Giáo trình toán dành cho học sinh du học quốc tế hệ IGCSE

Texas Instruments TI-84 Plus The TI-84 Plus uses a secondary function key We enter square roots by pressing For example, to enter.. The end bracket is used to tell the calculator we have[r]

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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

Endorsed by University of Cambridge International Examinations

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IGCSE CAMBRIDGE INTERNATIONAL MATHEMATICS (0607)

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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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

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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)

USING THE INTERACTIVE CD

Simply ‘click’ on the (or anywhere in the example box) to access the workedexample, 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

a a total of 5 b two numbers which are the same

2-D grid

There are 6 £ 6 = 36 possible outcomes

a P(total of 5) = 368 fthose with a g

b P(same numbers) = 1036 fthose circled g

Self Tutor

0 0 1 1 4 5 0

0 1 1 4 5

roll 2 roll 1

A die has the numbers , , , , and It is rolled Illustrate the sample space using

a -D grid Hence find the probability of getting:

0 0 1 1 4 52

twice

GRAPHICS CALCULATORS

The course assumes that each student will have a graphics calculator An introductory section ‘Graphicscalculator instructions’ appears on p 11 To help get students started, the section includes some basic instructionsfor the Texas Instruments TI-84 Plus and the Casio fx-9860G calculators

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

u n 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

¡!

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

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7 Table of contents

SYMBOLS AND NOTATION

H Difference of two squares factorisation 45

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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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9 Table of contents

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

G Working with functions

H Two variable analysis

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

You can return to this screen from any menu by pressing 2nd MODE When you are on this screen you can type in an expression and evaluate it using the ENTER key

Casio fx-9860g

Press MENU to access the Main Menu, and select RUN ¢MAT.

This is where most of the basic calculations are performed

When you are on this screen you can type in an expression and evaluate it using the EXE key

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12 Graphics calculator instructions

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

(¡) Simply press the sign change key and then type in the number.

For example, to enter¡7, press (¡) 7

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

For example, we enter 34 by typing 3 ¥ 4 If the fraction is part of a larger calculation, it is generally

wise to place this division in brackets, i.e., ( 3 ¥ 4 )

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.

For example, we can enter 234 as ( 11 ¥ 4 ) or ( 2 + 3 ¥ 4 )

Casio fx-9860g

To enter fractions we use the fraction key a b / c

For example, we enter 34 by typing 3 a b / c 4 and 23

4 by typing 2 a b / c 3 a b / c 4

Press SHIFT a b / c (abc $ d

c) to convert between mixed numbers and improper fractions

SIMPLIFYING FRACTIONS & RATIOS

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

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Graphics calculator instructions 13

To express the fraction 3556 in simplest form, press 35 ¥ 56 MATH 1

ENTER The result is 58

To express the ratio 23 : 11

4 in simplest form, press ( 2 ¥ 3 ) ¥ ( 1 + 1 ¥ 4 ) MATH 1 ENTER

The ratio is 8 : 15

Casio fx-9860g

To express the fraction 3556 in simplest form, press 35 a b / c 56 EXE The result is 58

To express the ratio 23 : 11

4 in simplest form, press 2 a b / c 3 ¥ 1 a b / c

1 a b / c 4 EXE The ratio is 8 : 15

ENTERING TIMES

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

To enter 2 hours 27 minutes, press 2 2nd APPS (ANGLE) 1:o 27 2nd APPS 2:0 This is equivalent to 2:45 hours.

To express 8:17 hours in terms of hours, minutes and seconds, press 8:17

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

Casio fx-9860g

To enter 2 hours 27 minutes, press OPTN F6 F5 (ANGL) 2 F4 (o000) 27

F4 (o000) EXE This is equivalent to 2:45 hours

To express 8:17 hours in terms of hours, minutes and seconds, press 8:17

hours, 10 minutes and 12 seconds

GROUPING SYMBOLS (BRACKETS)

Both the TI-84 Plus and Casio have bracket keys that look like ( and ) Brackets are regularly used in mathematics to indicate an expression which needs to be evaluated beforeother operations are carried out

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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

For example, to enter 4+12 we type 2 ¥ ( 4 + 1 ) If we typed 2 ¥ 4 + 1 the calculator

would think we meant 24+ 1

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

separately

POWER KEYS

Both the TI-84 Plus and Casio also have power keys that look like ^ We type the base first, press the

power key, then enter the index or exponent

For example, to enter 253 we type 25 ^ 3

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

Numbers can be squared on both TI-84 Plus and Casio using the special key x2

For example, to enter 252 we type 25 x 2

ROOTS

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

Keys).

The TI-84 Plus uses a secondary function key 2nd

We enter square roots by pressing 2nd x 2

For example, to enterp

36 we press 2nd x2 36 ) The end bracket is used to tell the calculator we have finished entering terms under the square root sign

Cube roots are entered by pressing MATH 4: p ( 3

For example, to enter p3

8 we press MATH 4 8 ) Higher roots are entered by pressing MATH 5: px .

For example, to enter p4

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Casio fx-9860g

The Casio uses a shift key SHIFT to get to its second functions

We enter square roots by pressing SHIFT x 2 For example, to enterp

36 we press SHIFT x 2 36

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

For example, to enterp

18 ¥ 2 we press SHIFT x 2 ( 18 ¥ 2 ) Cube roots are entered by pressing SHIFT ( For example, to enterp3

To evaluate log(47), press log 47 ) ENTER Since logab = log b

log a, we can use the base 10 logarithm to calculatelogarithms in other bases

To evaluate log311, we note that log311 =log 11

log 3 , so we press

log 11 ) ¥ log 3 ) ENTER

Casio fx-9860g

To evaluate log(47) press log 47 EXE

To evaluate log311, press SHIFT 4 (CATALOG), and select logab( You

can use the alpha keys to navigate the catalog, so in this example press I

to jump to “L” Press 3 , 11 ) EXE

ROUNDING OFF

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

To round to 2 decimal places, press MODE then H to scroll down to Float

Use the I button to move the cursor over the 2 and press ENTER Press

2nd MODE to return to the home screen

If you want to unfix the number of decimal places, press MODE H ENTER

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Casio fx-9860g

To round to 2 decimal places, select RUN ¢MAT from the Main Menu, and

press SHIFT MENU to enter the setup screen Scroll down to Display, and

press F1 (Fix) Press 2 EXE to select the number of decimal places Press

EXIT to return to the home screen

To unfix the number of decimal places, press SHIFT MENU to return to the setup screen, scroll down to

Display, and press F3 (Norm).

INVERSE TRIGONOMETRIC FUNCTIONS

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

and Alpha Keys).

The inverse trigonometric functions sin¡1, cos¡1and tan¡1 are the secondary functions of SIN , COS and

TAN respectively They are accessed by using the secondary function key 2nd

For example, if cos x =3

5, then x= cos¡1¡3

5

¢

To calculate this, press 2nd COS 3 ¥ 5 ) ENTER

Casio fx-9860g

The inverse trigonometric functions sin¡1, cos¡1 and tan¡1 are the secondary functions of sin , cos and

tan respectively They are accessed by using the secondary function key SHIFT

For example, if cos x =3

5, then x= cos¡1¡3

5

¢

To calculate this, press SHIFT cos ( 3 ¥ 5 ) EXE

STANDARD FORM

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

form, which is the form a£ 10n where 1 6 a < 10 and n is an integer

To evaluate 23003, press 2300 ^ 3 ENTER

The answer displayed is 1:2167e10, which means 1:2167 £ 1010

To evaluate 3

20 000, press 3 ¥ 20 000 ENTER .The answer displayed is 1:5e¡4, which means 1:5 £ 10¡4.

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

by pressing 2nd ,

For example, to evaluate 2:6 £ 1014

13 , press 2:6 2nd , 14 ¥ 13 ENTER The answer is 2£ 1013

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You can enter values in standard form using the EXP key For example, toevaluate 2:6 £ 1014

13 , press 2:6 EXP 14 ¥ 13 EXE .The answer is 2 £ 1013

The secondary function of each key is displayed in blue above the key It is accessed by pressing the 2nd

key, followed by the key corresponding to the desired secondary function For example, to calculatep

36,press 2nd x2 36 ) ENTER

The alpha function of each key is displayed in green above the key It is accessed by pressing the ALPHA

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

The shift function of each key is displayed in yellow above the key It is accessed by pressing the SHIFT

key followed by the key corresponding to the desired shift function

For example, to calculatep

36, press SHIFT x 2 36 EXE

The alpha function of each key is displayed in red above the key It is accessed by pressing the ALPHA

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

calculations Type in the number then press STO I ALPHA MATH (A)

ENTER

We can now add 10 to this value by pressing ALPHA MATH + 10 ENTER ,

or cube this value by pressing ALPHA MATH ^ 3 ENTER

Casio fx-9860g

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

calculations Type in the number then press I ALPHA X,µ,T (A) EXE

We can now add 10 to this value by pressing ALPHA X,µ,T + 10 EXE ,

or cube this value by pressing ALPHA X,µ,T ^ 3 EXE

ANS VARIABLE

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

in calculations by pressing 2nd (¡) .

For example, suppose you evaluate 3£ 4, and then wish to subtract this from

17 This can be done by pressing 17 ¡ 2nd ( ¡) ENTER

If you start an expression with an operator such as + , ¡ , etc, the previous

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

previous answer can be halved simply by pressing ¥ 2 ENTER

If you wish to view the answer in fractional form, press MATH 1 ENTER

Casio fx-9860g

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

in calculations by pressing SHIFT (¡) For example, suppose you evaluate

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

17 ¡ SHIFT (¡) EXE

If you start an expression with an operator such as + , ¡ , etc, the previous

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

previous answer can be halved simply by pressing ¥ 2 EXE

If you wish to view the answer in fractional form, press F J I D

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RECALLING PREVIOUS EXPRESSIONS

The ENTRY function recalls previously evaluated expressions, and is used by pressing 2nd ENTER This function is useful if you wish to repeat a calculation with a minor change, or if you have made an error

in typing

Suppose you have evaluated 100 +p

132 If you now want to evaluate 100 +p142, instead of retyping thecommand, it can be recalled by pressing 2nd ENTER

The change can then be made by moving the cursor over the 3 and changing it to a 4, then pressing ENTER

If you have made an error in your original calculation, and intended to calculate 1500 +p

132, again youcan recall the previous command by pressing 2nd ENTER

Move the cursor to the first 0

You can insert the digit 5, rather than overwriting the 0, by pressing 2nd DEL 5 ENTER

If you now want to evaluate 100 +p

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

Move the cursor between the 3 and the 2, then press DEL 4 to remove the 3 and change it to a 4 Press

EXE to re-evaluate the expression

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

Press STAT 1 to take you to the list editor screen.

To enter the dataf2, 5, 1, 6, 0, 8g into List 1, start by moving the cursor to the first entry of L 1 Press 2 ENTER 5 ENTER and so on until all thedata is entered

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Casio fx-9860g

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

To enter the dataf2, 5, 1, 6, 0, 8g into List 1, start by moving the cursor to

the first entry of List 1 Press 2 EXE 5 EXE and so on until all the data

is entered

DELETING LIST DATA

Pressing STAT 1 takes you to the list editor screen.

Move the cursor to the heading of the list you want to delete then press CLEAR ENTER

Casio fx-9860g

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

Move the cursor to anywhere on the list you wish to delete, then press F6 ( B) F4 (DEL-A) F1 (Yes).

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

this, move the cursor to the heading of L 2and press 2nd 1 (L 1) + 2 ENTER

Casio fx-9860g

Lists can be referenced using the List function, which is accessed by pressing SHIFT 1.

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

cursor to the heading of List 2 and press SHIFT 1 (List) 1 + 2 EXE

For Casio models without the List function, you can do this by pressing OPTN F1 (LIST) F1 (List) 1

+ 2 EXE

NUMBER SEQUENCES

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

This command is accessed by pressing 2nd STAT I to enter the OPS section of the List menu, then

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

of L 3, then press 2nd STAT I 5 to enter the seq command, followed by 2 X,T,µ,n , X,T,µ,n , 1 ,

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Casio fx-9860g

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

This command is accessed by pressing OPTN F1 (LIST) F5 (Seq).

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

of List 3, then press OPTN F1 F5 to enter a sequence, followed by 2 X,µ,T , X,µ,T , 1 , 4 ,

1 ) EXE 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

In this section we produce descriptive statistics and graphs for the data set: 5 2 3 3 6 4 5 3 7 5 7 1 8 9 5

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

19 To obtain descriptive statistics of the data set, press

To obtain a boxplot of the data, press 2nd Y= (STAT

9:ZoomStat to graph the boxplot with an appropriate

window

To obtain a vertical bar chart of the data, press 2nd Y= 1,

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

Press ZOOM 9:ZoomStat to draw the bar chart. Press

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

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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

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

and select Plot2 Set up Statplot2 in the same manner,

except set the XList to L 2 Press ZOOM 9:ZoomStat to

draw the side-by-side boxplots

Casio fx-9860g

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

To obtain the descriptive statistics, press F6 ( B) until the

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

press F2 (CALC) F1 (1 VAR).

To obtain a boxplot of the data, press EXIT EXIT F1

Press EXIT F1 (GPH1) to draw the boxplot

To obtain a vertical bar chart of the data, press EXIT F6

Press EXIT F2 (GPH2) to draw the bar chart (set Start

to 0, and Width to 1)

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, then press F6 (SET) F2 (GPH2) and set up

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

Press EXIT F4 (SEL), and turn on both StatGraph 1 and

boxplots

GRAPHING FUNCTIONS

Pressing Y= selects the Y= editor, where you can store functions to graph.

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

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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.Press GRAPH to draw a graph of the function

To view a table of values for the function, press 2nd GRAPH (TABLE) The

starting point and interval of the table values can be adjusted by pressing 2nd 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

by scrolling down to the function and pressing DEL F1 (Yes).

To graph the function y= x2¡ 3x ¡ 5, move the cursor to Y1 and press

X,µ,T x2 ¡ 3 X,µ,T ¡ 5 EXE This stores the function into Y1 Press

To view a table of values for the function, press MENU and select TABLE.

The function is stored in Y1, but not selected Press F1 (SEL) to select the

function, and F6 (TABL) to view the table You can adjust the table settings

by pressing EXIT and then F5 (SET) from the Table Function screen.

GRAPHING ABSOLUTE VALUE FUNCTIONS

You can perform operations involving absolute values by pressing MATH

I , which brings up the NUM menu, followed by 1: abs (

To graph the absolute value function y= j3x ¡ 6j, press Y= , move the

cursor to Y1, then press MATH I 1 3 X,T,µ,n ¡ 6 ) GRAPH

Casio fx-9860g

To graph the absolute value function y = j3x ¡ 6j, select GRAPH from

the Main Menu, move the cursor to Y1 and press OPTN F5 (NUM) F1

(Abs) ( 3 X,µ,T ¡ 6 ) EXE F6 (DRAW).

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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We can solve y= 11 ¡ 3x and y = 12 ¡ x2 simultaneously by finding

the point of intersection of these two lines

Press Y= , then store 11 ¡ 3x into Y 1 and 12 ¡ x

2 into Y 2 Press

GRAPH to draw a graph of the functions

To find their point of intersection, press 2nd TRACE (CALC) 5, which

selects 5:intersect Press ENTER twice to specify the functions Y 1 and Y 2

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

to move the cursor close to the point of intersection and press ENTER once

more

The solution x= 2, y = 5 is given

Casio fx-9860g

We can solve y = 11 ¡ 3x and y = 12 ¡ x2 simultaneously by

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

Main Menu, then store 11¡3x into Y1 and 12 ¡ x2 into Y2 Press F6

(DRAW) to draw a graph of the functions.

To find their point of intersection, press F5 (G-Solv) F5 (ISCT). The

solution x= 2, y = 5 is given

intersection can be found by pressing I

SOLVING f (x) = 0

In the special case when you wish to solve an equation of the form f(x) = 0, this can be done by

graphing y= f(x) and then finding when this graph cuts the x-axis

To solve x3 ¡ 3x2 + x + 1 = 0, press Y= and store

x3¡ 3x2+ x + 1 into Y 1 Press GRAPH to draw the graph

To find where this function first cuts the x-axis, press 2nd TRACE (CALC)

2, which selects 2:zero Move the cursor to the left of the first zero and press

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

Finally, move the cursor close to the first zero and press ENTER once more

The solution x¼ ¡0:414 is given

Repeat this process to find the remaining solutions x= 1 and x ¼ 2:414

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Casio fx-9860g

To solve x3¡ 3x2+ x + 1 = 0, select GRAPH from the Main Menu and

store x3¡ 3x2+ x + 1 into Y1 Press F6 (DRAW) to draw the graph.

To find where this function cuts the x-axis, press F5 (G-Solv) F1 (ROOT).

The first solution x¼ ¡0:414 is given

Press I to find the remaining solutions x = 1 and x¼ 2:414

TURNING POINTS

To find the turning point (vertex) of y= ¡x2+2x+3, press Y= and store

¡x2+ 2x + 3 into Y 1 Press GRAPH to draw the graph

From the graph, it is clear that the vertex is a maximum, so press 2nd TRACE (CALC) 4 to select 4:maximum.

Move the cursor to the left of the vertex and press ENTER , then move thecursor to the right of the vertex and press ENTER Finally, move the cursorclose to the vertex and press ENTER once more The vertex is (1, 4)

Casio fx-9860g

To find the turning point (vertex) of y= ¡x2+ 2x + 3, select GRAPH from the Main Menu and store

¡x2+ 2x + 3 into Y1 Press F6 (DRAW) to draw the graph

From the graph, it is clear that the vertex is a maximum, so to find the vertexpress F5 (G-Solv) F2 (MAX).

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:

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

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default setting of ¡10 6 x 6 10, ¡10 6 y 6 10:

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

manually by pressing WINDOW and setting the minimum and maximum

values for the x and y axes

Casio fx-9860g

The viewing window can be adjusted by pressing SHIFT F3 (V-Window).

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

or press F3 (STD) to obtain the standard viewing window

¡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

the values of Pearson’s correlation coefficient r and the coefficient of determination r2, which measure the

strength of the linear correlation between the two variables

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

y 5 8 10 13 16 18 20

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

given on page 19.

To produce a scatter diagram of the data, press 2nd Y=

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

Press ZOOM 9 : ZoomStat to draw the scatter diagram.

We will now find the line of best fit Press STAT I

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

Press 2nd 1 (L 1 ) , 2nd 2 (L 2 ) , VARS I 1 1 (Y1) This specifies

the lists L1 and L2as the lists which hold the data, and the line of best fit will

be pasted into the function Y1: Press ENTER to view the results

The line of best fit is given as y¼ 2:54x + 2:71: If the r and r2 values are

not shown, you need to turn on the Diagnostic by pressing 2nd 0 (CATALOG)

and selecting DiagnosticOn.

TWO VARIABLE ANALYSIS

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Press GRAPH to view the line of best fit

To find the line of best fit, press F1 (CALC) F2 (X).

We can see that the line of best fit is given as y ¼ 2:54x + 2:71, and wecan view the r and r2 values

Press F6 (DRAW) to view the line of best fit.

QUADRATIC AND CUBIC REGRESSION

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

To find the general term for the quadratic sequence ¡2, 5, 16, 31, 50, , wefirst 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.

Press STAT I 5: QuadReg, then 2nd 1 (L1) , 2nd 2 (L2) ENTER The result is a = 2, b = 1, c =¡5, which means the general term for thesequence is un = 2n2+ n ¡ 5

To find the general term for the cubic sequence ¡3, ¡9, ¡7, 9, 45, , we

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

Press STAT I 6: CubicReg, then 2nd 1 (L1) , 2nd 2 (L2) ENTER The result is a = 1, b =¡2, c = ¡7, d = 5, which means the generalterm for the sequence is un = n3¡ 2n2¡ 7n + 5

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Casio fx-9860G

To find the general term for the quadratic sequence ¡2, 5, 16, 31, 50, ,

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.

Press F2 (CALC) F3 (REG) F3 (Xˆ2).

The result is a = 2, b = 1, c =¡5, which means the general term for the

sequence is un= 2n2+ n ¡ 5

To find the general term for the cubic sequence ¡3, ¡9, ¡7, 9, 45, we enter

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

Press F2 (CALC) F3 (REG) F4 (Xˆ3).

The result is a = 1, b =¡2, c = ¡7, d = 5 (the calculator may not always

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

the general term for the sequence is un= n3¡ 2n2¡ 7n + 5

EXPONENTIAL REGRESSION

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

model of the form y= a £ bx which best fits the data.

y 7 11 20 26 45

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.

Press STAT I 0: ExpReg, then 2nd 1 (L1) , 2nd 2 (L2) ENTER

So, the exponential model which best fits the data is y¼ 5:13 £ 1:20x.

POWER REGRESSION

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

form y= a £ xb which best fits the data.

y 3 19 35 62

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.

Press STAT I , then scroll down to A: PwrReg and press ENTER

Press 2nd 1 (L1) , 2nd 2 (L2) ENTER

So, the power model which best fits the data is y¼ 3:01 £ x1:71.

Casio fx-9860g

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

Press F2 (CALC) F3 (REG) F6 F3 (Pwr).

So, the power model which best fits the data is y¼ 3:01 £ x1:71.

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Assumed Knowledge (Number)

F Ratio and proportion [1.5]

PRINTABLE CHAPTER

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G Interpreting graphs and tables [11.1]

(Geometry and graphs)

PRINTABLE CHAPTER

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C Difference of two squares [2.7]

D Perfect squares expansion [2.7]

F Algebraic common factors

G Factorising with common factors [2.8]

H Difference of two squares factorisation [2.8]

I Perfect squares factorisation [2.8]

J Expressions with four terms [2.8]

A square garden plot is surrounded by a path

50 cm wide Each side of the path is x m long,

The study of algebra is vital for many areas of mathematics We need it to manipulate equations, solve

problems for unknown variables, and also to develop higher level mathematical theories

In this chapter we consider the expansion of expressions which involve brackets, and the reverse process which is called factorisation.

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32 Algebra (Expansion and factorisation) (Chapter 1)

Consider the expression 2(x + 3) We say that 2 is the coefficient of the expression in the brackets We

can expand the brackets using the distributive law:

a(b + c) = ab + acThe distributive law says that we must multiply the coefficient by each term within the brackets, and add

the results

Geometric Demonstration:

The overall area is a(b + c)

However, this could also be found by adding the areas ofthe two small rectangles: ab+ ac

So, a(b + c) = ab + ac: fequating areasg

Expand the following:

Expand and simplify:

terms inside the following bracket.

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Algebra (Expansion and factorisation) (Chapter 1) 33

Consider the product (a + b)(c + d)

It has two factors, (a + b) and (c + d)

We can evaluate this product by using the distributive law several times

(a + b)(c + d) = a(c + d) + b(c + d)

= ac + ad + bc + bd

So, (a + b)(c + d) = ac + ad + bc + bdThe final result contains four terms:

ac is the product of the First terms of each bracket

ad is the product of the Outer terms of each bracket

bc is the product of the Inner terms of each bracket

bd is the product of the Last terms of each bracket

Expand and simplify: (x + 3)(x + 2):

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Expand and simplify: (2x + 1)(3x ¡ 2)

(2x + 1)(3x ¡ 2)

= 2x £ 3x + 2x £ ¡2 + 1 £ 3x + 1 £ ¡2

= 6x2¡ 4x + 3x ¡ 2

= 6x2¡ x ¡ 2

1 Consider the figure alongside:

Give an expression for the area of:

a rectangle 1 b rectangle 2

c rectangle 3 d rectangle 4

e the overall rectangle

What can you conclude?

2 Expand and simplify:

2 4

, what do you notice about the two middle terms?

Examples 5 6

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a (x + 2)(x ¡ 2) b (a ¡ 5)(a + 5) c (4 + x)(4 ¡ x)

d (2x + 1)(2x ¡ 1) e (5a + 3)(5a ¡ 3) f (4 + 3a)(4 ¡ 3a)

5 A square photograph has sides of length x cm

It is surrounded by a wooden frame with thedimensions shown Show that the area of therectangle formed by the outside of the frame isgiven by A = x2+ 10x + 24 cm2

a2 and b2are perfect squares and so a2¡ b2 is called the difference of two squares.

Notice that (a + b)(a ¡ b) = a2¡ ab + ab| {z }

the middle two terms add to zero

¡ b2= a2¡ b2

Thus, (a + b)(a ¡ b) = a2¡ b2

Geometric Demonstration:

Consider the figure alongside:

The shaded area

= area of large square ¡ area of small square

= a2¡ b2

Cutting along the dotted line and flipping (2) over,

we can form a rectangle

The rectangle’s area is (a + b)(a ¡ b):

) (a + b)(a ¡ b) = a2¡ b2

a-b

COMPUTER DEMO

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a (2x ¡ 3)(2x + 3) b (5 ¡ 3y)(5 + 3y) c (3x + 4y)(3x ¡ 4y)

g (2 ¡ 5y)(2 + 5y) h (3 + 4a)(3 ¡ 4a) i (4 + 3a)(4 ¡ 3a)

3 Expand and simplify using the rule (a + b)(a ¡ b) = a2¡ b2:

a (2a + b)(2a ¡ b) b (a ¡ 2b)(a + 2b) c (4x + y)(4x ¡ y)

d (4x + 5y)(4x ¡ 5y) e (2x + 3y)(2x ¡ 3y) f (7x ¡ 2y)(7x + 2y)

4 a Use the difference of two squares expansion to show that:

i 43 £ 37 = 402¡ 32 ii 24 £ 26 = 252¡ 12

b Evaluate without using a calculator:

Con was trying to multiply 19 £ 20 £ 21 without a calculator Aimee told him to ‘cube the middle

integer and then subtract the middle integer’ to get the answer

What to do:

1 Find 19 £ 20 £ 21 using a calculator

2 Find 203¡ 20 using a calculator Does Aimee’s rule seem to work?

3 Check that Aimee’s rule works for the following products:

a 4 £ 5 £ 6 b 9 £ 10 £ 11 c 49 £ 50 £ 51

4 Let the middle integer be x, so the other integers must be (x¡ 1) and (x + 1)

Find the product (x¡1)£x £(x +1) by expanding and simplifying Have you proved Aimee’srule?

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(a + b)2 and (a ¡ b)2 are called perfect squares.

Notice that (a + b)2= (a + b)(a + b)

= a2+ ab + ab + b2 fusing ‘FOIL’g

= a2+ 2ab + b2

Thus, we can state the perfect square expansion rule:

(a + b)2= a2+ 2ab + b2

We can remember the rule as follows:

Step 1: Square the first term.

Step 2: Add twice the product of the first and last terms.

Step 3: Add on the square of the last term.

Notice that (a ¡ b)2= (a + (¡b))2

= a2+ 2a(¡b) + (¡b)2

= a2¡ 2ab + b2

Once again, we have the square of the first term, twice the product of the first and last terms, and the square

of the last term

Expand and simplify using the perfect square expansion rule:

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Expand and simplify: a (2x2+ 3)2 b 5 ¡ (x + 2)2

1 Consider the figure alongside:

Give an expression for the area of:

a square 1 b rectangle 2 c rectangle 3

d square 4 e the overall square

What can you conclude?

2 Use the rule (a + b)2= a2+ 2ab + b2 to expand and simplify:

a a

b

1 3

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In this section we expand more complicated expressions by repeated use of the expansion laws

Consider the expansion of (a + b)(c + d + e):

2 terms in the first bracket £ 3 terms in the second bracket 6 terms in the expansion

Expand and simplify: (x + 3)(x2+ 2x + 4)(x + 3)(x2+ 2x + 4)

= x(x2+ 2x + 4) + 3(x2+ 2x + 4)

= x3+ 2x2+ 4x fall terms in the 2nd bracket £ xg+ 3x2+ 6x + 12 fall terms in the 2nd bracket £ 3g

= x3+ 5x2+ 10x + 12 fcollecting like termsg

Expand and simplify: (x + 2)3

= x3+ 6x2+ 12x + 8 fcollecting like termsg

a x(x + 1)(x + 3) b (x + 1)(x ¡ 3)(x + 2)

a x(x + 1)(x + 3)

= (x2+ x)(x + 3) fall terms in the first bracket £ xg

= x3+ 3x2+ x2+ 3x fexpanding the remaining factorsg

= x3+ 4x2+ 3x fcollecting like termsg

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b (x + 1)(x ¡ 3)(x + 2)

= (x2¡ 3x + x ¡ 3)(x + 2) fexpanding the first two factorsg

= (x2¡ 2x ¡ 3)(x + 2) fcollecting like termsg

= x3¡ 2x2¡ 3x + 2x2¡ 4x ¡ 6 fexpanding the remaining factorsg

= x3¡ 7x ¡ 6 fcollecting like termsg

Algebraic products are products which contain variables.

For example, 6c and 4x2y are both algebraic products

In the same way that whole numbers have factors, algebraic products are also made up of factors

For example, in the same way that we can write 60 as 2£ 2 £ 3 £ 5, we can write 2xy2 as 2£ x £ y £ y

To find the highest common factor of a group of numbers, we express the numbers as products of prime

factors The common prime factors are then found and multiplied to give the highest common factor (HCF)

We can use the same technique to find the highest common factor of a group of algebraic products

ALGEBRAIC COMMON FACTORS

Tiêu đề	Igcse Cambridge International Mathematics (0607) Extended
Tác giả	Keith Black, Alison Ryan, Michael Haese, Robert Haese, Sandra Haese, Mark Humphries
Trường học	Haese & Harris Publications
Chuyên ngành	Mathematics
Thể loại	Giáo trình
Năm xuất bản	2009
Thành phố	Adelaide

Định dạng
Số trang	752
Dung lượng	14,73 MB