxury bouquet

sold. Each day, Ryan sells all the bouquets he makes.

Using a graph paper insert (not included here), find the maximum profit Ryan can make in a day.

State the number of standard and luxury bouquets Ryan should sell in order to achieve the

maximum possible profit.

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3 A television company makes LCD and LED televisions. Both types of television require a number of component A and component B.

Each LCD television requires 2 of component A and 3 of component B.

Each LED television requires 4 of component A and 1 of component B.

Each day:

• the company has 50 of component A and 24 of component B available

• the company is to make at least 2 of each type of television, but no more than 20 in total.

The company sells each LCD television at a profit of £20 and each LED television at a profit of £25.

The company wants to find its minimum and maximum daily profit.

(a) Formulate the company’s situation as a linear programming problem.

(b) Illustrate the linear programming problem on a graph paper insert (not included here).

(c) (i) Find the company’s maximum daily profit.

(ii) State the number of each type of television the company should make in order to maximise its daily profit.

(d) Find the company’s minimum daily profit.


DD3 Use the Simplex algorithm for optimising (maximising and minimising) an objective function including the use of slack variables.


Teaching guidance


• identify when the simplex algorithm should be applied (ie for situations that have three or more basic variables)

• introduce and use slack variables to convert inequalities into equalities (only less than or equal to inequalities will be used during assessment)

• use the simplex algorithm to maximise an objective function subject to linear constraints

• use the simplex algorithm to minimise an objective function P by maximising the objective function Q = –P.

Examples 1 Each day, a factory makes three types of hinge: basic, standard and luxury. The hinges

produced need three different components: type A, type B and type C.

Basic hinges need 2 components of type A, 3 components of type B and 1 component of type C.

Standard hinges need 4 components of type A, 2 components of type B and 3 components of type C.

Luxury hinges need 3 components of type A, 4 components of type B and 5 components of type C.

Each day, there are 360 components of type A available, 270 of type B and 450 of type C.

Each day, the factory must use at most 1000 components in total.

In order to be cost-efficient, the factory needs to maximise the number of hinges that are made in a single day.

Find the number of each type of hinge that the factory should produce in order to meet its needs.


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2 Each year, Farmer Giles buys some goats, pigs and sheep.

The total of the number of pigs and the number of sheep that he buys must not be greater than 150.

He must buy at least as many pigs as goats.

Each goat costs £16, each pig costs £8 and each sheep costs £24.

He has £3120 to spend on the animals.

At the end of the year, Giles sells all of the animals. He makes a profit of £70 on each goat, £30 on each pig and £50 on each sheep.

Giles wants to maximise his total profit.

(a) Formulate Giles’s situation as a linear programming problem.

(b) Find Giles’s maximum profit for this year and the number of each animal that he must buy to obtain this maximum profit.

3 A factory makes and sells three different kinds of novelty box: red, blue and green.

Each box contains three different types of toy: A, B and C.

The following table shows how many of each type of toy the different kinds of box contain.

Kind of novelty box

Number of toy A

Number of toy B

Number of toy C

Cost per novelty box made (£)

Red 2 3 4 4

Blue 3 1 3 3

Green 4 5 2 6

Each day, the maximum number of each type of toy available to be packed is 360 type A, 300 type B and 400 type C.

Each day, the factory makes x red boxes, y blue boxes and z green boxes.

The factory wants to minimise the total cost of making the novelty boxes.

(a) Formulate the factory’s situation as a linear programming problem.

(b) Find the number of each novelty box that the company should make each day in order to minimise the total cost of making the novelty boxes.


DD4 Interpret a Simplex tableau.


Teaching guidance


• understand and use the stopping condition for the simplex algorithm (ie the objective row being non-negative)

• find the optimal value of the objective function from a simplex tableau

• identify any basic or slack variables that have zero values

• find the values of any basic or slack variables that have non-zero values.

Examples 1 A student is solving a linear programming problem using the simplex algorithm. The student

obtains the following tableau.

P x y z r s t value

1 0 2 0 k – 10 0 6 15

0 1 1 0 – 32

0 0 25

0 0 – 14

1 23

0 – 13

23

0 0 52

0 34

1 – 12

52

(a) Given that no further iterations of the simplex algorithm are required, find a condition that k

must satisfy.

(b) Given that k = 25, find:

(i) the optimal value for the objective function

(ii) the values of all basic variables

(iii) the values of all slack variables.

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2 A student is solving a linear programming problem using the simplex algorithm. The student obtains the following tableau.

P x y r s value

1 k2 – 10k + 16 0 k – 5 0 45

0 1 1 – 32

0 12

0 34

0 23

1 30

The simplex tableau is optimal.

Find the allowed values of k .

3 A student is solving a linear programming problem using the simplex algorithm. The student

obtains the following tableau.

P x y r s value

1 k2 – 12k + 32 0 k – 7 2 60

0 0 1 – 32

0 25

0 23

0 34

1 20

The simplex tableau is optimal. The optimal value of the basic variable x is non-zero, whereas

the optimal value of the slack variable r is zero.

Find the value of k .


DE Critical path analysis

DE1 Construct, represent and interpret a precedence (activity) network using activity-on-node.


Teaching guidance


• interpret a table of activities showing an order of precedence

• construct an activity network from a table of activities showing an order of precedence, labelling each node with the activity label, the earliest start time, the duration and the latest finish time.

Notes

The activity-on-arc method is not acceptable in this specification. It is a valid method, but we will expect students to know and use only the activity-on-node method.

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Examples 1 A major project has been divided into a number of tasks, as shown in the table. The time taken

to complete each task is also shown.

Activity Immediate predecessor(s) Duration (hours)

A – 3

B A 3

C A 4

D B, C 6

E B, C 5

F C 2

G C 1

H A 15

I D, E 4

J F 6

K G 10

L H, I, J, K 1

Construct an activity network for the project.


2 A cleaning project is to be undertaken. The table shows the activities involved.


A – 4

B – 6

C A, B 7

D C 9

E C 10

F B 3

G D, E 6

H F, G 5

I G 3

J H, I 2


3 A construction company is involved in a building project. The table shows the activities involved.


A – 5

B A 2

C A 3

D B, C 4

E C 2

F C 3

G D, E 4

H F 1

I G, H 3


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DE2 Determine earliest and latest start and finish times for an activity network.


Teaching guidance


• determine the earliest start times for each activity in the activity network by using the fact that an activity can only start once all of the preceding activities have been completed

• determine the latest finish times for each activity in the activity network by using the fact that an activity should not unnecessarily increase the earliest start time of a succeeding activity.

Examples 1 A small building project is to be undertaken. The following precedence table shows each activity

and its duration.


A – 3

B A 4

C A 2

D B 3

E D 11

F D 4

G C, D 5

H F, G 2

I E, H 2

(a) Construct an activity network for the project.

(b) Find the earliest start time for each activity.

(c) Find the latest finish time for each activity.


2 Figure 1 shows an activity network for a building project.

Figure 1

On Figure 1:

(a) find the earliest start time for each activity

(b) find the latest finish time for each activity.

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3 Figure 1 shows an activity for a network for a major building project.

Figure 1

(a) On Figure 1:

(i) find the earliest start time for each activity

(ii) find the latest finish time for each activity.


DE3 Identify critical activities, critical paths and the float of non-critical activities.


Teaching guidance


• identify all critical activities in an activity network

• identify all critical paths in an activity network and state the corresponding minimum completion time for the project

• determine the float of a non-critical activity.

Examples

1 A construction is to be undertaken. The table shows the activities involved.


A – 2

B A 5

C A 8

D B 8

E B 10

F B 4

G C, F 7

H D, E 4

I G, H 3




(d) Find the critical path.

(e) State the float time for each non-critical activity.

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2 A construction project is to be undertaken. The table shows the activities involved.


A – 2

B – 1

C A 3

D A, B 2

E B 4

F C 1

G C, D, E 3

H E 5

I F, G 2

J H, I 3




(d) State the minimum completion time for the building project.

(e) Identify the critical path(s).


3 Figure 1 shows an activity diagram for a building project. The time needed for each activity is

given in days.

Figure 1

(a) On Figure 1:



(b) (i) Find the critical path.

(ii) State the minimum completion time for the building project.

(c) Find the activity with the greatest float time and state the value of its float.

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DE4 Refine models and understand the implications of possible changes in the context of critical path analysis.


Teaching guidance


• determine the impact of a change to the problem on the minimum completion time and which activities are critical

• understand the limitations of critical path analysis and suggest ways in which a problem may be refined to make it more realistic.


Examples 1 Figure 1 shows an activity diagram for a home renovation project. The time needed for each

activity is given in hours.

Figure 1

(a) On Figure 1:



(b) Determine which activities, if delayed by any amount of time, would increase the project completion time.

(c) State one limitation of the project used for the home renovation.

Explain how this limitation affects the project.

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2 An activity diagram for a project is shown below. The duration of each activity is given in weeks. The earliest start time and the latest finish time for each activity are shown in Figure 1.

Figure 1

(a) Find the values of x, y and z.

(b) State the critical path.

(c) Some of the activities can be speeded up at an additional cost. The following table lists the activities that can be speeded up together with the minimum possible duration of these activities. The table also shows the additional cost of reducing the duration of each of these activities by one week.

Activity Additional cost per week (£)

Minimum completion time (weeks)

E 8000 1

F 7000 4

G 6000 5

The company wants to complete the project as soon as possible.

(i) Find which activities should be speeded up. For each such activity, state, with justification, the reduction in the number of weeks.

(ii) Hence state the revised minimum time for the completion of the whole project.

(iii) Calculate the total additional cost that the company would incur in meeting this revised completion time.


3 Figure 1 shows an activity diagram for a project. The duration required for each activity is given

in hours. The project is to be completed in the minimum time.

Figure 1

(a) On Figure 1:

(ii) find the earliest start time for each activity


(b) Find:

(ii) the critical path

(ii) the float time of activity E.

(c) Given that activities H and K will both overrun by 10 hours, find the new minimum completion time for the project.

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DE5 Construct and interpret Gantt (cascade) charts and resource histograms.


Teaching guidance


• construct a Gantt chart by drawing each activity on its own horizontal line, including showing the float of non-critical activities

• interpret a Gantt chart

• construct and interpret resource histograms, where the vertical axis represents the resource and the horizontal axis represents the time.


Examples 1 A group of workers is involved in a decorating project. The table shows the activities involved.

Each worker can perform any of the given activities.

Activity A B C D E F G H I J K L

Duration (days) 2 5 6 7 9 4 3 2 3 2 3 1

Number of workers required 6 3 5 2 5 2 4 4 5 3 2 4

The activity network for the project is shown in Figure 1.

Figure 1

(a) On Figure 1:



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(b) Given that each activity starts as early as possible and assuming that there is no limit to the number of workers available, draw a resource histogram for the project on Figure 2, indicating clearly which activities are taking place at any given time.

Figure 2


2 Figure 1, shows an activity diagram for a project. Each activity requires one worker. The

duration required for each activity is given in hours.

Figure 1

(a) On Figure 1:



(b) Find the critical path.

(c) Draw a Gantt chart on Figure 2 to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.

Figure 2

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3 Figure 1 below shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.

Figure 1

Draw a resource histogram on Figure 2 to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.

Figure 2


DE6 Carry out resource levelling (using heuristic procedures) and evaluate problems where resources are restricted.


Teaching guidance

Students should understand and use resource levelling to ensure a resource is not used beyond a given value during a project, even if this means increasing the project completion time.

Examples 1 A group of workers is involved in a decorating project. The table shows the activities involved.

Each worker can perform any of the given activities.

Activity A B C D E F G H I J K L

Duration (days) 2 5 6 7 9 4 3 2 3 2 3 1

Number of workers required 6 3 5 2 5 2 4 4 5 3 2 4

The activity network for the project is shown in Figure 1.

Figure 1

(a) On Figure 1:



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(b) Given that each activity starts as early as possible and assuming that there is no limit to the number of workers available, draw a resource histogram for the project on Figure 2, indicating clearly which activities are taking place at any given time.

Figure 2

(c) It is later discovered that there are only 8 workers available at any time.

(i) Use resource levelling to construct a new resource histogram on Figure 3, showing how the project can be completed with the minimum extra time.

Figure 3

(ii) State the minimum extra time required.


2 Figure 1 shows the activity network and the duration, in days, of each activity for a particular

building project carried out by Elton Construction Ltd.

Figure 1

(a) On Figure 1:



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(b) The number of workers required to carry out each activity is shown in the table.

Activity A B C D E F G H I J

Number of workers required 2 2 3 2 3 2 1 3 5 2

Given that each activity starts as early as possible and assuming that there is no limit to the

number of workers available, draw a resource histogram for the project on Figure 2, indicating clearly which activities take place at any given time.

Figure 2


(c) Elton Construction Ltd only employs 5 workers.

(i) Construct a new resource histogram for the project on Figure 3 which uses at most 5 workers.

Figure 3

(ii) State the minimum time that Elton Construction Ltd could take to complete the

building project.

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3 A group of workers is involved in a building project. The table shows the activities involved. Each worker can perform any of the given activities.

Activity Immediate predecessor(s) Duration (days) Number of workers required

A – 3 5

B A 8 2

C A 7 3

D B, C 8 4

E C 10 2

F C 3 3

G D, E 3 4

H F 6 1

I G, H 2 3




(d) State the minimum completion time for the building project.

(e) Identify the critical path.

(f) Given that each activity starts as early as possible and assuming there is no limit to the number of workers available, draw a resource histogram for the project on Figure 1.

Figure 1


(g) It is later discovered that there are only 7 workers available at any time.

Explain why the project will overrun.

Determine which activities need to be delayed so that the project can be completed with the minimum extra time.

State the minimum extra time required.

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DF Game theory for zero-sum games

DF1 Understand, interpret and construct pay-off matrices.


Teaching guidance


• understand that player 1 is the name commonly given to the player whose strategies comprise the rows of the pay-off matrix

• understand that player 2 is the name commonly given to the player whose strategies comprise the columns of the pay-off matrix

• understand that pay-off matrices are written from the perspective of player 1, so that positive values show gains and negative values show losses for player 1

• understand and state that a zero-sum game between 2 players means that the sum of player 1’s gain + player 2’s gain for each pair of strategies is equal to zero, and use this to rewrite the pay-off matrix to show the gains and losses for player 2.

Examples 1 Two players, Gary and Carrie, play a zero-sum game. The pay-off matrix for the game is shown

below.

Carrie

Strategy X Y Z

Gary A 3 1 –2

B –2 3 1

C 1 –2 3

Write down a pay-off matrix from the perspective of Carrie.


2 Two players, Andy and Maisie, play a zero-sum game. The pay-off matrix for the game is shown

below.

Maisie

Strategy X Y Z

Andy A 0 4 –2

B –2 –3 4

C 1 –3 3

The units for the values in the pay-off matrix are pounds (£).

(a) State what is meant by a two-player zero-sum game.

(b) Andy plays strategy B and Maisie plays strategy Z.

State the gain or loss for Maisie.

3 Two players, Jen and Jeff, play a zero-sum game. The pay-off matrix for the game is shown

below.

Jeff

Strategy I II III

Jen I 1 4 –2

II –1 0 1

III 1 –1 0

(a) State what is meant by a two-player zero-sum game.

(b) On four consecutive trials of the game, Jen plays strategy III and Jeff plays strategy II.

Find the gain or loss for Jen.

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DF2 Find play-safe strategies and the value of the game.


Teaching guidance


• identify the minimum gain for each strategy for players 1 and 2 (the row minima and column maxima)

• find the maximum minimum gain for each player (the max(row minima) and min(col maxima))

• state the play-safe strategy (or strategies) for each player, ie the strategy that minimises the potential loss for each player

• understand that if the value of max(row minima) is equal to the value of min(col maxima), then the value of the game is equal to max(row minima)

• understand that, unless stated otherwise, the value of the game refers to the expected gain for player 1 for each game played.

Examples

1 Two players, Gary and Carrie, play a zero-sum game. The pay-off matrix for the game is shown below.

Carrie

Strategy X Y Z

Gary A 3 1 –3

B –2 4 1

C 1 –1 2

Find the play-safe strategy for each player.


2 Two players, Andy and Maisie, play a zero-sum game. The pay-off matrix for the game is shown

below.

Maisie

Strategy X Y Z

Andy A 0 4 –2

B –2 –3 –3

C 1 –3 –4

The units for the values in the pay-off matrix are pounds (£).

(a) Find the play-safe strategy for each player.

(b) Find the value of the game.


below.

Jeff

Strategy I II III

Jen I 2 4 –2

II –2 –5 –3

III 4 –3 –4

Find the value of the game for Jeff.

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DF3 Prove the existence or non-existence of a stable solution.


Teaching guidance


• state and use the condition for a stable solution to exist, that is min(col maxima) = max(row minima)

• use the result min(col maxima) ≠ max(row minima) to conclude that a stable solution does not exist.

Examples 1 Two players, Andy and Maisie, play a zero-sum game. The pay-off matrix for the game is shown

below.

Maisie

Strategy X Y Z

Andy A 0 4 –2

B –2 –3 –3

C 1 –3 –4

Prove that the zero-sum game between Andy and Maisie has a stable solution.



below.

Jeff

Strategy I II III

Jen I 1 4 –2

II –1 0 1

III 1 –1 0

Determine whether or not a stable solution exists for the game played by Jen and Jeff.

3 Two players, Gary and Carrie, play a zero-sum game. The pay-off matrix for the game is shown

below, where k is an integer.

Carrie

Strategy X Y Z

Gary A 3 –3 –3

B –1 k 1

C 1 –4 2

A stable solution for the game exists.

Find the allowed value(s) of k .

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DF4 Identify and make use of dominated strategies.


Teaching guidance


• identify dominance in a pay-off matrix for player 1

• identify dominance in a pay-off matrix for player 2

• use dominance to remove strategies from a pay-off matrix, thereby reducing the size of the pay-off matrix.

Examples

1 Two players, Dan and Christine, play a zero-sum game. The pay-off matrix for the game is shown below.

Christine

Strategy I II III

Dan I 1 4 –2

II –1 0 –2

III 1 –1 0

(a) Explain why Dan should only choose to play two of his three strategies.

(b) Which strategy should Christine never play?

Explain your answer.


2 Two players, Sidney and Raj, play a zero-sum game. The pay-off matrix for the game is shown

below.

Raj

Strategy X Y Z

Sidney A 1 4 5

B 2 –2 0

C 0 3 5

(a) Reduce the pay-off matrix to a 2 × 2 pay-off matrix, making clear your reasoning for doing

so.

(b) Using your answer to part (a), write a 2 × 2 pay-off matrix from the perspective of Raj.

3 Two players, Albert and Granville, play a zero-sum game. The pay-off matrix for the game is

shown below.

Granville

Strategy W X Y Z

A 1 4 5 –3

Albert B 2 –2 0 2

C 0 3 5 k

D –1 3 1 –3

State the value(s) of k :

(a) if strategy C dominates strategy D

(b) if strategy A dominates strategy C

(c) if strategy Z dominates strategy W.

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DF5 Find optimal mixed strategies for a game including use of graphical methods.


Teaching guidance


• understand that games that do not have a stable solution can have the value of the game for player 1 maximised by an optimal mixed strategy

• identify and remove any dominated strategies before attempting to find an optimal mixed strategy

• introduce probability variables for player 1 or player 2

• find the expected gain for player 1 (player 2) for each of player 2’s (player 1’s) strategies

• plot the expected gains on a graph, identify the optimal vertex at which the value of the game is maximised and find the value of the probability variable

• find the optimal mixed strategy for player 1 and player 2

• find the value of the game for player 1 or player 2.

Examples 1 Mark and Owen play a zero-sum game. The game is represented by the following pay-off matrix

for Mark.

Owen

Strategy D E F

Mark A 4 1 –1

B 3 –2 –2

C –2 0 3

The value of the game is 0.6.

Find the optimal mixed strategy for Owen.


2 Kate and Pippa play a zero-sum game. The game is represented by the following pay-off matrix

for Kate.

Pippa

Strategy D E F

Kate A –2 0 3

B 3 –2 –2

C 4 1 –1

(a) Find the optimal mixed strategy for Kate.


(c) Find the optimal mixed strategy for Pippa.

3 Roza plays a zero-sum game against a computer. The game is represented by the following

pay-off matrix for Roza.

Computer

Strategy C1 C2 C3

Roza R1 3 4 –3

R2 –2 –1 5

(a) Find the optimal mixed strategy for Roza.


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DF6 Convert and solve higher order games to linear programming problems.


Teaching guidance


• identify when a zero-sum game between two players cannot be solved graphically (ie when the pay-off matrix is 3 × 3 or larger and cannot be reduced by dominance)

• modify the pay-off matrix in such situations by adding a constant term to each entry in the matrix to ensure each entry is positive

• introduce (where necessary) and use a probability variable for each of player 1’s strategies, such as p, q and r

• write each expected gain for player 1 as an inequality between the probability variables and the value of the game V

• formulate the game theory problem as a linear programming problem to maximise the value of the game (minus the constant term that may have been added to each entry in the matrix) subject to the inequalities found from the expected gains, as well as p + q + r ≤ 1 and p, q, r ≥ 0

Examples 1 John and his son William play a zero-sum game. The pay-off matrix for the game is shown

below.

William

Strategy X Y Z

John A 6 2 5

B 2 5 2

C 2 4 4

John chooses to play strategy A with probability p, strategy B with probability q and strategy C

with probability r.

Formulate the problem of finding the value of the game V as a linear programming problem.


2 Tom and Clare play a zero-sum game. The pay-off matrix for the game is shown below.

Clare

Strategy X Y Z

Tom P 3 –1 2

Q –1 2 –1

R –1 1 1

Tom chooses to play strategy P with probability p, strategy Q with probability q and strategy R

with probability r.

(a) Formulate the problem of finding the value of the game V as a linear programming

problem.

(b) Find the value of the probabilities p, q and r.

(c) Find the value of the game V.

3 Chester and Seiji play a zero-sum game. The pay-off matrix for the game is shown below.

Seiji

Strategy S T U

Chester P 7 1 6

Q 2 6 1

R 3 3 5

(a) Formulate the problem of finding the value of the game as a linear programming problem.

(b) Determine the value of the game.

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DG Binary operations and group theory

DG1 Understand and use binary operations including use of modular arithmetic and matrix multiplication.


Teaching guidance


• apply a given binary operation to two numbers or elements of a set

• find the sum and product of two numbers modulo n, where n is a positive integer, including use of the abbreviation (mod n)

• understand and use the notation +n to represent addition modulo n, where n is a positive integer

• understand and use the notation ×n to represent multiplication modulo n, where n is a positive integer

• find the product of two matrices using matrix multiplication.

Examples 1 Find the value of 5 +12 10

2 Determine the value of 5 ×12 10

3 The binary operation ◊ is given by a ◊ b = a2 + b2 – 2ab where a b∈,

(a) Find the value of 3 ◊ 4

(b) Find a relationship between a and b such that a ◊ b = 0

4 The binary operation ■ is given by x ■ y = x2 + y + 2 (mod 16) where ,x y∈

(a) Find the value of 5 ◊ 3

(b) Find two integers a and b such that a ■ b = 0


5 The binary operation ∗ is given by M ∗ N = MN + M – N where M and N are 2 × 2 matrices.

Show that, for any 2 × 2 matrix M , (M ∗ I) ∗ I = aM + bI where a and b are integers to be determined.

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DG2 Understand, use and prove the commutativity of a binary operation.


Teaching guidance


• understand that a binary operation on a set is commutative if reversing the order of operation between any two elements does not affect the result (ie a ∗ b = b ∗ a for a binary operation ∗ for all pairs a and b)

• use the condition for the commutativity of a binary operation to prove whether or not a binary operation is commutative.

Examples

1 The binary operation ∗ is given by M ∗ N = MN + M – N where M and N are 2 × 2 matrices.

Explain whether or not the binary operation ∗ is commutative.


2 The binary operation ■ is given by x ■ y = x2 + y + 2 where x y∈,

Explain whether or not the binary operation ■ is commutative.


3 The binary operation ◊ is given by a ◊ b = a2 + b2 – 2ab where a b∈,

Explain whether or not the binary operation ◊ is commutative.


4 The binary operation is defined as xy

yx y x−= +

1 1 1 where x and y are real numbers.

Prove that the binary operation is not commutative.



DG3 Understand, use and prove the associativity of a binary operation.


Teaching guidance


• understand that a binary operation is associative if the order of operation of the binary operation does not matter so long as the order of elements does not change (ie (a ∗ b) ∗ c = a ∗ (b ∗ c) for a binary operation ∗ and for all elements a, b and c)

• use the condition for the associativity of a binary operation to prove whether or not a binary operation is associative.

Examples

1 The binary operation ∗ is defined as A ∗ B = AB + A – B where A and B are 2 × 2 matrices.

(a) Show that (A ∗ B) ∗ C = ABC + AB + AC – BC + A – B – C

(b) Determine whether or not the binary operation ∗ is associative.


2 The binary operation ◊ is defined as a ◊ b = a2 + b2 – ab where a b∈,

Prove that the binary operation ◊ is not associative.


3 The binary operation is defined as x y = x + y + 1 (mod 2) where x and y are real numbers.

Prove that the binary operation is associative.


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DG4 Construct a Cayley Table for a given set and binary operation.


Teaching guidance

Students should be able to find all the entries for a Cayley table given a set and a binary operation.

Examples 1 The set S is given by S = {0,2,4}

Construct a Cayley table for the set S under the operation addition modulo 6.

2 A binary operation is commutative and associative when operating on the elements of the set A = {w, x, y, z}

Complete the following Cayley table for the binary operation and the set A.

w x y z

w w x y z

x w z

y x

z x


3 The binary operation ⊗ when used with the set of elements {a, b, c, d, e} results in the following

Cayley table.

⊗ a b c d e

a a b c d e

b c a b e d

c b d e a c

d d e a c b

e e c d b a

(a) Prove that the binary operation ⊗ is not commutative.

(b) Prove that the binary operation ⊗ is not associative.

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DG5 Understand and prove the existence of an identity element for a given set under a given binary operation.


Teaching guidance


• use the condition for an identity element with respect to a binary operation (ie e ∗ x = x and x ∗ e = x) to find an identity element for a given set

• use the condition for an identity element with respect to a binary operation (ie e ∗x = x and x ∗ e = x) to prove an element is an identity element for a given set.

Examples 1 The binary operation ∗ is given by A ∗ B = AB + A – B where A and B are 2 × 2 matrices.

Prove that the null matrix

0 00 0

is an identity with respect to the binary operation ∗.

2 The binary operation is defined as x y = x + y + 1 (mod 2) where x y∈,

(a) Find an identity element of the set

with respect to the binary operation .


(b) Prove that the set

has infinitely many identity elements with respect to the binary operation .

3 The binary operation ◊ is given by a ◊ b = a + b – 5ab where a b∈, .

Find an identity element in the set

with respect to the binary operation ◊.



DG6 Find the inverse of an element belonging to a given set under a given binary operation.


Teaching guidance

Students should be able to use the condition for an inverse of an element under a binary operation (ie x ∗x-1 = e and x-1 ∗ x = e) to find an inverse of element belonging to a given set.

Examples 1 The binary operation ◊ is given by a ◊ b = a + b – ab where ,a b∈

(a) Show that an identity element in the set

with respect to the binary operation ◊ is 0.


(b) Find the inverse of the element 2 under the binary operation ◊.

2 The set S is defined as S = {1,2,3,4,5,6,7,8,9,10,11}.

Find the inverse of the element 5 in the set S under multiplication modulo 12.


3 The binary operation is defined as x y = x + y + 1 where ,x y∈

Find, in terms of x, the inverse of the element x∈under the binary operation


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DG7 Understand and use the language of groups including: order, period, subgroup, proper, trivial, non-trivial.


Teaching guidance


• understand that the order of a group is equal to the number of elements within the group

• use index notation to represent repeated application of the binary operation of the group

• understand that the period (or order) of an element, x, of a group is equal to the smallest non-negative integer n such that xn = e, where e is the identity.

• understand and use that a subgroup of a group is a group that has the same binary operation as the (parent) group, is closed under the binary operation, has the same identity element as the (parent) group, and each element of the subgroup has its inverse in the subgroup

• understand that a group is a subgroup of itself

• understand that a proper subgroup is any subgroup that is not the (parent) group itself (the trivial subgroup is a proper subgroup)

• understand that a trivial subgroup of a group is the one-element subgroup containing the identity element of the (parent) group

• understand that a non-trivial subgroup is any subgroup of a group that is not the trivial subgroup

• identify any subgroups of a group.

Examples 1 The set S, defined as S = {1,2,3,4,5,6,7,8,9,10}, forms a group G under the binary operation

multiplication modulo 11.

(a) State the order of the group G.

(b) Determine the period of the element 10 of the group G.

2 Show that the group ( ),+ is a subgroup of ( ),+ .


3 A group G is formed by the set A = {w, x, y, z} under the binary operation .

The following Cayley table shows the outcome for each pair of elements of the set A under the binary operation .

w x y z

w w x y z

x x w z y

y y z x w

z z y w x

Find all of the proper subgroups of G.

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DG8 Understand and use the group axioms: closure, identity, inverses and associativity, including use of Cayley tables.


Teaching guidance


• understand what it means for a set to be closed under a binary operation, including identifying this property in a Cayley table

• understand what it means for a set to have an identity under a binary operation, including identifying this element in a Cayley table

• understand what it means for each element of a set to have an inverse in the set with respect to a binary operation, including identifying the inverse of an element in a Cayley table

• understand the group axioms; ie that a group is a set under a binary operation such that it is closed under the binary operation, has an identity with respect to the binary operation, each element has an inverse (in the group) with respect to the binary operation, and that the binary operation is associative when operating on all combinations of the elements (in the group)

• use the group axioms to prove that a given set and binary operation form a group

• use the group axioms to show that a given set and binary operation do not form a group

• understand and use the notation for groups, eg (S, ∗), where S is a set and ∗ is a binary operation.

Examples 1 The set A is defined as A = {w, x, y, z}

The following Cayley table shows the outcome for each pair of elements of the set A under the binary operation

w x y z

w w x y z

x x w z y

y y z x w

z z y w x

Prove that the set 𝐴 forms a group under the binary operation


2 Prove that the set of non-negative integers does not form a group under the binary operation of

addition.

3 A student proposes that the set A = {2,5,8,10} forms a group under multiplication modulo 15.

Determine whether or not the student is correct.

4 The Klein group V = ({a, b, c, d}∗) has the corresponding Cayley table

∗ a b c d

a a b c d

b b a d c

c c d a b

d d c b a

For the Klein group V:

(a) state the period of each element

(b) find the inverse of the element c.

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DG9 Recognise and use finite and infinite groups and their subgroups, including: groups of symmetries of regular polygons, cyclic groups and abelian groups.


Teaching guidance


• recognise and use finite groups, and identify any subgroups

• recognise and use infinite groups (for example, the group of all invertible 2 × 2 matrices under the operation of matrix multiplication) and identify any subgroups

• recognise and use the groups of symmetries (reflections and rotations) of regular polygons (the dihedral groups recognise the cyclic group of order n as the group of rotational symmetries of a regular n-gon

• understand that an abelian group is a group with the additional property of a commutative binary operation between the elements of the group.

Examples

1 M is the set of all invertible 2 × 2 matrices, that is : ad bd ca b ca bc d

∈ = − ≠

, , , 0,M

(a) Show that M forms a group under the binary operation of matrix multiplication.

(b) State whether or not the group formed by M is an abelian group.

(c) Show that the set , = −

1 0 1 00 1 0 1

N is a subgroup of the group formed by M under the

binary operation of matrix multiplication.

2 J is the set of complex numbers J = {1, –1, i, –i}.

(a) Show that J forms a group under the binary operation of multiplication.

(b) Identify all subgroups of (J, × ).

(c) State, with a reason, whether or not (J, × ) is an abelian group.


3 Prove that the set of rotational symmetries of the square given by the matrices

− − − −

1 0 0 1 1 0 0 10 1 1 0 0 1 1 0

, , , forms a group under matrix multiplication.

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DG10 Understand and use Lagrange’s theorem.


Teaching guidance


• understand Lagrange’s theorem for groups and their subgroups, ie for any finite group G, the order of every subgroup of G divides the order of G

• use Lagrange’s theorem to prove that no subgroups of a particular order can exist for a given group.

Examples 1 Determine the only possible orders of the subgroups of a group which has an order of 81.


2 Prove that no subgroups of order 12 exist for a group of order 196.


3 Which of the following groups cannot be a subgroup of G, which has an order of 12?

Circle your answer.

A, order 3 B, order 4 C, order 5 D, order 6

4 A student states that ({1, –1, i}, × ) is a subgroup of the group ({1, –1, i, – i}, × ).

Explain whether or not the student is correct.



DG11 Identify and use the generators of a group.


Teaching guidance


• use a generator with a given binary operation to find the entire set of elements that belong to a group

• identify a generator within a group (the group must be a cyclic group)

• understand and use the notation G = ( g , ∗), where the element g∈G generates the group G under the binary operation ∗

Notes

The only groups considered here are cyclic groups so that the group can be generated by a single element. Students should recognise that different elements of the group can be generators.

Examples

1 Determine the order of the group K = ( )+72 ,

2 The cyclic group G under the binary operation has the Cayley table

w x y z

w w x y z

x x w z y

y y z x w

z z y w x

(a) Find a generator of G.

(b) Find all proper subgroups of G.

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3 The group R is defined as R = ( )×74 ,

Construct a Cayley table for R.

4 The group of rotational symmetries of the square G is generated by the 2 × 2 matrix −

0 11 0

under matrix multiplication.

(a) Determine each element of G.

(b) State the order of G.


DG12 Recognise and find isomorphism between groups of finite order.


Teaching guidance


• understand and use the fact that isomorphic groups must have the same order

• identify isomorphism between cyclic groups by using the generator of each group

• understand and use the fact that an isomorphism maps the identity of one group to the identity of the other group

• find and state the isomorphism between groups as a one-to-one mapping between the elements of the two isomorphic groups, including using Cayley tables

• recognise and use the notation ≅ to denote isomorphism between groups.

Examples 1 The group R is defined as R = ( )×74 , and the group S is defined as S = { }( ),+30,1,2

Prove that R ≅ S.


2 The group J is defined as ({1, –1, i, – i},× ) and the group N is defined as N = ( ),×72 .

Determine whether or not J is isomorphic to N.


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3 The group G under the binary operation has the Cayley table

w x y z

w w x y z

x x w z y

y y z x w

z z y w x

The group V under the binary operation ∗ has the Cayley table

∗ a b c d

a a b c d

b b a d c

c c d a b

d d c b a

Show that G is not isomorphic to V.

4 The group of rotational symmetries of the square G is generated by the 2 × 2 matrix −

1 11 0

under matrix multiplication.

The group J is defined as ({1, –1, i, – i}, × ).

Prove that G ≅ J by finding a one-to-one mapping between the elements of G and the elements of J.


A1 Appendix 1 Mathematical notation for AS and A-level qualifications in Maths and Further Maths

The tables below set out the notation that must be used by AS and A-level Mathematics and Further Mathematics specifications. Students will be expected to understand this notation without need for further explanation. AS students will be expected to understand notation that relates to AS content, and will not be expected to understand notation that relates only to A-level content.

1 Set notation

1.1 ∈ is an element of

1.2 ∉ is not an element of

1.3 ⊆ is a subset of

1.4 ⊂ is a proper subset of

1.5 { }1 2, ,...x x the set with elements x1, x2, …

1.6 { }: ...x the set of all x such that …

1.7 ( )n A the number of elements in set A

1.8 ∅ the empty set

1.9 ε the universal set

1.10 ′A the complement of the set A

1.11 the set of natural numbers, { }1,2,3,...

1.12

the set of integers, { }0, 1, 2, 3,...± ± ±

1.13 +

the set of positive integers, { }1,2,3,...

1.14 0+

the set of non-negative integers, { }0,1,2,3,...

1.15

the set of real numbers

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1.16 the set of rational numbers, : , + ∈ ∈

p p qq

1.17 ∪ union

1.18 ∩ intersection

1.19 ( ),x y the ordered pair x, y

1.20 [ ],a b the closed interval { }:∈ ≤ ≤x a x b

1.21 [ ),a b the interval { }:∈ ≤ <x a x b

1.22 ( ],a b the interval { }:∈ < ≤x a x b

1.23 ( ),a b the open interval { }:∈ < <x a x b

1 Set notation (Further Mathematics only)

1.24

the set of complex numbers

2 Miscellaneous symbols

2.1 = is equal to

2.2 ≠ is not equal to

2.3 ≡ is identical to or is congruent to

2.4 ≈ is approximately equal to

2.5 ∞ infinity

2.6 ∝ is proportional to

2.7 ∴ therefore

2.8

because

2.9 < is less than

2.10 ≤ is less than or equal to, is not greater than

2.11 > is greater than


2.12 ≥ is greater than or equal to, is not less than

2.13 ⇒p q p implies q (if p then q)

2.14 ⇐p q p is implied by q (if q then p)

2.15 ⇔p q p implies and is implied by q (p is equivalent to q)

2.16 a first term of an arithmetic or geometric sequence

2.17 l last term of an arithmetic sequence

2.18 d common difference of an arithmetic sequence

2.19 r common ratio of a geometric sequence

2.20 Sn sum to n terms of a sequence

2.21 ∞S sum to infinity of a sequence

3 Operations

3.1 a + b a plus b

3.2 a – b a minus b

3.3 ×a b , ab, a.b a multiplied by b

3.4 ÷a b , ab

a divided by b

3.5 1∑

n

ii=

a a1 + a2 + … an

3.6 1∏

n

ii=

a a1 × a2 × … an

3.7 a the non-negative square root of a

3.8 a the modulus of a

3.9 n! n factorial: n! = n × (n – 1) ×… ×2 ×1, ∈n ; 0! = 1

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3.10

nr

, nrC , n rC

the binomial coefficient ( )

!! !−

nr n r

for 0, +∈ ≤n r , r n

or ( ) ( )!!

1 1...− − +n n n rr

for 0, +∈ ∈ n r

4 Functions

4.1 f(x) the value of the function f at x

4.2 f : x

y the function f maps the element x to the element y

4.3 f–1 the inverse function of the function f

4.4 gf the composite function of f and g which is defined by

( )gf( = g f() )x x

4.5 x→lim

a f(x) the limit of f(x) as x tends to a

4.6 ∆x , δx an increment of x

4.7 ddyx

the derivative of y with respect to x

4.8 nd

d n

yx

the nth derivative of y with respect to x

4.9 ( )f ( ), f ( ),... f ( ),′ ′′ nx x x the first, second, …, nth derivatives of f(x) with respect to x

4.10 x, x, ... the first, second, … derivatives of x with respect to t

4.11 d∫ y x the indefinite integral of y with respect to x

4.12 db

y x∫a the definite integral of y with respect to x between the limits x = a and x = b


5 Exponentials and logarithmic functions

5.1 e base of natural logarithms

5.2 ex, exp x exponential function of x

5.3 loga x logarithm to the base a of x

5.4 ln x, loge x natural logarithm of x

6 Trigonometric functions

6.1

sin, cos, tancosec, sec, cot

the trigonometric functions

6.2

–1 –1 –1sin , cos , tanarcsin, arccos, arctan

the inverse trigonometric functions

6.3 ° degrees

6.4 rad radians

6 Trigonometric functions (Further Mathematics only)

6.5

–1 –1 –1cosec , sec , cotarccosec, arcsec, arccot

the inverse trigonometric functions

6.6

sinh, cosh, tanhcosech, sech, coth

the hyperbolic functions

6.7

–1 –1 –1

–1 –1 –1

sinh , cosh , tanh

cosec h , sec h , coth

arsinh, arcosh, artanharcosech, arsec h, arcoth

the inverse hyperbolic functions

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7 Complex numbers (Further Mathematics only)

7.1 i, j square root of –1

7.2 x + iy complex number with real part x and imaginary part y

7.3 ( )ir θ θ+cos sin modulus argument form of a complex number with modulus r and argument θ

7.4 z a complex number, z = x + iy = ( )ir θ θ+cos sin

7.5 Re(z) the real part of z, Re(z) =x

7.6 lm(z) the imaginary part of z, lm(z) = y

7.7 z the modulus of z, 2 2=z x + y

7.8 arg(z) the argument of z, arg(z) = θ, – π< θ ≤ π

7.9 z* the complex conjugate of z, x – iy

8 Matrices (Further Mathematics only)

8.1 M a matrix M

8.2 0 zero matrix

8.3 I identity matrix

8.4 M–1 the inverse of the matrix M

8.5 MT the transpose of the matrix M

8.6 ∆ , det M or M the determinant of the square matrix M

8.7 Mr image of column vector r under the transformation associated with the matrix M


9 Vectors

9.1 a, a , a

the vector a, a , a

; these alternatives apply throughout section 9

9.2 AB

the vector represented in magnitude and direction by the directed line segment AB

9.3 a a unit vector in the direction of a

9.4 i, j, k unit vectors in the directions of the Cartesian coordinate axes

9.5 a , a the magnitude of a

9.6 AB

, AB the magnitude of AB

9.7

ab

, +i ja b column vector and corresponding unit vector notation

9.8 r position vector

9.9 s displacement vector

9.10 v velocity vector

9.11 a acceleration vector

9 Vectors (Further Mathematics only)

9.12 a.b the scalar product of a and b

10 Differential equations (Further Mathematics only)

10.1 ω angular speed

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11 Probability and statistics

11.1 A, B, C, etc events

11.2 ∪A B union of the events A and B

11.3 ∩A B intersection of the events A and B

11.4 ( )P A probability of the event A

11.5 ′A complement of the event A

11.6 ( )P |A B probability of the event A conditional on the event B

11.7 X, Y, R, etc random variables

11.8 x, y, r, etc values of the random variables X, Y, R, etc

11.9 x1, x2, … values of observations

11.10 f1, f 2, … frequencies with which the observations x1, x2, …occur

11.11 p(x), P(X = x) probability function of the discrete random variable X

11.12 p1, p2, … probabilities of the values x1, x2, … of the discrete random variable X

11.13 E(X) expectation of the random variable X

11.14 Var(X) variance of the random variable X

11.15

has the distribution

11.16 B(n, p) binomial distribution with parameters n and p, where n is the number of trials and p is the probability of success in a trial

11.17 q q = 1 – p for binomial distribution

11.18 ( )2N ,µ σ Normal distribution with mean µ and variance 2σ

11.19 ( )N 0,1Z standard Normal distribution

11.20 φ probability density function of the standardised Normal variable with distribution ( )0,1N

11.21 Φ corresponding cumulative distribution function

11.22 µ population mean


11.23 2σ population variance

11.24 σ population standard deviation

11.25 x sample mean

11.26 s2 sample variance

11.27 s sample standard deviation

11.28 H0 null hypothesis

11.29 H1 alternative hypothesis

11.30 r product moment correlation coefficient for a sample

11.31 ρ product moment correlation coefficient for a population

12 Mechanics

12.1 kg kilogram(s)

12.2 m metre(s)

12.3 km kilometre(s)

12.4 m/s, m s–1 metre(s) per second (velocity)

12.5 m/s2, m s–2 metre(s) per second per second (acceleration)

12.6 F force or resultant force

12.7 N newton

12.8 Nm newton metre (moment of force)

12.9 t time

12.10 s displacement

12.11 u initial velocity

12.12 v velocity or final velocity

12.13 a acceleration

12.14 g acceleration due to gravity

12.15 µ coefficient of friction

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A2 Appendix 2 Mathematical formulae and identities

Students must use the following formulae and identities for AS and A-level Mathematics and Further Mathematics, without these formulae and identities being provided, either in these forms or in equivalent forms. These formulae and identities may only be provided where they are the starting point for a proof or as a result to be proved.

Pure mathematics

Quadratic equations ax2 + bx + c = 0 has roots 2 4

2− ± −b b ac

a

Laws of indices ≡x y x+ya a a –÷ ≡x y x ya a a

( ) ≡x y xya a a

Laws of logarithms log= ⇔nax a n = x for a > 0 and x > 0

( )log log log≡a a ax + y xy

log log log

− ≡

a a axx yy

log log ( )≡ ka ak x x

Coordinate geometry A straight line, gradient m passing through (x1, y1) has equation

1 1( )y y m x x− = −

Straight lines with gradients m1 and m2 are perpendicular when m1m2 = –1

Sequences General term of an arithmetic progression:

un = a + (n – 1)d

General term of a geometric progression: un = arn – 1


Trigonometry In the triangle ABC:

sine rule: sin sin sin A B C

= =a b c

cosine rule: 2 2 2 2 cos A= + −a b c bc

area: 1 sin 2

Cab

A+ A ≡2 2cos sin 1 2 2sec 1 tan≡A + A

2 2cosec 1 cot≡A + A

sin2 2sin cos≡A A A 2 2cos2 sincos≡ −A A A

2tan2tan

2tan1

≡−

AAA

Mensuration Circumference (C) and area (A) of a circle, radius r and diameter d.

2= π πC r = d 2= πA r

Pythagoras’ Theorem: In any right-angled triangle, where a, b and c are the lengths of the sides and c is the hypotenuse:

2 2 2c = a +b

Area of a trapezium: 12

( )+a b h where a and b are the lengths of the

parallel sides and h is their perpendicular separation Volume of a prism = area of cross section × length

For a circle or radius r, where an angle at the centre of θ radians subtends an arc of length s and encloses an associated sector of area A:

s = r θ

212

θ=A r

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Calculus and differential equations Differentiation

Function Derivative

xn nxn–1

sin kx k cos kx

cos kx –k sin kx

ekx kekx

ln x 1x

f(x) + g(x) f g( ) ( )+′ ′x x

f(x)g(x) f g gf( ) ( ) ( ) ( )+′ ′x x x x

( )f g( )x ( )f g g( ) ( )′ ′x x

Integration

Function Derivative

xn 11, 1

1+ + ≠ −

+nx c n

n

cos kx 1sin +kx c

k

sin kx 1cos +− kx c

k

ekx e1+kx c

k

1x

ln 0+ ≠x c, x

f g( ) ( )+′ ′x x f g c( ) ( )+ +x x

( )f g g( ) ( )′ ′x x ( )f g c( ) +x

Area under a curve d 0( )= ≥∫b

a

y x y


Vectors x y z x y z+ + = + +2 2 2( )i j k

Mechanics

Forces and equilibrium Weight = mass x g

Friction: µ≤F R

Newton’s second law in the form: F = ma

Kinematics For motion in a straight line with variable acceleration:

ddrv =t

d dd d

2

2=v ra =t t

d∫r = v t

d∫v = a t

Statistics

The mean of a set of data =∑ ∑∑

x fxx =

n f

The standard Normal variable

µσ−XZ = where ( )2N ,µ σX

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