The Free High School Science Texts: Textbooks for High School Students Mathematics

The Free High School Science Texts: Textbooks for High School Students Mathematics
FHSST Authors
The Free High School Science Texts:
Textbooks for High School Students
Studying the Sciences
Mathematics
Grades 10 - 12
Version 0
September 17, 2008
ii
iii
Copyright 2007 “Free High School Science Texts”
Permission is granted to copy, distribute and/or modify this document under the
terms of the GNU Free Documentation License, Version 1.2 or any later version
published by the Free Software Foundation; with no Invariant Sections, no FrontCover Texts, and no Back-Cover Texts. A copy of the license is included in the
section entitled “GNU Free Documentation License”.
STOP!!!!
Did you notice the FREEDOMS we’ve granted you?
Our copyright license is different! It grants freedoms
rather than just imposing restrictions like all those other
textbooks you probably own or use.
• We know people copy textbooks illegally but we would LOVE it if you copied
our’s - go ahead copy to your hearts content, legally!
• Publishers revenue is generated by controlling the market, we don’t want any
money, go ahead, distribute our books far and wide - we DARE you!
• Ever wanted to change your textbook? Of course you have! Go ahead change
ours, make your own version, get your friends together, rip it apart and put
it back together the way you like it. That’s what we really want!
• Copy, modify, adapt, enhance, share, critique, adore, and contextualise. Do it
all, do it with your colleagues, your friends or alone but get involved! Together
we can overcome the challenges our complex and diverse country presents.
• So what is the catch? The only thing you can’t do is take this book, make
a few changes and then tell others that they can’t do the same with your
changes. It’s share and share-alike and we know you’ll agree that is only fair.
• These books were written by volunteers who want to help support education,
who want the facts to be freely available for teachers to copy, adapt and
re-use. Thousands of hours went into making them and they are a gift to
everyone in the education community.
iv
FHSST Core Team
Mark Horner ; Samuel Halliday ; Sarah Blyth ; Rory Adams ; Spencer Wheaton
FHSST Editors
Jaynie Padayachee ; Joanne Boulle ; Diana Mulcahy ; Annette Nell ; René Toerien ; Donovan
Whitfield
FHSST Contributors
Rory Adams ; Prashant Arora ; Richard Baxter ; Dr. Sarah Blyth ; Sebastian Bodenstein ;
Graeme Broster ; Richard Case ; Brett Cocks ; Tim Crombie ; Dr. Anne Dabrowski ; Laura
Daniels ; Sean Dobbs ; Fernando Durrell ; Dr. Dan Dwyer ; Frans van Eeden ; Giovanni
Franzoni ; Ingrid von Glehn ; Tamara von Glehn ; Lindsay Glesener ; Dr. Vanessa Godfrey ; Dr.
Johan Gonzalez ; Hemant Gopal ; Umeshree Govender ; Heather Gray ; Lynn Greeff ; Dr. Tom
Gutierrez ; Brooke Haag ; Kate Hadley ; Dr. Sam Halliday ; Asheena Hanuman ; Neil Hart ;
Nicholas Hatcher ; Dr. Mark Horner ; Mfandaidza Hove ; Robert Hovden ; Jennifer Hsieh ;
Clare Johnson ; Luke Jordan ; Tana Joseph ; Dr. Jennifer Klay ; Lara Kruger ; Sihle Kubheka ;
Andrew Kubik ; Dr. Marco van Leeuwen ; Dr. Anton Machacek ; Dr. Komal Maheshwari ;
Kosma von Maltitz ; Nicole Masureik ; John Mathew ; JoEllen McBride ; Nikolai Meures ;
Riana Meyer ; Jenny Miller ; Abdul Mirza ; Asogan Moodaly ; Jothi Moodley ; Nolene Naidu ;
Tyrone Negus ; Thomas O’Donnell ; Dr. Markus Oldenburg ; Dr. Jaynie Padayachee ;
Nicolette Pekeur ; Sirika Pillay ; Jacques Plaut ; Andrea Prinsloo ; Joseph Raimondo ; Sanya
Rajani ; Prof. Sergey Rakityansky ; Alastair Ramlakan ; Razvan Remsing ; Max Richter ; Sean
Riddle ; Evan Robinson ; Dr. Andrew Rose ; Bianca Ruddy ; Katie Russell ; Duncan Scott ;
Helen Seals ; Ian Sherratt ; Roger Sieloff ; Bradley Smith ; Greg Solomon ; Mike Stringer ;
Shen Tian ; Robert Torregrosa ; Jimmy Tseng ; Helen Waugh ; Dr. Dawn Webber ; Michelle
Wen ; Dr. Alexander Wetzler ; Dr. Spencer Wheaton ; Vivian White ; Dr. Gerald Wigger ;
Harry Wiggins ; Wendy Williams ; Julie Wilson ; Andrew Wood ; Emma Wormauld ; Sahal
Yacoob ; Jean Youssef
Contributors and editors have made a sincere effort to produce an accurate and useful resource.
Should you have suggestions, find mistakes or be prepared to donate material for inclusion,
please don’t hesitate to contact us. We intend to work with all who are willing to help make
this a continuously evolving resource!
www.fhsst.org
v
vi
Contents
I
Basics
1
1 Introduction to Book
1.1
II
3
The Language of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . .
Grade 10
3
5
2 Review of Past Work
7
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
What is a number? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3
Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.4
Letters and Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.5
Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.6
Multiplication and Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.7
Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.8
Negative Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.9
2.8.1
What is a negative number? . . . . . . . . . . . . . . . . . . . . . . . . 10
2.8.2
Working with Negative Numbers . . . . . . . . . . . . . . . . . . . . . . 11
2.8.3
Living Without the Number Line . . . . . . . . . . . . . . . . . . . . . . 12
Rearranging Equations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.10 Fractions and Decimal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.11 Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.12 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.12.1 Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.12.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.12.3 Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.12.4 Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.13 Mathematical Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.14 Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.15 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Rational Numbers - Grade 10
23
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2
The Big Picture of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
vii
CONTENTS
CONTENTS
3.4
Forms of Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5
Converting Terminating Decimals into Rational Numbers . . . . . . . . . . . . . 25
3.6
Converting Repeating Decimals into Rational Numbers . . . . . . . . . . . . . . 25
3.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.8
End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Exponentials - Grade 10
29
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3
Laws of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3.1
Exponential Law 1: a0 = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3.2
Exponential Law 2: am × an = am+n . . . . . . . . . . . . . . . . . . . 30
4.3.3
Exponential Law 3: a−n =
4.3.4
4.4
m
1
an , a
n
6= 0 . . . . . . . . . . . . . . . . . . . . 31
Exponential Law 4: a ÷ a = am−n . . . . . . . . . . . . . . . . . . . 32
4.3.5
Exponential Law 5: (ab)n = an bn . . . . . . . . . . . . . . . . . . . . . 32
4.3.6
Exponential Law 6: (am )n = amn . . . . . . . . . . . . . . . . . . . . . 33
End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Estimating Surds - Grade 10
37
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2
Drawing Surds on the Number Line (Optional) . . . . . . . . . . . . . . . . . . 38
5.3
End of Chapter Excercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Irrational Numbers and Rounding Off - Grade 10
41
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2
Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.3
Rounding Off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.4
End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7 Number Patterns - Grade 10
7.1
45
Common Number Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.1.1
Special Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7.2
Make your own Number Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7.3
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.3.1
7.4
Patterns and Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8 Finance - Grade 10
53
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.2
Foreign Exchange Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.3
8.2.1
How much is R1 really worth? . . . . . . . . . . . . . . . . . . . . . . . 53
8.2.2
Cross Currency Exchange Rates
8.2.3
Enrichment: Fluctuating exchange rates . . . . . . . . . . . . . . . . . . 57
. . . . . . . . . . . . . . . . . . . . . . 56
Being Interested in Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
viii
CONTENTS
8.4
Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8.4.1
8.5
8.6
8.7
CONTENTS
Other Applications of the Simple Interest Formula . . . . . . . . . . . . . 61
Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8.5.1
Fractions add up to the Whole . . . . . . . . . . . . . . . . . . . . . . . 65
8.5.2
The Power of Compound Interest . . . . . . . . . . . . . . . . . . . . . . 65
8.5.3
Other Applications of Compound Growth . . . . . . . . . . . . . . . . . 67
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8.6.1
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8.6.2
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
9 Products and Factors - Grade 10
71
9.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.2
Recap of Earlier Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.2.1
Parts of an Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.2.2
Product of Two Binomials . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.2.3
Factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
9.3
More Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
9.4
Factorising a Quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9.5
Factorisation by Grouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
9.6
Simplification of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
9.7
End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
10 Equations and Inequalities - Grade 10
83
10.1 Strategy for Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
10.2 Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
10.3 Solving Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10.4 Exponential Equations of the form ka(x+p) = m . . . . . . . . . . . . . . . . . . 93
10.4.1 Algebraic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.5 Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
10.6 Linear Simultaneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
10.6.1 Finding solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
10.6.2 Graphical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
10.6.3 Solution by Substitution
. . . . . . . . . . . . . . . . . . . . . . . . . . 101
10.7 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
10.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
10.7.2 Problem Solving Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 104
10.7.3 Application of Mathematical Modelling
. . . . . . . . . . . . . . . . . . 104
10.7.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 106
10.8 Introduction to Functions and Graphs . . . . . . . . . . . . . . . . . . . . . . . 107
10.9 Functions and Graphs in the Real-World . . . . . . . . . . . . . . . . . . . . . . 107
10.10Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
ix
CONTENTS
CONTENTS
10.10.1 Variables and Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
10.10.2 Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
10.10.3 The Cartesian Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
10.10.4 Drawing Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
10.10.5 Notation used for Functions
. . . . . . . . . . . . . . . . . . . . . . . . 110
10.11Characteristics of Functions - All Grades . . . . . . . . . . . . . . . . . . . . . . 112
10.11.1 Dependent and Independent Variables . . . . . . . . . . . . . . . . . . . 112
10.11.2 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.11.3 Intercepts with the Axes . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.11.4 Turning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.11.5 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.11.6 Lines of Symmetry
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.11.7 Intervals on which the Function Increases/Decreases . . . . . . . . . . . 114
10.11.8 Discrete or Continuous Nature of the Graph . . . . . . . . . . . . . . . . 114
10.12Graphs of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
10.12.1 Functions of the form y = ax + q . . . . . . . . . . . . . . . . . . . . . 116
10.12.2 Functions of the Form y = ax2 + q . . . . . . . . . . . . . . . . . . . . . 120
10.12.3 Functions of the Form y =
a
x
+ q . . . . . . . . . . . . . . . . . . . . . . 125
10.12.4 Functions of the Form y = ab(x) + q . . . . . . . . . . . . . . . . . . . . 129
10.13End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
11 Average Gradient - Grade 10 Extension
135
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
11.2 Straight-Line Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
11.3 Parabolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
11.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
12 Geometry Basics
139
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
12.2 Points and Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
12.3 Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
12.3.1 Measuring angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
12.3.2 Special Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
12.3.3 Special Angle Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
12.3.4 Parallel Lines intersected by Transversal Lines . . . . . . . . . . . . . . . 143
12.4 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
12.4.1 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
12.4.2 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
12.4.3 Other polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
12.4.4 Extra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
12.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
12.5.1 Challenge Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
x
CONTENTS
13 Geometry - Grade 10
CONTENTS
161
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
13.2 Right Prisms and Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
13.2.1 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
13.2.2 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
13.3 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
13.3.1 Similarity of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
13.4 Co-ordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
13.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
13.4.2 Distance between Two Points . . . . . . . . . . . . . . . . . . . . . . . . 172
13.4.3 Calculation of the Gradient of a Line . . . . . . . . . . . . . . . . . . . . 173
13.4.4 Midpoint of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
13.5 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
13.5.1 Translation of a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
13.5.2 Reflection of a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
13.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
14 Trigonometry - Grade 10
189
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
14.2 Where Trigonometry is Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
14.3 Similarity of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
14.4 Definition of the Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 191
14.5 Simple Applications of Trigonometric Functions . . . . . . . . . . . . . . . . . . 195
14.5.1 Height and Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
14.5.2 Maps and Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
14.6 Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 199
14.6.1 Graph of sin θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
14.6.2 Functions of the form y = a sin(x) + q . . . . . . . . . . . . . . . . . . . 200
14.6.3 Graph of cos θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
14.6.4 Functions of the form y = a cos(x) + q
. . . . . . . . . . . . . . . . . . 202
14.6.5 Comparison of Graphs of sin θ and cos θ . . . . . . . . . . . . . . . . . . 204
14.6.6 Graph of tan θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
14.6.7 Functions of the form y = a tan(x) + q . . . . . . . . . . . . . . . . . . 205
14.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
15 Statistics - Grade 10
211
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
15.2 Recap of Earlier Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
15.2.1 Data and Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . 211
15.2.2 Methods of Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . 212
15.2.3 Samples and Populations . . . . . . . . . . . . . . . . . . . . . . . . . . 213
15.3 Example Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
xi
CONTENTS
CONTENTS
15.3.1 Data Set 1: Tossing a Coin . . . . . . . . . . . . . . . . . . . . . . . . . 213
15.3.2 Data Set 2: Casting a die . . . . . . . . . . . . . . . . . . . . . . . . . . 213
15.3.3 Data Set 3: Mass of a Loaf of Bread . . . . . . . . . . . . . . . . . . . . 214
15.3.4 Data Set 4: Global Temperature . . . . . . . . . . . . . . . . . . . . . . 214
15.3.5 Data Set 5: Price of Petrol . . . . . . . . . . . . . . . . . . . . . . . . . 215
15.4 Grouping Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
15.4.1 Exercises - Grouping Data
. . . . . . . . . . . . . . . . . . . . . . . . . 216
15.5 Graphical Representation of Data . . . . . . . . . . . . . . . . . . . . . . . . . . 217
15.5.1 Bar and Compound Bar Graphs . . . . . . . . . . . . . . . . . . . . . . . 217
15.5.2 Histograms and Frequency Polygons . . . . . . . . . . . . . . . . . . . . 217
15.5.3 Pie Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
15.5.4 Line and Broken Line Graphs . . . . . . . . . . . . . . . . . . . . . . . . 220
15.5.5 Exercises - Graphical Representation of Data
. . . . . . . . . . . . . . . 221
15.6 Summarising Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
15.6.1 Measures of Central Tendency . . . . . . . . . . . . . . . . . . . . . . . 222
15.6.2 Measures of Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
15.6.3 Exercises - Summarising Data
. . . . . . . . . . . . . . . . . . . . . . . 228
15.7 Misuse of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
15.7.1 Exercises - Misuse of Statistics . . . . . . . . . . . . . . . . . . . . . . . 230
15.8 Summary of Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
15.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
16 Probability - Grade 10
235
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
16.2 Random Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
16.2.1 Sample Space of a Random Experiment . . . . . . . . . . . . . . . . . . 235
16.3 Probability Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
16.3.1 Classical Theory of Probability . . . . . . . . . . . . . . . . . . . . . . . 239
16.4 Relative Frequency vs. Probability . . . . . . . . . . . . . . . . . . . . . . . . . 240
16.5 Project Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
16.6 Probability Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
16.7 Mutually Exclusive Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
16.8 Complementary Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
16.9 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
III
Grade 11
17 Exponents - Grade 11
249
251
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
17.2 Laws of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
√
m
17.2.1 Exponential Law 7: a n = n am . . . . . . . . . . . . . . . . . . . . . . 251
17.3 Exponentials in the Real-World . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
17.4 End of chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
xii
CONTENTS
CONTENTS
18 Surds - Grade 11
18.1 Surd Calculations . . . . . . . . . .
√
√ √
18.1.1 Surd Law 1: n a n b = n ab
√
p
n
a
18.1.2 Surd Law 2: n ab = √
. .
n
b
√
m
18.1.3 Surd Law 3: n am = a n . .
255
. . . . . . . . . . . . . . . . . . . . . . . . 255
. . . . . . . . . . . . . . . . . . . . . . . . 255
. . . . . . . . . . . . . . . . . . . . . . . . 255
. . . . . . . . . . . . . . . . . . . . . . . . 256
18.1.4 Like and Unlike Surds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
18.1.5 Simplest Surd form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
18.1.6 Rationalising Denominators . . . . . . . . . . . . . . . . . . . . . . . . . 258
18.2 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
19 Error Margins - Grade 11
261
20 Quadratic Sequences - Grade 11
265
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
20.2 What is a quadratic sequence? . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
20.3 End of chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
21 Finance - Grade 11
271
21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
21.2 Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
21.3 Simple Depreciation (it really is simple!) . . . . . . . . . . . . . . . . . . . . . . 271
21.4 Compound Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
21.5 Present Values or Future Values of an Investment or Loan . . . . . . . . . . . . 276
21.5.1 Now or Later . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
21.6 Finding i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
21.7 Finding n - Trial and Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
21.8 Nominal and Effective Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . 280
21.8.1 The General Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
21.8.2 De-coding the Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 282
21.9 Formulae Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
21.9.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
21.9.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
21.10End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
22 Solving Quadratic Equations - Grade 11
287
22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
22.2 Solution by Factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
22.3 Solution by Completing the Square . . . . . . . . . . . . . . . . . . . . . . . . . 290
22.4 Solution by the Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . 293
22.5 Finding an equation when you know its roots . . . . . . . . . . . . . . . . . . . 296
22.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
xiii
CONTENTS
CONTENTS
23 Solving Quadratic Inequalities - Grade 11
301
23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
23.2 Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
23.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
24 Solving Simultaneous Equations - Grade 11
307
24.1 Graphical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
24.2 Algebraic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
25 Mathematical Models - Grade 11
313
25.1 Real-World Applications: Mathematical Models . . . . . . . . . . . . . . . . . . 313
25.2 End of Chatpter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
26 Quadratic Functions and Graphs - Grade 11
321
26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
26.2 Functions of the Form y = a(x + p)2 + q
. . . . . . . . . . . . . . . . . . . . . 321
26.2.1 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
26.2.2 Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
26.2.3 Turning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
26.2.4 Axes of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
26.2.5 Sketching Graphs of the Form f (x) = a(x + p)2 + q . . . . . . . . . . . 325
26.2.6 Writing an equation of a shifted parabola . . . . . . . . . . . . . . . . . 327
26.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
27 Hyperbolic Functions and Graphs - Grade 11
329
27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
27.2 Functions of the Form y =
a
x+p
+q
. . . . . . . . . . . . . . . . . . . . . . . . 329
27.2.1 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
27.2.2 Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
27.2.3 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
27.2.4 Sketching Graphs of the Form f (x) =
a
x+p
+ q . . . . . . . . . . . . . . 333
27.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
28 Exponential Functions and Graphs - Grade 11
335
28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
28.2 Functions of the Form y = ab(x+p) + q . . . . . . . . . . . . . . . . . . . . . . . 335
28.2.1 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
28.2.2 Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
28.2.3 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
28.2.4 Sketching Graphs of the Form f (x) = ab(x+p) + q . . . . . . . . . . . . . 338
28.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
29 Gradient at a Point - Grade 11
341
29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
29.2 Average Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
29.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
xiv
CONTENTS
30 Linear Programming - Grade 11
CONTENTS
345
30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
30.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
30.2.1 Decision Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
30.2.2 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
30.2.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
30.2.4 Feasible Region and Points . . . . . . . . . . . . . . . . . . . . . . . . . 346
30.2.5 The Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
30.3 Example of a Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
30.4 Method of Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
30.5 Skills you will need . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
30.5.1 Writing Constraint Equations . . . . . . . . . . . . . . . . . . . . . . . . 347
30.5.2 Writing the Objective Function . . . . . . . . . . . . . . . . . . . . . . . 348
30.5.3 Solving the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
30.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
31 Geometry - Grade 11
357
31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
31.2 Right Pyramids, Right Cones and Spheres . . . . . . . . . . . . . . . . . . . . . 357
31.3 Similarity of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
31.4 Triangle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
31.4.1 Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
31.5 Co-ordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
31.5.1 Equation of a Line between Two Points . . . . . . . . . . . . . . . . . . 368
31.5.2 Equation of a Line through One Point and Parallel or Perpendicular to
Another Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
31.5.3 Inclination of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
31.6 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
31.6.1 Rotation of a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
31.6.2 Enlargement of a Polygon 1 . . . . . . . . . . . . . . . . . . . . . . . . . 376
32 Trigonometry - Grade 11
381
32.1 History of Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
32.2 Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 381
32.2.1 Functions of the form y = sin(kθ) . . . . . . . . . . . . . . . . . . . . . 381
32.2.2 Functions of the form y = cos(kθ) . . . . . . . . . . . . . . . . . . . . . 383
32.2.3 Functions of the form y = tan(kθ) . . . . . . . . . . . . . . . . . . . . . 384
32.2.4 Functions of the form y = sin(θ + p) . . . . . . . . . . . . . . . . . . . . 385
32.2.5 Functions of the form y = cos(θ + p) . . . . . . . . . . . . . . . . . . . 386
32.2.6 Functions of the form y = tan(θ + p) . . . . . . . . . . . . . . . . . . . 387
32.3 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
32.3.1 Deriving Values of Trigonometric Functions for 30◦ , 45◦ and 60◦ . . . . . 389
32.3.2 Alternate Definition for tan θ . . . . . . . . . . . . . . . . . . . . . . . . 391
xv
CONTENTS
CONTENTS
32.3.3 A Trigonometric Identity . . . . . . . . . . . . . . . . . . . . . . . . . . 392
32.3.4 Reduction Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
32.4 Solving Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 399
32.4.1 Graphical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
32.4.2 Algebraic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
32.4.3 Solution using CAST diagrams . . . . . . . . . . . . . . . . . . . . . . . 403
32.4.4 General Solution Using Periodicity . . . . . . . . . . . . . . . . . . . . . 405
32.4.5 Linear Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . 406
32.4.6 Quadratic and Higher Order Trigonometric Equations . . . . . . . . . . . 406
32.4.7 More Complex Trigonometric Equations . . . . . . . . . . . . . . . . . . 407
32.5 Sine and Cosine Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
32.5.1 The Sine Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
32.5.2 The Cosine Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
32.5.3 The Area Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
32.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
33 Statistics - Grade 11
419
33.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
33.2 Standard Deviation and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . 419
33.2.1 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
33.2.2 Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
33.2.3 Interpretation and Application . . . . . . . . . . . . . . . . . . . . . . . 423
33.2.4 Relationship between Standard Deviation and the Mean . . . . . . . . . . 424
33.3 Graphical Representation of Measures of Central Tendency and Dispersion . . . . 424
33.3.1 Five Number Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
33.3.2 Box and Whisker Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 425
33.3.3 Cumulative Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
33.4 Distribution of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
33.4.1 Symmetric and Skewed Data . . . . . . . . . . . . . . . . . . . . . . . . 428
33.4.2 Relationship of the Mean, Median, and Mode . . . . . . . . . . . . . . . 428
33.5 Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
33.6 Misuse of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
33.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
34 Independent and Dependent Events - Grade 11
437
34.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
34.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
34.2.1 Identification of Independent and Dependent Events
. . . . . . . . . . . 438
34.3 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
IV
Grade 12
35 Logarithms - Grade 12
443
445
35.1 Definition of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
xvi
CONTENTS
CONTENTS
35.2 Logarithm Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
35.3 Laws of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
35.4 Logarithm Law 1: loga 1 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
35.5 Logarithm Law 2: loga (a) = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
35.6 Logarithm Law 3: loga (x · y) = loga (x) + loga (y) . . . . . . . . . . . . . . . . . 448
35.7 Logarithm Law 4: loga xy = loga (x) − loga (y) . . . . . . . . . . . . . . . . . 449
35.8 Logarithm Law 5: loga (xb ) = b loga (x) . . . . . . . . . . . . . . . . . . . . . . . 450
√
35.9 Logarithm Law 6: loga ( b x) = logab(x) . . . . . . . . . . . . . . . . . . . . . . . 450
35.10Solving simple log equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
35.10.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
35.11Logarithmic applications in the Real World . . . . . . . . . . . . . . . . . . . . . 454
35.11.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
35.12End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
36 Sequences and Series - Grade 12
457
36.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
36.2 Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
36.2.1 General Equation for the nth -term of an Arithmetic Sequence . . . . . . 458
36.3 Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
36.3.1 Example - A Flu Epidemic . . . . . . . . . . . . . . . . . . . . . . . . . 459
36.3.2 General Equation for the nth -term of a Geometric Sequence . . . . . . . 461
36.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
36.4 Recursive Formulae for Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 462
36.5 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
36.5.1 Some Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
36.5.2 Sigma Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
36.6 Finite Arithmetic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
36.6.1 General Formula for a Finite Arithmetic Series . . . . . . . . . . . . . . . 466
36.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
36.7 Finite Squared Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
36.8 Finite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
36.8.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
36.9 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
36.9.1 Infinite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . 471
36.9.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
36.10End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
37 Finance - Grade 12
477
37.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
37.2 Finding the Length of the Investment or Loan . . . . . . . . . . . . . . . . . . . 477
37.3 A Series of Payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
37.3.1 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
xvii
CONTENTS
CONTENTS
37.3.2 Present Values of a series of Payments . . . . . . . . . . . . . . . . . . . 479
37.3.3 Future Value of a series of Payments . . . . . . . . . . . . . . . . . . . . 484
37.3.4 Exercises - Present and Future Values . . . . . . . . . . . . . . . . . . . 485
37.4 Investments and Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
37.4.1 Loan Schedules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
37.4.2 Exercises - Investments and Loans . . . . . . . . . . . . . . . . . . . . . 489
37.4.3 Calculating Capital Outstanding . . . . . . . . . . . . . . . . . . . . . . 489
37.5 Formulae Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
37.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
37.5.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
37.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
38 Factorising Cubic Polynomials - Grade 12
493
38.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
38.2 The Factor Theorem
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
38.3 Factorisation of Cubic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 494
38.4 Exercises - Using Factor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 496
38.5 Solving Cubic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
38.5.1 Exercises - Solving of Cubic Equations . . . . . . . . . . . . . . . . . . . 498
38.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
39 Functions and Graphs - Grade 12
501
39.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
39.2 Definition of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
39.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
39.3 Notation used for Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
39.4 Graphs of Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
39.4.1 Inverse Function of y = ax + q . . . . . . . . . . . . . . . . . . . . . . . 503
39.4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
39.4.3 Inverse Function of y = ax2
. . . . . . . . . . . . . . . . . . . . . . . . 504
39.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
39.4.5 Inverse Function of y = ax . . . . . . . . . . . . . . . . . . . . . . . . . 506
39.4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
39.5 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
40 Differential Calculus - Grade 12
509
40.1 Why do I have to learn this stuff? . . . . . . . . . . . . . . . . . . . . . . . . . 509
40.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
40.2.1 A Tale of Achilles and the Tortoise . . . . . . . . . . . . . . . . . . . . . 510
40.2.2 Sequences, Series and Functions . . . . . . . . . . . . . . . . . . . . . . 511
40.2.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
40.2.4 Average Gradient and Gradient at a Point . . . . . . . . . . . . . . . . . 516
40.3 Differentiation from First Principles . . . . . . . . . . . . . . . . . . . . . . . . . 519
xviii
CONTENTS
CONTENTS
40.4 Rules of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
40.4.1 Summary of Differentiation Rules . . . . . . . . . . . . . . . . . . . . . . 522
40.5 Applying Differentiation to Draw Graphs . . . . . . . . . . . . . . . . . . . . . . 523
40.5.1 Finding Equations of Tangents to Curves
. . . . . . . . . . . . . . . . . 523
40.5.2 Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
40.5.3 Local minimum, Local maximum and Point of Inflextion . . . . . . . . . 529
40.6 Using Differential Calculus to Solve Problems . . . . . . . . . . . . . . . . . . . 530
40.6.1 Rate of Change problems . . . . . . . . . . . . . . . . . . . . . . . . . . 534
40.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535
41 Linear Programming - Grade 12
539
41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
41.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
41.2.1 Feasible Region and Points . . . . . . . . . . . . . . . . . . . . . . . . . 539
41.3 Linear Programming and the Feasible Region . . . . . . . . . . . . . . . . . . . 540
41.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
42 Geometry - Grade 12
549
42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
42.2 Circle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
42.2.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
42.2.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
42.2.3 Theorems of the Geometry of Circles . . . . . . . . . . . . . . . . . . . . 550
42.3 Co-ordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
42.3.1 Equation of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
42.3.2 Equation of a Tangent to a Circle at a Point on the Circle . . . . . . . . 569
42.4 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
42.4.1 Rotation of a Point about an angle θ . . . . . . . . . . . . . . . . . . . . 571
42.4.2 Characteristics of Transformations . . . . . . . . . . . . . . . . . . . . . 573
42.4.3 Characteristics of Transformations . . . . . . . . . . . . . . . . . . . . . 573
42.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
43 Trigonometry - Grade 12
577
43.1 Compound Angle Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
43.1.1 Derivation of sin(α + β) . . . . . . . . . . . . . . . . . . . . . . . . . . 577
43.1.2 Derivation of sin(α − β) . . . . . . . . . . . . . . . . . . . . . . . . . . 578
43.1.3 Derivation of cos(α + β) . . . . . . . . . . . . . . . . . . . . . . . . . . 578
43.1.4 Derivation of cos(α − β) . . . . . . . . . . . . . . . . . . . . . . . . . . 579
43.1.5 Derivation of sin 2α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
43.1.6 Derivation of cos 2α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
43.1.7 Problem-solving Strategy for Identities . . . . . . . . . . . . . . . . . . . 580
43.2 Applications of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . 582
43.2.1 Problems in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 582
xix
CONTENTS
CONTENTS
43.2.2 Problems in 3 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 584
43.3 Other Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
43.3.1 Taxicab Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
43.3.2 Manhattan distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
43.3.3 Spherical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
43.3.4 Fractal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
43.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
44 Statistics - Grade 12
591
44.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
44.2 A Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
44.3 Extracting a Sample Population . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
44.4 Function Fitting and Regression Analysis . . . . . . . . . . . . . . . . . . . . . . 594
44.4.1 The Method of Least Squares
. . . . . . . . . . . . . . . . . . . . . . . 596
44.4.2 Using a calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
44.4.3 Correlation coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
44.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
45 Combinations and Permutations - Grade 12
603
45.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
45.2 Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
45.2.1 Making a List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
45.2.2 Tree Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604
45.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604
45.3.1 The Factorial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 604
45.4 The Fundamental Counting Principle . . . . . . . . . . . . . . . . . . . . . . . . 604
45.5 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
45.5.1 Counting Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
45.5.2 Combinatorics and Probability . . . . . . . . . . . . . . . . . . . . . . . 606
45.6 Permutations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
45.6.1 Counting Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
45.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
45.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
V
Exercises
613
46 General Exercises
615
47 Exercises - Not covered in Syllabus
617
A GNU Free Documentation License
619
xx
Chapter 31
Geometry - Grade 11
31.1
Introduction
Activity :: Extension : History of Geometry
Work in pairs or groups and investigate the history of the development of geometry in the last 1500 years. Describe the various stages of development and how
different cultures used geometry to improve their lives.
The works of the following people or cultures can be investigated:
1. Islamic geometry (c. 700 - 1500)
A Thabit ibn Qurra
B Omar Khayyam
C Sharafeddin Tusi
2. Geometry in the 17th - 20th centuries (c. 700 - 1500)
31.2
Right Pyramids, Right Cones and Spheres
A pyramid is a geometric solid that has a polygon base and the base is joined to an apex.
Examples of pyramids are shown in Figure 31.1.
Figure 31.1: Examples of a square pyramid, a triangular pyramid and a cone.
Method: Surface Area of a Pyramid
The surface area of a pyramid is calculated by adding the area of each face together.
357
31.2
CHAPTER 31. GEOMETRY - GRADE 11
Worked Example 127: Surface Area
Question: If a cone
√ has a height of h and a base of radius r, show that the surface
area is πr2 + πr r2 + h2 .
Answer
Step 1 : Draw a picture
a
h
r
h
r
Step 2 : Identify the faces that make up the cone
The cone has two faces: the base and the walls. The base is a circle of radius r and
the walls can be opened out to a sector of a circle.
a
2πr = circumference
This curved surface can be cut into many thin triangles with height close to a (a
is called a slant height). The area of these triangles will add up to 12 ×base×height
which is 12 × 2πr × a = πra
Step 3 : Calculate a
a can be calculated by using the Theorem of Pythagoras. Therefore:
p
a = r 2 + h2
Step 4 : Calculate the area of the circular base
Ab = πr2
Step 5 : Calculate the area of the curved walls
Aw
= πra
p
= πr r2 + h2
Step 6 : Calculate surface area A
A =
Ab + Aw
=
πr2 + πr
p
Method:
Volume of a Pyramid
The volume of a pyramid is found by:
V =
1
A·h
3
where A is the area of the base and h is the height.
358
r 2 + h2
CHAPTER 31. GEOMETRY - GRADE 11
31.2
A cone is a pyramid, so the volume of a cone is given by
V =
1 2
πr h.
3
A square pyramid has volume
V =
1 2
a h
3
where a is the side length.
Worked Example 128: Volume of a Pyramid
Question: What is the volume of a square pyramid, 3cm high with a side length
of 2cm?
Answer
Step 1 : Determine the correct formula
The volume of a pyramid is
1
V = A · h,
3
which for a square base means
V =
1
a · a · h.
3
3cm
b
2cm
2cm
Step 2 : Substitute the given values
=
=
=
1
·2·2·3
3
1
· 12
3
4 cm3
We accept the following formulae for volume and surface area of a sphere (ball).
Surface area =
Volume =
Exercise: Surface Area and Volume
359
4πr2
4 3
πr
3
31.3
CHAPTER 31. GEOMETRY - GRADE 11
1. Calculate the volumes and surface areas of the following solids: *Hint for (e):
find the perpendicular height using Pythagoras.
c)
b)
d)
e)
3
b
a)
6b
b
4
13
5
14
24
b
24
7
a hemisphere
a sphere
a cone
a pyramid with
a square base
a hemisphere on
top of a cone
2. Water covers approximately 71% of the Earth’s surface. Taking the radius of
the Earth to be 6378 km, what is the total area of land (area not covered by
water)?
3.
A right triangular pyramid is placed on top of a
right triangular prism. The prism has an equilateral triangle of side length 20 cm as a base,
and has a height of 42 cm. The pyramid has a
height of 12 cm.
A Find the total volume of the object.
B Find the area of each face of the pyramid.
C Find the total surface area of the object.
31.3
Similarity of Polygons
In order for two polygons to be similar the following must be true:
1. All corresponding angles must be congruent.
2. All corresponding sides must be in the same proportion to each other.
A
If
P
E
T
Q
B
1. Â = P̂ ; B̂ = Q̂; Ĉ = R̂; D̂ = Ŝ;
Ê = T̂
and
2.
S
D
AB
PQ
=
BC
QR
=
CD
RS
then the polygons
PQRST are similar.
R
C
Worked Example 129: Similarity of Polygons
Question:
360
=
DE
ST
=
ABCDE
EA
TP
and
CHAPTER 31. GEOMETRY - GRADE 11
31.4
R
Q
x
3-
x
Polygons PQTU and PRSU are similar.
Find the value of x.
P
T
3
U
1
S
Answer
Step 1 : Identify corresponding sides
Since the polygons are similar,
PQ
PR
∴
TU
SU
3
1
=
x
=
x + (3 − x)
x
=
∴
3
∴x =
31.4
Triangle Geometry
31.4.1
Proportion
3
9
Two line segments are divided in the same proportion if the ratios between their parts are equal.
AB
x
kx
DE
= =
=
BC
y
ky
EF
∴ the line segments are in the same proportion
C
B
x
A
y
D
kx
E
ky
F
If the line segments are proportional, the following also hold
1. AC · F E = CB · DF
2.
CB
AC
=
FE
DF
3.
AB
BC
=
DE
FE
and
BC
AB
=
FE
DE
4.
AB
AC
=
DE
DF
and
AC
AB
=
DF
DE
• Triangles with equal heights have areas which are in the same proportion to each other as
the bases of the triangles.
361
31.4
CHAPTER 31. GEOMETRY - GRADE 11
h1
area △ABC
∴
area △DEF
= h2
1
BC × h1
BC
=
= 21
EF
EF
×
h
2
2
A
D
h1
h2
B
E
C
F
• A special case of this happens when the bases of the triangles are equal:
Triangles with equal bases between the same parallel lines have the same area.
area △ABC =
A
1
· h · BC = area △DBC
2
D
h
B
C
• Triangles on the same side of the same base, with equal areas, lie between parallel lines.
If area △ ABC = area △ BDC,
then AD k BC.
A
D
B
C
Theorem 1. Proportion Theorem:A line drawn parallel to one side of a triangle divides the other
two sides proportionally.
E
A
D
A
A
h1
h2
E
D
B
D
C
B
C B
E
362
C
CHAPTER 31. GEOMETRY - GRADE 11
31.4
Given:△ABC with line DE k BC
R.T.P.:
AD
AE
=
DB
EC
Proof:
Draw h1 from E perpendicular to AD, and h2 from D perpendicular to AE.
Draw BE and CD.
area △ADE
area △BDE
=
area △ADE
area △CED
=
but area △BDE =
area △ADE
∴
=
area △BDE
AD
=
∴
DB
∴ DE divides AB and AC proportionally.
1
2 AD · h1
1
2 DB · h1
1
2 AE · h2
1
2 EC · h2
=
AD
DB
=
AE
EC
area △CED (equal base and height)
area △ADE
area △CED
AE
EC
Similarly,
AD
AB
AB
BD
=
=
AE
AC
AC
CE
Following from Theorem 1, we can prove the midpoint theorem.:
Theorem 2. Midpoint Theorem: A line joining the midpoints of two sides of a triangle is parallel
to the third side and equal to half the length of the third side.
Proof:
This is a special case of the Proportionality Theorem (Theorem 1).
A
If AB = BD and AC = AE,
then DE k BC and BC = 2DE.
B
C
E
D
Theorem 3. Similarity Theorem 1:Equiangular triangles have their sides in proportion and are
therefore similar.
A
b
D
b
G
H
E
B
C
Given:△ABC and △DEF with  = D̂; B̂ = Ê; Ĉ = F̂
363
F
31.4
CHAPTER 31. GEOMETRY - GRADE 11
R.T.P.:
AB
AC
=
DE
DF
Construct: G on AB, so that AG = DE
H on AC, so that AH = DF
Proof: In △’s AGH and DEF
∴
∴
∴
∴
∴
∴
AG = DE; AH = DF
(const.)
 = D̂
△AGH ≡ △DEF
(given)
(SAS)
AĜH = Ê = B̂
GH k BC
AH
AG
=
AB
AC
DF
DE
=
AB
AC
△ABC ||| △DEF
(corres. ∠’s equal)
(proportion theorem)
(AG = DE; AH = DF)
Important: ||| means “is similar to”
Theorem 4. Similarity Theorem 2:Triangles with sides in proportion are equiangular and therefore similar.
A
h1
h2
D
B
E
C
Given:△ABC with line DE such that
AD
AE
=
DB
EC
R.T.P.:DE k BC; △ADE ||| △ABC
Proof:
Draw h1 from E perpendicular to AD, and h2 from D perpendicular to AE.
Draw BE and CD.
364
CHAPTER 31. GEOMETRY - GRADE 11
area △ADE
area △BDE
area △ADE
area △CED
AD
but
DB
area △ADE
∴
area △BDE
∴ area △BDE
∴ DE k BC
∴ AD̂E
and AÊD
=
=
=
=
=
31.4
1
2 AD · h1
1
2 DB · h1
1
2 AE · h2
1
2 EC · h2
=
AD
DB
=
AE
EC
AE
(given)
EC
area △ADE
area △CED
area △CED
(same side of equal base DE, same area)
= AB̂C (corres ∠’s)
= AĈB
∴ △ADE and △ABC are equiangular
∴ △ADE ||| △ABC (AAA)
Theorem 5. Pythagoras’ Theorem:The square on the hypotenuse of a right angled triangle is
equal to the sum of the squares on the other two sides.
Given:△ ABC with  = 90◦
A
1
2
1 2
B
C
R.T.P.:BC 2 = AB 2 + AC 2
Proof:
Let Ĉ
= x
∴ Â2
∴ Â1
= 90◦ − x (∠ ’s of a △ )
= x
B̂
D̂1
= 90◦ − x (∠ ’s of a △ )
= D̂2 = Â = 90◦
∴ △ABD |||△CBA and △CAD |||△ CBA (AAA)
AB
CA
BD
CD
AD
AD
∴
and
=
=
=
=
CB
BA
CA
CB
CA
BA
∴ AB 2 = CB × BD and AC 2 = CB × CD
∴ AB 2 + AC 2
i.e. BC 2
=
=
CB(BD + CD)
CB(CB)
=
=
CB 2
AB 2 + AC 2
Worked Example 130: Triangle Geometry 1
Question: In △ GHI, GH k LJ; GJ k LK and
365
JK
KI
= 35 . Determine
HJ
KI .
31.4
CHAPTER 31. GEOMETRY - GRADE 11
G
L
I
K
J
H
Answer
Step 1 : Identify similar triangles
ˆ
LIJ
J L̂I
∴ △LIJ
ˆ
LIK
K L̂I
∴ △LIK
=
ˆ
GIH
= H ĜI
||| △GIH
(Corres. ∠s)
(Equiangular △s)
ˆ
GIJ
=
= J ĜI
||| △GIJ
(Corres. ∠s)
(Equiangular △s)
Step 2 : Use proportional sides
HJ
JI
GL
and
LI
GL
LI
JK
KI
5
3
5
3
=
=
=
∴
HJ
JI
=
(△LIJ ||| △GIH)
(△LIK ||| △GIJ)
Step 3 : Rearrange to find the required ratio
HJ
KI
HJ
JI
×
JI
KI
5 8
×
3 3
40
9
=
=
=
Worked Example 131: Triangle Geometry 2
Question: PQRS is a trapezium, with PQ k RS. Prove that PT · TR = ST · TQ.
P
Q
1
2
1 2
T
2
2
1
1
R
366
S
CHAPTER 31. GEOMETRY - GRADE 11
31.4
Answer
Step 1 : Identify similar triangles
Pˆ1
Sˆ1
=
(Alt. ∠s)
Q̂1 = R̂1
∴ △P T Q ||| △ST R
(Alt. ∠s)
(Equiangular △s)
Step 2 : Use proportional sides
PT
=
TQ
∴ PT · TR =
ST
TR
ST · T Q
(△ PTQ ||| △ STR)
Exercise: Triangle Geometry
1. Calculate SV
S
V
10
b b
20
U
35
T
2.
CB
YB
= 23 . Find
DS
SB .
D
A
S
Z
X
C
Y
B
3. Given the following figure with the following lengths, find AE, EC and BE.
BC = 15 cm, AB = 4 cm, CD = 18 cm, and ED = 9 cm.
C
A
E
D
B
4. Using the following figure and lengths, find IJ and KJ.
HI = 26 m, KL = 13 m, JL = 9 m and HJ = 32 m.
367
31.5
CHAPTER 31. GEOMETRY - GRADE 11
J
K
H
L
I
5. Find FH in the following figure.
E
36
42
D
G
21
b
b
F
H
6. BF = 25 m, AB = 13 m, AD = 9 m, DF = 18m.
Calculate the lengths of BC, CF, CD, CE and EF, and find the ratio
DE
AC .
A
D
B
C
E
F
7. If LM k JK, calculate y.
J
2
L
K
y
y-2
M
7
I
31.5
Co-ordinate Geometry
31.5.1
Equation of a Line between Two Points
There are many different methods of specifying the requirements for determining the equation of
a straight line. One option is to find the equation of a straight line, when two points are given.
Assume that the two points are (x1 ; y1 ) and (x2 ; y2 ), and we know that the general form of the
equation for a straight line is:
368
CHAPTER 31. GEOMETRY - GRADE 11
31.5
y = mx + c
(31.1)
So, to determine the equation of the line passing through our two points, we need to determine
values for m (the gradient of the line) and c (the y-intercept of the line). The resulting equation
is
y − y1 = m(x − x1 )
(31.2)
where (x1 ; y1 ) are the co-ordinates of either given point.
Extension: Finding the second equation for a straight line
This is an example of a set of simultaneous equations, because we can write:
y1
y2
=
=
mx1 + c
mx2 + c
(31.3)
(31.4)
We now have two equations, with two unknowns, m and c.
Subtract (31.3) from (31.4) y2 − y1
∴
m
Re-arrange (31.3) to obtain c
y1
c
= mx2 − mx1
y2 − y1
=
x2 − x1
= mx1 + c
= y1 − mx1
(31.5)
(31.6)
(31.7)
(31.8)
Now, to make things a bit easier to remember, substitute (31.7) into (31.1):
y
which can be re-arranged to: y − y1
=
mx + c
=
=
mx + (y1 − mx1 )
m(x − x1 )
(31.9)
(31.10)
(31.11)
Important: If you are asked to calculate the equation of a line passing through two points,
use:
y2 − y1
m=
x2 − x1
to calculate m and then use:
y − y1 = m(x − x1 )
to determine the equation.
For example, the equation of the straight line passing through (−1; 1) and (2; 2) is given by first
calculating m
m
y2 − y1
x2 − x1
2−1
2 − (−1)
1
3
=
=
=
and then substituting this value into
y − y1 = m(x − x1 )
to obtain
y − y1
=
1
(x − x1 ).
3
369
31.5
CHAPTER 31. GEOMETRY - GRADE 11
Then substitute (−1; 1) to obtain
y − (1) =
So, y = 13 x +
4
3
y−1
=
y
=
y
=
1
(x − (−1))
3
1
1
x+
3
3
1
1
x+ +1
3
3
1
4
x+
3
3
passes through (−1; 1) and (2; 2).
3
(2;2)
b
2
y = 31 x +
(-1;1)
b
−3
−2
4
3
1
−1
1
2
3
Figure 31.2: The equation of the line passing through (−1; 1) and (2; 2) is y = 13 x + 43 .
Worked Example 132: Equation of Straight Line
Question: Find the equation of the straight line passing through (−3; 2) and (5; 8).
Answer
Step 1 : Label the points
(x1 ; y1 ) =
(x2 ; y2 ) =
(−3; 2)
(5; 8)
Step 2 : Calculate the gradient
m
=
=
=
=
=
y2 − y1
x2 − x1
8−2
5 − (−3)
6
5+3
6
8
3
4
Step 3 : Determine the equation of the line
370
CHAPTER 31. GEOMETRY - GRADE 11
y − y1
31.5
=
y − (2) =
=
y
=
=
=
m(x − x1 )
3
(x − (−3))
4
3
(x + 3) + 2
4
3
3
x+ ·3+2
4
4
9 8
3
x+ +
4
4 4
17
3
x+
4
4
Step 4 : Write the final answer
The equation of the straight line that passes through (−3; 2) and (5; 8) is y =
3
17
4x + 4 .
31.5.2
Equation of a Line through One Point and Parallel or Perpendicular to Another Line
Another method of determining the equation of a straight-line is to be given one point, (x1 ; y1 ),
and to be told that the line is parallel or perpendicular to another line. If the equation of the
unknown line is y = mx + c and the equation of the second line is y = m0 x + c0 , then we know
the following:
If the lines are parallel, then
m
If the lines are perpendicular, then m × m0
= m0
= −1
(31.12)
(31.13)
Once we have determined a value for m, we can then use the given point together with:
y − y1 = m(x − x1 )
to determine the equation of the line.
For example, find the equation of the line that is parallel to y = 2x − 1 and that passes through
(−1; 1).
First we determine m. Since the line we are looking for is parallel to y = 2x − 1,
m=2
The equation is found by substituting m and (−1; 1) into:
y − y1
y−1
= m(x − x1 )
= 2(x − (−1)
y
y
= 2x + 2 + 1
= 2x + 3
y−1
y−1
31.5.3
= 2(x + 1)
= 2x + 2
Inclination of a Line
In Figure 31.4(a), we see that the line makes an angle θ with the x-axis. This angle is known as
the inclination of the line and it is sometimes interesting to know what the value of θ is.
371
31.5
CHAPTER 31. GEOMETRY - GRADE 11
3
2
(-1;1)
−3
−2
y = 2x + 3
b
−1
y = 2x − 1
1
1
2
3
−1
−2
Figure 31.3: The equation of the line passing through (−1; 1) and parallel to y = 2x − 1 is
y = 2x + 3. It can be seen that the lines are parallel to each other. You can test this by using
your ruler and measuring the distance between the lines at different points.
f (x) = 4x − 4
3
3
∆y
2
1
g(x) = 2x − 2
2
1
∆x
θg
θ
1
2
3
1
(a)
θf
2
3
4
(b)
Figure 31.4: (a) A line makes an angle θ with the x-axis. (b) The angle is dependent on the
gradient. If the gradient of f is mf and the gradient of g is mg then mf > mg and θf > θg .
372
CHAPTER 31. GEOMETRY - GRADE 11
31.6
Firstly, we note that if the gradient changes, then the value of θ changes (Figure 31.4(b)), so
we suspect that the inclination of a line is related to the gradient. We know that the gradient is
a ratio of a change in the y-direction to a change in the x-direction.
m=
∆y
∆x
But, in Figure 31.4(a) we see that
tan θ
=
∴m
=
∆y
∆x
tan θ
For example, to find the inclination of the line y = x, we know m = 1
∴ tan θ
∴θ
=
=
1
45◦
Exercise: Co-ordinate Geometry
1. Find the equations of the following lines
A
B
C
D
E
through points (−1; 3) and (1; 4)
through points (7; −3) and (0; 4)
parallel to y = 21 x + 3 passing through (−1; 3)
perpendicular to y = − 21 x + 3 passing through (−1; 2)
perpendicular to 2y + x = 6 passing through the origin
2. Find the inclination of the following lines
A
B
C
D
E
y = 2x − 3
y = 13 x − 7
4y = 3x + 8
y = − 23 x + 3 (Hint: if m is negative θ must be in the second quadrant)
3y + x − 3 = 0
3. Show that the line y = k for any constant k is parallel to the x-axis. (Hint:
Show that the inclination of this line is 0◦ .)
4. Show that the line x = k for any constant k is parallel to the y-axis. (Hint:
Show that the inclination of this line is 90◦ .)
31.6
Transformations
31.6.1
Rotation of a Point
When something is moved around a fixed point, we say that it is rotated. What happens to the
coordinates of a point that is rotated by 90◦ or 180◦ around the origin?
Activity :: Investigation : Rotation of a Point by 90◦
373
31.6
CHAPTER 31. GEOMETRY - GRADE 11
Complete the table, by filling in the coordinates of the points shown in the figure.
Point x-coordinate y-coordinate
A
B
C
D
E
F
G
H
What do you notice about the x-coordinates?
What do you notice about the y-coordinates?
What would happen to the coordinates of
point A, if it was rotated to the position of
point C? What about point B rotated to the
position of D?
bC bB
Db
Eb
F
b b
G
bA
b
H
Activity :: Investigation : Rotation of a Point by 180◦
Complete the table, by filling in the coordinates of the points shown in the figure.
Point x-coordinate y-coordinate
A
B
C
D
E
F
G
H
What do you notice about the x-coordinates?
What do you notice about the y-coordinates?
What would happen to the coordinates of
point A, if it was rotated to the position of
point E? What about point F rotated to the
position of B?
Db
Eb
bC bB
bF b
G
From these activities you should have come to the following conclusions:
374
bA
b
H
CHAPTER 31. GEOMETRY - GRADE 11
31.6
y
P(x; y)
b
P’(y; -x)
• 90◦ clockwise rotation:
The image of a point P(x; y) rotated clockwise
through 90◦ around the origin is P’(y; −x).
We write the rotation as (x; y) → (y; −x).
b
x
y
b
P(x; y)
x
P”(-y; x)
b
• 90◦ anticlockwise rotation:
The image of a point P(x; y) rotated anticlockwise through 90◦ around the origin is P’(−y; x).
We write the rotation as (x; y) → (−y; x).
y
b
P(x; y)
• 180◦ rotation:
The image of a point P(x; y) rotated through
180◦ around the origin is P’(−x; −y).
We write the rotation as (x; y) → (−x; −y).
x
b
P”’(-x; -y)
Exercise: Rotation
1. For each of the following rotations about the origin:
(i) Write down the rule.
(ii) Draw a diagram showing the direction of rotation.
A OA is rotated to OA′ with A(4;2) and A′ (-2;4)
B OB is rotated to OB′ with B(-2;5) and B′ (5;2)
C OC is rotated to OC′ with C(-1;-4) and C′ (1;4)
2. Copy ∆XYZ onto squared paper. The co-ordinates are given on the picture.
A Rotate ∆XYZ anti-clockwise through an angle of 90◦ about the origin to
give ∆X′ Y′ Z′ . Give the co-ordinates of X′ , Y′ and Z′ .
B Rotate ∆XYZ through 180◦ about the origin to give ∆X′′ Y′′ Z′′ . Give the
co-ordinates of X′′ , Y′′ and Z′′ .
375
31.6
CHAPTER 31. GEOMETRY - GRADE 11
X(4;4)
Z(-4;-1)
Y(-1;-4)
31.6.2
Enlargement of a Polygon 1
When something is made larger, we say that it is enlarged. What happens to the coordinates of
a polygon that is enlarged by a factor k?
Activity :: Investigation : Enlargement of a Polygon
Complete the table, by filling in the coordinates of the points shown in the figure.
Point x-coordinate y-coordinate
A
B
C
D
E
F
G
H
What do you notice about the x-coordinates?
What do you notice about the y-coordinates?
What would happen to the coordinates of
point A, if the square ABCD was enlarged
by a factor 2?
b
b
F
b1
B
−1b
−1C
bG
Activity :: Investigation : Enlargement of a Polygon 2
376
b
E
A
b1D
bH
CHAPTER 31. GEOMETRY - GRADE 11
31.6
7
I’
6
5
H’
4
I
3
H
2
J’
K’
1
K
J
0
1
0
2
3
4
5
6
7
8
9
In the figure quadrilateral HIJK has been enlarged by a factor of 2 through the origin
to become H’I’J’K’. Complete the following table.
Co-ordinate
H = (;)
I = (;)
J = (;)
K = (;)
Co-ordinate’
H’ = (;)
I’ = (;)
J’ = (;)
K’ + (;)
Length
OH =
OI =
OJ =
OK =
Length’
OH’ =
OI’ =
OJ’ =
OK’ =
What conclusions can you draw about
1. the co-ordinates
2. the lengths when we enlarge by a factor of 2?
We conclude as follows:
Let the vertices of a triangle have co-ordinates S(x1 ; y1 ), T(x2 ; y2 ), U(x3 ; y3 ). △S’T’U’ is an
enlargement through the origin of △STU by a factor of c (c > 0).
• △STU is a reduction of △S’T’U’ by a factor of c.
• △S’T’U’ can alternatively be seen as an reduction through the origin of △STU by a factor
of 1c . (Note that a reduction by 1c is the same as an enlargement by c).
• The vertices of △S’T’U’ are S’(cx1 ; cy1 ), T’(cx2 ,cy2 ), U’(cx3 ,cy3 ).
• The distances from the origin are OS’ = cOS, OT’ = cOT and OU’ = cOU.
9
8
T’
7
6
5
S’
4
T
3
U’
2
S
U
1
0
0
1
2
3
4
5
377
6
7
8
9
10
11
31.6
CHAPTER 31. GEOMETRY - GRADE 11
Exercise: Transformations
1. 1) Copy polygon STUV onto squared paper and then answer the following
questions.
3
S
2
T
1
0
-3
-2
-1
0
1
2
3
4
5
-1
V
-2
U
-3
A What are the co-ordinates of polygon STUV?
B Enlarge the polygon through the origin by a constant factor of c = 2. Draw
this on the same grid. Label it S’T’U’V’.
C What are the co-ordinates of the vertices of S’T’U’V’ ?
2. △ABC is an enlargement of △A’B’C’ by a constant factor of k through the
origin.
A What are the co-ordinates of the vertices of △ABC and △A’B’C’ ?
B Giving reasons, calculate the value of k.
C If the area of △ABC is m times the area of △A’B’C’, what is m?
5
A
4
3
A’
B
2
B’
1
0
-4
-5
-3
-2
-1
0
-1
-2
C’
-3
-4
C
-5
378
1
2
3
4
5
CHAPTER 31. GEOMETRY - GRADE 11
31.6
5
M
4
3
2
N
P
1
Q
0
-2
-1
0
1
2
3
4
5
-1
3.
-2
A What are the co-ordinates of the vertices of polygon MNPQ?
B Enlarge the polygon through the origin by using a constant factor of c = 3,
obtaining polygon M’N’P’Q’. Draw this on the same set of axes.
C What are the co-ordinates of the new vertices?
D Now draw M”N”P”Q” which is an anticlockwise rotation of MNPQ by 90◦
around the origin.
E Find the inclination of OM”.
379
31.6
CHAPTER 31. GEOMETRY - GRADE 11
380
Appendix A
GNU Free Documentation License
Version 1.2, November 2002
c 2000,2001,2002 Free Software Foundation, Inc.
Copyright 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
Everyone is permitted to copy and distribute verbatim copies of this license document, but
changing it is not allowed.
PREAMBLE
The purpose of this License is to make a manual, textbook, or other functional and useful document “free” in the sense of freedom: to assure everyone the effective freedom to copy and
redistribute it, with or without modifying it, either commercially or non-commercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while
not being considered responsible for modifications made by others.
This License is a kind of “copyleft”, which means that derivative works of the document must
themselves be free in the same sense. It complements the GNU General Public License, which
is a copyleft license designed for free software.
We have designed this License in order to use it for manuals for free software, because free
software needs free documentation: a free program should come with manuals providing the
same freedoms that the software does. But this License is not limited to software manuals; it
can be used for any textual work, regardless of subject matter or whether it is published as a
printed book. We recommend this License principally for works whose purpose is instruction or
reference.
APPLICABILITY AND DEFINITIONS
This License applies to any manual or other work, in any medium, that contains a notice placed
by the copyright holder saying it can be distributed under the terms of this License. Such a
notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under
the conditions stated herein. The “Document”, below, refers to any such manual or work. Any
member of the public is a licensee, and is addressed as “you”. You accept the license if you
copy, modify or distribute the work in a way requiring permission under copyright law.
A “Modified Version” of the Document means any work containing the Document or a portion
of it, either copied verbatim, or with modifications and/or translated into another language.
A “Secondary Section” is a named appendix or a front-matter section of the Document that deals
exclusively with the relationship of the publishers or authors of the Document to the Document’s
overall subject (or to related matters) and contains nothing that could fall directly within that
overall subject. (Thus, if the Document is in part a textbook of mathematics, a Secondary
Section may not explain any mathematics.) The relationship could be a matter of historical
connection with the subject or with related matters, or of legal, commercial, philosophical,
ethical or political position regarding them.
619
APPENDIX A. GNU FREE DOCUMENTATION LICENSE
The “Invariant Sections” are certain Secondary Sections whose titles are designated, as being
those of Invariant Sections, in the notice that says that the Document is released under this
License. If a section does not fit the above definition of Secondary then it is not allowed to be
designated as Invariant. The Document may contain zero Invariant Sections. If the Document
does not identify any Invariant Sections then there are none.
The “Cover Texts” are certain short passages of text that are listed, as Front-Cover Texts or
Back-Cover Texts, in the notice that says that the Document is released under this License. A
Front-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words.
A “Transparent” copy of the Document means a machine-readable copy, represented in a format
whose specification is available to the general public, that is suitable for revising the document
straightforwardly with generic text editors or (for images composed of pixels) generic paint
programs or (for drawings) some widely available drawing editor, and that is suitable for input
to text formatters or for automatic translation to a variety of formats suitable for input to text
formatters. A copy made in an otherwise Transparent file format whose markup, or absence of
markup, has been arranged to thwart or discourage subsequent modification by readers is not
Transparent. An image format is not Transparent if used for any substantial amount of text. A
copy that is not “Transparent” is called “Opaque”.
Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo
input format, LATEX input format, SGML or XML using a publicly available DTD and standardconforming simple HTML, PostScript or PDF designed for human modification. Examples of
transparent image formats include PNG, XCF and JPG. Opaque formats include proprietary
formats that can be read and edited only by proprietary word processors, SGML or XML for
which the DTD and/or processing tools are not generally available, and the machine-generated
HTML, PostScript or PDF produced by some word processors for output purposes only.
The “Title Page” means, for a printed book, the title page itself, plus such following pages as
are needed to hold, legibly, the material this License requires to appear in the title page. For
works in formats which do not have any title page as such, “Title Page” means the text near the
most prominent appearance of the work’s title, preceding the beginning of the body of the text.
A section “Entitled XYZ” means a named subunit of the Document whose title either is precisely
XYZ or contains XYZ in parentheses following text that translates XYZ in another language.
(Here XYZ stands for a specific section name mentioned below, such as “Acknowledgements”,
“Dedications”, “Endorsements”, or “History”.) To “Preserve the Title” of such a section when
you modify the Document means that it remains a section “Entitled XYZ” according to this
definition.
The Document may include Warranty Disclaimers next to the notice which states that this
License applies to the Document. These Warranty Disclaimers are considered to be included by
reference in this License, but only as regards disclaiming warranties: any other implication that
these Warranty Disclaimers may have is void and has no effect on the meaning of this License.
VERBATIM COPYING
You may copy and distribute the Document in any medium, either commercially or non-commercially,
provided that this License, the copyright notices, and the license notice saying this License applies
to the Document are reproduced in all copies, and that you add no other conditions whatsoever
to those of this License. You may not use technical measures to obstruct or control the reading
or further copying of the copies you make or distribute. However, you may accept compensation
in exchange for copies. If you distribute a large enough number of copies you must also follow
the conditions in section A.
You may also lend copies, under the same conditions stated above, and you may publicly display
copies.
COPYING IN QUANTITY
If you publish printed copies (or copies in media that commonly have printed covers) of the
Document, numbering more than 100, and the Document’s license notice requires Cover Texts,
620
APPENDIX A. GNU FREE DOCUMENTATION LICENSE
you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: FrontCover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also
clearly and legibly identify you as the publisher of these copies. The front cover must present the
full title with all words of the title equally prominent and visible. You may add other material on
the covers in addition. Copying with changes limited to the covers, as long as they preserve the
title of the Document and satisfy these conditions, can be treated as verbatim copying in other
respects.
If the required texts for either cover are too voluminous to fit legibly, you should put the first
ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent
pages.
If you publish or distribute Opaque copies of the Document numbering more than 100, you must
either include a machine-readable Transparent copy along with each Opaque copy, or state in or
with each Opaque copy a computer-network location from which the general network-using public
has access to download using public-standard network protocols a complete Transparent copy of
the Document, free of added material. If you use the latter option, you must take reasonably
prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this
Transparent copy will remain thus accessible at the stated location until at least one year after
the last time you distribute an Opaque copy (directly or through your agents or retailers) of that
edition to the public.
It is requested, but not required, that you contact the authors of the Document well before
redistributing any large number of copies, to give them a chance to provide you with an updated
version of the Document.
MODIFICATIONS
You may copy and distribute a Modified Version of the Document under the conditions of
sections A and A above, provided that you release the Modified Version under precisely this
License, with the Modified Version filling the role of the Document, thus licensing distribution
and modification of the Modified Version to whoever possesses a copy of it. In addition, you
must do these things in the Modified Version:
1. Use in the Title Page (and on the covers, if any) a title distinct from that of the Document,
and from those of previous versions (which should, if there were any, be listed in the History
section of the Document). You may use the same title as a previous version if the original
publisher of that version gives permission.
2. List on the Title Page, as authors, one or more persons or entities responsible for authorship
of the modifications in the Modified Version, together with at least five of the principal
authors of the Document (all of its principal authors, if it has fewer than five), unless they
release you from this requirement.
3. State on the Title page the name of the publisher of the Modified Version, as the publisher.
4. Preserve all the copyright notices of the Document.
5. Add an appropriate copyright notice for your modifications adjacent to the other copyright
notices.
6. Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the
Addendum below.
7. Preserve in that license notice the full lists of Invariant Sections and required Cover Texts
given in the Document’s license notice.
8. Include an unaltered copy of this License.
9. Preserve the section Entitled “History”, Preserve its Title, and add to it an item stating
at least the title, year, new authors, and publisher of the Modified Version as given on the
Title Page. If there is no section Entitled “History” in the Document, create one stating
the title, year, authors, and publisher of the Document as given on its Title Page, then
add an item describing the Modified Version as stated in the previous sentence.
621
APPENDIX A. GNU FREE DOCUMENTATION LICENSE
10. Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document
for previous versions it was based on. These may be placed in the “History” section. You
may omit a network location for a work that was published at least four years before the
Document itself, or if the original publisher of the version it refers to gives permission.
11. For any section Entitled “Acknowledgements” or “Dedications”, Preserve the Title of the
section, and preserve in the section all the substance and tone of each of the contributor
acknowledgements and/or dedications given therein.
12. Preserve all the Invariant Sections of the Document, unaltered in their text and in their
titles. Section numbers or the equivalent are not considered part of the section titles.
13. Delete any section Entitled “Endorsements”. Such a section may not be included in the
Modified Version.
14. Do not re-title any existing section to be Entitled “Endorsements” or to conflict in title
with any Invariant Section.
15. Preserve any Warranty Disclaimers.
If the Modified Version includes new front-matter sections or appendices that qualify as Secondary
Sections and contain no material copied from the Document, you may at your option designate
some or all of these sections as invariant. To do this, add their titles to the list of Invariant
Sections in the Modified Version’s license notice. These titles must be distinct from any other
section titles.
You may add a section Entitled “Endorsements”, provided it contains nothing but endorsements
of your Modified Version by various parties–for example, statements of peer review or that the
text has been approved by an organisation as the authoritative definition of a standard.
You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25
words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only
one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through
arrangements made by) any one entity. If the Document already includes a cover text for the
same cover, previously added by you or by arrangement made by the same entity you are acting
on behalf of, you may not add another; but you may replace the old one, on explicit permission
from the previous publisher that added the old one.
The author(s) and publisher(s) of the Document do not by this License give permission to use
their names for publicity for or to assert or imply endorsement of any Modified Version.
COMBINING DOCUMENTS
You may combine the Document with other documents released under this License, under the
terms defined in section A above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them
all as Invariant Sections of your combined work in its license notice, and that you preserve all
their Warranty Disclaimers.
The combined work need only contain one copy of this License, and multiple identical Invariant
Sections may be replaced with a single copy. If there are multiple Invariant Sections with the
same name but different contents, make the title of each such section unique by adding at the
end of it, in parentheses, the name of the original author or publisher of that section if known,
or else a unique number. Make the same adjustment to the section titles in the list of Invariant
Sections in the license notice of the combined work.
In the combination, you must combine any sections Entitled “History” in the various original
documents, forming one section Entitled “History”; likewise combine any sections Entitled “Acknowledgements”, and any sections Entitled “Dedications”. You must delete all sections Entitled
“Endorsements”.
622
APPENDIX A. GNU FREE DOCUMENTATION LICENSE
COLLECTIONS OF DOCUMENTS
You may make a collection consisting of the Document and other documents released under
this License, and replace the individual copies of this License in the various documents with a
single copy that is included in the collection, provided that you follow the rules of this License
for verbatim copying of each of the documents in all other respects.
You may extract a single document from such a collection, and distribute it individually under
this License, provided you insert a copy of this License into the extracted document, and follow
this License in all other respects regarding verbatim copying of that document.
AGGREGATION WITH INDEPENDENT WORKS
A compilation of the Document or its derivatives with other separate and independent documents
or works, in or on a volume of a storage or distribution medium, is called an “aggregate” if the
copyright resulting from the compilation is not used to limit the legal rights of the compilation’s
users beyond what the individual works permit. When the Document is included an aggregate,
this License does not apply to the other works in the aggregate which are not themselves derivative
works of the Document.
If the Cover Text requirement of section A is applicable to these copies of the Document, then if
the Document is less than one half of the entire aggregate, the Document’s Cover Texts may be
placed on covers that bracket the Document within the aggregate, or the electronic equivalent
of covers if the Document is in electronic form. Otherwise they must appear on printed covers
that bracket the whole aggregate.
TRANSLATION
Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section A. Replacing Invariant Sections with translations requires
special permission from their copyright holders, but you may include translations of some or
all Invariant Sections in addition to the original versions of these Invariant Sections. You may
include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and
the original versions of those notices and disclaimers. In case of a disagreement between the
translation and the original version of this License or a notice or disclaimer, the original version
will prevail.
If a section in the Document is Entitled “Acknowledgements”, “Dedications”, or “History”, the
requirement (section A) to Preserve its Title (section A) will typically require changing the actual
title.
TERMINATION
You may not copy, modify, sub-license, or distribute the Document except as expressly provided
for under this License. Any other attempt to copy, modify, sub-license or distribute the Document
is void, and will automatically terminate your rights under this License. However, parties who
have received copies, or rights, from you under this License will not have their licenses terminated
so long as such parties remain in full compliance.
FUTURE REVISIONS OF THIS LICENSE
The Free Software Foundation may publish new, revised versions of the GNU Free Documentation
License from time to time. Such new versions will be similar in spirit to the present version, but
may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/.
623
APPENDIX A. GNU FREE DOCUMENTATION LICENSE
Each version of the License is given a distinguishing version number. If the Document specifies
that a particular numbered version of this License “or any later version” applies to it, you have the
option of following the terms and conditions either of that specified version or of any later version
that has been published (not as a draft) by the Free Software Foundation. If the Document does
not specify a version number of this License, you may choose any version ever published (not as
a draft) by the Free Software Foundation.
ADDENDUM: How to use this License for your documents
To use this License in a document you have written, include a copy of the License in the document
and put the following copyright and license notices just after the title page:
c YEAR YOUR NAME. Permission is granted to copy, distribute and/or
Copyright modify this document under the terms of the GNU Free Documentation License,
Version 1.2 or any later version published by the Free Software Foundation; with no
Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the
license is included in the section entitled “GNU Free Documentation License”.
If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the “with...Texts.”
line with this:
with the Invariant Sections being LIST THEIR TITLES, with the Front-Cover Texts being LIST,
and with the Back-Cover Texts being LIST.
If you have Invariant Sections without Cover Texts, or some other combination of the three,
merge those two alternatives to suit the situation.
If your document contains nontrivial examples of program code, we recommend releasing these
examples in parallel under your choice of free software license, such as the GNU General Public
License, to permit their use in free software.
624
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement