A Level Further Mathematics for AQA: Discrete Student Book (AS/A Level)

268 Pages·2017·40.536 MB·English

by  Jan Dangerfield

Brighter Thinking
A Level Further Mathematics
for AQA
Discrete Student Book (AS/A Level)
Jan Dangerfield
Contents
Introduction
How to use this resource
1 Graphs
1: The language of graphs
2: Properties of graphs
3: Planarity and isomorphism
Mixed practice 1
2 Networks
1: Network optimisation problems using spanning trees
2: Route inspection problems
3: Travelling salesperson problem
Mixed practice 2
3 Network flows
1: Flows and cuts
2: Variations on flow problems
Mixed practice 3
4 Linear programming
1: Formulating constrained optimisation problems
2: Graphical solutions
3: The simplex algorithm
Mixed practice 4
Cross-topic review exercise 1
5 Critical path analysis
1: Analysing precedence networks
2: Scheduling
Mixed practice 5
6 Game theory for zero-sum games
1: Games with stable solutions
2: Mixed strategies
Mixed practice 6
7 Binary operations
1: Properties of binary operations
2: Using Cayley tables
Mixed practice 7
8 Groups
1: The group axioms
2: Generators of a group
Mixed practice 8
Cross-topic review exercise 2
Practice paper
Set notation
Answers
Acknowledgements
Copyright
Introduction
You have probably been told that mathematics is very useful, yet it can often seem like a lot of techniques
that just have to be learnt to answer examination questions. You are now getting to the point where you
will start to see where some of these techniques can be applied in solving real problems. However, we
hope that, as well as seeing how maths can be useful, anyone working through this resource will realise
that it can also be incredibly frustrating, surprising and ultimately beautiful.
The book is woven around three key themes from the new curriculum.
Proof
Maths is valued because it trains you to think logically and communicate precisely. At a high level, maths is
far less concerned about answers and more about the clear communication of ideas. It is not about being
neat – although that might help! It is about creating a coherent argument that other people can easily
follow but find difficult to refute. Have you ever tried looking at your own work? If you cannot follow it
yourself it is unlikely anybody else will be able to understand it. In maths we communicate using a variety
of means – feel free to use combinations of diagrams, words and algebra to aid your argument. And once
you have attempted a proof, try presenting it to your peers. Look critically (but positively) at some other
people’s attempts. It is only through having your own attempts evaluated and trying to find flaws in other
proofs that you will develop sophisticated mathematical thinking. This is why we have included lots of
common errors in our Work it out boxes – just in case your friends don’t make any mistakes!
Problem solving
Maths is valued because it trains you to look at situations in unusual, creative ways, to persevere and to
evaluate solutions along the way. We have been heavily influenced by a great mathematician and maths
educator George Polya, who believed that students were not just born with problem-solving skills – they
were developed by seeing problems being solved and reflecting on their solutions before trying similar
problems. You may not realise it but good mathematicians spend most of their time being stuck. You need
to spend some time on problems you can’t do, trying out different possibilities. If after a while you have not
cracked it then look at the solution and try a similar problem. Don’t be disheartened if you cannot get it
immediately – in fact, the longer you spend puzzling over a problem the more you will learn from the
solution. You may never need to integrate a rational function in the future, but we firmly believe that the
problem-solving skills you will develop by trying it can be applied to many other situations.
Modelling
Maths is valued because it helps us solve real-world problems. However, maths describes ideal situations
and the real world is messy! Modelling is about deciding on the important features needed to describe the
essence of a situation and turning that into a mathematical form, then using it to make predictions,
compare to reality and possibly improve the model. In many situations the technical maths is actually the
easy part – especially with modern technology. Deciding which features of reality to include or ignore and
anticipating the consequences of these decisions is the hard part. Yet it is amazing how some fairly drastic
assumptions – such as pretending a car is a single point or that people’s votes are independent – can result
in models that are surprisingly accurate.
More than anything else this book is about making links – links between the different chapters, the topics
covered and the themes above, links to other subjects and links to the real world. We hope that you will
grow to see maths as one great complex but beautiful web of interlinking ideas.
Maths is about so much more than examinations, but we hope that if you absorb these ideas (and do
plenty of practice!) you will find maths examinations a much more approachable and possibly even
enjoyable experience. However, always remember that the results of what you write down in a few hours
by yourself in silence under exam conditions are not the only measure you should consider when judging
your mathematical ability – it is only one variable in a much more complicated mathematical model!
How to use this resource
Throughout this resource you will notice particular features that are designed to aid your learning. This
section provides a brief overview of these features.
In this chapter you will learn how to:
use the language of graphs
recognise and use some special types of graph
use graphs to model problems.
If you are following the A Level course you will also learn how to:
understand and use the term planarity
understand and use the term isomorphism.
Learning objectives
A short summary of the content that you will learn in each chapter.
Before you start...
GCSE and A Level Mathematics You should be familiar with the 1 Which of these
Student Book 1 terms integer, rational number, real numbers belong to the
number and the notation for these set , the set of
sets. positive integers?
Further Mathematics Student Book You should know how to carry out 2 Calculate:
1, Chapter 7 matrix addition and matrix
a
multiplication.
b
Before you start
Points you should know from your previous learning and questions to check that you’re ready to start the
chapter.
WORKED EXAMPLE
The left-hand side shows you how to set out your working. The right-hand side explains the more
difficult steps and helps you understand why a particular method was chosen.
WORK IT OUT
Identify the correct solution and find the mistakes in the two incorrect solutions.
Key point
A summary of the most important methods, facts and formulae.
Common error
Specific mistakes that are often made. These typically appear next to the point in the Worked
example where the error could occur.
Tip
Useful guidance, including ways of calculating or checking answers and using technology.
Each chapter ends with a Checklist of learning and understanding and a Mixed practice exercise,
which includes past paper questions marked with the icon .
Between some chapters, you will find extra sections that bring together topics in a more synoptic way.
CROSS-TOPIC REVIEW EXERCISE
Questions covering topics from across the preceding chapters, testing your ability to apply what
you have learned.
You will find practice paper questions towards the end of the resource, as well as a glossary of key terms
(picked out in colour within the chapters) and answers.
Maths is all about making links, which is why throughout this resource you will find signposts emphasising
connections between different topics, applications and suggestions for further research.
Rewind
Reminders of where to find useful information from earlier in your study.
Fast forward
Links to topics that you may cover in greater detail later in your study.
Did you know?
Interesting or historical information and links with other subjects to improve your awareness
about how mathematics contributes to society.
Colour-coding of exercises
The questions in the exercises are designed to provide careful progression, ranging from basic fluency to
practice questions. They are uniquely colour-coded, as shown here.
1 Show that, for the graph in question 1, the sum of the degrees of the vertices is twice the number
of edges.
2 A graph has vertices and edges. What is the sum of the degrees of the vertices?
3 A graph is drawn with vertices labelled , , , , and . An edge is drawn between two vertices if
the larger is a multiple of the smaller.
a Draw this graph.
b List a cycle in this graph.
c Write down a trail that starts at and travels through every vertex in the graph once.
4 Explain why it is impossible to draw a graph with exactly five vertices that have degrees , , ,
and .
5 A simple graph has six vertices. The degrees of the vertices are , , , , and .
6 Explain why there is no simple graph with exactly four vertices with degrees , , and .
7 There are different trees with vertices. Draw an example of each and give its degrees.
8 A connected graph has vertices and edges. Explain why the graph must be simple.
9 A connected graph is semi-Eulerian if exactly two of its vertices are of odd degree.
a A graph is drawn with vertices and edges. What is the sum of the degrees of the vertices?
b Draw a simple semi-Eulerian graph with exactly vertices and edges, in which exactly one of
the vertices has degree .
c Draw a simple semi-Eulerian graph with exactly vertices that is also a tree.
d A simple graph has vertices. The graph has two vertices of degree . Explain why the graph can
have no vertex of degree .
10 A graph has vertices. The vertex degrees are .
a How many edges does the graph have?
b Explain how the vertex degrees show that the graph is not simple-connected.
c How many different graphs fit this description?
Black – practice questions that come in several parts, each with subparts i and ii. You only need
attempt subpart i at first; subpart ii is essentially the same question, which you can use for further
practice if you got part i wrong, for homework, or when you revisit the exercise during revision.
Green – practice questions at a basic level.
Blue – practice questions at an intermediate level.
Red – practice questions at an advanced level.
Purple – challenging questions that apply the concept of the current chapter across other areas of
maths.
Yellow – designed to encourage reflection and discussion.
– indicates content that is for A Level students only.
– indicates a question that requires a calculator.
1 Graphs
In this chapter you will learn how to:
​• use the language of graphs
​• recognise and use some special types of graph
​• use graphs to model problems.
If you are following the A Level course you will also learn how to:
​• understand and use the term planarity
​• understand and use the term isomorphism.
Before you start…
GCSE Mathematics You should be able to use a mapping to 1 What relationship is shown
represent relationships. in this mapping?
Section 1: The language of graphs
Key point 1.1
A graph is a set of vertices connected by edges.
​• A vertex is shown as a point on a graph. Vertices are sometimes labelled, and sometimes not.
​• An edge is a line or curve with a vertex at each end.
WORKED EXAMPLE 1.1
a List the vertices in this graph.
b List the edges in the graph.
a The vertices are , ,
Not all graphs will have labelled vertices.
, and .
b and You can identify an edge by the vertices at its ends. Edge is the
. same as edge for example.
Key point 1.2
​• A walk is a set of edges joined end to end, so the end vertex of one edge is the start vertex of
the next.
​• A trail is a walk in which no edges are repeated. Vertices can be repeated in a trail, although
often they are not.
​• A cycle is a trail that starts and finishes at the same vertex. Other than the start being the
same as the finish, vertices are not repeated in a cycle.
Fast forward
The terms trail and cycle will be used in route inspection problems and the travelling salesperson
problem in Chapter 2.
WORKED EXAMPLE 1.2
Using the graph in Worked example 1.1:
a give an example of a walk that passes through every vertex
b give two examples of trails, one with a repeated vertex and one with no repeated vertices
c explain why is not a trail
d give an example of a cycle
e explain why is not a cycle.
a e.g. This is not a trail because the edge is repeated.
b An example of a trail with a repeated
This trail repeats the vertex .
vertex is .