Cambridge Additional Mathematics

504 Pages·2014·17.09 MB·English

by  Michael Haese

Haese Mathematics
specialists in mathematics publishing
Endorsed by Cambridge International Examinations
Cambridge
Additional
Mathematics
IGCSE® (0606)
O Level (4037)
Michael Haese
Sandra Haese
Mark Humphries
Chris Sangwin
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CAMBRIDGEADDITIONALMATHEMATICS(0606)(4037)
MichaelHaese B.Sc.(Hons.),Ph.D.
SandraHaese B.Sc.
MarkHumphries B.Sc.(Hons.)
ChrisSangwin M.A.,M.Sc.,Ph.D.
HaeseMathematics
152RichmondRoad,Marleston, SA5033,AUSTRALIA
Telephone: [phone], Fax: [phone]
Email: [e-mail]
Web: www.haesemathematics.com.au
NationalLibraryofAustraliaCardNumber&ISBN 978-1-921972-42-3
©Haese&HarrisPublications2014
PublishedbyHaeseMathematics
152RichmondRoad,Marleston, SA5033,AUSTRALIA
FirstEdition 2014
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Acknowledgements: The publishers acknowledge the cooperation of Oxford University Press, Australia,
for the reproduction of material originally published in textbooks produced in association with
HaeseMathematics.
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any accidental infringement where copyright has proved untraceable. They would be pleased to come to a
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FOREWORD
This book has been written to cover the ‘Cambridge O Level Additional Mathematics (4037)’ and the
‘CambridgeIGCSE®AdditionalMathematics(0606)’coursesoveraone-yearperiod.
These syllabuses enable learners to extend the mathematics skills, knowledge, and understanding developed
in the Cambridge IGCSE or O Level Mathematics courses, and use skills in the context of more advanced
techniques.
ThesyllabuseshaveaPureMathematicsonlycontentwhichenableslearnerstoacquireasuitablefoundation
in mathematics for further study in the subject. Knowledge of the content of the Cambridge IGCSE or O
LevelMathematicssyllabus(oranequivalentsyllabus)isassumed.
Learnerswhosuccessfullycompletethesecoursesgainlifelongskills,including:
• thefurtherdevelopmentofmathematicalconceptsandprinciples
(cid:129) anabilitytosolveproblems,presentsolutionslogically,andinterpretresults.
This book is an attempt to cover, in one volume, the content outlined in the Cambridge O LevelAdditional
Mathematics (4037) and Cambridge IGCSE Additional Mathematics (0606) syllabuses. The book can be
usedasapreparationforGCEAdvancedLevelMathematics. ThebookhasbeenendorsedbyCambridge.
Toreflecttheprinciplesonwhichthecourseisbased,wehaveattemptedtoproduceabookandCDpackage
that embraces understanding and problem solving in order to give students different learning experiences.
Reviewexercisesappearattheendofeachchapter. Answersaregivenattheendofthebook,followedbyan
index.
The interactive CD contains Self Tutor software (see p. 5), geometry and graphics software,
demonstrations and simulations. The CD also contains the text of the book so that students can load it on a
homecomputerandkeepthetextbookatschool.
The examinations for Cambridge Additional Mathematics are in the form of two papers. Many of the
problems in this textbook have been written to reflect the style of the examination questions.The questions,
workedsolutionsandcommentsthatappearinthebookandCDwerewrittenbytheauthors.
Thebookcanbeusedasaschemeofworkbutitisexpectedthattheteacherwillchoosetheorderoftopics.
Exercises in the book range from routine practice and consolidation of basic skills, to problem solving
exercisesthatarequitedemanding.
Inthischangingworldofmathematicseducation,webelievethatthecontextualapproachshowninthisbook
will enhance the students’ understanding, knowledge and appreciation of mathematics, and its universal
application.
Wewelcomeyourfeedback.
Email: [e-mail]
Web: www.haesemathematics.com.au PMH,SHH,MH,CS
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ABOUT THE AUTHORS
Michael Haese completed a BSc at the University of Adelaide, majoring in Infection and
Immunity, and Applied Mathematics. He then completed Honours in Applied Mathematics,
and a PhD in high speed fluid flows. He has a keen interest in education and a desire to
see mathematics come alive in the classroom through its history and relationship with other
subject areas. Michael has been the principal editor for Haese Mathematics since 2008.
SandraHaesecompletedaBScattheUniversityofAdelaide,majoringinPureMathematics
and Statistics. She taught mathematics at Underdale High School and later at Westminster
School in Adelaide. In 1979, Sandra’s husband Bob Haese began to write textbooks for
mathematics students at high schools, and Sandra assumed the role of proof reader. She
continues to work for Haese Mathematics as an editor and proof reader, and she produces
muchoftheaudioworkfortheSelfTutorsoftware.In2007shewasawardedLifeMembership
of the Mathematics Association of South Australia.
Mark Humphries completed an honours degree in Pure Mathematics and an Economics
degree at the University of Adelaide. His mathematical interests include public key
cryptography and number theory. He has been working at Haese Mathematics since 2006.
ChrisSangwincompletedaBAinMathematicsattheUniversityofOxford,andanMScand
PhD in Mathematics at the University of Bath. He spent thirteen years in the Mathematics
Department at the University of Birmingham,and from 2000 - 2011 was seconded half time
to the UK Higher Education Academy “Maths Stats and OR Network” to promote learning
and teaching of university mathematics. He was awarded a National Teaching Fellowship in
2006, and is now a Senior Lecturer in MathematicsEducation in the MathematicsEducation
Centre at Loughborough University. His learning and teaching interests include automatic
assessment of mathematics using computer algebra, and problem solving using the Moore
method and similar student-centredapproaches.
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USING THE INTERACTIVE CD
TheinteractiveStudentCDthatcomeswiththisbookisdesignedforthosewho
wThaentCtoDuitciloisnettheacthanpoploeagrysitnhtreoaucghhinogutatnhdelebaoronkindgeMnoattehseamnaaticctsiv.elinkontheCD. demonstrationsH(cid:129)garapehinsgseoftwaMre(cid:129)asimtuhlatieonsm(cid:129)reavistioincgsames(cid:129)printouts
IINNTTEERRAACCTTIIVVEESSTTUUDDEENNTTCCDD
SimplyclickontheiconwhenrunningtheCDtoaccessalargerangeofinteractive
featuresthatincludes: includSeselfTutor www.haesemathematics.com.au EEInnCxtadaemmornribnasrateiidtodigonebnasly
(cid:129) printableworksheets CaAmbdriddgeitional IGCSE®(0606),
OLevel(4037)
(cid:129) graphingpackages Mathematics
INTERACTIVELINK
(cid:129) demonstrations ©2014
(cid:129) simulations
(cid:129) revisiongames
(cid:129) SELFTUTOR
SELF TUTOR is an exciting feature of this book.
The Self Tutor icon on each worked example denotes an active link on the CD.
Simply ‘click’ on the Self Tutor (or anywhere in the example box) to access the worked
example, with a teacher’s voice explaining each step necessary to reach the answer.
Play any line as often as you like. See how the basic processes come alive using movement and
colour on the screen.
Ideal for students who have missed lessons or need extra help.
Example 10 Self Tutor
Find the two angles µ on the unit circle, with 06µ 62¼, such that:
a cosµ = 1 b sinµ = 3 c tanµ =2
3 4
a cos¡1(1)¼1:23 b sin¡1(3)¼0:848 c tan¡1(2)¼1:11
3 4
y y y
1 1 1
Er
-1 O 1 x -1 O 1 x -1 O 1 x
Qe
-1 -1 -1
) µ ¼1:23 or 2¼¡1:23 ) µ ¼0:848 or ¼¡0:848 ) µ ¼1:11 or ¼+1:11
) µ ¼1:23 or 5:05 ) µ ¼0:848 or 2:29 ) µ ¼1:11 or 4:25
SeeChapter8,Theunitcircleandradianmeasure,page209
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6
SYMBOLS AND NOTATION USED IN THIS BOOK
N the set of natural numbers, f1, 2, 3, ....g
Z the set of integers, f0, §1, §2, §3, ....g
Z+ the set of positive integers, f1, 2, 3, ....g
Q the set of rational numbers
Q + the set of positive rational numbers, fx2Q , x>0g
R the set of real numbers
R+ the set of positive real numbers, fx2R, x>0g
[a, b] the closed interval fx2R :a6x6bg
[a, b) the interval fx2R :a6x<bg
(a, b] the interval fx2R :a<x6bg
(a, b) the open interval fx2R :a<x<bg
fx , x , ....g the set with elements x , x , ....
1 2 1 2
n(A) the number of elementsin the finite set A
fx: .... the set of all x such that
2 is an element of
2= is not an elementof
? or f g the empty set
U the universal set
[ union
\ intersection
µ is a subset of
½ is a proper subset of
* is not a subset of
6½ is not a proper subset of
A0 the complementof the set A
an1, pna a to the power of 1, nth root of a (if a>0 then pna>0)
n
1 p p
a2, a a to the power 1, square root of a (if a>0 then a>0)
2 ½
x for x>0, x2R
jxj the modulusor absolutevalue of x, that is
¡x for x<0, x2R
´ identity or is equivalentto
¼ is approximatelyequal to
n! n factorialfor n2N (0!=1)
¡ ¢
n the binomialcoefficient n! for n, r2N, 06r6n
r
r!(n¡r)!
> is greater than
¸ or > is greater than or equal to
< is less than
· or 6 is less than or equal to
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7
Pn
ai a1+a2+::::+an
i=1
f functionf
f : x7!y f is a functionunder which x is mapped to y
f(x) the image of x under the function f
f¡1 the inverse functionof the functionf
g±f, gf the compositefunctionof f and g
lim f(x) the limit of f(x) as x tends to a
x!a
dy
the derivativeof y with respect to x
dx
d2y
the second derivativeof y with respect to x
dx2
f0(x) the derivativeof f(x) with respect to x
f00(x) the second derivativeof f(x) with respect to x
R
y dx the indefiniteintegral of y with respect to x
R
b
y dx the definite integral of y with respect to x for values of x betweena and b
a
e base of natural logarithms
ex exponentialfunctionof x
lgx logarithmof x to base 10
lnx natural logarithm of x
log x logarithmto the base a of x
a
sin, cos, tan, the circular functions
cosec, sec, cot
A(x, y) the point A in the plane with Cartesian coordinatesx and y
8
<the line segmentwith endpoints A and B
AB the distance from A to B
:
the line containingpoints A and B
b
A the angle at A
b
CAB the angle between CA and AB
4ABC the triangle whose vertices are A, B, and C
a the vector a
¡!
AB the vector representedin magnitudeand directionby the directed line segment
from A to B
jaj the magnitudeof vector a
¡! ¡!
jABj the magnitudeof AB
i, j unit vectors in the directionsof the Cartesian coordinateaxes
M a matrix M
M¡1 the inverse of the square matrix M
detM the determinantof the square matrix M
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8 Table of contents
TABLE OF CONTENTS G Exponentialfunctions 118
H Thenaturalexponentialex 123
Reviewset4A 125
Reviewset4B 127
SYMBOLSANDNOTATION
5 LOGARITHMS 129
USEDINTHISBOOK 6
A Logarithmsinbase10 130
B Logarithmsinbasea 133
1 SETSANDVENNDIAGRAMS 11
C Lawsoflogarithms 135
A Sets 12 D Logarithmicequations 138
B Intervalnotation 15 E Naturallogarithms 142
C Relations 17 F Solvingexponentialequations
D Complementsofsets 18 usinglogarithms 145
E Propertiesofunionandintersection 20 G Thechangeofbaserule 147
F Venndiagrams 21 H Graphsoflogarithmicfunctions 149
G Numbersinregions 26 Reviewset5A 152
H ProblemsolvingwithVenndiagrams 28 Reviewset5B 154
Reviewset1A 31
Reviewset1B 33 6 POLYNOMIALS 155
A Realpolynomials 156
2 FUNCTIONS 35
B Zeros,roots,andfactors 162
A Relationsandfunctions 36 C TheRemaindertheorem 167
B Functionnotation 40 D TheFactortheorem 169
C Domainandrange 43 Reviewset6A 173
D Themodulusfunction 46 Reviewset6B 173
E Compositefunctions 49
F Signdiagrams 51 7 STRAIGHTLINEGRAPHS 175
G Inversefunctions 54
A Equationsofstraightlines 177
Reviewset2A 60
B Intersectionofstraightlines 183
Reviewset2B 61
C Intersectionofastraightlineandacurve 186
D Transformingrelationshipsto
3 QUADRATICS 63
straightlineform 187
A Quadraticequations 65 E Findingrelationshipsfromdata 192
B Quadraticinequalities 72 Reviewset7A 197
C Thediscriminantofaquadratic 73 Reviewset7B 199
D Quadraticfunctions 75
E Findingaquadraticfromitsgraph 87 8 THEUNITCIRCLEANDRADIAN
F Wherefunctionsmeet 91 MEASURE 201
G Problemsolvingwithquadratics 93
A Radianmeasure 202
H Quadraticoptimisation 95
B Arclengthandsectorarea 205
Reviewset3A 98
C Theunitcircleandthetrigonometricratios 208
Reviewset3B 99
D Applicationsoftheunitcircle 213
E Multiplesof "y and "r 217
4 SURDS,INDICES,AND
F Reciprocaltrigonometricratios 221
EXPONENTIALS 101
Reviewset8A 221
A Surds 102 Reviewset8B 222
B Indices 107
C Indexlaws 108 9 TRIGONOMETRICFUNCTIONS 225
D Rationalindices 111
A Periodicbehaviour 226
E Algebraicexpansionandfactorisation 113
B Thesinefunction 230
F Exponentialequations 116
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Table of contents 9
C Thecosinefunction 236 I Derivativesofexponentialfunctions 355
D Thetangentfunction 238 J Derivativesoflogarithmicfunctions 359
E Trigonometricequations 240 K Derivativesoftrigonometricfunctions 361
F Trigonometricrelationships 246 L Secondderivatives 363
G Trigonometricequationsinquadraticform 250 Reviewset13A 365
Reviewset9A 251 Reviewset13B 366
Reviewset9B 252
14 APPLICATIONSOFDIFFERENTIAL
10 COUNTINGANDTHEBINOMIAL CALCULUS 367
EXPANSION 255
A Tangentsandnormals 369
A Theproductprinciple 256 B Stationarypoints 375
B Countingpaths 258 C Kinematics 380
C Factorialnotation 259 D Ratesofchange 388
D Permutations 262 E Optimisation 393
E Combinations 267 F Relatedrates 399
F Binomialexpansions 270 Reviewset14A 402
G TheBinomialTheorem 273 Reviewset14B 405
Reviewset10A 277
Reviewset10B 278 15 INTEGRATION 409
A Theareaunderacurve 410
11 VECTORS 279
B Antidifferentiation 415
A Vectorsandscalars 280 C Thefundamentaltheoremofcalculus 417
B Themagnitudeofavector 284 D Integration 422
C Operationswithplanevectors 285 E Rulesforintegration 424
D Thevectorbetweentwopoints 289 F Integrating f(ax+b) 428
E Parallelism 292 G Definiteintegrals 431
F Problemsinvolvingvectoroperations 294 Reviewset15A 434
G Lines 296 Reviewset15B 435
H Constantvelocityproblems 298
Reviewset11A 302 16 APPLICATIONSOFINTEGRATION 437
Reviewset11B 303
A Theareaunderacurve 438
B Theareabetweentwofunctions 440
12 MATRICES 305
C Kinematics 444
A Matrixstructure 307 Reviewset16A 449
B Matrixoperationsanddefinitions 309 Reviewset16B 450
C Matrixmultiplication 315
D Theinverseofa2×2matrix 323 ANSWERS 453
E Simultaneouslinearequations 328
Reviewset12A 330 INDEX 503
Reviewset12B 331
13 INTRODUCTIONTODIFFERENTIAL
CALCULUS 333
A Limits 335
B Ratesofchange 336
C Thederivativefunction 340
D Differentiationfromfirstprinciples 342
E Simplerulesofdifferentiation 344
F Thechainrule 348
G Theproductrule 351
H Thequotientrule 353
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