A2 Pure Mathematics (C3 and C4)

426 Pages·2004·4.566 MB·English

by  Roger Porkess, Val Hanrahan, Peter Secker, et al

THIRD
EDITION
A2 Pure Mathematics
Catherine Berry
Val Hanrahan
Roger Porkess
Peter Secker
Series Editor: Roger Porkess
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MEI Structured Mathematics
Mathematics is not only a beautiful and exciting subject in its own right but also
one that underpins many other branches of learning. It is consequently
fundamental to the success of a modern economy.
MEI Structured Mathematics is designed to increase substantially the number of
people taking the subject post-GCSE, by making it accessible, interesting and
relevant to a wide range of students.
It is a credit accumulation scheme based on 45 hour modules which may be taken
individually or aggregated to give Advanced Subsidiary (AS) and Advanced GCE
(A Level) qualifications in Mathematics, Further Mathematics and related
subjects (like Statistics). The modules may also be used to obtain credit towards
other types of qualification.
The course is examined by OCR (previously the Oxford and Cambridge Schools
Examination Board) with examinations held in January and June each year.
MEI Structured Mathematics
NM NC
FP1 FP2 FP3
AM
C2 C3
G
C C1 C4 DE H
S E
E M1 M2 M3 M4
S1 S2 S3 S4
FAM
D2
C, FP -Pure mathematics D -Decision mathematics
D1 M -Mechanics N -Numerical analysis
S -Statistics DE -Differential Equations
DC FSMQs Additional mathematics, Foundations of Advanced
Mathematics
This is one of the series of books written to support the course. Its position
within the whole scheme can be seen in the diagram above.
iii
MathematicsinEducationandIndustryisacurriculumdevelopmentbodywhich
aimstopromotethelinksbetweenEducationandIndustryinMathematicsat
secondarylevel,andtoproducerelevantexaminationandteachingsyllabusesand
supportmaterial.Sinceitsfoundationinthe1960s,MEIhasprovidedsyllabusesfor
GCSE(orOLevel),AdditionalMathematicsandALevel.
FormoreinformationaboutMEIStructuredMathematicsorothersyllabusesand
materials,writetoMEIOffice,AlbionHouse,MarketPlace,Westbury,Wiltshire,
BA133DEorvisitwww.mei.org.uk.
iv
Introduction
The twelve chapters of this book cover the pure mathematics required for the A2
subject criteria. The material is divided into the two units (or modules) for MEI
Structured Mathematics: C3, Methods for Advanced Mathematicsand C4,
Applications of Advanced Mathematics. It is the second in a series of pure
mathematics books for AS and A Levels in Mathematics and Further Mathematics.
Since their total content is the same, this book also covers the requirements of all
the other specifications for A2 Mathematics, and it is also suitable for other
courses at this level.
Throughout the series the emphasis is on understanding rather than mere
routine calculations, but the varied exercises do nonetheless provide plenty of
scope for practising basic techniques. Extensive on-line support is available via
the MEI site, www.mei.org.uk.
This book is part of the third edition of this series and is written on the
assumption that you have already studied AS Mathematics. Much of its content
was previously in Pure Mathematics 2and3but it has now been reorganised to
meet the requirements of the new specification being first taught in September
2004. Thanks are due to Val Hanrahan for her work in preparing the new edition
and for her original contributions. Thanks are also due to the various
examination boards who have given permission for their past questions to be
included in the exercises.
Roger Porkess
Series Editor
v
Key to symbols in this book
●
? This symbol means that you may want to discuss a point with your
teacher. If you are working on your own there are answers in the back of
the book. It is important, however, that you have a go at answering the
questions before looking up the answers if you are to understand the
mathematics fully.
! This is a warning sign. It is used where a common mistake,
misunderstanding or tricky point is being described.
This is the ICT icon. It indicates where you should use a graphic calculator
or a computer.
●
This symbol invites you to join in a discussion about proof.The answers to
these questions are given in the back of the book.
●
This symbol and a dotted line down the right-hand side of the page
indicates material which is beyond the criteria for the unit but which is
included for completeness.
Harder questions are indicated with stars. Many of these go beyond the
usual examination standard.
vi
Contents
C3 Methods for Advanced Mathematics
1
1 Proof
2
Proof by direct argument 2
Proof by exhaustion 3
Proof by contradiction 4
Disproof by the use of a counter-example 6
2 Natural logarithms and exponentials
8
A new function 9
The natural logarithm function 11
The exponential function 12
3 Functions
19
The language of functions 19
Using transformations to sketch the curves of functions 25
Composite functions 36
Inverse functions 39
Inverse trigonometrical functions 45
Even, odd and periodic functions 49
The modulus function 56
Curve sketching 60
4 Techniques for differentiation
63
The chain rule 63
The product rule 68
The quotient rule 71
Differentiating an inverse function 77
Differentiating natural logarithms and exponentials 82
Differentiating sinx and cosx 91
Differentiating functions defined implicitly 96
vii
5 Techniques for integration
103
Integration by substitution 103
Integrals involving the exponential function 110
Integrals involving the natural logarithm function 111
Extending the domain for logarithmic integrals 113
Integrating sinx and cosx 123
Integration by parts 125
6 Numerical solution of equations
135
Interval estimation – change of sign methods 136
Fixed point iteration 143
Rearranging the equation f(x) = 0 into the form x = g(x) 143
The Newton–Raphson method 150
C4 Applications of Advanced Mathematics
155
7 Algebra
156
The general binomial expansion 157
Review of algebraic fractions 166
Partial fractions 173
8 Trigonometry
183
Reciprocal trigonometrical functions 184
Compound-angle formulae 187
Double-angle formulae 192
The factor formulae 197
The forms r cos(θ ± α), r sin(θ ± α) 201
Small-angle approximations 210
The general solutions of trigonometrical equations 218
Using trigonometrical identities in integration 220
9 Parametric equations
224
Graphs from parametric equations 226
Finding the equation by eliminating the parameter 227
The parametric equation of a circle 231
The parametric equations of other standard curves 232
Parametric differentiation 238
viii
10 Further techniques for integration
253
Finding volumes by integration 254
The use of partial fractions in integration 261
General integration 266
Integrals you cannot do 269
11 Vectors
275
Vectors 275
Co-ordinate geometry using vectors: two dimensions 289
Co-ordinate geometry using vectors: three dimensions 303
12 Differential equations
335
Forming differential equations from rates of change 336
Solving differential equations 341
Answers
358
Chapter 1 358
Chapter 2 358
Chapter 3 359
Chapter 4 369
Chapter 5 376
Chapter 6 380
Chapter 7 383
Chapter 8 386
Chapter 9 392
Chapter 10 397
Chapter 11 403
Chapter 12 410
Index
413
ix