igcse math extended toan tieng anh THCS THPT

However, to help get you started, we have included here some basic instructions for the Texas Instruments TI-84 Plus and the Casio fx-9860G calculators.. Texas Instruments TI-84 Plus To

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(0607) Extended

Haese and Harris Publications

specialists in mathematics publishing

IGCSE

Keith Black Alison Ryan Michael Haese Robert Haese Sandra Haese Mark Humphries

Cambridge International

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Keith Black B.Sc.(Hons.), Dip.Ed.

Haese & Harris Publications

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 978-1-921500-04-6

Published by Raksar Nominees Pty Ltd

3 Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIA

Cartoon artwork by John Martin Artwork and cover design by Piotr Poturaj

Fractal artwork on the cover copyright by Jaros aw Wierny,

Computer software by David Purton, Troy Cruickshank and Thomas Jansson

Typeset in Australia by Susan Haese and Charlotte Sabel (Raksar Nominees) Typeset in Times Roman 10 /11

This textbook and its accompanying CD have been endorsed by University of Cambridge International

Examinations (CIE) They have been developed independently of the International Baccalaureate Organization

(IBO) and are not connected with or endorsed by, the IBO

Except as permitted by the Copyright Act (any fair dealing for the purposes ofprivate 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 & Harris

Publications

: 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

: The publishers acknowledge the cooperation of Oxford University Press, Australia, for the

Haese & Harris Publications

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 (URL’s) given in this book were valid at the time of printing Whilethe 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

ł www.fractal.art.pl

This book is copyright

Copying for educational purposes

Acknowledgements

Disclaimer

info@haeseandharris.com.auwww.haeseandharris.com.auWeb:

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This book has been written to cover the ‘ ’course over a two-year period.

The new course was developed by University of Cambridge International Examinations (CIE) in consultationwith teachers in international schools around the world It has been designed for schools that want theirmathematics teaching to focus more on investigations and modelling, and to utilise the powerful technology

of graphics calculators

The course springs from the principles that students should develop a good foundation of mathematical skillsand that they should learn to develop strategies for solving open-ended problems It aims to promote apositive attitude towards Mathematics and a confidence that leads to further enquiry Some of the schoolsconsulted by CIE were IB schools and as a result, Cambridge International Mathematics integratesexceptionally well with the approach to the teaching of Mathematics in IB schools

This book is an attempt to cover, in one volume, the content outlined in the Cambridge InternationalMathematics (0607) syllabus References to the syllabus are made throughout but the book can be used as afull course in its own right, as a preparation for GCE Advanced Level Mathematics or IB DiplomaMathematics, for example The book has been endorsed by CIE but it has been developed independently ofthe Independent Baccalaureate Organization and is not connected with, or endorsed by, the IBO

To reflect the principles on which the new course is based, we have attempted to produce a book and CDpackage that embraces technology, problem solving, investigating and modelling, in order to give studentsdifferent learning experiences There are non-calculator sections as well as traditional areas of mathematics,especially algebra An introductory section ‘Graphics calculator instructions’ appears on p 11 It is intended

as a basic reference to help students who may be unfamiliar with graphics calculators Two chapters of

‘assumed knowledge’ are accessible from the CD: ‘Number’ and ‘Geometry and graphs’ (see pp 29 and 30).They can be printed for those who want to ensure that they have the prerequisite levels of understanding forthe course To reflect one of the main aims of the new course, the last two chapters in the book are devoted tomulti-topic questions, and investigations and modelling Review exercises appear at the end of each chapterwith some ‘Challenge’ questions for the more able student Answers are given at the end of the book,followed by an index

demonstrations and simulations, and the two printable chapters on assumed knowledge The CD also containsthe text of the book so that students can load it on a home computer and keep the textbook at school

The Cambridge International Mathematics examinations are in the form of three papers: one a non-calculatorpaper, another requiring the use of a graphics calculator, and a third paper containing an investigation and amodelling question All of these aspects of examining are addressed in the book

The book can be used as a scheme of work but it is expected that the teacher will choose the order of topics.There are a few occasions where a question in an exercise may require something done later in the book butthis has been kept to a minimum Exercises in the book range from routine practice and consolidation ofbasic skills, to problem solving exercises that are quite demanding

In this changing world of mathematics education, we believe that the contextual approach shown in thisbook, with the associated use of technology, will enhance the students’ understanding, knowledge andappreciation of mathematics, and its universal application

We welcome your feedback

Self Tutor

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The authors and publishers would like to thank University of Cambridge International Examinations (CIE)

for their assistance and support in the preparation of this book Exam questions from past CIE exam papers

are reproduced by permission of the University of Cambridge Local Examinations Syndicate The University

of Cambridge Local Examinations Syndicate bears no responsibility for the example answers to questions

taken from its past question papers which are contained in this publication

In addition we would like to thank the teachers who offered to read proofs and who gave advice and support:

Simon Bullock, Philip Kurbis, Richard Henry, Johnny Ramesar, Alan Daykin, Nigel Wheeler, Yener Balkaya,

and special thanks is due to Fran O'Connor who got us started

The publishers wish to make it clear that acknowledging these teachers, does not imply any endorsement of

this book by any of them, and all responsibility for the content rests with the authors and publishers

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The interactive Student CD that comes with this book is designed for those whowant to utilise technology in teaching and learning Mathematics.

The CD icon that appears throughout the book denotes an active link on the CD

Simply click on the icon when running the CD to access a large range of interactivefeatures that includes:

spreadsheetsprintable worksheetsgraphing packagesgeometry softwaredemonstrationssimulationsprintable chaptersSELF TUTORFor those who want to ensure they have the prerequisite levels of understanding for this new course, printablechapters of assumed knowledge are provided for Number (see p 29) and Geometry and Graphs (see p 30)

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 andcolour 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 Self Tutor icon on each worked example denotes an active link on the CD

INTERACTIVE LINK

Example 8

2-D grid

Self Tutor

0 0 1 1 4 5 0

0 1 1 4 5

roll 2 roll 1

a -D grid Hence find the probability of getting:

SeeChapter 25 Probability, , p.516

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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 j x > 0, x 2 Q g

R the set of real numbers

R + the set of positive real numbers,

fx j x > 0, x 2 R g

fx1, x 2 , g the set with elements x1, x 2 ,

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

? or f g the empty (null) set

U the universal set

· or 6 is less than or equal to

un the nth term of a sequence or series

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

logax logarithm to the base a of x sin, cos, tan the circular functions A(x, y) the point A in the plane with Cartesian

coordinates x and y AB

CbAB the angle between CA and AB

¢ABC 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

j a j the magnitude of vector a

j¡!AB j the magnitude of ¡!

AB P(A) probability of event A P(A 0 ) probability of the event “not A”

x 1 , x 2 , observations of a variable

f 1 , f 2 , frequencies with which the observations

x 1 , x 2 , x 3 , occur

x mean of the values x 1 , x 2 ,

§f sum of the frequencies f 1 , f 2 ,

r Pearson’s correlation coefficient

r 2 coefficient of determination

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SYMBOLS AND NOTATION

GRAPHICS CALCULATOR

ASSUMED KNOWLEDGE (NUMBER) 29

ASSUMED KNOWLEDGE (GEOMETRY AND GRAPHS) 30

1 ALGEBRA (EXPANSION AND FACTORISATION) 31

H Difference of two squares factorisation 45

4 LINES, ANGLES AND POLYGONS 93

5 GRAPHS, CHARTS AND TABLES 111

6 EXPONENTS AND SURDS 123

TABLE OF CONTENTS

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E Gradient of parallel and

C Equations of lines

A Labelling sides of a right angled triangle 314

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B Multiplying and dividing algebraic

C Adding and subtracting algebraic

J Mutually exclusive and

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sincosin

A Solving one variable inequalities with

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Graphics calculator instructions

In this course it is assumed that you have a graphics calculator If you learn how to operate your calculator

successfully, you should experience little difficulty with future arithmetic calculations

There are many different brands (and types) of calculators Different calculators do not have exactly thesame keys It is therefore important that you have an instruction booklet for your calculator, and use itwhenever you need to

However, to help get you started, we have included here some basic instructions for the Texas Instruments

TI-84 Plus and the Casio fx-9860G calculators Note that instructions given may need to be modified

slightly for other models

GETTING STARTED

Texas Instruments TI-84 Plus

The screen which appears when the calculator is turned on is the home screen This is where most basic

calculations are performed

Casio fx-9860g

This is where most of the basic calculations are performed

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Most modern calculators have the rules for Order of Operations built into them This order is sometimes

referred to as BEDMAS

This section explains how to enter different types of numbers such as negative numbers and fractions, and

how to perform calculations using grouping symbols (brackets), powers, and square roots It also explains

how to round off using your calculator

NEGATIVE NUMBERS

To enter negative numbers we use the sign change key On both the TI-84 Plus and Casio this looks like

FRACTIONS

On most scientific calculators and also the Casio graphics calculator there is a special key for entering

fractions No such key exists for the TI-84 Plus, so we use a different method.

To enter common fractions, we enter the fraction as a division

To enter mixed numbers, either convert the mixed number to an improper fraction and enter as a common

fraction or enter the fraction as a sum.

Casio fx-9860g

SIMPLIFYING FRACTIONS & RATIOS

Graphics calculators can sometimes be used to express fractions and ratios in simplest form.

BASIC CALCULATIONS

A

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ENTER The result is 58

ENTERING TIMES

In questions involving time, it is often necessary to be able to express time in terms of hours, minutes andseconds

2nd APPS 4: IDMS ENTER This is equivalent to 8 hours, 10 minutes and 12 seconds

Casio fx-9860g

hours, 10 minutes and 12 seconds

GROUPING SYMBOLS (BRACKETS)

Brackets are regularly used in mathematics to indicate an expression which needs to be evaluated beforeother operations are carried out

BASIC FUNCTIONS

B

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For example, to enter 2£ (4 + 1) we type 2 £ ( 4 + 1 )

We also use brackets to make sure the calculator understands the expression we are typing in

In general, it is a good idea to place brackets around any complicated expressions which need to be evaluated

separately

POWER KEYS

power key, then enter the index or exponent

Note that there are special keys which allow us to quickly evaluate squares

ROOTS

To enter roots on either calculator we need to use a secondary function (see Secondary Function and Alpha

Keys).

The end bracket is used to tell the calculator we have finished entering terms under the square root sign

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Casio fx-9860g

If there is a more complicated expression under the square root sign you should enter it in brackets

method we use depends on the brand of calculator

logarithms in other bases

log 11 ) ¥ log 3 ) ENTER

Casio fx-9860g

ROUNDING OFF

You can use your calculator to round off answers to a fixed number of decimal places

to highlight Float

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Casio fx-9860g

INVERSE TRIGONOMETRIC FUNCTIONS

To enter inverse trigonometric functions, you will need to use a secondary function (see Secondary Function

and Alpha Keys).

5

¢

Casio fx-9860g

5

¢

STANDARD FORM

If a number is too large or too small to be displayed neatly on the screen, it will be expressed in standard

You can enter values in standard form using the EE function, which is accessed

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Casio fx-9860g

36,

key followed by the key corresponding to the desired letter The main purpose of the alpha keys is to store

values into memory which can be recalled later Refer to the Memory section.

Casio fx-9860g

key followed by the key corresponding to the desired shift function

key followed by the key corresponding to the desired letter The main purpose of the alpha keys is to storevalues which can be recalled later

Utilising the memory features of your calculator allows you to recall calculations you have performedpreviously This not only saves time, but also enables you to maintain accuracy in your calculations

SPECIFIC STORAGE TO MEMORY

Values can be stored into the variable letters A, B, , Z using either calculator Storing a value in memory

is useful if you need that value multiple times

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Suppose we wish to store the number 15:4829 for use in a number of

ENTER

Casio fx-9860g

Suppose we wish to store the number 15:4829 for use in a number of

ANS VARIABLE

The variable Ans holds the most recent evaluated expression, and can be used

answer Ans is automatically inserted ahead of the operator For example, the

Casio fx-9860g

The variable Ans holds the most recent evaluated expression, and can be used

3 £ 4, and then wish to subtract this from 17 This can be done by pressing

17 ¡ SHIFT (¡) EXE .

answer Ans is automatically inserted ahead of the operator For example, the

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RECALLING PREVIOUS EXPRESSIONS

This function is useful if you wish to repeat a calculation with a minor change, or if you have made an error

in typing

132, again you

Move the cursor to the first 0

142, instead of retyping the command, it can be recalled by pressingthe left cursor key

Lists are used for a number of purposes on the calculator They enable us to enter sets of numbers, and weuse them to generate number sequences using algebraic rules

CREATING A LIST

data is entered

LISTS

E

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Casio fx-9860g

Selecting STAT from the Main Menu takes you to the list editor screen.

is entered

DELETING LIST DATA

Casio fx-9860g

Selecting STAT from the Main Menu takes you to the list editor screen.

REFERENCING LISTS

Lists can be referenced by using the secondary functions of the keypad numbers 1–6

For example, suppose you want to add 2 to each element of List 1 and display the results in List 2 To do

Casio fx-9860g

For example, if you want to add 2 to each element of List 1 and display the results in List 2, move the

+ 2 EXE

NUMBER SEQUENCES

You can create a sequence of numbers defined by a certain rule using the seq command.

selecting 5:seq.

For example, to store the sequence of even numbers from 2 to 8 in List 3, move the cursor to the heading

4 ) ENTER

This evaluates 2x for every value of x from 1 to 4

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Casio fx-9860g

You can create a sequence of numbers defined by a certain rule using the seq command.

For example, to store the sequence of even numbers from 2 to 8 in List 3, move the cursor to the heading

This evaluates 2x for every value of x from 1 to 4 with an increment of 1

Your graphics calculator is a useful tool for analysing data and creating statistical graphs

Enter the data set into List 1 using the instructions on page

19 To obtain descriptive statistics of the data set, press

STAT I 1:1-Var Stats 2nd 1 (L 1 ) ENTER

9:ZoomStat to graph the boxplot with an appropriate

window

and change the type of graph to a vertical bar chart as shown

WINDOW and set the Xscl to 1, then GRAPH to redrawthe bar chart

STATISTICAL GRAPHS

F

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We will now enter a second set of data, and compare it to

the first

Enter the data set 9 6 2 3 5 5 7 5 6 7 6 3 4 4 5 8 4 into List

2, press 2nd Y= 1, and change the type of graph back to a

boxplot as shown Move the cursor to the top of the screen

and select Plot2 Set up Statplot2 in the same manner,

draw the side-by-side boxplots

Casio fx-9860g

Enter the data into List 1 using the instructions on page 19.

GRPH icon is in the bottom left corner of the screen, then

to 0, and Width to 1)

We will now enter a second set of data, and compare it to

the first

StatGraph 2 to draw a boxplot of this data set as shown.

StatGraph 2 Press F6 (DRAW) to draw the side-by-side

boxplots

GRAPHING FUNCTIONS

Delete any unwanted functions by scrolling down to the function and pressing

CLEAR

WORKING WITH FUNCTIONS

G

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To graph the function y= x2¡ 3x ¡ 5, move the cursor to Y 1, and press

X,T,µ,n x2 ¡ 3 X,T,µ,n ¡ 5 ENTER This stores the function into Y 1

WINDOW (TBLSET).

Casio fx-9860g

Selecting GRAPH from the Main Menu takes you to the Graph Function

screen, where you can store functions to graph Delete any unwanted functions

X,µ,T x 2 ¡ 3 X,µ,T ¡ 5 EXE This stores the function into Y1 Press

F6 (DRAW) to draw a graph of the function.

GRAPHING ABSOLUTE VALUE FUNCTIONS

Casio fx-9860g

FINDING POINTS OF INTERSECTION

It is often useful to find the points of intersection of two graphs, for instance, when you are trying to solvesimultaneous equations

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the point of intersection of these two lines

GRAPH to draw a graph of the functions

as the functions you want to find the intersection of, then use the arrow keys

more

Casio fx-9860g

find-ing the point of intersection of these two lines Select GRAPH from the

(DRAW) to draw a graph of the functions.

Note: If there is more than one point of intersection, the remaining points of

ENTER , then move the cursor to the right of the first zero and press ENTER

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Casio fx-9860g

TURNING POINTS

TRACE (CALC) 4 to select 4:maximum.

Casio fx-9860g

From the graph, it is clear that the vertex is a maximum, so to find the vertex

The vertex is (1, 4)

ADJUSTING THE VIEWING WINDOW

When graphing functions it is important that you are able to view all the important features of the graph

As a general rule it is best to start with a large viewing window to make sure all the features of the graphare visible You can then make the window smaller if necessary

Some useful commands for adjusting the viewing window include:

ZOOM 0:ZoomFit : This command scales the y-axis to fit the minimum

and maximum values of the displayed graph withinthe current x-axis range

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ZOOM 6:ZStandard : This command returns the viewing window to the

If neither of these commands are helpful, the viewing window can be adjusted

values for the x and y axes

Casio fx-9860g

You can manually set the minimum and maximum values of the x and y axes,

¡10 6 x 6 10, ¡10 6 y 6 10:

LINE OF BEST FIT

We can use our graphics calculator to find the line of best fit connecting two variables We can also find

strength of the linear correlation between the two variables

We will examine the relationship between the variables x and y for the data:

Enter the x values into List 1 and the y values into List 2 using the instructions

given on page 19.

(STAT PLOT) 1, and set up Statplot 1 as shown.

4:LinReg(ax+b) to select the linear regression option from the CALC menu.

and selecting DiagnosticOn.

TWO VARIABLE ANALYSIS

H

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Press GRAPH to view the line of best fit.

Casio fx-9860G

Enter the x values into List 1 and the y values into List 2 using the instructions given on page 19.

diagram

QUADRATIC AND CUBIC REGRESSION

You can use quadratic or cubic regression to find the formula for the general term of a quadratic or cubicsequence

first notice that we have been given 5 members of the sequence We therefore

enter the numbers 1 to 5 into L1, and the members of the sequence into L2.

enter the numbers 1 to 5 into L1 and the members of the sequence into L2.

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Casio fx-9860G

we first notice that we have been given 5 members of the sequence Enter the

numbers 1 to 5 into List 1, and the members of the sequence into List 2.

the numbers 1 to 5 into List 1 and the members of the sequence into List 2.

give the result exactly as is the case with c and d in this example) Therefore

EXPONENTIAL REGRESSION

When we have data for two variables x and y, we can use exponential regression to find the exponential

We will examine the exponential relationship between x and y for the data:

Enter the x values into L1 and the y values into L2.

POWER REGRESSION

When we have data for two variables x and y, we can use power regression to find the power model of the

We will examine the power relationship between x and y for the data:

Enter the x values into L1 and the y values into L2.

Casio fx-9860g

Enter the x values into List 1 and the y values into List 2.

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Assumed Knowledge (Number)

PRINTABLE CHAPTER

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Assumed Knowledge (Number)

The set of natural numbers is endless, so we call it an infinite set.

All decimal numbers that terminate or recur are rational numbers

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2 Assumed Knowledge (Number)

The set of irrational numbers includes all real numbers which cannot be written in the form

PRIMES AND COMPOSITES

The factors of a positive integer are the positive integers which divide exactly into it, leaving no remainder.

For example, the factors of 10 are: 1, 2, 5 and 10

A positive integer is a prime number if it has exactly two factors, 1 and itself.

A positive integer is a composite number if it has more than two factors.

6 is composite as it has four factors: 1, 2, 3 and 6

1 is neither prime nor composite

If we are given a positive integer, we can use the following procedure to see if it is prime:

Step 1: Find the square root of the number

Step 2: Divide the whole number in turn by all known primes less than its square root

If the division is never exact, the number is a prime

The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19,

A perfect square or square number is an integer which can be written as the square of another integer.

A perfect cube is an integer which can be written as the cube of another integer.

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Assumed Knowledge (Number) 3

¡11

6 0

p

¡2

RULES FOR THE ORDER OF OPERATIONS

² Perform the operations within brackets first.

² Calculate any part involving exponents.

² Starting from the left, perform all divisions and

multiplications as you come to them.

² Starting from the left, perform all additions and

subtractions as you come to them.

RULES FOR THE USE OF BRACKETS

² If an expression contains one set of brackets or group symbols, work that part first.

² If an expression contains two or more sets of grouping symbols one inside the other, work the

grouping symbols

EXERCISE A

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[12 + (9 ¥ 3)] ¡ 11

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7 + 8

further than Dean can Kevin throw?

on 205 days in one year Find the total distance Chen travels on the bus to and from school forthe year

2 containers of ice cream weighing 1:5 kg each Find the total weight of these items

How many hours will he need to work?

Numbers can be expressed as products of their factors

Factors that are prime numbers are called prime factors The prime factors of any number can be found by

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COMMON FACTORS AND HCF

Notice that 2 is a factor of both 24 and 42 We say that 2 is a common factor of 24 and 42.

3 is also a common factor of 24 and 42, which means the product 2 £ 3 = 6 is another common factor

A common factor is a number that is a factor of two or more other numbers.

The highest common factor (HCF) is the largest factor that is common to two or more numbers.

To find the highest common factor of a group of numbers it is often best to express the numbers as products

of prime factors Then the common prime factors can be found and multiplied to give the HCF

Find the highest common factor of 36 and 81

A multiple of any positive integer is obtained by multiplying it by another positive integer.

them as 7, 14, 21, 28, 35,

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The lowest common multiple or LCM of two or more positive integers is the smallest multiple

which is common to all of them

) the common multiples of 3 and 4 are: 12, 24, 36, of which 12 is the LCM

same instant After how many seconds will all three again chime simultaneously?

EXERCISE C

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A common fraction consists of two whole numbers, a

numerator and a denominator, separated by a bar symbol.

LOWEST COMMON DENOMINATOR

The lowest common denominator (LCD) of two or more numerical fractions is the lowest

common multiple of their denominators

ADDITION AND SUBTRACTION

To add (or subtract) two fractions we convert them to equivalent fractions with a

common denominator We then add (or subtract) the new numerators

6 3

3¡ 12 5

4 5

numerator denominator

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MULTIPLICATION

To multiply two fractions, we first cancel any common factors in the numerator and denominator We

then multiply the numerators together and the denominators together

2£7 2

4 or 121

4

DIVISION

To divide by a fraction, we multiply the number by the reciprocal of the fraction we are dividing by.

3¥2 3

a 3 ¥2

3

1¥2 3

1£3 2

3¥2 3

3£3 2

1 1

EXERCISE D

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