Mathematics HL (Core)

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Transcript of Mathematics HL (Core)

untitledSpecialists in mathematics publishing
third edition
David Martin
Robert Haese
Sandra Haese
Michael Haese
Mark Humphries
David Martin
Robert Haese
Sandra Haese
Michael Haese
Mark Humphries
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David Martin B.A.,B.Sc.,M.A.,M.Ed.Admin. Robert Haese B.Sc. Sandra Haese B.Sc. Michael Haese B.Sc.(Hons.),Ph.D. Mark Humphries B.Sc.(Hons.)
First Edition 2004 2005 three times , 2006, 2007
Second Edition 2008 2009 , 2010, 2011
Reprinted (with minor corrections)
Reprinted (with minor corrections)
Haese Mathematics 3 Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIA Telephone: +61 8 8355 9444, Fax: +61 8 8355 9471 Email:
National Library of Australia Card Number & ISBN 978-1-921972-11-9
© Haese Mathematics 2012
Published by Haese Mathematics. 3 Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIA
Typeset in Times Roman .
The textbook and its accompanying CD have been developed independently of the International Baccalaureate Organization (IBO). The textbook and CD are in no way connected with, or endorsed by, the IBO.
. Except as permitted by the Copyright Act (any fair dealing for the purposes of private study, research, criticism or review), no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Enquiries to be made to Haese Mathematics.
: Where copies of part or the whole of the book are made under Part VB of the Copyright Act, the law requires that the educational institution or the body that administers it has given a remuneration notice to Copyright Agency Limited (CAL). For information, contact the Copyright Agency Limited.
: While every attempt has been made to trace and acknowledge copyright, the authors and publishers apologise for any accidental infringement where copyright has proved untraceable. They would be pleased to come to a suitable agreement with the rightful owner.
: All the internet addresses (URLs) given in this book were valid at the time of printing. While the authors and publisher regret any inconvenience that changes of address may cause readers, no responsibility for any such changes can be accepted by either the authors or the publisher.
Third Edition 2012
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Mathematics for the International Student: Mathematics HL has been written to embrace the syllabus for the two-year Mathematics HL Course, to be first examined in 2014. It is not our intention to define the course. Teachers are encouraged to use other resources. We have developed this book independently of the International Baccalaureate Organization (IBO) in consultation with many experienced teachers of IB Mathematics. The text is not endorsed by the IBO.
Syllabus references are given at the beginning of each chapter. The new edition reflects the new Mathematics HL syllabus. Discussion topics for the Theory of Knowledge have been included in this edition. See page 12 for a summary.
In response to the introduction of a calculator-free examination paper, the review sets at the end of each chapter have been categorised as ‘calculator’ or ‘non-calculator’. Also, the final chapter contains over 200 examination-style questions, categorised as ‘calculator’ or ‘non-calculator’. These questions should provide more difficult challenges for advanced students.
Comprehensive graphics calculator instructions for Casio fx-9860G Plus, TI-84 Plus and (see page 16) and, occasionally, where additional help may be needed, more detailed instructions are available from icons located throughout the book. The extensive use of graphics calculators and computer packages throughout the book enables students to realise the importance, application, and appropriate use of technology. No single aspect of technology has been favoured. It is as important that students work with a pen and paper as it is that they use their graphics calculator, or use a spreadsheet or graphing package on computer.
This package is language rich and technology rich. The combination of textbook and interactive Student CD will foster the mathematical development of students in a stimulating way. Frequent use of the interactive features on the CD is certain to nurture a much deeper understanding and appreciation of mathematical concepts. The CD also offers for every worked example.
is accessed via the CD – click anywhere on any worked example to hear a teacher’s voice explain each step in that worked example. This is ideal for catch-up and revision, or for motivated students who want to do some independent study outside school hours.
For students who may not have a good understanding of the necessary background knowledge for this course, we have provided printable pages of information, examples, exercises, and answers on the Student CD – see ‘Background knowledge’ (page 16).
The interactive features of the CD allow immediate access to our own specially designed geometry software, graphing software and more. Teachers are provided with a quick and easy way to demonstrate concepts, and students can discover for themselves and re-visit when necessary.
Casio fx-CG20, TI- spire are accessible as printable pages on the CD
It is not our intention that each chapter be worked through in full. Time constraints may not allow for this. Teachers must select exercises carefully, according to the abilities and prior knowledge of their students, to make the most efficient use of time and give as thorough coverage of work as possible. Investigations throughout the book will add to the discovery aspect of the course and enhance student understanding and learning.
In this changing world of mathematics education, we believe that the contextual approach shown in this book, with the associated use of technology, will enhance the students’ understanding, knowledge and appreciation of mathematics, and its universal application.
We welcome your feedback.
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Cartoon artwork by John Martin. Artwork by Piotr Poturaj and Benjamin Fitzgerald.
Cover design by Piotr Poturaj.
Computer software by Thomas Jansson, Troy Cruickshank, Ashvin Narayanan, Adrian Blackburn, Edward Ross and Tim Lee.
Typeset in Australia by Charlotte Frost.
Editorial review by Catherine Quinn and Tim Knight.
Additional questions by Sriraman R. Iyer, Aditya Birla World Academy, Mumbai, India.
The authors and publishers would like to thank all those teachers who offered advice and encouragement on this book. Many of them read the page proofs and offered constructive comments and suggestions. These teachers include: Jeff Kutcher, Peter Blythe, Gail Smith, Dhruv Prajapati, Peter McCombe, and Chris Carter. To anyone we may have missed, we offer our apologies.
The publishers wish to make it clear that acknowledging these individuals does not imply any endorsement of this book by any of them, and all responsibility for the content rests with the authors and publishers.
Refer to our website for guidance in using this textbook in HL and SL combined classes.
This is a companion to the textbook. It offers coverage of each of the following options:
Topic 7 –Statistics and probability
Topic 8 – Sets, relations and groups
Topic 9 – Calculus
Topic 10 – Discrete mathematics
In addition, coverage of the Linear algebra and Geometry topics for students undertaking the IB Diploma course is presented on the CD that accompanies the book.
Mathematics HL (Core)

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The interactive CD is ideal for independent study.
Students can revisit concepts taught in class and undertake their own revision and practice. The CD also has the text of the book, allowing students to leave the textbook at school and keep the CD at home.
By clicking on the relevant icon, a range of interactive features can be accessed:
: Detailed instructions are available on the
CD, as printable pages (see page 16). Click on the icon for
TI-84 Plus, or instructions.
Interactive links to spreadsheets, graphing and geometry software, computer demonstrations and simulations
Casio fx-9860G
Graphics calculator instructions
Simply ‘click’ on the (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.
Self Tutor
SELF TUTOR is an exciting feature of this book.
The icon on each worked example denotes an active link on the CD.Self Tutor
See , , p. 438Chapter 15 Vector applications
A line passes through the point A(1, 5) and has direction vector
µ 3 2
Describe the line using:
a a vector equation b parametric equations c a Cartesian equation.
a The vector equation is r = a + ¸b where
a = ¡! OA =
b x = 1 + 3¸ and y = 5 + 2¸, ¸ 2 R
c Now ¸ = x¡ 1
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B Scientific notation (standard form) CD
C Number systems and set notation CD
D Algebraic simplification CD
G Product expansion CD
K Congruence and similarity CD
L Pythagoras’ theorem CD
M Coordinate geometry CD
Matrices CD
Casio fx-9860G PLUS CD
A Quadratic equations 19
C The sum and product of the roots 27
D Quadratic functions 28
F Where functions meet 40
G Problem solving with quadratics 43
H Quadratic optimisation 45
B Function notation 55
D Composite functions 62
F Sign diagrams 66
G Inequalities (inequations) 70
I Rational functions 78
J Inverse functions 81
K Graphing functions 87
Review set 2A 90
Review set 2B 91
Review set 2C 93
C Rational exponents 101
E Exponential equations 106
F Exponential functions 108
Review set 3A 119
Review set 3B 120
Review set 3C 121
B Logarithms in base 127
C Laws of logarithms 130
D Natural logarithms 134
F The change of base rule 139
G Graphs of logarithmic functions 141
H Growth and decay 145
Review set 4A 147
Review set 4B 148
Review set 4C 149
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B Translations 154
C Stretches 156
D Reflections 157
G The reciprocal of a function 165
H Modulus functions 166
Review set 5A 167
Review set 5B 169
Review set 5C 170
B Complex numbers 176
C Real polynomials 183
E Polynomial theorems 193
Review set 6A 209
Review set 6B 210
Review set 6C 211
A Number sequences 214
C Arithmetic sequences 218
D Geometric sequences 222
B Counting paths 246
C Factorial notation 247
Review set 8A 263
B The principle of mathematical induction 269
Review set 9A 277
Review set 9B 277
Review set 9C 278
A Radian measure 280
C The unit circle and the trigonometric ratios 286
D Applications of the unit circle 292
E Negative and complementary angle formulae 295
F Multiples of and 297
Review set 10A 301
Review set 10B 302
Review set 10C 303
D Using the sine and cosine rules 317
Review set 11A 321
Review set 11B 323
Review set 11C 323
A Periodic behaviour 326
C Modelling using sine functions 336
D The cosine function 339
E The tangent function 341
F General trigonometric functions 344
G Reciprocal trigonometric functions 346
H Inverse trigonometric functions 348
Review set 12A 350
Review set 12B 351
Review set 12C 351
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C Trigonometric relationships 364
F Trigonometric equations in quadratic form 376
G Trigonometric series and products 377
Review set 13A 379
Review set 13B 380
Review set 13C 382
B Geometric operations with vectors 387
C Vectors in the plane 394
D The magnitude of a vector 396
E Operations with plane vectors 398
F The vector between two points 401
G Vectors in space 404
H Operations with vectors in space 408
I Parallelism 412
Review set 14A 428
Review set 14B 430
Review set 14C 431
B Area 436
E Constant velocity problems 444
F The shortest distance from a line to a point 447
G Intersecting lines 451
I Planes 460
K Intersecting planes 467
Review set 15A 472
Review set 15B 474
Review set 15C 476
B Modulus 483
D Euler’s form 495
E De Moivre’s theorem 497
F Roots of complex numbers 500
G Miscellaneous problems 504
Review set 16A 504
Review set 16B 505
Review set 16C 506
C Trigonometric limits 515
F Differentiation from first principles 523
Review set 17A 526
Review set 17B 526
Review set 17C 527
B The chain rule 534
C The product rule 537
D The quotient rule 540
E Implicit differentiation 542
I Derivatives of inverse trigonometric functions 555
J Second and higher derivatives 557
Review set 18A 559
Review set 18B 560
Review set 18C 561
B Increasing and decreasing functions 570
C Stationary points 575
Review set 19A 587
Review set 19B 588
Review set 19C 589
IB_HL-3ed magentacyan yellow black
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C Optimisation 606
B Antidifferentiation 634
D Integration 640
F Integrating 648
I Miscellaneous integration 660
J Definite integrals 661
Review set 21A 667
Review set 21B 668
Review set 21C 669
C Kinematics 681
E Solids of revolution 690
Review set 22A 697
Review set 22B 699
Review set 22C 701
B Measuring the centre of data 709
C Variance and standard deviation 721
Review set 23A 728
Review set 23B 729
Review set 23C 731
A Experimental probability 735
B Sample space 740
C Theoretical probability 741
E Compound events 747
F Tree diagrams 751
H Sets and Venn diagrams 757
I Laws of probability 763
J Independent events 767
combinations 769
C Expectation 786
E Properties of E(X) and Var(X) 792
F The binomial distribution 795
G The Poisson distribution 805
Review set 25A 808
Review set 25B 809
Review set 25C 811
C Probabilities using a calculator 823
D The standard normal distribution
( -distribution) 826
Review set 26A 835
Review set 26B 836
Review set 26C 837
A Non-calculator questions 840
B Calculator questions 852
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N the set of positive integers and zero,
f0, 1, 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,
fx jx > 0 , x 2 Q g R the set of real numbers
R + the set of positive real numbers,
fx jx > 0 , x 2 R g C the set of complex numbers,
fa + bi j a, b 2 R g i
jzj the modulus of z
arg z the argument of z
Re z the real part of z
Im z the imaginary part of z
fx1, x2, ....g the set with elements x1, x2, ....
n(A) the number of elements in the finite set A
fx j .... the set of all x such that
2 is an element of
=2 is not an element of
? the empty (null) set
U the universal set
µ is a subset of
A0 the complement of the set A
a 1
(if a > 0 then n p a > 0)
a 1
2 , square root of a
(if a > 0 then p a > 0)
jxj the modulus or absolute value of x, that is½ x for x > 0, x 2 R
¡x for x < 0, x 2 R
´ identity or is equivalent to
¼ is approximately equal to
> is greater than
< is less than
is not greater than
¥ is not less than
[a, b] the closed interval a 6 x 6 b
] a, b [ the open interval a < x < b
un the nth term of a sequence or series
d the common difference of an arithmetic sequence
r the common ratio of a geometric sequence
Sn the sum of the first n terms of a sequence, u1 + u2 + :::: + un
S1 or S the sum to infinity of a sequence, u1 + u2 + ::::
nP i=1
¢ n!
r!(n¡ r)!
f : A ! B f is a function under which each element of set A has an image in set B
f : x 7! y f is a function under which x is mapped to y
f(x) the image of x under the function f
f¡1 the inverse function of the function f
f ± g the composite function of f and g
lim x!a
f(x) the limit of f(x) as x tends to a
f 0(x) the derivative of f(x) with respect to x
dx2 the second derivative of y with respect to x
f 00(x) the second derivative of f(x) with respect to x
dxn the nth derivative of y with respect to x
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y dx the definite integral of y with respect to x between the limits x = a and x = b
ex exponential function of x
loga x logarithm to the base a of x
lnx the natural logarithm of x, loge x
sin, cos, tan the circular functions
arcsin , arccos , arctan
csc, sec, cot the reciprocal circular functions
cis µ cos µ + i sin µ
A(x, y) the point A in the plane with Cartesian
coordinates x and y
[AB] the line segment with end points A and B
AB the length of [AB]
(AB) the line containing points A and BbA the angle at A
[CAB or CbAB the angle between [CA] and [AB]
4ABC the triangle whose vertices are A, B, and C
v the vector v ¡! AB the vector represented in magnitude and
direction by the directed line segment
from A to B
a the position vector ¡! OA
i, j, k unit vectors in the directions of the Cartesian coordinate axes
j a j the magnitude of vector a
j ¡! AB j the magnitude of
¡! AB
v ² w the scalar product of v and w
v £ w the vector product of v and w
I the identity matrix
P0(A) probability of the event “not A”
P(A j B) probability of the event A given B
x1, x2, .... observations of a variable
f1, f2, .... frequencies with which the observations x1, x2, x3, .... occur
px probability distribution function P(X = x)
of the discrete random variable X
f(x) probability density function of the
continuous random variable X
variable X
¹ population mean
sn standard deviation of the sample
s 2 n¡1 unbiased estimate of the population
n and p
N(¹, ¾2) normal distribution with mean ¹ and
variance ¾2
X » B(n, p) the random variable X has…