Advanced Mathematics 3

553 Pages·1985·32.451 MB·English

by  C W Celia, A T F Nice, K F Elliott (auth.), Dr C. Plumpton (eds.)

Advanced
mathematics
3
Advanced
mathematics
3
C W Celia
formerly PrincipalLecturer in Mathematics,
City ofLondon Polytechnic
A T F Nice
Mathematics Department, Lady Eleanor Holles
School, Hampton;formerly PrincipalLecturer
in Mathematics, Middlesex Polytechnic
K F Elliott
formerly Head ofthe DivisionofMathematics
Education, Derby Lonsdale College ofHigher
Education
Consultant Editor: Dr C. Plumpton, Moderator
in Mathematics, University ofLondon School
Examinations Board:formerly Reader in
Engineering Mathematics;Queen Mary
College, London
M
MACMILLAN
© C.W.Celia,A.T.F.NiceandK.F.Elliott 1985
Allrightsreserved. Noreproduction,copyortransmission
ofthispublication maybemade withoutwrittenpermission.
Noparagraph ofthispublicationmaybereproduced,copiedor
transmitted savewithwrittenpermissionorinaccordancewith
theprovisionsoftheCopyright,DesignsandPatentsAct 1988,
orunderthetermsofanylicencepermitting limitedcopying
issuedbytheCopyrightLicensingAgency,33-4AlfredPlace,
London WClE7DP.
Anypersonwhodoes anyunauthorisedactinrelationto
thispublicationmaybeliabletocriminalprosecution and
civilclaimsfordamages.
Firstedition 1985
Reprinted 1986,1991
Publishedby
MACMILLAN EDUCATIONLTD
Houndmills,Basingstoke, Hampshire RG212XS
andLondon
Companiesandrepresentatives
throughout theworld
BritishLibrary Cataloguing inPublication Data
Celia, C.W.
Advancedmathematics.
3
I. Mathematics-1961
I.Title II.Nice,A.T.F.
III.Elliott,K.F.
510 QA39.2
ISBN978-0-333-34827-7 ISBN 978-1-349-06711-4(eBook)
DOI 10.1007/978-1-349-06711-4
/9
Contents
Preface Vll
1 The idea of proof
Propositions Implication Necessary and sufficient conditions
Relationship analysis Data sufficiency Solution of equations
Methods ofproof
2 Complex numbers 20
De Moivre's theorem Square roots and cube roots The nth roots
of unity Conjugate complex numbers The exponential function
Straight lines and circles in the Argand diagram The transforma-
tions w = az + b, w = liz, w = (az + b)/(cz + d)
3 Hyperbolic functions 72
Properties of functions Coshx Sinhx Tanhx Summary of
properties Inverse hyperbolic functions Applications to integra-
tion
4 Differential equations 100
Separable equations First-order linear differential equations
Linear differential equations with constant coefficients Integral
curves Applications Vector differential equations
5 Polar coordinates 162
The straight line The circle The angle cP between tangent and
radius vector Curve sketching Area ofa sector
6 Sequences and series: convergence 186
Convergence ofsequences and ofseries Tests for convergence of
seriesofpositive terms Comparisontest Ratiotest Convergence
of integrals Maclaurin's series Taylor series Repeated dif
ferentiation Leibnitz's theorem
7 Further integration and applications 232
Definite integrals Reduction formulae Length of arc Area of
surface of revolution Mean values First moments Theorems
of Pappus
8 Equations and inequalities 272
Roots of equations Further inequalities Inequalities in two
variables The calculus applied to inequalities
9 Numerical solution of differential equations 308
Isoclines The Euler method The modified Euler method
The Taylor series method Second-order equations Order of
convergence
10 Coordinate geometry 343
The rectangular hyperbola xy = c2 Tangent and normal The
hyperbola x2/a2 - y2/b2 = I Tangent and normal The ellipse
x2/a2 + y2/b2 = 1 Tangent and normal Change of axes The
equations ax? + 2hxy + by? = 0 and ax? + 2hxy + by? = 1
Tangents at the origin
11 Algebraic structure 396
Groups Symmetry groups Subgroups and cyclic groups Per
mutation groups Isomorphism Rings and fields
12 Vector spaces 422
Linear dependence Vector spaces
13 Determinants and matrices 435
Determinants of order 2 and order 3 Properties of matrices
Theinverseofamatrix Linear transformationsinthree dimensions
Orthogonal matrices Eigenvectors of a matrix Diagonalisation
of a matrix Quadratic forms
14 The vector product 496
The vector product Distributive law The vector product in com
ponent form Perpendiculardistance from a point to a straight line
The scalar triple product The vector triple product
Notation 519
Formulae 520
Answers 524
Index 543
vi Contents
Preface
This isthe third ofa series ofbooks written for students preparing for A level
Mathematics. Book I covers the essential core ofsixth-form mathematics now
accepted bythe GCE Boards,whileBook 2covers the applied mathematics, i.e.
the numerical methods, mechanics and probability, contained in most single
subject mathematics syllabuses.
Book 3covers the additional puremathematicsneeded bystudentstakingthe
double subject Mathematics and Further Mathematics, and by those taking
Pure Mathematics as a single subject.
Itmust beemphasisedthat thereaderisexpected to befamiliarwith the useof
a calculator, so that numerical work presents no difficulties.
The material isarranged underwell-known headingsand isorganised sothat
the teacher is free to follow his or her own preferred order oftreatment. The
chaptercontentsare listed andan indexisalso provided to make iteasyfor both
theteacherand the studentto referback rapidlytoany particulartopic.Forease
of reference, a listofthe notation used isgivenat the back ofthe book together
with a list offormulae.
The approach in Book 3 is the same as in Books I and 2, each topic being
developed mainly through worked examples. There is a brief introduction to
each new piece ofwork followed by worked examples and numerous simple
exercises to build up the student's technical skills and to reinforce his or her
understanding. Itishoped that this approach willenable the individual student
workingon hisor her own to make effectiveuseofthe booksand the teacherto
usethem with mixed ability groups. At the end ofeach chapter there are many
miscellaneousexamples, taken largely from pastAlevelexaminationpapers.In
addition to their value as examination preparation, these miscellaneous
examples are intended to give the student the opportunity to apply the
techniquesacquired from the exercisesthroughout thechapterto aconsiderable
range of problems ofthe appropriate standard.
We are most grateful to the University of London University School
Examinations Board (L), the Associated ExaminingBoard(AEB), the Univers
ityofCambridgeLocal ExaminationSyndicate(C)and the Joint Matriculation
Board (JMB) for giving us permission to use questions from their past
examination papers.
Weare also grateful to the staffofMacmillanEducationfor the patiencethey
have shown and the help they have given us in the preparation ofthese books.
c.
W. Celia
A. T. F. Nice
K. F. Elliott
viii Preface
1 The idea of proof
1.1 Propositions
Any mathematical proofdepends on valid arguments, and so a discussion of
proofmust consider logical relations between statements.
A statement which is either true or false is known as a proposition. A true
proposition has a truth value T;a false proposition has a truth value F.
Two propositionsp and qcan be combined to form compound propositions.
The conjunctionp /\ q, read as'p and q', istrue when and only whenp and q
are both true.
The disjunction p Vq, read as 'p or q', is true when eitherp or q is true, or
when both are true.
The symbol - p, read as 'notp', stands for the negation ofp. Whenp istrue,
- p is false, and when p is false, p is true.
-s
The truthvaluesofp/\q,PVqand - p fordifferenttruthvaluesofpand qare
shown in the truth table below:
P q p/\q pVq
T T T T F
T F F T F
F T F T T
F F F F T
Two propositionsare said to beequal ifthey have thesame truth values inall
possible cases. For example, p /\ q and q /\ Pare clearly equal.
Considerthe propositions(- p) V(- q)and - (p /\q).Whenp and qare both
true,each ofthesepropositionsisfalse.In allothercases,they are both true.As
the following truth table shows, these two propositions are equal:
P q (-p) V(-q) -(p/\q)
T T F F
T F T T
F T T T
F F T T
Example
Letp be the proposition that,in the triangle ABC, AB = CA, and let q be the
proposition that AB = BC.
When p and qare both true, AB = BC = CA. Hence whenpf\ qistrue, the
triangle isequilateral.
When either p or q is true, or both are true, at least two ofthe sides ofthe
triangle are equal. Hence when p Vq is true, the triangle iscertainly isosceles,
and possibly equilateral.
The proposition('"p) V('"q) istrue when eitherAB #- CA or AB #-BC, or
both, i.e. when the triangle is not equilateral.
The proposition",(pf\ q)isthenegationofthepropositionthatAB = BC =
CA, i.e. the negation ofthe proposition that the triangle is equilateral.
Open statements
A statement involving a variable x iscalled an open statement.
An open statement becomes a propositionwhen x isgivena particularvalue.
Thus the equation x2 - 5x + 4 = 0, where x E IR, is an open statement.
Thisstatementistrue whenx = Ior whenx = 4,but otherwiseitisfalse.The
set{I, 4} isknown asthe truthsetoftheequation. Similarly,the truthsetofthe
inequality x2 < 1,where x E IR, is the set {x:-l < x < I}.
Let P and Qbe the truth sets of the open statements p and q.
Thenthe truthset ofpf\ qistheintersection P1\ Q,and the truthsetofp Vq
is the union P u Q.
Anequationinx which istrue for allvalues ofx iscalledan identity. Its truth
set will be the set IR. For example,
x2 - I == (x + I)(x - I),
sin 2x == 2 sin x cos x,
x4 + 1 == (x2 + J 2x + I)(x2 J 2x + I).
-
Note that the equals sign is replaced by the identity sign.
Exercises 1.1
1 Given thatpisthe proposition'Jackistall' and thatqisthe proposition 'Jill
is slim', express the following propositions in terms ofp and q:
(a) Either Jack is short or Jill is slim (or both),
(b) Jack is tall and Jill is not slim,
(c) Jill is slim and Jack is not tall,
(d) Either Jill is not slim or Jack is tall (or both).
2 Draw up truth tables for",p f\ q and p V '"q.
3 Show that the propositions '"p Vq and '"(p f\ '"q) are equal.
4 Given that x is real, find the truth sets ofthe propositionsp f\ q and p Vq
when p and q are the following open statements:
2 Advanced mathematics 3