Karen Morrison and Lucille Dunne
Cambridge IGCSE®
Mathematics
Extended Practice Book
cambridge university press
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Singapore, São Paulo, Delhi, Mexico City
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
www.cambridge.org
Information on this title: www.cambridge.org/9781107672727
© Cambridge University Press 2013
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2013
Printed and bound in the United Kingdom by the MPG Books Group
A catalogue record for this publication is available from the British Library
ISBN-13 978-1-107-67272-7 Paperback
Cover image: Seamus Ditmeyer/Alamy
Cambridge University Press has no responsibility for the persistence or
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IGCSE® is the registered trademark of Cambridge International Examinations.
Contents
Introduction
v
Unit 1
Chapter 1: Reviewing number concepts
1.1 Different types of numbers
1.2 Multiples and factors
1.3 Prime numbers
1.4 Powers and roots
1.5 Working with directed numbers
1.6 Order of operations
1.7 Rounding numbers
1
1
2
3
3
4
4
5
Chapter 2: Making sense of algebra
2.1 Using letters to represent
unknown values
2.2 Substitution
2.3 Simplifying expressions
2.4 Working with brackets
2.5 Indices
7
7
8
8
9
9
Chapter 3: Lines, angles and shapes
3.1 Lines and angles
3.2 Triangles
3.3 Quadrilaterals
3.4 Polygons
3.5 Circles
3.6 Construction
12
12
15
16
18
19
19
Chapter 4: Collecting, organising and
displaying data
4.1 Collecting and classifying data
4.2 Organising data
4.3 Using charts to display data
22
22
23
24
Chapter 7: Perimeter, area and volume
7.1 Perimeter and area in two dimensions
7.2 Three-dimensional objects
7.3 Surface areas and volumes of solids
39
39
43
44
Chapter 8: Introduction to probability
8.1 Basic probability
8.2 Theoretical probability
8.3 The probability that an event does not
happen
8.4 Possibility diagrams
8.5 Combining independent and mutually exclusive
events
48
48
49
Chapter 12: Averages and measures of spread
12.1 Different types of average
12.2 Making comparisons using averages
and ranges
12.3 Calculating averages and ranges for
frequency data
12.4 Calculating averages and ranges for
grouped continuous data
12.5 Percentiles and quartiles
75
75
Unit 2
Chapter 5: Fractions
5.1 Equivalent fractions
5.2 Operations on fractions
5.3 Percentages
5.4 Standard form
5.5 Estimation
29
29
29
30
32
33
Chapter 6: Equations and transforming formulae
6.1 Further expansions of brackets
6.2 Solving linear equations
6.3 Factorising algebraic expressions
6.4 Transformation of a formula
35
35
35
36
37
50
51
52
Unit 3
Chapter 9: Sequences and sets
9.1 Sequences
9.2 Rational and irrational numbers
9.3 Sets
54
54
55
56
Chapter 10: Straight lines and quadratic equations
10.1 Straight lines
10.2 Quadratic expressions
59
59
61
Chapter 11: Pythagoras’ theorem and
similar shapes
11.1 Pythagoras’ theorem
11.2 Understanding similar triangles
11.3 Understanding similar shapes
11.4 Understanding congruence
66
66
68
69
70
Contents
76
77
78
79
iii
Unit 4
Chapter 13: Understanding measurement
13.1 Understanding units
13.2 Time
13.3 Upper and lower bounds
13.4 Conversion graphs
13.5 More money
Chapter 14: Further solving of equations and
inequalities
14.1 Simultaneous linear equations
14.2 Linear inequalities
14.3 Regions in a plane
14.4 Linear programming
14.5 Completing the square
14.6 Quadratic formula
14.7 Factorising quadratics where the coefficient of
x2 is not 1
14.8 Algebraic fractions
81
81
82
84
85
86
89
89
91
92
93
94
95
Chapter 15: Scale drawings, bearings and
trigonometry
15.1 Scale drawings
15.2 Bearings
15.3 Understanding the tangent, cosine
and sine ratios
15.4 Solving problems using trigonometry
15.5 Angles between 0° and 180°
15.6 The sine and cosine rules
15.7 Area of a triangle
15.8 Trigonometry in three dimensions
99
99
100
101
105
105
105
108
108
Chapter 16: Scatter diagrams
and correlation
16.1 Introduction to bivariate data
111
111
Chapter 19: Symmetry and loci
19.1 Symmetry in two dimensions
19.2 Symmetry in three dimensions
19.3 Symmetry properties of circles
19.4 Angle relationships in circles
19.5 Locus
126
126
127
128
129
131
96
96
Unit 5
Chapter 17: Managing money
17.1 Earning money
17.2 Borrowing and investing money
17.3 Buying and selling
114
114
115
116
Chapter 18: Curved graphs
18.1 Plotting quadratic graphs (the parabola)
18.2 Plotting reciprocal graphs (the hyperbola)
18.3 Using graphs to solve quadratic equations
18.4 Using graphs to solve simultaneous linear
and non-linear equations
18.5 Other non-linear graphs
18.6 Finding the gradient of a curve
119
119
121
122
122
123
124
Chapter 20: Histograms and frequency distribution
diagrams
135
20.1 Histograms
135
20.2 Cumulative frequency
137
Unit 6
Chapter 21: Ratio, rate and proportion
21.1 Working with ratio
21.2 Ratio and scale
21.3 Rates
21.4 Kinematic graphs
21.5 Proportion
21.6 Direct and inverse proportion in algebraic
terms
21.7 Increasing and decreasing amounts by a
given ratio
139
139
140
141
141
144
146
Chapter 22: More equations, formulae and
functions
22.1 Setting up equations to solve problems
22.2 Using and transforming formulae
22.3 Functions and function notation
149
149
150
151
145
Chapter 23: Transformations and matrices
23.1 Simple plane transformations
23.2 Vectors
23.3 Further transformations
23.4 Matrices and matrix transformation
23.5 Matrices and transformations
153
153
158
161
163
164
Chapter 24: Probability using tree diagrams
24.1 Using tree diagrams to show outcomes
24.2 Calculating probability from tree diagrams
169
169
169
Answers
Example practice papers can be found online, visit education.cambridge.org/extendedpracticebook
iv
Contents
171
Introduction
This highly illustrated practice book has been written by experienced teachers to help students
revise the Cambridge IGCSE Mathematics (0580) Extended syllabus. Packed full of exercises, the
only narrative consists of helpful bulleted lists of key reminders and useful hints in the margins
for students needing more support.
There is plenty of practice offered via ‘drill’ exercises throughout each chapter. These consist of
progressive and repetitive questions that allow the student to practise methods applicable to
each subtopic. At the end of each chapter there are ‘Mixed exercises’ that bring together all the
subtopics of a chapter in such a way that students have to decide for themselves what methods to
use. The answers to all of these questions are supplied at the back of the book. This encourages
students to assess their progress as they go along, choosing to do more or less practice as
required.
The book has been written with a clear progression from start to finish, with some later chapters
requiring knowledge learned in earlier chapters. There are useful signposts throughout that link
the content of the chapters, allowing the individual to follow their own course through the book:
where the content in one chapter might require knowledge from a previous chapter, a comment
is included in a ‘Rewind’ box; and where content will be practised in more detail later on, a
comment is included in a ‘Fast forward’ box. Examples of both are included below:
FAST FORWARD
REWIND
You learned how to plot lines from
equations in chapter 10.
Remember ‘coefficient’ is the
number in the term.
Tip
It is essential that you
remember to work out
both unknowns. Every
pair of simultaneous linear
equations will have a pair
of solutions.
You will learn much more about
sets in chapter 9. For now, just think
of a set as a list of numbers or other
items that are often placed inside
curly brackets.
Other helpful guides in the margin of the book are as follows:
Hints: these are general comments to remind students of important or key information that is
useful when tackling an exercise, or simply useful to know. They often provide extra information
or support in potentially tricky topics.
Tip: these are tips that relate to good practice in examinations, and also just generally in
mathematics! They cover common pitfalls based on the authors’ experiences of their students,
and give students things to be wary of or to remember in order to score marks in the exam.
The Extended Practice Book mirrors the chapters and subtopics of the Cambridge IGCSE
Mathematics Core and Extended Coursebook written by Karen Morrison and Nick Hamshaw
(9781107606272). However, this book has been written such that it can be used without the
coursebook; it can be used as a revision tool by any student regardless of what coursebook they
are using. Various aspects of the Core syllabus are also revised for complete coverage.
Also in the Cambridge IGCSE Mathematics series:
Cambridge IGCSE Mathematics Core and Extended Coursebook (9781107606272)
Cambridge IGCSE Mathematics Core Practice Book (9781107609884)
Cambridge IGCSE Mathematics Teacher’s Resource CD-ROM (9781107627529)
Introduction
v
1
Reviewing number concepts
1.1 Different types of numbers
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Exercise 1.1
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Unit 1: Number
1
1 Reviewing number concepts
1.2 Multiples and factors
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To find the LCM of a set of
numbers, you can list the multiples
of each number until you find the
first multiple that is in the lists for
all of the numbers in the set.
FAST FORWARD
You will use LCM again when
you work with fractions to find the
lowest common denominator
of two or more fractions. See
chapter 5. You need to work out whether
to use LCM or HCF to find the
answers. Problems involving LCM
usually include repeating events.
Problems involving HCF usually
involve splitting things into smaller
pieces or arranging things in equal
groups or rows.
Exercise 1.2 A
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(b) BOE
(f) BOE
(c) BOE
(g) BOE
(d) BOE
(h) BOE
(c) BOE
(g) BOE
(d) BOE
(h) BOE
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Exercise 1.2 B
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(a) 'JOEUIFBSFBPGUIFMBSHFTUQPTTJCMFUJMFTJODN
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2
Unit 1: Number
1 Reviewing number concepts
1.3 Prime numbers
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Exercise 1.3
You can use a tree diagram or
division to find the prime factors
of a composite whole number.
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1.4 Powers and roots
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3
3
FAST FORWARD
Powers greater than 3 are dealt
with in chapter 2. See topic 2.5
indices. Exercise 1.4
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9
(a)
(e)
(i)
(m)
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16
(b)
9 16 ( 25 )
2
36
4
(f)
27 − 3 1 (j)
100 ÷ 4 (n)
1
4
×
(c)
+
(g)
2
(o)
(d)
64 + 36 9
(h)
169 − 144
(l)
16 × 3 27
16
1+
(k)
( 13 ) 64 + 36 3
9
16
1 − 3 −125
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Unit 1: Number
3
1 Reviewing number concepts
1.5 Working with directed numbers
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Exercise 1.5
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(c) −¡$PS−¡$
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1.6 Order of operations
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Remember the order of operations
using BODMAS:
Tip
Brackets
Of
Divide
Multiply
Add
Subtract
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FAST FORWARD
The next section will remind you
of the rules for rounding
numbers. 4
Exercise 1.6
Unit 1: Number
1 $BMDVMBUFBOEHJWFZPVSBOTXFSDPSSFDUUPUXPEFDJNBMQMBDFT
(a) +×
(b)
+ ×
(d) +×
(e) ×−
(c) ×−÷
(f)
÷ +
1 Reviewing number concepts
5 27
1.4 × 1.35
89
(m) 8 9 −
10.4
5.34 3.315
4 03
11.5
(k)
2 9 − 1.43
12.6 1 98
−
(n)
8 3 4 62
(p) − 3
(q) −
(g) 1.453 +
76
32
(h)
(j)
(s)
4 072 8.2 − 4.09
(t) 6 8 +
( )
2
(v) 6 1 + 2.1 28 16
1. 4 1. 2
−
69 93
(w) 6.4 (1.2
.22 + 1 92 )2 (i)
(l)
6 54
− 1 08
23
0.23 4.26
1.32 3.43
(o) −
(r)
(
)
2
16.8
− 1 01
93
(
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5
(x)
)
2
(4 8 − 916 ) × 4 3
1.7 Rounding numbers
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Exercise 1.7
FAST FORWARD
Rounding is very useful when you
have to estimate an answer. You
will deal with this in more detail in
chapter 5. 1 3PVOEUIFTFOVNCFSTUP
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(a) (e) (b) (f) (c) (g) (d) (h) Unit 1: Number
5
1 Reviewing number concepts
Mixed exercise
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−
3
4
−
3
1
2
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(a)
(b)
(c)
(e)
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8IBUJTUIFIJHIFTUOVNCFSUIBUJTBGBDUPSPGCPUIBOE
3 8SJUFFBDIOVNCFSBTBQSPEVDUPGJUTQSJNFGBDUPST
(a) (b) (c) 4 "NJSBTUBSUTBOFYFSDJTFQSPHSBNNFPOUIFSEPG.BSDI4IFEFDJEFTTIFXJMMTXJNFWFSZ
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TBNFEBZ
5 4UBUFXIFUIFSFBDIFRVBUJPOJTUSVFPSGBMTF
(a) Ÿ ¨ (b) ¨ o (c) 30 + 10 o
30
(d) 6 4JNQMJGZ
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100 ÷ 4 (c)
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3
(d) 3+
7 $BMDVMBUF(JWFZPVSBOTXFSDPSSFDUUPUXPEFDJNBMQMBDFT
(a) 5.4 × 12.2 4. 1
2
(b) 12.2 2
39
(d) 3.8 × 12.6 4 35
(e)
2.8 × 4.22 3.32 × 6.22
12.65
(c) 2 04 + 1.7 × 4.3
(
(f) 2.55 − 3.1 +
05
5
)
2
8 3PVOEFBDIOVNCFSUPUISFFTJHOJĕDBOUĕHVSFT
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6
Unit 1: Number
2
Making sense of algebra
2.1 Using letters to represent unknown values
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Exercise 2.1
Tip
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(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
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3 ćSFFQFPQMFXJOBQSJ[FPGx
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Unit 1: Algebra
7
2 Making sense of algebra
2.2 Substitution
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'PSFYBNQMF ZPVNBZCFUPMEUPFWBMVBUFxXIFOx=−5PEPUIJTZPVXPSLPVU× − =−
Exercise 2.2
1
1 ćFGPSNVMBGPSĕOEJOHUIFBSFB A PGBUSJBOHMFJTA = bh XIFSFbJTUIFMFOHUIPGUIFCBTF
2
BOEhJTUIFQFSQFOEJDVMBSIFJHIUPGUIFUSJBOHMF
'JOEUIFBSFBPGBUSJBOHMFJG
REWIND
Remember that the BODMAS rules
always apply in these calculations. Take special care when substituting
negative numbers. If you replace x
with −3 in the expression 4x, you
will obtain 4 × −3 = −12, but in
the expression −4x, you will obtain
−4 × −3 = 12.
(a)
(b)
(c)
(d)
UIFCBTFJTDNBOEUIFIFJHIUJTDN
UIFCBTFJTNBOEUIFIFJHIUJTN
UIFCBTFJTDNBOEUIFIFJHIUJTIBMGBTMPOHBTUIFCBTF
UIFIFJHIUJTDNBOEUIFCBTFJTUIFDVCFPGUIFIFJHIU
2 &WBMVBUFxyo xoy XIFOxBOEyo
3 (JWFOUIBUa boBOEco FWBMVBUF a b oc
n3
mn n
m2 5 ćFOVNCFSPGHBNFTUIBUDBOCFQMBZFEBNPOHxDPNQFUJUPSTJOBDIFTTUPVSOBNFOUJTHJWFO
CZUIFFYQSFTTJPO 12 xo 12 x
4 8IFOmBOEno XIBUJTUIFWBMVFPGm3o
(a) )PXNBOZHBNFTXJMMCFQMBZFEJGUIFSFBSFDPNQFUJUPST
(b) )PXNBOZHBNFTXJMMCFQMBZFEJGUIFSFBSFDPNQFUJUPST
2.3 Simplifying expressions
t 5PTJNQMJGZBOFYQSFTTJPOZPVBEEPSTVCUSBDUMJLFUFSNT
t -JLFUFSNTBSFUIPTFUIBUIBWFFYBDUMZUIFTBNFWBSJBCMFT JODMVEJOHQPXFSTPGWBSJBCMFT t :PVDBOBMTPNVMUJQMZBOEEJWJEFUPTJNQMJGZFYQSFTTJPOT#PUIMJLFBOEVOMJLFUFSNTDBOCFNVMUJQMJFEPSEJWJEFE
Remember, like terms must have
exactly the same variables with
exactly the same indices. So 3x
and 2x are like terms but 3x2 and
2x are not like terms.
Exercise 2.3
1 4JNQMJGZUIFGPMMPXJOHFYQSFTTJPOT
(a) 3x 2 + 6 x 8
3 (b) x 2 y + 3x
ac + 3ba
3x 2 y 2 yx (c) 2ab − 44ac
(d) x 2 + 2xx 4 + 3x
3x 2
Remember, multiplication can be
done in any order so, although it is
better to put variable letters in a term
in alphabetical order, ab = ba. So,
3ab + 2ba can be simplified to 5ab.
Remember,
x × x = x2
y × y × y = y3
x÷x=1
8
Unit 1: Algebra
2
y 3x − 1
(e) −6m × 5n
2
(g) −2 xy × −3 y (h) −2 xy × 2 x (k) 33abc 11ca
y 2y
(o)
×
x x
x 2
(s)
× 4 3y
45mn 20n
xy y
(p)
× 2 x
(t) 3x × 9 x 5
2
(l)
(i) 12ab ÷ 3a
80 xy 2
12 x 2 y
(q) 5a × 3a 4
(m)
(f) 3xy 2 x (j) 12 x ÷ 48 xy
3
(n) −36 x
−12 xy
−2 y
(r) 7 ×
5
2 Making sense of algebra
2.4 Working with brackets
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UPTJNQMJGZUIFFYQSFTTJPOGVMMZ
t *OHFOFSBMUFSNTa b + c = ab + ac
Exercise 2.4
Remember the rules for multiplying
integers:
1 3FNPWFUIFCSBDLFUTBOETJNQMJGZXIFSFQPTTJCMF
(a) 2x ( x 2)
(b) ( y 3) x (c) ( x ) 3x
)( x )
(e) ( x )(
(f) 2 ( x 1) (1 x ) (g) x ( x 2 − 2x
2 x 1)
(h) − x ( − ) + 2 ( + ) − 4
+×+=+
−×−=+
+×−=−
If the quantity in front of a bracket
is negative, the signs of the terms
inside the bracket will change when
the brackets are expanded.
(d) −2 x − ( − )
2 3FNPWFUIFCSBDLFUTBOETJNQMJGZXIFSFQPTTJCMF
(a) 2 x ( 12 x + 14 )
(b) −3
(d) ( x + y ) − ( x
1
2
1
y)
(e)
(
− y ) − 2 x ( y − 2 x ) (c) −2 x
− −
(f) 2x (2x − 2) x ( x 2)
(3 − 2x ) (
5
)
(g) x (1 x ) x (2 x − 5) 2 x (1 3x )
2.5 Indices
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4
Exercise 2.5 A
Tip
.FNPSJTFUIJTTVNNBSZ
PGUIFindex laws:
x m x n = x m+ n xm xn = xm n
( )n x mn
x 0 = 1 1
x −m = m
x m
m
1 m
xn
xn = n x
( ) ( )
1 4JNQMJGZ
(a)
x 4 y × y2 x6
x 4 y5 (b)
2 x 2 y 4 3x 3 y
2 xy 4
(c)
2 x 5 y 4 2 xy 3
2 x 2 y 5 3x 2 y 3
(d)
x3 y7 x2 y8
× 3
xy 4
x y (e)
2 x 7 y 2 10 x 8 y 4
×
4 x 3 y 7 2x 3 y 2
(f)
x9 y6 x3 y2
÷
x 4 y2 x5 y
10 x 5 y 2 3x 3 y
(g)
÷
9 x 6 y 6 5x 7 y 4
(2 x y ) × ( x y )
( y x ) 3(x y )
4
(j)
7 y 3 x 2 5x 6 y 2
(h)
÷
5 y5 x 4 7x5 y3
3
2
2
3
3
4
4
2
2
3
2
2
5
(i)
2
⎛ 2⎞ ⎛ 5⎞
(k) ⎜ x 4 ⎟ × ⎜ x 2 ⎟ ⎝y ⎠ ⎝y ⎠
(x y ) × (x y ) (x y )
(5x y ) ÷ ⎛ 2xy
3
3
(l)
3
2
4x7 y6
3
3
4
2
3
⎞
⎜⎝ 5x 2 y 4 ⎟⎠
3
Unit 1: Algebra
2
9
2 Making sense of algebra
Tip
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TUBUFTQPTJUJWFJOEJDFT
POMZ ZPVDBOVTFUIFMBX
5
1
−4
x − m = m TPUIBU5x = 4 x
x
y
4JNJMBSMZ −2 = x 2 y
x
2 4JNQMJGZFBDIFYQSFTTJPOBOEHJWFZPVSBOTXFSVTJOHQPTJUJWFJOEJDFTPOMZ
x5 y 4
(a) −33 2
x y (x )
÷
−1
⎛ ⎞
(d) ⎜ x3 ⎟
⎝y ⎠
2
4
−10
y2
(e) x 2 ÷ ⎛⎜ 3 ⎞⎟
( y− ) ⎝ x ⎠
y −3
(2 x y )
(y x )
−33
x −4 y 3 x 7 y −5
(b) 2 −1 × −4 3
x y
x y (c)
2
3
1
2
2
(
2
)
( )
2
x −2 y 6
⎛ x4 y 1 ⎞
(f) ⎜ 5 3 ⎟ ×
⎝ x y ⎠ 2 xy 3 −2
−4
3 4JNQMJGZ
1
1
1
1
(a) x 4 × x 4
1
(b) x 3 × x 5
(e)
(64 x )
(f)
(8 x y )
1
3
(i)
( )
x2
x3
(j)
(
1
3
(a)
(x ) × xx
(b)
(x ) × xx
(d)
(x y )
(e)
⎛ x2 y3 ⎞ 2
1 ÷ ⎜
3 4⎟
x2 ⎝ x y ⎠
(b)
−24
(−2)4
6
8
1
x2
1
2
×
9
x6 y3
(c)
x2
1
x3
(g)
) (
× x 8y
10
)
1
2
(d)
xy
( )
1
1
2
×
1
3
1
3
1
⎛ x6 ⎞ 2
(h) ⎜ 2 ⎟
⎝y ⎠
8
(k) xy 3
(x )
1
x2 y2
xy 4
4 4JNQMJGZ
Tip
"QQMZUIFJOEFYMBXTBOE
XPSLJOUIJTPSEFS
TJNQMJGZBOZUFSNTJO
CSBDLFUT
BQQMZUIFNVMUJQMJDBUJPO
MBXUPOVNFSBUPSTBOE
UIFOUPEFOPNJOBUPST
DBODFMOVNCFSTJG
ZPVDBO
BQQMZUIFEJWJTJPO
MBXJGUIFTBNF
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EFOPNJOBUPS
FYQSFTTZPVSBOTXFS
VTJOHQPTJUJWFJOEJDFT
t
t
t
t
t
1
2
2
3
2
3
2
3
1
3
4
1
Unit 1: Algebra
(c)
1
2
1
y3
2
1
( x y ) × (x y )
1
2
2
1
2
3
4
4
1
3
1
1
⎛ xy 4 ⎞ 4
4
(f) x 3 × ⎜ 3 2 ⎟
y2 ⎝ x y ⎠
3
Exercise 2.5 B
1 &WBMVBUF
(a) (
4
)(
)2 −1
(e) 256 4 (i)
( )
1
8 −3
27
−4
(f) 125 3 (j)
(c)
63
( 3)4
(d) 8 3
(g)
()
(h)
5
1 −2
4
1
()
2
1 −3
8
( )
1
8 −2
18
2 $BMDVMBUF
(a)
(b)
(c)
(d)
o o oŸ3
o oo Ÿo
o o Ÿ o 3
o 4o o 6
3 4PMWFGPSx
(a) x=
(e) x=
10
3
5
1
2
x3 y3
×
xy 2
5
2
(b) 3x=
(f) ox= 251 (c) 3x= 811 (g) 3x−=
(d) x
(h) x+=
2 Making sense of algebra
Mixed exercise
1 8SJUFFBDIPGUIFGPMMPXJOHBTBOBMHFCSBJDFYQSFTTJPO6TFxUPSFQSFTFOUAUIFOVNCFS
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
"OVNCFSJODSFBTFECZ
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'JWFUJNFTBOVNCFS
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"RVBSUFSPGBOVNCFS
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2 %FUFSNJOFUIFWBMVFPGxoxJG
(a) x=
(b) x=−
(c) x= 13
3 &WBMVBUFFBDIFYQSFTTJPOJGa=− b=BOEc=
b (c − a )
−2a + 3b
a b2
(b)
(c)
(a)
2ab
b a
c a2
3 2 ( − 1)
(d)
(e) a 3b2 − 2a 2 + a 4b2 ac 3
c − a (b − 1)
4 4JNQMJGZFBDIPGUIFGPMMPXJOHFYQSFTTJPOTBTGVMMZBTQPTTJCMF
(b) x 2 + 4x
4 x x 2
(a) 3a 4b + 6a 3b
(d) 2 (
(c) −2a 2b(2a 2 − 3b2 )
10 x 2 5xy
(f)
2x
3) − ( x 4 ) 2 x 2 (e) 16 x 2 y ÷ 4 y 2 x 5 &YQBOEBOETJNQMJGZJGQPTTJCMF
(a) 2 ( 4 x 3) 3( x + 1)
(c) x ( x + ) + 3x − 3( x −
(b) 3 (2 x + 3) 2 ( 4 3x )
(d) x 2 ( x + 3) 2 x 3 ( x − 5)
)
6 4JNQMJGZ(JWFBMMBOTXFSTXJUIQPTJUJWFJOEJDFTPOMZ
7
(a) 15x 2
18 x 5
(b) 5x × 3x7 x
(d) (2 xy
⎛ 4x ⎞
(e) ⎜ 5 ⎟
⎝ y ⎠
(g)
3
)
2 4
(2xy ) (3x y )
−2
3
2
3
(c)
−
(h)
4
−3
⎛ x −3 y 3 ⎞
÷ ⎜ 2 −1 ⎟
⎝x y ⎠
2
4
8
(f)
(x y )
(x y ) ×
(xy )
(c)
(x
3
(x y )
2 ( xy )
(x )
(x )
3
2
2
2
3
4
2
3
3
2
7 4JNQMJGZFBDIPGUIFGPMMPXJOHFYQSFTTJPOT
(a)
(125x y )
3
2
1
3
(b)
( )
1
4
(x y )
6
3
1
2
1
×
1
x2 y2
x3 y2
2
y
3
) × (x y)
1
2
−4
5
2
1
2
x2 y
⎛ x3 y3 ⎞
÷ 3 5
(d) ⎜
1 ⎟
2x y
⎝ xy 3 ⎠
Unit 1: Algebra
11
3
Lines, angles and shapes
3.1 Lines and angles
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Exercise 3.1 A
1 -PPLBUUIFDMPDLGBDFPOUIFMFę$BMDVMBUFUIFGPMMPXJOH
10
11
12
1
9
2
3
8
4
7
6
5
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(i) PDMPDL
(ii) IPVST (iii) BN
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4 8IBUJTUIFDPNQMFNFOUPGFBDIUIFGPMMPXJOHBOHMFT
(a) °
12
Unit 1: Shape, space and measures
(b) x°
(c)
ox °
3 Lines, angles and shapes
5 8IBUJTUIFTVQQMFNFOUPGFBDIPGUIFGPMMPXJOHBOHMFT
(a) °
(d) ox °
Tip
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(c) x°
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Exercise 3.1 B
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Q
M
a
Remember, give reasons
for statements. Use these
abbreviations:
(b) °
(e) ox °
O
37°
N
P
ˆ = 85°
QOR
Comp ∠s
R
Supp ∠s
∠s on line
2 $BMDVMBUFUIFWBMVFPGxJOFBDIPGUIFGPMMPXJOHĕHVSFT
∠s round point
(a) (b)
Vertically opposite ∠s
A
E
27°
3x + 25°
D
O
x 112°
x
6x
5x + 35°
C
P
B
Exercise 3.1 C
1 'JOEUIFWBMVFTPGUIFBOHMFTNBSLFExBOEyJOFBDIEJBHSBN
Remember, give reasons
for statements. Use these
abbreviations to refer to types of
angles:
Alt ∠s
(b)
(a)
N
M
Corr ∠s
Co-int ∠s
O
y
x
P
R
y
B
57°
72°
S
95°
Q
E
M
P
T
C x
F
81° G
N
Q
H
(c)
(d)
A
K
E
Bx
I
C 65°
D
F
108°
G
y
J
x xx
x
x
x x x
H
Unit 1: Shape, space and measures
13
3 Lines, angles and shapes
2 $BMDVMBUFUIFWBMVFPGxJOFBDIPGUIFGPMMPXJOHĕHVSFT(JWFSFBTPOTGPSZPVSBOTXFST
(a)
(b)
E
112°
A
F
S
B
Q
M
x
G
C
N
57°
T
D
H
(c) A
30°
(d) A
B
C
x
C
D
D
E 60°
A
x
C
B
14
Unit 1: Shape, space and measures
D
50°
G
45°
E
F
126°
108°
F
B
(e)
E
x
F
3 Lines, angles and shapes
3.2 Triangles
t 4DBMFOFUSJBOHMFTIBWFOPFRVBMTJEFTBOEOPFRVBMBOHMFT
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Exercise 3.2
REWIND
You may also need to apply the
angle relationships for points,
lines and parallel lines to find the
missing angles in triangles. 1 'JOEUIFBOHMFTNBSLFEXJUIMFUUFST(JWFSFBTPOTGPSBOZTUBUFNFOUT
(a)
29°
48°
(b)
(c)
113°
37°
c
31°
a
62°
(d)
53°
d
b
(e)
e
(f)
32°
61°
(g)
21°
54°
(h)
x
(i)
x
x
y
x
30°
102°
57°
y
Unit 1: Shape, space and measures
15
3 Lines, angles and shapes
2 $BMDVMBUFUIFWBMVFPGxBOEIFODFĕOEUIFTJ[FPGUIFNBSLFEBOHMFT
(a)
(b)
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A
x
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C
D
(c)
(d)
M
M
N x
O
x N
O
18°
P
P
3 *O6ABC ∠A¡ ∠BxBOE∠Cx.
Tip
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3.3 Quadrilaterals
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t
16
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Unit 1: Shape, space and measures
3 Lines, angles and shapes
Exercise 3.3
REWIND
The angle relationships for parallel
lines will apply when aquadrilateral
has parallel sides. 1 &BDIPGUIFGPMMPXJOHTUBUFNFOUTBQQMJFTUPPOFPSNPSFRVBESJMBUFSBMT'PSFBDIPOF OBNF
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(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
"MMTJEFTBSFFRVBMJOMFOHUI
"MMBOHMFTBSFFRVBMJOTJ[F
ćFEJBHPOBMTBSFUIFTBNFMFOHUI
ćFEJBHPOBMTCJTFDUFBDIPUIFS
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0QQPTJUFBOHMFTBSFFRVBMJOTJ[F
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ćFEJBHPOBMTCJTFDUUIFPQQPTJUFBOHMFT
0OFEJBHPOBMEJWJEFTUIFRVBESJMBUFSBMJOUPUXPJTPTDFMFTUSJBOHMFT
2 $PQZUIFEJBHSBNTCFMPX 'JMMJOUIFTJ[FTPGBMMUIFBOHMFT
a
f
e
d
g
f
b
63° a
b
c
ed
c
3 $BMDVMBUFUIFTJ[FPGUIFNBSLFEBOHMFTJOUIFGPMMPXJOHĕHVSFT(JWFSFBTPOTPSTUBUFUIF
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(a)
A 79°
(b) A
B
B
32°
(c)
A
38°
x
D
D
D 58°
(d)
M
C
E
C
N
x
B
O
(e)
(f)
3x
F
C
N
M
c
d
68°
O
x
Q 65°
P
2x
R e
Q
a
b P
4 2VBESJMBUFSBMPQRS IBT∠P∠S¡BOE∠R∠Q.
$BMDVMBUFUIFTJ[FPG
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(b) ∠R
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5 "LJUFPMNOIBTEJBHPOBMTPNBOEMOUIBUJOUFSTFDUBUQ∠QMN¡BOE∠PNO¡
$BMDVMBUFUIFTJ[FPG
(a) ∠MNP
(b) ∠MNO
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Unit 1: Shape, space and measures
17
3 Lines, angles and shapes
3.4 Polygons
t "QPMZHPOJTBUXPEJNFOTJPOBMTIBQFXJUIUISFFPSNPSFTJEFT1PMZHPOTBSFOBNFEBDDPSEJOHUIFOVNCFSPGTJEFT
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o
o
o
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t
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Exercise 3.4
Tip
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° UPXPSLPVUUIFTJ[F
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1 'PSFBDIPGUIFGPMMPXJOHĕOE
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(a) BSFHVMBSPDUBHPO
(b) BSFHVMBSEFDBHPO
(c) BSFHVMBSTJEFEQPMZHPO
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JOUFSJPSBOHMF
3 ćFJOUFSJPSBOHMFPGBSFHVMBSQPMZHPOJT°)PXNBOZTJEFTEPFTUIFQPMZHPOIBWF
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(a)
(b)
120° 125°
161°
C
98°
50
x
100°
x – 10°
B
H
56° 100° y
17°
D
x–
130°
84°
x
°
110°
(c) A
x
E
44°
x
100°
F
18
Unit 1: Shape, space and measures
C
125° G
3 Lines, angles and shapes
3.5 Circles
t "DJSDMFJTBTFUPGQPJOUTFRVJEJTUBOUGSPNBĕYFEDFOUSF)BMGBDJSDMFJTBTFNJDJSDMF
t ćFQFSJNFUFSPGBDJSDMFJTDBMMFEJUTDJSDVNGFSFODF
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t "ATMJDFPGBDJSDMF NBEFCZUXPSBEJJBOEUIFBSDCFUXFFOUIFNPOUIFDJSDVNGFSFODF JTDBMMFEBTFDUPS
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Exercise 3.5
1 %SBXBDJSDMFXJUIBSBEJVTPGDNBOEDFOUSFO#ZESBXJOHUIFQBSUTBOEMBCFMMJOHUIFN JOEJDBUFUIFGPMMPXJOHPOZPVSEJBHSBN
(a)
(b)
(c)
(d)
(e)
BTFDUPSXJUIBOBOHMFPG¡
DIPSEDE
MON UIFEJBNFUFSPGUIFDJSDMF
BUBOHFOUUIBUUPVDIFTUIFDJSDMFBUM
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3.6 Construction
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t ćFEJBHSBNTCFMPXTIPXZPVIPXUPCJTFDUBOBOHMFBOEIPXUPESBXUIFQFSQFOEJDVMBSCJTFDUPSPGBMJOF
A
How to bisect an angle.
Tip
"MXBZTTUBSUXJUIBSPVHI
TLFUDI-BCFMZPVSSPVHI
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B
How to draw the perpendicular bisector of a line.
Exercise 3.6
=°"DDVSBUFMZCJTFDUUIFBOHMF
1 %SBXBOHMFABC
2 $POTUSVDUΔABCXJUIAC=DN CB=DNBOEAB=DN
3 $POTUSVDUΔMNOXJUIMN=DN NO=DNBOEMO=DN$POTUSVDUUIF
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MFOHUIPGXYUPUIFOFBSFTUNJMMJNFUSF
Unit 1: Shape, space and measures
19
3 Lines, angles and shapes
4 $POTUSVDUΔDEFXJUIDE=NN FE=NNBOEDF=NN
(a) 8IBUUZQFPGUSJBOHMFJTDEF
(b) #JTFDUFBDIBOHMFPGUIFUSJBOHMF%PUIFBOHMFCJTFDUPSTNFFUBUUIFTBNFQPJOU
5 "DDVSBUFMZDPOTUSVDUBTRVBSFPGTJEFDN8IBUJTUIFMFOHUIPGBEJBHPOBMPGUIFTRVBSF
Mixed exercise
(a)
(b)
G
23°
A
1 'JOEUIFWBMVFPGUIFNBSLFEBOHMFTJOFBDIPGUIFGPMMPXJOH
F
(c)
B
A
x
98°
C
D
71° C
D
x
D
(d)
(e)
y
104°
32°
(f)
y
110°
x
30°
(g)
40°
68°
B
C
x
E
E
A
111°
16°
(h)
y
x
y
110°
x
z
2 'PSFBDITIBQFDPNCJOBUJPOĕOEUIFTJ[FPGBOHMFxćFTIBQFTJOQBSUT B BOE C BSF
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(a)
(b)
x
(c)
A
x
x
50°
C 70°
120°
50° D
E
60°
20
Unit 1: Shape, space and measures
B
110°
B
3 Lines, angles and shapes
3 6TFUIFEJBHSBNPGUIFDJSDMFXJUIDFOUSFOUPBOTXFSUIFTFRVFTUJPOT
A
O
B
D
C
(a) 8IBUBSFUIFDPSSFDUNBUIFNBUJDBMOBNFTGPS
(i) DO
(ii) AB
(iii) AC
(b) 'PVSSBEJJBSFTIPXOPOUIFEJBHSBN/BNFUIFN
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(d) %
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QPJOUB.
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5 $POTUSVDUΔABCXJUIAB=BC=AC=DN$POTUSVDUUIFQFSQFOEJDVMBSCJTFDUPSPGAB
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Unit 1: Shape, space and measures
21
4
Collecting, organising
and displaying data
4.1 Collecting and classifying data
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1
2
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Exercise 4.1
ćFGPMMPXJOHUBCMFPGEBUBXBTDPMMFDUFEBCPVUUFOTUVEFOUTJOBIJHITDIPPM4UVEZUIFUBCMFBOE
UIFOBOTXFSUIFRVFTUJPOTBCPVUUIFEBUB
Student
1
2
3
4
5
6
7
8
9
10
Gender
'
'
M
M
M
'
M
'
'
M
Shoe size
7
7
8
7
Mass (kg)
Height (m)
Eye colour
#S
(S
(S
#S
#S
#S
#S
(S
#M
#S
Hair colour
#M
#M
#MP
#S
#S
#S
#M
#M
#M
#M
No. of brothers/
sisters
1
1
(a)
(b)
(c)
(d)
(e)
22
Unit 1: Data handling
8IJDIPGUIFTFEBUBDBUFHPSJFTBSFRVBMJUBUJWF
8IJDIPGUIFTFEBUBDBUFHPSJFTBSFRVBOUJUBUJWF
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4 Collecting, organising and displaying data
4.2 Organising data
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Exercise 4.2
In data handling, the word
frequency means the number
of times a score or observation
occurs.
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IPVSćFTFBSFIJTSFTVMUT
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Phone calls
Tally
Frequency
1
2 /JLBDPVOUFEUIFOVNCFSPGNPTRVJUPFTJOIFSCFESPPNGPSOJHIUTJOBSPXBOEHPUUIFTF
SFTVMUT
(a) $PQZBOEDPNQMFUFUIJTGSFRVFODZUBCMFUPPSHBOJTFUIFEBUB
Number of
mosquitoes
0
1
2
3
4
5
6
Frequency
(b) %PFT/JLBIBWFBNPTRVJUPQSPCMFN (JWFBSFBTPOGPSZPVSBOTXFS
Unit 1: Data handling
23
4 Collecting, organising and displaying data
Score
Frequency
3 ćFTFBSFUIFQFSDFOUBHFTDPSFTPGTUVEFOUTJOBOFYBNJOBUJPO
o
o
o
o
o
o
o
(a)
(b)
(c)
(d)
(e)
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ć
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Student
1
2
3
4
5
6
7
8
9
10
Gender
'
'
M
M
M
'
M
'
'
M
Eye colour
#S
(S
(S
#S
#S
#S
#S
(S
#M
#S
Hair colour
#M
#M
#MP
#S
#S
#S
#M
#M
#M
#M
No. of siblings
(brothers/sisters)
1
1
(a) $PQZBOEDPNQMFUFUIJTUXPXBZUBCMFVTJOHEBUBGSPNUIFUBCMF
Eye colour
Brown
Blue
Green
Male
Female
(b) %SBXBOEDPNQMFUFUXPTJNJMBSUXPXBZUBCMFTPGZPVSPXOUPTIPXUIFIBJSDPMPVSBOE
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4.3 Using charts to display data
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24
Unit 1: Data handling
4 Collecting, organising and displaying data
Exercise 4.3
1 4UVEZUIFEJBHSBNDBSFGVMMZBOEBOTXFSUIFRVFTUJPOTBCPVUJU
Number of students in each year
Year 8
Year 9
Year 10
Key
8IBUUZQFPGDIBSUJTUIJT
8IBUEPFTUIFDIBSUTIPX
8IBUEPFTFBDIGVMMTZNCPMSFQSFTFOU
)PXBSFTUVEFOUTTIPXOPOUIFDIBSU
)PXNBOZTUVEFOUTBSFUIFSFJO:FBS
8
IJDIZFBSHSPVQIBTUIFNPTUTUVEFOUT )PXNBOZBSFUIFSFJOUIJT
ZFBSHSPVQ
(g) %PZPVUIJOLUIFTFBSFBDDVSBUFPSSPVOEFEĕHVSFT 8IZ
(a)
(b)
= 30 students (c)
(d)
(e)
(f)
Year 11
Year 12
Tip
$PNQPVOECBSDIBSUT
TIPXUXPPSNPSFTFUTPG
EBUBPOUIFTBNFQBJSPG
BYFT"LFZJTOFFEFEUP
TIPXXIJDITFUFBDICBS
SFQSFTFOUT
City
Population (millions)
Tokyo
Seoul
Mexico City New York
Mumbai
%SBXBQJDUPHSBNUPTIPXUIJTEBUB
3 4UVEZUIFUXPCBSDIBSUTCFMPX
(a) 8IBUEPFTDIBSU"TIPX
(b) )PXNBOZCPZTBSFUIFSFJO$MBTT"
(c) )PXNBOZTUVEFOUTBSFUIFSF
JO"BMUPHFUIFS
(d) 8IBUEPFTDIBSU#TIPX
A. Number of students in 10A
20
18
16
Frequency
$IPPTFTZNCPMTUIBUBSF
FBTZUPESBXBOEUPEJWJEF
JOUPQBSUT*GJUJTOPUHJWFO DIPPTFBTVJUBCMFTDBMFGPS
ZPVSTZNCPMTTPZPVEPOU
IBWFUPESBXUPPNBOZ
2 ćFUBCMFTIPXTUIFQPQVMBUJPO JONJMMJPOT PGĕWFPGUIFXPSMETMBSHFTUDJUJFT
14
Frequency
Tip
12
10
8
6
4
(e) 8IJDITQPSUJTNPTUQPQVMBSXJUICPZT
(f) 8IJDITQPSUJTNPTUQPQVMBSXJUIHJSMT
(g) )
PXNBOZTUVEFOUTDIPTFCBTLFUCBMM
BTUIFJSGBWPVSJUFTQPSU
14
12
10
8
6
4
2
0
B. Favourite sport of students in 10A
Key
boys
girls
soccer
basketball athletics
Sport
2
0
boys girls
Gender
Unit 1: Data handling
25
4 Collecting, organising and displaying data
4 ćFUBCMFCFMPXTIPXTUIFUZQFPGGPPEUIBUBHSPVQPGTUVEFOUTJOBIPTUFMDIPTFGPSCSFBLGBTU
Cereal
Hot porridge
Bread
Girls
8
Boys
(a) %SBXBTJOHMFCBSDIBSUUPTIPXUIFDIPJDFPGDFSFBMBHBJOTUCSFBE
(b) %SBXBDPNQPVOECBSDIBSUUPTIPXUIFCSFBLGBTUGPPEDIPJDFGPSHJSMTBOECPZT
5 +ZPUJSFDPSEFEUIFOVNCFSBOEUZQFPGWFIJDMFTQBTTJOHIFSIPNFJO#BOHBMPSF
4IFESFXUIJTQJFDIBSUUPTIPXIFSSFTVMUT
Tip
5PXPSLPVUUIF
QFSDFOUBHFUIBUBOBOHMFJO
BQJFDIBSUSFQSFTFOUT VTF
UIFGPSNVMB
n
× 100
Traffic passing my home
handcarts
bicycles
taxis
(a)
(b)
(c)
(d)
motorcycles
360
XIFSFnJTUIFTJ[FPGUIF
BOHMF
cars
trucks
8IJDIUZQFPGWFIJDMFXBTNPTUDPNNPO
8IBUQFSDFOUBHFPGUIFWFIJDMFTXFSFUVLUVLT
)PXNBOZUSVDLTQBTTFE+ZPUJTIPNF
8IJDIUZQFTPGWFIJDMFTXFSFMFBTUDPNNPO
buses
tuk-tuks
6 *OBO*($4&FYBNUIFSFTVMUTGPSTUVEFOUTXFSFBUUBJOFEBO"HSBEF BUUBJOFE
B#HSBEF BUUBJOFEB$HSBEF BUUBJOFEB%HSBEFBOEUIFSFTUBUUBJOFE&HSBEF
PS‫ڀ‬MPXFS
(a)
(b)
(c)
(d)
3FQSFTFOUUIJTJOGPSNBUJPOPOBQJFDIBSU
)PXNBOZTUVEFOUTBUUBJOFEBO"
)PXNBOZTUVEFOUTBUUBJOFEB%PSMPXFS
8IJDIHSBEFXBTBUUBJOFECZNPTUPGUIFTUVEFOUT
7 ćFHSBQITCFMPXSFQSFTFOUUIFBWFSBHFNPOUIMZUFNQFSBUVSFBOEUIFBWFSBHFNPOUIMZ
SBJOGBMMJOUIFEFTFSUJO&HZQU
"WFSBHFNPOUIMZUFNQFSBUVSF
"WFSBHFNPOUIMZSBJOGBMM
20
140
10
120
Rainfall (mm)
Temperature (°C)
30
0
–10
–20
–30
–40
100
80
60
40
J
F
M
A
M
J J
Month
A
S
O
N
D
20
J
F
M
(a) 8IBUJTUIFNBYJNVNUFNQFSBUVSF
(b) *OXIBUNPOUITJTUIFBWFSBHFUFNQFSBUVSFBCPWF¡$
26
Unit 1: Data handling
A
M
J J
Month
A
S
O
N
D
4 Collecting, organising and displaying data
(c)
(d)
(e)
(f)
(g)
Mixed exercise
*T&HZQUJOUIFOPSUIFSOPSTPVUIFSOIFNJTQIFSF
*TUIFUFNQFSBUVSFFWFSCFMPXGSFF[JOHQPJOU
8IBUJTUIFBWFSBHFSBJOGBMMJO/PWFNCFS
*OXIJDINPOUIJTUIFBWFSBHFSBJOGBMMNN
-PPLJOHBUCPUIHSBQIT XIBUDBOZPVTBZBCPVUUIFSBJOGBMMXIFOUIFUFNQFSBUVSFT
BSFIJHI
1 .JLBDPMMFDUFEEBUBBCPVUIPXNBOZDIJMESFOEJČFSFOUGBNJMJFTJOIFSDPNNVOJUZIBE
ćFTFBSFIFSSFTVMUT
(a)
(b)
(c)
(d)
(e)
(f)
)PXEPZPVUIJOL.JLBDPMMFDUFEUIFEBUB
*TUIJTEBUBEJTDSFUFPSDPOUJOVPVT 8IZ
*TUIJTEBUBRVBMJUBUJWFPSRVBOUJUBUJWF 8IZ
%SBXVQBGSFRVFODZUBCMF XJUIUBMMJFT UPPSHBOJTFUIFEBUB
3FQSFTFOUUIFEBUBPOBQJFDIBSU
%
SBXBCBSDIBSUUPDPNQBSFUIFOVNCFSPGGBNJMJFTUIBUIBWFUISFFPSGFXFSDIJMESFO
XJUIUIPTFUIBUIBWFGPVSPSNPSFDIJMESFO
2 .ST4BODIF[CBLFTBOETFMMTDPPLJFT0OFXFFLTIFTFMMTQFBOVUDSVODIJFT DIPDPMBUF
DVQTBOEDPDPOVUNVODIJFT%SBXBQJDUPHSBNUPSFQSFTFOUUIJTEBUB
3 4UVEZUIFDIBSUJOUIFNBSHJO
Number of phones per 100 people
100
Mobile phones and land lines, per 100 people
90
80
70
60
50
40
30
20
10
0
Canada USA Germany Denmark UK
Key
Mobile phones
(a) 8IBUEPZPVDBMMUIJTUZQFPGDIBSU
(b) 8IBUEPFTUIFDIBSUTIPX
(c) $
BOZPVUFMMIPXNBOZQFPQMFJOFBDIDPVOUSZIBWFBNPCJMFQIPOF
GSPNUIJTDIBSU &YQMBJOZPVSBOTXFS
(d) *OXIJDIDPVOUSJFTEPBHSFBUFSQSPQPSUJPOPGUIFQFPQMFIBWFBMBOE
MJOFUIBOBNPCJMFQIPOF
(e) *OXIJDIDPVOUSJFTEPNPSFQFPQMFIBWFNPCJMFQIPOFTUIBO
MBOEMJOFT
(f) *OXIJDIDPVOUSZEPNPSFUIBOPGUIFQPQVMBUJPOIBWFBMBOE
MJOFBOEBNPCJMFQIPOF
(g) 8IBUEPZPVUIJOLUIFCBSTXPVMEMPPLMJLFGPSZPVSDPVOUSZ 8IZ
Sweden Italy
Land lines
Unit 1: Data handling
27
4 Collecting, organising and displaying data
4 6TFUIFUBCMFPGEBUBGSPN&YFSDJTF SFQFBUFECFMPX BCPVUUIFUFOTUVEFOUTGPSUIJT
RVFTUJPO
Student
1
7
8
Gender
'
'
M
M
M
'
M
'
'
M
Height (m)
Shoe size
7
7
8
7
Mass (kg)
Eye colour
#S
(S
(S
#S
#S
#S
#S
(S
#M
#S
Hair colour
#M
#M
#MP
#S
#S
#S
#M
#M
#M
#M
No. of brothers/sisters
1
1
(a) %SBXBQJFDIBSUUPTIPXUIFEBUBBCPVUUIFOVNCFSPGTJCMJOHT
(b) 3FQSFTFOUUIFIFJHIUPGTUVEFOUTVTJOHBOBQQSPQSJBUFDIBSU
(c) %SBXBDPNQPVOECBSDIBSUTIPXJOHFZFBOEIBJSDPMPVSCZHFOEFS
5 "NZCPVHIUBOFX7BVYIBMM$PSTBJO*UTWBMVFJTTIPXOJOUIFUBCMFCFMPX
Year
FAST FORWARD
In part (b), the ‘percentage
depreciation’ requires you to
first calculate how much the
car’s value decreased in the year
she had it, and then calculate
this as a percentage of the
original value. You will see more
about percentage decrease in
chapter 5. 28
Unit 1: Data handling
Value of car
(a) %SBXBMJOFHSBQIUPSFQSFTFOUUIJTJOGPSNBUJPO
(b) 8IBUJTUIFQFSDFOUBHFEFQSFDJBUJPOJOUIFĕSTUZFBSTIFPXOFEUIFDBS
(c) 6TFZPVSHSBQIUPFTUJNBUFUIFWBMVFPGUIFDBSJO
5
Fractions
5.1 Equivalent fractions
t &RVJWBMFOUNFBOT AIBTUIFTBNFWBMVF
t 5PĕOEFRVJWBMFOUGSBDUJPOTFJUIFSNVMUJQMZCPUIUIFOVNFSBUPSBOEEFOPNJOBUPSCZUIFTBNFOVNCFS
PSEJWJEFCPUIUIFOVNFSBUPSBOEEFOPNJOBUPSCZUIFTBNFOVNCFS
You can cross multiply to make
an equation and then solve it. For
example:
1
x
=
2
28
2 x = 28
x = 14
Exercise 5.1
1 'JOEUIFNJTTJOHWBMVFJOFBDIQBJSPGFRVJWBMFOUGSBDUJPOT
(a)
2 26
=
5 x
(b)
5 120
=
7
x
(c)
6 66
=
5 x
(d)
11 143
=
9
x
(e)
5 80
=
3 x
(f)
8
x
=
12 156
5.2 Operations on fractions
t 5PNVMUJQMZGSBDUJPOT NVMUJQMZOVNFSBUPSTCZOVNFSBUPSTBOEEFOPNJOBUPSTCZEFOPNJOBUPST
.JYFEOVNCFSTTIPVMECFSFXSJUUFOBTJNQSPQFSGSBDUJPOTCFGPSFNVMUJQMZJOHPSEJWJEJOH
t 5PBEEPSTVCUSBDUGSBDUJPOTDIBOHFUIFNUPFRVJWBMFOUGSBDUJPOTXJUIUIFTBNFEFOPNJOBUPS UIFOBEE PSTVCUSBDU UIFOVNFSBUPSTPOMZ
t 5PEJWJEFCZBGSBDUJPO JOWFSUUIFGSBDUJPO UVSOJUVQTJEFEPXO BOEDIBOHFUIF ÷ TJHOUPB×TJHO
t 6OMFTTZPVBSFTQFDJĕDBMMZBTLFEGPSBNJYFEOVNCFS HJWFBOTXFSTUPDBMDVMBUJPOTXJUIGSBDUJPOTBTQSPQFSPS
JNQSPQFSGSBDUJPOTJOUIFJSTJNQMFTUGPSN
Tip
*GZPVDBOTJNQMJGZUIF
GSBDUJPOQBSUĕSTUZPVXJMM
IBWFTNBMMFSOVNCFST
UPNVMUJQMZUPHFUUIF
JNQSPQFSGSBDUJPO
Exercise 5.2
1 3FXSJUFFBDINJYFEOVNCFSBTBOJNQSPQFSGSBDUJPOJOJUTTJNQMFTUGPSN
(a) 3 5 40
(b) 1 12 22
3
(e) 14 (f) 2
4
(c) 11 24 30
(d) 3
75
100
35
45
2 $BMDVMBUF
Remember: you can cancel to
simplify when you are multiplying
fractions; and the word ‘of’ means ×.
(a) 1
4
5
12
1
1
2
3
(e) 2 4 × (i)
8
9
of 81
(b)
9
× 7
13
(f)
1 12
1
× ×2
5 19
2
2
1
(j)
3
(c) 3
of 4 2
(g)
(k)
1
3
1
2
1
2
4
(d) 2
of 360 of 9
16
50
(h)
(l)
3
4
3
4
1
3
of
2
2
5
2
7
of 2
1
3
Unit 2: Number
29
5 Fractions
3 $BMDVMBUF HJWJOHZPVSBOTXFSBTBGSBDUJPOJOTJNQMFTUGPSN
Tip
:PVDBOVTFBOZDPNNPO
EFOPNJOBUPSCVUJUJT
FBTJFSUPTJNQMJGZJGZPV
VTFUIFMPXFTUPOF
(a)
1
6
3
8
+ (b)
(e) 2
1
8
1 (i)
5
7
1 2
9
10
−
7
12
1
7
(f) 4
3
10
1
3
(j) 1 − (c)
3
4
3 1
2
4 1
+ 7 3
(g) 1
7
3
1
13
− 4
5
1
3
17
3
(k) 2 −
(d) 2
1
2
(h) 3
9
10
2
(l)
1 −
4
9
13
3
2⎞
3⎠
5×
3
1
3
7
8
4 $BMDVMBUF
1
7
8
(a) 8 ÷ (b) 12 ÷ (d)
(e)
3
2 18
÷ 9 30
(g) 1
14
26
÷
10
13
8 4
÷ 9 5
(h) 3
(c)
6
15
7
8
÷ 12
3
7
2
(f) 1
2
3
5 (i) 5
1
5
2
9
1
3
10
5 4JNQMJGZUIFGPMMPXJOH
REWIND
The order of operations rules
(BODMAS) that were covered in
chapter 1 apply here too. 2
3
1
3
(a) 4 + × (d) 2
7
8
(b) 2
⎛ 8 1 − 6 3⎞ ⎝ 4
8⎠
5 15
5 1
(g) ⎛ ÷ ⎞ − ⎛ × ⎞ ⎝ 8 4 ⎠ ⎝ 6 5⎠
(e)
1
8
5 1
×
6 4
⎛2 1 − 7⎞
⎝ 5 8⎠
5
8
(c)
2
7
3
5
1
(f) ⎛ 5 ÷ − ⎞ ×
⎝ 11 12 ⎠ 6
1
3
+ × 2
3
3
(h) ⎛ 2
4− ⎞ ×
⎝ 3
10 ⎠ 17
3 ⎛2
× +6
7 ⎝3
(i)
2 1⎞ 2
⎛
7÷ − ×
⎝
9 3⎠ 3
7
12
6 .ST8FTUIBTEPMMBSTJOIFSBDDPVOU4IFTQFOET PGUIJT
(a) )PXNVDIEPFTTIFTQFOE
(b) )PXNVDIEPFTTIFIBWFMFę
3
4
7 *UUBLFTBCVJMEFS PGBOIPVSUPMBZUJMFT
1
2
(a) )PXNBOZUJMFTXJMMIFMBZJO4 IPVST
(b) )
PXMPOHXJMMJUUBLFIJNUPDPNQMFUFBĘPPSOFFEJOHUJMFT
5.3 Percentages
t 1FSDFOUNFBOTQFSIVOESFE"QFSDFOUBHFJTBGSBDUJPOXJUIBEFOPNJOBUPSPG
t 5PXSJUFPOFRVBOUJUZBTBQFSDFOUBHFPGBOPUIFS FYQSFTTJUBTBGSBDUJPOBOEUIFODPOWFSUUIFGSBDUJPOUPBQFSDFOUBHF
t 5PĕOEBQFSDFOUBHFPGBRVBOUJUZ NVMUJQMZUIFQFSDFOUBHFCZUIFRVBOUJUZ
t 5PJODSFBTFPSEFDSFBTFBOBNPVOUCZBQFSDFOUBHF ĕOEUIFQFSDFOUBHFBNPVOUBOEBEEPSTVCUSBDUJUGSPNUIF
PSJHJOBMBNPVOU
Exercise 5.3 A
1 &YQSFTTUIFGPMMPXJOHBTQFSDFOUBHFT3PVOEZPVSBOTXFSTUPPOFEFDJNBMQMBDF
(a) 16 (e) 30
Unit 2: Number
(b) 85 (f) 93
(c) 312
(g) (d) (h) 5 Fractions
2 &YQSFTTUIFGPMMPXJOHQFSDFOUBHFTBTDPNNPOGSBDUJPOTJOUIFJSTJNQMFTUGPSN
(a) (b) (c) (d) (e) 3 &YQSFTTUIFGPMMPXJOHEFDJNBMTBTQFSDFOUBHFT
(a) (e) Tip
8IFOĕOEJOHBQFSDFOUBHF
PGBRVBOUJUZ ZPVSBOTXFS
XJMMIBWFBVOJUBOEOPUB
QFSDFOUBHFTJHOCFDBVTF
ZPVBSFXPSLJOHPVUBO
BNPVOU
(b) (f) (c) (d) 4 $BMDVMBUF
(a) PG LH
(e) PG
(i) PG
(b) PG
(c) PG MJUSFT (d) PG NM
(f) PGb
(g) PG LN (h) PGHSBNT
(j) PGDVCJDNFUSFT
5 $BMDVMBUFUIFQFSDFOUBHFJODSFBTFPSEFDSFBTFBOEDPQZBOEDPNQMFUFUIFUBCMF
3PVOEZPVSBOTXFSTUPPOFEFDJNBMQMBDF
Original amount
New amount
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Percentage increase or
decrease
6 *ODSFBTFFBDIBNPVOUCZUIFHJWFOQFSDFOUBHF
(a) JODSFBTFECZ
(c) JODSFBTFECZ
(e) JODSFBTFECZ
(b) JODSFBTFECZ
(d) JODSFBTFECZ
(f) JODSFBTFECZ
7 %FDSFBTFFBDIBNPVOUCZUIFHJWFOQFSDFOUBHF
(a) EFDSFBTFECZ
(c) EFDSFBTFECZ
(e) EFDSFBTFECZ
(b) EFDSFBTFECZ
(d) EFDSFBTFECZ
(f) EFDSFBTFECZ
Exercise 5.3 B
1 UJDLFUTXFSFBWBJMBCMFGPSBOJOUFSOBUJPOBMDSJDLFUNBUDIPGUIFUJDLFUTXFSFTPME
XJUIJOBEBZ)PXNBOZUJDLFUTBSFMFę
2 .ST3BKBIPXOTPGBDPNQBOZ*GUIFDPNQBOZJTTVFT TIBSFT IPXNBOZTIBSFT
TIPVMETIFHFU
1
3 "CVJMEJOH XIJDIDPTUUPCVJME JODSFBTFEJOWBMVFCZ 3 8IBUJTUIFCVJMEJOH
2
XPSUIOPX
4 "QMBZFSTDPSFEPVUPGUIFQPJOUTJOBCBTLFUCBMMNBUDI8IBUQFSDFOUBHFPGUIFQPJOUT
EJEIFTDPSF
Unit 2: Number
31
5 Fractions
5 "DPNQBOZIBTBCVEHFUPGGPSQSJOUJOHCSPDIVSFTćFNBSLFUJOHEFQBSUNFOUIBT
TQFOUPGUIFCVEHFUBMSFBEZ)PXNVDINPOFZJTMFęJOUIFCVEHFU
6 +PTIDVSSFOUMZFBSOTQFSNPOUI*GIFSFDFJWFTBOJODSFBTFPG XIBUXJMMIJTOFX
NPOUIMZFBSOJOHTCF
7 "DPNQBOZBEWFSUJTFTUIBUJUTDPUUBHFDIFFTFJTGBUGSFF*GUIJTJTDPSSFDU IPXNBOZ
HSBNTPGGBUXPVMEUIFSFCFJOBHSBNUVCPGUIFDPUUBHFDIFFTF
8 4BMMZFBSOTQFSTIJę)FSCPTTTBZTTIFDBOFJUIFSIBWFNPSFQFSTIJęPSB
JODSFBTF8IJDIJTUIFCFUUFSPČFS Exercise 5.3 C
Tip
'JOEJOHBOPSJHJOBM
BNPVOUJOWPMWFTSFWFSTF
QFSDFOUBHFT/PUFUIBU
UIFSFBSFEJČFSFOUXBZTPG
TBZJOHAPSJHJOBMBNPVOU TVDIBTPMEBNPVOU QSFWJPVTBNPVOU BNPVOU
CFGPSFUIFJODSFBTFPS
EFDSFBTF BOETPPO
1 .JTIBQBJEGPSB%7%TFUBUBPČTBMF8IBUXBTUIFPSJHJOBMQSJDFPGUIF%7%TFU
2 *OBMBSHFTDIPPMTUVEFOUTBSFJO(SBEFćJTJTPGUIFTDIPPMQPQVMBUJPO
(a) )PXNBOZTUVEFOUTBSFUIFSFJOUPUBMJOUIFTDIPPM
(b) )PXNBOZTUVEFOUTBSFJOUIFSFTUPGUIFTDIPPM
3 4VLJ UIFXBJUSFTT IBTIFSXBHFTJODSFBTFECZ)FSOFXXBHFTBSF8IBUXBT
IFSXBHFCFGPSFUIFJODSFBTF
4 ćJTTVNNFS BOBNVTFNFOUQBSLJODSFBTFEJUTFOUSZQSJDFTCZUPćJTTVNNFS UIFOVNCFSPGQFPQMFFOUFSJOHUIFQBSLESPQQFECZGSPNUIFQSFWJPVTTVNNFSUP
(a) 8IBUXBTUIFFOUSZQSJDFUIFQSFWJPVTTVNNFS
(b) )PXNBOZWJTJUPSTEJEUIFQBSLIBWFUIFQSFWJPVTTVNNFS
(c) *GUIFSVOOJOHDPTUTPGUIFBNVTFNFOUQBSLSFNBJOFEUIFTBNFBTUIFQSFWJPVTTVNNFS
BOEUIFZNBEFBQSPĕUPOUIFFOUSZGFFTJOUIJTTVNNFS IPXNVDIXBTUIFJSQSPĕU
BNPVOUJOEPMMBST
5.4 Standard form
t "OVNCFSJOTUBOEBSEGPSNJTXSJUUFOBTBOVNCFSCFUXFFOBOENVMUJQMJFECZSBJTFEUPBQPXFSFH a × t 4UBOEBSEGPSNJTBMTPDBMMFETDJFOUJĕDOPUBUJPO
t 5PXSJUFBOVNCFSJOTUBOEBSEGPSN
t
o QMBDFBEFDJNBMQPJOUBęFSUIFĕSTUTJHOJĕDBOUEJHJU
o DPVOUUIFOVNCFSPGQMBDFPSEFSTUIFĕSTUTJHOJĕDBOUEJHJUIBTUPNPWFUPHFUGSPNUIJTOFXOVNCFSUPUIF
PSJHJOBMOVNCFS UIJTHJWFTUIFQPXFSPG
o JGUIFTJHOJĕDBOUEJHJUIBTNPWFEUPUIFMFę OPUFUIJTlooksMJLFUIFEFDJNBMQPJOUIBTNPWFEUPUIFSJHIU UIF
QPXFSPGJTQPTJUJWF CVUJGUIFTJHOJĕDBOUEJHJUIBTNPWFEUPUIFSJHIU PSEFDJNBMUPUIFMFę UIFQPXFSPGJT
OFHBUJWF
5PXSJUFBOVNCFSJOTUBOEBSEGPSNBTBOPSEJOBSZOVNCFS NVMUJQMZUIFEFDJNBMGSBDUJPOCZUPUIFHJWFOQPXFS
Tip
.BLFTVSFZPVLOPXIPX
ZPVSDBMDVMBUPSEFBMTXJUI
TUBOEBSEGPSN
32
k
Unit 2: Number
Exercise 5.4 A
1 8SJUFUIFGPMMPXJOHOVNCFSTJOTUBOEBSEGPSN
(a) (e) (i) (b) (f) (j) (c) (g) (k) (d) (h) (l) 5 Fractions
2 8SJUFUIFGPMMPXJOHBTPSEJOBSZOVNCFST
If the number part of your
standard form answer is a whole
number, there is no need to add a
decimal point.
REWIND
Remember, the first significant
figure is the first non-zero digit
from the left. (a) × (e) × −
(i) × −
(b) × (f) × −
(c) × (g) × −
(d) × −
(h) × Exercise 5.4 B
1 $BMDVMBUF HJWJOHZPVSBOTXFSTJOTUBOEBSEGPSNDPSSFDUUPUISFFTJHOJĕDBOUĕHVSFT
(a)
(d) × (g)
(b) ÷ 9500 0.00054
(e)
× (c) ÷ (f) 4525 × 8760
0.00002
(h)
5 25 108 (i)
3
9 1 × 10 −8
2 4JNQMJGZFBDIPGUIFGPMMPXJOH(JWFZPVSBOTXFSJOTUBOEBSEGPSN
(b)
(a) × × × −
−
(d) × × × (e)
(g) × ÷ × (h)
× × × (c)
× × × (f)
× − ÷ × − (i)
× × ÷ × 3
9 1 × 10 −8
3 ćFSVOIBTBNBTTPGBQQSPYJNBUFMZ×UPOOFTćFQMBOFU.FSDVSZIBTBNBTTPG
BQQSPYJNBUFMZ×UPOOFT
(a) 8IJDIIBTUIFHSFBUFSNBTT
(b) )PXNBOZUJNFTIFBWJFSJTUIFHSFBUFSNBTTDPNQBSFEXJUIUIFTNBMMFSNBTT
4 -JHIUUSBWFMTBUBTQFFEPG × NFUSFTQFSTFDPOEćF&BSUIJTBOBWFSBHFEJTUBODFPG
× NGSPNUIF4VOBOE1MVUPJTBOBWFSBHF × NGSPNUIF4VO
(a) 8
PSLPVUIPXMPOHJUUBLFTMJHIUGSPNUIF4VOUPSFBDI&BSUI JOTFDPOET (JWFZPVS
BOTXFSJOCPUIPSEJOBSZOVNCFSTBOETUBOEBSEGPSN
(b) )PXNVDIMPOHFSEPFTJUUBLFGPSUIFMJHIUUPSFBDI1MVUP (JWFZPVSBOTXFSJOCPUI
PSEJOBSZOVNCFSTBOETUBOEBSEGPSN
5.5 Estimation
t &TUJNBUJOHJOWPMWFTSPVOEJOHWBMVFTJOBDBMDVMBUJPOUPOVNCFSTUIBUBSFFBTZUPXPSLXJUI VTVBMMZXJUIPVU
UIFOFFEGPSBDBMDVMBUPS t "OFTUJNBUFBMMPXTZPVUPDIFDLUIBUZPVSDBMDVMBUJPOTNBLFTFOTF
Exercise 5.5
Remember, the symbol ≈ means
‘is approximately equal to’.
1 6TFXIPMFOVNCFSTUPTIPXXIZUIFTFFTUJNBUFTBSFDPSSFDU
(a) × ≈
(b) × ≈
(c) × ≈ (d) ÷ ≈
2 &TUJNBUFUIFBOTXFSTUPFBDIPGUIFTFDBMDVMBUJPOTUPUIFOFBSFTUXIPMFOVNCFS
(a) + o + (c)
9 3 × 7. 6 5 9 × 0 95
(b)
+ ÷ o
(d) 8 92 × 8 98
Unit 2: Number
33
5 Fractions
Mixed exercise
1 &TUJNBUFUIFBOTXFSUPFBDIPGUIFTFDBMDVMBUJPOTUPUIFOFBSFTUXIPMFOVNCFS
(a) × (c) 36.4 6.32
9.987
(b) ÷ 64.25 × 3.0982
(d)
2 4JNQMJGZ
(a)
160
200
48
72
(b)
(c)
36
54
3 $BMDVMBUF
(a)
4
9
(f) 2
×
1
3
3
8
(b) 84 ×
1
2
9 3
4
3
(g) ⎛ 4 ⎞
⎝ 4⎠
2
(c)
5 1
÷ 9 3
(d)
(h) 9 1 1 7
5
9
9 3
−
11 4
(i) 9
1
5
7
5
9
8
1 ×2
(e)
5
24
+
7
16
(j)
4
÷
18
5
25
+
⎛ 2⎞
⎝ 3⎠
2
4 +PTIVBJTQBJEQFSIPVS)FOPSNBMMZXPSLTBIPVSXFFL
(a) &TUJNBUFIJTXFFLMZFBSOJOHTUPUIFOFBSFTUEPMMBS
(b) &TUJNBUFIJTBOOVBMFBSOJOHT
5 ćFWBMVFPGBQMPUPGMBOEJODSFBTFECZUP8IBUXBTJUTQSFWJPVTWBMVF
6 *OBOFMFDUJPO PGUIFCBMMPUQBQFSTXFSFSFKFDUFEBTTQPJMFEWPUFT
(a) )PXNBOZWPUFTXFSFTQPJMFE
(b) 0
GUIFSFTUPGUIFWPUFT XFSFGPS$BOEJEBUF")PXNBOZWPUFTEJE$BOEJEBUF"
SFDFJWF
7 "CBCZIBEBNBTTPG LHXIFOTIFXBTCPSO"ęFSXFFLT IFSNBTTIBEJODSFBTFEUP
LH&YQSFTTUIJTBTBQFSDFOUBHFJODSFBTF DPSSFDUUPPOFEFDJNBMQMBDF
8 1MVUPJT × NGSPNUIF4VO
(a) &YQSFTTUIJTJOLJMPNFUSFT HJWJOHZPVSBOTXFSJOTUBOEBSEGPSN
(b) * OBDFSUBJOQPTJUJPO UIF&BSUIJT × LNGSPNUIF4VO*G1MVUP UIF&BSUIBOE
UIF4VOBSFJOBTUSBJHIUMJOFJOUIJTQPTJUJPO BOECPUIQMBOFUTBSFUIFTBNFTJEFPGUIF
SVO DBMDVMBUFUIFBQQSPYJNBUFEJTUBODF JOLN CFUXFFOUIF&BSUIBOE1MVUP(JWFZPVS
BOTXFSJOTUBOEBSEGPSN
9 "MJHIUZFBSJTUIFEJTUBODFUIBUMJHIUUSBWFMTJOPOFZFBS LN
(a) 8SJUFPOFMJHIUZFBSJOTUBOEBSEGPSN
(b) ć
F4VOJTMJHIUZFBSTGSPN&BSUI&YQSFTTUIJTEJTUBODFJOMJHIUZFBSTJO
TUBOEBSEGPSN
(c) 1
SPYJNBDFOUBVSJ BTUBS JTMJHIUZFBSTGSPN&BSUI)PXNBOZLJMPNFUSFTJTUIJT (JWFZPVSBOTXFSJOTUBOEBSEGPSN
34
Unit 2: Number
6
Equations and transforming
formulae
6.1 Further expansions of brackets
t &YQBOENFBOTUPSFNPWFUIFCSBDLFUTCZNVMUJQMZJOHPVU
t &BDIUFSNJOTJEFUIFCSBDLFUNVTUCFNVMUJQMJFECZUIFUFSNPVUTJEFUIFCSBDLFU
t /FHBUJWFUFSNTJOGSPOUPGUIFCSBDLFUTXJMMBČFDUUIFTJHOTPGUIFFYQBOEFEUFSNT
Exercise 6.1
Remember:
+×−=−
1 &YQBOEBOETJNQMJGZJGQPTTJCMF
−×+=−
+×+=+
(a)
(d)
(g)
(j)
−+−=+
− x+y x −y −a − x+ (b)
(e)
(h)
(k)
− a−b −x x+y − x+y x x−y (c)
(f)
(i)
(l)
− −x+y)
− x−
x− x−
−x x−y)
2 &YQBOEBOETJNQMJGZBTGBSBTQPTTJCMF
(a) x−y +x − 1
(d) − x ( −
2
1
(g) − x (
4
−
) − 2y(
+
)+2−(
2
(b) −x y− − x+xy (c) − x+y − x y−
)
(e) xy− +y − −x (f) x −y −y −x)
− 3)
(h) − x−y +x x−y (i)
−
1
2
(
−
)+3−(
+ 7)
6.2 Solving linear equations
t 5PTPMWFBOFRVBUJPO ZPVĕOEUIFWBMVFPGUIFVOLOPXOMFUUFS WBSJBCMF UIBUNBLFTUIFFRVBUJPOUSVF
t *GZPVBEEPSTVCUSBDUUIFTBNFOVNCFS PSUFSN UPCPUITJEFTPGUIFFRVBUJPO ZPVQSPEVDFBOFRVJWBMFOUFRVBUJPO
BOEUIFTPMVUJPOSFNBJOTVODIBOHFE
t *GZPVNVMUJQMZPSEJWJEFFBDIUFSNPOCPUITJEFTPGUIFFRVBUJPOCZUIFTBNFOVNCFS PSUFSN ZPVQSPEVDFBO
FRVJWBMFOUFRVBUJPOBOEUIFTPMVUJPOSFNBJOTVODIBOHFE
Exercise 6.2
In this exercise leave answers as
fractions rather than decimals,
where necessary.
1 4PMWFUIFTFFRVBUJPOTGPSx4IPXUIFTUFQTJOZPVSXPSLJOH
(a) x+=x+
(e) x−=x+
(i) x+=x−
(b) x+=x+
(f) x−=−x
(j) x−=x−
(c) x−=x+
(g) −x=x+
(k) x+=x−
(d) x−=x+
(h) +x=x−
(l) x+=x−
Unit 2: Algebra
35
6 Equations and transforming formulae
When an equation has brackets it
is usually best to expand them first.
To remove the denominators of
fractions in an equation, multiply
each term on both sides by the
common denominator.
2 4PMWFUIFTFFRVBUJPOTGPSx
(a)
(e)
(i)
(l)
x− =
− x− =−
x+= x− x− − x−
(b) x+ =
(f) −x =
(j) x−= x+ = x−
(c) x+ =
(d) x− =
(g) x+ =x
(h) x+ =x
(k) x+ − x− =
3 4PMWFUIFTFFRVBUJPOTGPSx
(a) x − 3 6 2
(b) x + 2 = 11
3
(e) x − 2 = 5 3
2x 1
(i)
= x
5
(f) 12 x − 1 = 9 5
2 3
= x − 6
(j)
5
(m)
2x x
− = 7
3 2
(n) −2
(c) 4 x = 16 6
5 2
= −1
(g)
3
(k)
10 x + 2
= 6− x
3
(d) 28 − x = 12
6
5 2x
(h)
= −1
4
(l)
x x
− =3
2 5
( + 4)
= x+7
2
6.3 Factorising algebraic expressions
t ćFĕSTUTUFQJOGBDUPSJTJOHJTUPJEFOUJGZBOEAUBLFPVU"--DPNNPOGBDUPST
t $PNNPOGBDUPSTDBOCFOVNCFST WBSJBCMFT CSBDLFUTPSBDPNCJOBUJPOPGUIFTF
t 'BDUPSJTJOHJTUIFPQQPTJUFPGFYQBOEJOHoXIFOZPVGBDUPSJTFZPVQVUCSBDLFUTCBDLJOUPUIFFYQSFTTJPO
Exercise 6.3
Remember, x2 means x × x, so x is
a factor of x2
Find the HCF of the numbers first.
Then find the HCF of the variables,
if there is one, in alphabetical
order.
1 'JOEUIFIJHIFTUDPNNPOGBDUPSPGFBDIQBJS
(a)
(d)
(g)
(j)
xBOE
aBOEab
xyBOExyz
x yzBOExyz
(b)
(e)
(h)
(k)
BOEx
xyBOEyz
pqBOEpq
abBOEab
(c)
(f)
(i)
(l)
aBOEb
abBOEab
abcBOEab
xyBOExy
2 'BDUPSJTFBTGVMMZBTQPTTJCMF
(a) x+
(e) ab+a
(i) xy−yz
(b) +y
(f) x −y
(j) x −xy
(c) a−
(g) xyz−xz
(d) x−xy
(h) ab−bc
(c) x+x
(g) x−x
(k) ab−bD
(d) x−x
(h) xy−xy
(l) ab−ab
3 'BDUPSJTFUIFGPMMPXJOH
Remember, if one of the terms is
exactly the same as the common
factor, you must put a 1 where the
term would appear in the bracket.
(a) x+x
(e) ab+b
(i) abc−abc
(b) a−a
(f) xy−xy
(j) x−xy
4 3FNPWFBDPNNPOGBDUPSUPGBDUPSJTFFBDIPGUIFGPMMPXJOHFYQSFTTJPOT
(a)
(c)
(e)
(g)
(i)
(k)
36
Unit 2: Algebra
x +y + +y a+b −a a+b x −y + −y +y −x y+ x x− − x− x x+ +y +x (b)
(d)
(f)
(h)
(j)
(l)
x y− + y−
a a−b − a−b)
x x− + x−
a b−c − c−b)
x x−y − x−y)
x−y −x x−y)
6 Equations and transforming formulae
6.4 Transformation of a formula
t "GPSNVMBJTBHFOFSBMSVMF VTVBMMZJOWPMWJOHTFWFSBMWBSJBCMFT GPSFYBNQMF UIFBSFBPGBSFDUBOHMF A=bh.
t "WBSJBCMFJTDBMMFEUIFTVCKFDUPGUIFGPSNVMBXIFOJUJTPOJUTPXOPOPOFTJEFPGUIFFRVBMTTJHO
t :PVDBOUSBOTGPSNBGPSNVMBUPNBLFBOZWBSJBCMFUIFTVCKFDU:PVVTFUIFTBNFSVMFTUIBUZPVVTFEUPTPMWF
FRVBUJPOT
Exercise 6.4 A
1 .BLFmUIFTVCKFDUJGD=km
2 .BLFcUIFTVCKFDUJGy=mx+c
3 (JWFOUIBUP=ab−c NBLFbUIFTVCKFDUPGUIFGPSNVMB
4 (JWFOUIBUa=bx+c NBLFbUIFTVCKFDUPGUIFGPSNVMB
Tip
1BZBUUFOUJPOUPUIFTJHOT
XIFOZPVUSBOTGPSNB
GPSNVMB
5 .BLFaUIFTVCKFDUPGFBDIGPSNVMB
(a) a+b=c
(b) a−b=c
(e) bc−a=d
(f) bc −a=−d
(i) abc−d=e
(j) cab+d=ef
(m) c a−b =d
(n) d a+b =c
(c) ab−c=d
2a b
= d
(g)
c
(k)
ab
+ de = f c
(d) ab+c=d
c ba
(h)
=e
d
(l) c +
ab
=e
d
Exercise 6.4 B
1 ćFQFSJNFUFSPGBSFDUBOHMFDBOCFHJWFOBTP= l + b XIFSFPJTUIFQFSJNFUFS lJTUIF
MFOHUIBOEbJTUIFCSFBEUI
(a) .BLFbUIFTVCKFDUPGUIFGPSNVMB
(b) 'JOEbJGUIFSFDUBOHMFIBTBMFOHUIPGDNBOEBQFSJNFUFSPGDN
Tip
*GZPVBSFHJWFOBWBMVFGPS
ɀ ZPVNVTUVTFUIFHJWFO
WBMVFUPBWPJEDBMDVMBUPS
BOESPVOEJOHFSSPST
Tip
*ORVFTUJPOTTVDIBT
RVFTUJPO JUNBZCF
IFMQGVMUPESBXBEJBHSBN
UPTIPXXIBUUIFQBSUTPG
UIFGPSNVMBSFQSFTFOU
2 ćFDJSDVNGFSFODFPGBDJSDMFDBOCFGPVOEVTJOHUIFGPSNVMBC=ɀr XIFSFrJTUIFSBEJVTPG
UIFDJSDMF
(a) .BLFrUIFTVCKFDUPGUIFGPSNVMB
(b) 'JOEUIFSBEJVTPGBDJSDMFPGDJSDVNGFSFODFDN6TFɀ=
(c) 'JOEUIFEJBNFUFSPGBDJSDMFPGDJSDVNGFSFODFDN6TFɀ=
h (a b)
XIFSFhJTUIFEJTUBODF
3 ćFBSFBPGBUSBQF[JVNDBOCFGPVOEVTJOHUIFGPSNVMBA =
2
CFUXFFOUIFQBSBMMFMTJEFTBOEaBOEbBSFUIFMFOHUITPGUIFQBSBMMFMTJEFT#ZUSBOTGPSNJOH
UIFGPSNVMBBOETVCTUJUVUJPO ĕOEUIFMFOHUIPGb JOBUSBQF[JVNPGBSFBDNXJUI
a=DNBOEh=DN
4 "OBJSMJOFVTFTUIFGPSNVMBTP BUPSPVHIMZFTUJNBUFUIFUPUBMNBTTPGQBTTFOHFST
BOEDIFDLFECBHTQFSĘJHIUJOLJMPHSBNT T JTUIFUPUBMNBTT P JTUIFOVNCFSPGQBTTFOHFST
BOEBJTUIFOVNCFSPGCBHT
(a) 8IBUNBTTEPFTUIFBJSMJOFBTTVNFGPS
(i) BQBTTFOHFS (ii) BDIFDLFECBH
Unit 2: Algebra
37
6 Equations and transforming formulae
(b) &TUJNBUFUIFUPUBMNBTTGPSQBTTFOHFSTFBDIXJUIUXPDIFDLFECBHT
(c) .BLFBUIFTVCKFDUPGUIFGPSNVMB
(d) $BMDVMBUFUIFUPUBMNBTTPGUIFCBHTJGUIFUPUBMNBTTPGQBTTFOHFSTBOEDIFDLFECBHT
POBĘJHIUJTUPOOFT
5 8IFOBOPCKFDUJTESPQQFEGSPNBIFJHIU UIFEJTUBODF m JONFUSFTUIBUJUIBTGBMMFODBOCF
SFMBUFEUPUIFUJNFJUUBLFTGPSJUUPGBMM t JOTFDPOETCZUIFGPSNVMBmt
(a) .BLFt UIFTVCKFDUPGUIFGPSNVMB
(b) $BMDVMBUFUIFUJNFJUUBLFTGPSBOPCKFDUUPGBMMGSPNBEJTUBODFPGN
Mixed exercise
1 4PMWFGPSx
(a) x−=−
(b) x+=−
(c)
2x 4
= 2 7
(f) x−= x+ (g) 3x 7 = 1 4 x 4
8
2 .BLFxUIFTVCKFDUPGFBDIGPSNVMB
nx + p
(a) m = nxp − r
(b) m =
q
3 &YQBOEBOETJNQMJGZXIFSFQPTTJCMF
(e) x−=−x
Remember to inspect your answer
to see if there are any like terms.
If there are, add and/or subtract
them to simplify the expression.
(a) x− +
(d) −y −y −y
(g) −x x− +x
(d) 5 =
1 4x
5
(h) 3(2 x − 5) = x + 1
5
2
(b) −x x− (e) x− + x+ (h) x x+ −x x−
(c) − x−y+
(f) x x− + x−
(b) x−y
(e) xy+xy
(h) x x+y −x x+y)
(c) −x−
(f) x−y +x x−y 4 'BDUPSJTFGVMMZ
(a) x−
(d) xy−x
(g) x +x − x+
5 (JWFOUIBU GPSBSFDUBOHMF BSFB=MFOHUI×CSFBEUI XSJUFBOFYQSFTTJPOGPSUIFBSFBPGFBDI
SFDUBOHMF&YQBOEFBDIFYQSFTTJPOGVMMZ
(a)
x−7
(b)
4
REWIND
Use geometric properties to make
the equations (see chapter 3). 2x
(c)
19x
x+9
6 6TFUIFJOGPSNBUJPOJOFBDIEJBHSBNUPNBLFBOFRVBUJPOBOETPMWFJUUPĕOEUIFTJ[F
PGFBDIBOHMF
2x + 8
12x – 45
180 – 3x
38
Unit 2: Algebra
(d)
4x – 10
x
4x + 15
6x – 45
7
Perimeter, area and volume
7.1 Perimeter and area in two dimensions
t 1FSJNFUFSJTUIFUPUBMEJTUBODFBSPVOEUIFPVUTJEFPGBTIBQF:PVDBOĕOEUIFQFSJNFUFSPGBOZTIBQF
CZBEEJOHVQUIFMFOHUITPGUIFTJEFT
t ćFQFSJNFUFSPGBDJSDMFJTDBMMFEUIFDJSDVNGFSFODF6TFUIFGPSNVMBC = πdPSC = 2πrUPĕOEUIFDJSDVNGFSFODF
PGBDJSDMF
t "SFBJTUIFUPUBMTQBDFDPOUBJOFEXJUIJOBTIBQF6TFUIFTFGPSNVMBFUPDBMDVMBUFUIFBSFBPGEJČFSFOUTIBQFT
o
o
o
o
o
o
bh
USJBOHMFA =
2
TRVBSFA = s2
SFDUBOHMFA = bh
QBSBMMFMPHSBNA = bh
SIPNCVTA = bh
1
LJUF A = (produc f diagonals)
2
o USBQF[JVNA =
t
(
fp
2
ll l d )h
o DJSDMFA = πr2
:PVDBOXPSLPVUUIFBSFBPGDPNQMFYTIBQFTJOBGFXTUFQT%JWJEFDPNQMFYTIBQFTJOUPLOPXOTIBQFT
8PSLPVUUIFBSFBPGFBDIQBSUBOEUIFOBEEUIFBSFBTUPHFUIFSUPĕOEUIFUPUBMBSFB
Exercise 7.1 A
1 'JOEUIFQFSJNFUFSPGFBDITIBQF
(a)
(b)
32 mm
(c)
11.25 cm
19 mm
28 mm
45 mm
(d)
(e)
21 mm
14 mm
(f)
1.5 cm
6.8 cm
5.3 cm
3.4 cm
4.9 cm
92
mm
7.2 cm
69 mm
Unit 2: Shape, space and measures
39
7 Perimeter, area and volume
2 'JOEUIFQFSJNFUFSPGUIFTIBEFEBSFBJOFBDIPGUIFTFTIBQFT6TFπ = JOZPVSDBMDVMBUJPOT
Tip
(a)
'PSBTFNJDJSDMF UIF
QFSJNFUFSJODMVEFTIBMG
UIFDJSDVNGFSFODFQMVT
UIFMFOHUIPGUIFEJBNFUFS
*GZPVBSFOPUHJWFOUIF
WBMVFPGπ VTFUIFπLFZ
POZPVSDBMDVMBUPS3PVOE
ZPVSBOTXFSTUPUISFF
TJHOJĕDBOUĕHVSFT
(b)
(c)
(d)
7 cm
5m
(e)
(f)
21 mm
4.5 m
4m
(g)
3m
8 mm
60°
8 cm
3 "TRVBSFĕFMEIBTBQFSJNFUFSPGN8IBUJTUIFMFOHUIPGPOFPGJUTTJEFT
4 'JOEUIFDPTUPGGFODJOHBSFDUBOHVMBSQMPUNMPOHBOENXJEFJGUIFDPTUPGGFODJOHJT
QFSNFUSF
Tip
.BLFTVSFBMM
NFBTVSFNFOUTBSFJOUIF
TBNFVOJUTCFGPSFZPVEP
BOZDBMDVMBUJPOT
5 "OJTPTDFMFTUSJBOHMFIBTBQFSJNFUFSPGDN$BMDVMBUFUIFMFOHUIPGFBDIPGUIFFRVBMTJEFT
JGUIFSFNBJOJOHTJEFJTNNMPOH
6 )PXNVDITUSJOHXPVMEZPVOFFEUPGPSNBDJSDVMBSMPPQXJUIBEJBNFUFSPGDN
7 ćFSJNPGBCJDZDMFXIFFMIBTBSBEJVTPGDN
(a) 8IBUJTUIFDJSDVNGFSFODFPGUIFSJN
(b) ćFUZSFUIBUHPFTPOUPUIFSJNJTDNUIJDL$BMDVMBUFUIFDJSDVNGFSFODFPGUIFXIFFM
XIFOUIFUZSFJTĕUUFEUPJU
Remember, give your answer in
square units.
Exercise 7.1 B
1 'JOEUIFBSFBPGFBDIPGUIFTFTIBQFT
(a)
12.5 cm
17 cm
(b)
1.7 m
(c)
90 cm
14 cm 19 cm
(d)
21 cm
15 cm
19 cm
20 cm
35 cm
11 cm
21 cm
12 cm
5 cm
40
(g)
72 mm
13 cm
Unit 2: Shape, space and measures
112 mm
41 mm
67 mm
(h)
(i)
8 cm
15 cm
10 cm
(f)
25 cm
12 cm
7 cm
cm
10
(e)
11 cm
7 cm
17 cm
7 Perimeter, area and volume
2 'JOEUIFBSFBPGFBDITIBQF6TFπ = JOZPVSDBMDVMBUJPOT(JWFZPVSBOTXFSTDPSSFDUUP
UXPEFDJNBMQMBDFT
(a)
(b)
(c)
(d)
100 mm
(e)
300°
20° 10 cm
140 mm
8 cm
14 mm
8PSLPVUBOZNJTTJOH
EJNFOTJPOTPOUIF
ĕHVSFVTJOHUIFHJWFO
EJNFOTJPOTBOEUIF
QSPQFSUJFTPGTIBQFT
3 'JOEUIFBSFBPGUIFTIBEFEQBSUJOFBDIPGUIFTFĕHVSFT4IPXZPVSXPSLJOHDMFBSMZJOFBDI
DBTF"MMEJNFOTJPOTBSFHJWFOJODFOUJNFUSFT
(a)
(b) 2
6
6
3
(f)
1
13
14
14
6
36
13
(g)
(h)
14
3
%JWJEFJSSFHVMBSTIBQFT
JOUPLOPXOTIBQFTBOE
DPNCJOFUIFBSFBTUPHFU
UIFUPUBMBSFB
35
20
(i)
20
25
5
20
6
45
30
30
40
2
90
90
95
15
Tip
18
7
8
12
9
5
12
9
(d)
22
2
21
(e)
(c)
6
4 'JOEUIFBSFBPGUIFGPMMPXJOHĕHVSFTHJWJOHZPVSBOTXFSTDPSSFDUUPUXPEFDJNBMQMBDFT
6TFπ = GPSQBSU(f)
(a)
2.5 cm
m
6c
(b)
(c)
9 cm 4.5 cm
11 cm
2 cm
7 cm
4.8 cm
5 cm
(d)
(e)
4 cm
3
cm
6.5 cm
5.5 cm
cm
6.3
(f)
15 cm
32.4 cm
40 cm
8 cm
40 mm
Tip
50 mm
30 mm
20 cm
5 "N × NSFDUBOHVMBSSVHJTQMBDFEPOUIFĘPPSJOBN × NSFDUBOHVMBSSPPN
)PXNVDIPGUIFĘPPSJTOPUDPWFSFECZUIFSVH
6 ćFBSFBPGBSIPNCVTPGTJEFDNJTNN2%FUFSNJOFUIFIFJHIUPGUIFSIPNCVT
Unit 2: Shape, space and measures
41
7 Perimeter, area and volume
Exercise 7.1 C
6TFɀGPSUIFTFRVFTUJPOT
1 $BMDVMBUFUIFMFOHUIPGUIFBSDAB TVCUFOEFECZUIFHJWFOBOHMF JOFBDIPGUIFTFDJSDMFT
(a)
(b)
A
21 mm
O
(c)
A
O
120°
O
10 cm
B
40°
A
B
12 mm
B
2 ćFEJBHSBNTIPXTBDSPTTTFDUJPOPGUIF&BSUI5XPDJUJFT XBOEY MJFPOUIFTBNF
MPOHJUVEF(JWFOUIBUUIFSBEJVTPGUIF&BSUIJTLN DBMDVMBUFUIFEJTUBODF XY CFUXFFO
UIFUXPDJUJFT
637
60°
1k
m
Y
X
3 $BMDVMBUFUIFTIBEFEBSFBPGFBDIDJSDMF
(a)
12 cm
(b)
60°
(c)
20 cm
150°
18 mm
4 "MBSHFDJSDVMBSQJ[[BIBTBEJBNFUFSPGDNćFQJ[[BSFTUBVSBOUDVUTJUTQJ[[BTJOUPFJHIU
FRVBMTMJDFT$BMDVMBUFUIFTJ[FPGFBDITMJDFJODN2
42
Unit 2: Shape, space and measures
7 Perimeter, area and volume
7.2 Three-dimensional objects
t "OZTPMJEPCKFDUJTUISFFEJNFOTJPOBMćFUISFFEJNFOTJPOTPGBTPMJEBSFMFOHUI CSFBEUIBOEIFJHIU
t ćFOFUPGBTPMJEJTBUXPEJNFOTJPOBMEJBHSBN*UTIPXTUIFTIBQFPGBMMGBDFTPGUIFTPMJEBOEIPXUIFZBSFBUUBDIFE
UPFBDIPUIFS*GZPVGPMEVQBOFU ZPVHFUBNPEFMPGUIFTPMJE
Exercise 7.2
1 8IJDITPMJETXPVMECFNBEFGSPNUIFGPMMPXJOHGBDFT
(a)
(b)
×2
×6
(c)
×2
×2
(d)
×4
×1
×8
2 %FTDSJCFUIFTPMJEZPVDPVMEQSPEVDFVTJOHFBDIPGUIFGPMMPXJOHOFUT
(a)
(b)
(c)
3 4LFUDIBQPTTJCMFOFUGPSFBDIPGUIFGPMMPXJOHTPMJET
(a)
(b)
(c)
(d)
Unit 2: Shape, space and measures
43
7 Perimeter, area and volume
7.3 Surface areas and volumes of solids
t ćFTVSGBDFBSFBPGBUISFFEJNFOTJPOBMPCKFDUJTUIFUPUBMBSFBPGBMMJUTGBDFT
t ćFWPMVNFPGBUISFFEJNFOTJPOBMPCKFDUJTUIFBNPVOUPGTQBDFJUPDDVQJFT
t :PVDBOĕOEUIFWPMVNFPGBDVCFPSDVCPJEVTJOHUIFGPSNVMB V = l × b × h XIFSFlJTUIFMFOHUI bJTUIFCSFBEUIBOE
hJTUIFIFJHIUPGUIFPCKFDU
t "QSJTNJTBUISFFEJNFOTJPOBMPCKFDUXJUIBVOJGPSNDSPTTTFDUJPO UIFFOEGBDFTPGUIFTPMJEBSFJEFOUJDBMBOE
t
QBSBMMFM *GZPVTMJDFUISPVHIUIFQSJTNBOZXIFSFBMPOHJUTMFOHUI BOEQBSBMMFMUPUIFFOEGBDFT ZPVXJMMHFUB
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:PVDBOĕOEUIFWPMVNFPGBOZQSJTN JODMVEJOHBDZMJOEFS CZNVMUJQMZJOHUIFBSFBPGJUTDSPTTTFDUJPOCZUIF
EJTUBODFCFUXFFOUIFQBSBMMFMGBDFTćJTJTFYQSFTTFEJOUIFGPSNVMB V = al XIFSFaJTUIFBSFBPGUIFCBTFBOElJTUIF
MFOHUIPGUIFQSJTN:PVOFFEUPVTFUIFBQQSPQSJBUFBSFBGPSNVMBGPSUIFTIBQFPGUIFDSPTTTFDUJPO
t 'JOEUIFWPMVNFPGBDPOFVTJOHUIFGPSNVMB V = 13 πr h XIFSFhJTUIFQFSQFOEJDVMBSIFJHIU5PĕOEUIFDVSWFE
2
TVSGBDFBSFBVTFUIFGPSNVMB TVSGBDFBSFBπrl XIFSFlJTUIFTMBOUIFJHIUPGUIFDPOF
area of base × h
'JOEUIFWPMVNFPGBQZSBNJEVTJOHUIFGPSNVMB V =
XIFSFhJTUIFQFSQFOEJDVMBSIFJHIU
3
t
t 'JOEUIFWPMVNFPGBTQIFSFVTJOHUIFGPSNVMB V = 43 πr 5PĕOEUIFTVSGBDFBSFBVTFUIFGPSNVMB TVSGBDF
3
BSFBπr2
Exercise 7.3 A
6TFπ = GPSBOZTIBQFTJOWPMWJOHDJSDMFTJOUIJTFYFSDJTF
Tip
%SBXJOHUIFOFUTPGUIF
TPMJETNBZIFMQZPVXPSL
PVUUIFTVSGBDFBSFBPG
FBDITIBQF
1 $BMDVMBUFUIFTVSGBDFBSFBPGFBDITIBQF
(a)
0.5 mm
1.2 mm
(b)
18.4 m
0.4 mm
12 m
14 m
(c)
(d)
8m
4m
12 mm
1.5 cm
2 "XPPEFODVCFIBTTJYJEFOUJDBMTRVBSFGBDFT FBDIPGBSFBDN2
(a) 8IBUJTUIFTVSGBDFBSFBPGUIFDVCF
(b) 8IBUJTUIFIFJHIUPGUIFDVCF
Remember,
1 m2 = 10 000 cm2
44
3 .ST/JOJJTPSEFSJOHXPPEFOCMPDLTUPVTFJOIFSNBUITDMBTTSPPN
ćFCMPDLTBSFDVCPJETXJUIEJNFOTJPOTDN × DN × DN
(a) $BMDVMBUFUIFTVSGBDFBSFBPGPOFCMPDL
(b) .ST/JOJOFFETCMPDLT8IBUJTUIFUPUBMTVSGBDFBSFBPGBMMUIFCMPDLT
(c) 4 IFEFDJEFTUPWBSOJTIUIFCMPDLT"UJOPGWBSOJTIDPWFSTBOBSFBPGN2
)PXNBOZUJOTXJMMTIFOFFEUPWBSOJTIBMMUIFCMPDLT
Unit 2: Shape, space and measures
7 Perimeter, area and volume
4 $BMDVMBUFUIFWPMVNFPGFBDIQSJTN
Tip
ćFMFOHUIPGUIFQSJTNJT
UIFEJTUBODFCFUXFFOUIF
UXPQBSBMMFMGBDFT8IFO
BQSJTNJTUVSOFEPOUPJUT
GBDF UIFMFOHUINBZMPPL
MJLFBIFJHIU8PSLPVUUIF
BSFBPGUIFDSPTTTFDUJPO
FOEGBDF CFGPSFZPV
BQQMZUIFWPMVNFGPSNVMB
(a)
(b)
45 mm
4 cm
(d)
65 mm
3 cm
80 mm
50 mm
(c)
2 cm
m
40 c
A = 28 cm2
8 cm
20 mm
(e)
8 cm
10 cm
(f)
(g)
(h)
1.25 m
4m
12 cm
1.2 m
1.2 m
25
12 cm
cm
1.25 m
5m
1.2
5 "QPDLFUEJDUJPOBSZJTDNMPOH DNXJEFBOEDNUIJDL
$BMDVMBUFUIFWPMVNFPGTQBDFJUUBLFTVQ
6 (a) 'JOEUIFWPMVNFPGBMFDUVSFSPPNUIBUJTNMPOH NXJEFBOENIJHI
(b) 4BGFUZSFHVMBUJPOTTUBUFUIBUEVSJOHBOIPVSMPOHMFDUVSFFBDIQFSTPOJOUIFSPPNNVTU
IBWFN3PGBJS$BMDVMBUFUIFNBYJNVNOVNCFSPGQFPQMFXIPDBOBUUFOEBOIPVS
MPOHMFDUVSF
7 "DZMJOESJDBMUBOLJTNIJHIXJUIBOJOOFSSBEJVTPGDN$BMDVMBUFIPXNVDIXBUFSUIF
UBOLXJMMIPMEXIFOGVMM
8 "NBDIJOFTIPQIBTGPVSEJČFSFOUSFDUBOHVMBSQSJTNTPGWPMVNFNN3$PQZBOEĕMMJO
UIFQPTTJCMFEJNFOTJPOTGPSFBDIQSJTNUPDPNQMFUFUIFUBCMF
Volume (mm3)
64 000
64 000
Length (mm)
Breadth (mm)
64 000
64 000
Height (mm)
Exercise 7.3 B
6TFɀGPSBOZTIBQFTJOWPMWJOHDJSDMFTJOUIJTFYFSDJTF
1 'JOEUIFWPMVNFPGUIFGPMMPXJOHTPMJET(JWFZPVSBOTXFSTDPSSFDUUPUXPEFDJNBMQMBDFT
1.2 cm
3.5 cm
(a)
(b)
(c)
20 m
(d)
(e)
3 cm
3.5 cm
2.7 cm
8 cm
36 mm
12 cm
Unit 2: Shape, space and measures
40 mm
45
7 Perimeter, area and volume
2 (JWFZPVSBOTXFSTUPUIJTRVFTUJPOJOTUBOEBSEGPSNUPUISFFTJHOJĕDBOUĕHVSFT
ćF&BSUIIBTBOBWFSBHFSBEJVTPGLN
(a) "TTVNJOHUIF&BSUIJTNPSFPSMFTTTQIFSJDBM DBMDVMBUF
(i) JUTWPMVNF
(ii) JUTTVSGBDFBSFB
(b) * GPGUIFTVSGBDFBSFBPGUIF&BSUIJTDPWFSFECZUIFPDFBOT DBMDVMBUFUIFBSFBPGMBOE
POUIFTVSGBDF
Mixed exercise
6TFɀGPSBOZTIBQFTJOWPMWJOHDJSDMFTJOUIJTFYFSDJTF
1 "DJSDVMBSQMBUFPOBTUPWFIBTBEJBNFUFSPGDNćFSFJTBNFUBMTUSJQBSPVOEUIFPVUTJEF
PGUIFQMBUF
(a) $BMDVMBUFUIFTVSGBDFBSFBPGUIFUPQPGUIFQMBUF
(b) $BMDVMBUFUIFMFOHUIPGUIFNFUBMTUSJQ
2 8IBUJTUIFSBEJVTPGBDJSDMFXJUIBOBSFBPGDN2
3 $BMDVMBUFUIFTIBEFEBSFBJOFBDIĕHVSF
170 mm
40 mm
(c)
120 mm
150 mm
2 cm
(b)
50 mm
192 mm
6 cm
(a)
320 mm
(d)
5 cm
(e)
8 cm
4 cm
6 cm
6 cm
12 cm
(g)
12 cm
30°
10°
46
Unit 2: Shape, space and measures
(f)
5 cm
5 cm
1 cm
7 cm
5 cm
2 cm
3 cm
6 cm
3 cm
5 cm
7 Perimeter, area and volume
4 MNOPJTBUSBQF[JVNXJUIBOBSFBPGDN2$BMDVMBUFUIFMFOHUIPGNO
12 m
M
N
10 m
P
O
5 4UVEZUIFUXPQSJTNT
20 mm
A
40 mm
120 mm
B
15 mm
20 mm
(a) 8IJDIPGUIFUXPQSJTNTIBTUIFTNBMMFSWPMVNF
(b) 8IBUJTUIFEJČFSFODFJOWPMVNF
(c) 4 LFUDIBOFUPGUIFDVCPJE:PVSOFUEPFTOPUOFFEUPCFUPTDBMF CVUZPVNVTUJOEJDBUF
UIFEJNFOTJPOTPGFBDIGBDFPOUIFOFU
(d) $BMDVMBUFUIFTVSGBDFBSFBPGFBDIQSJTN
6 )PXNBOZDVCFTPGTJEFDNDBOCFQBDLFEJOUPBXPPEFOCPYNFBTVSJOHDNCZ
DNCZDN 7 'JOEUIFEJČFSFODFCFUXFFOUIFWPMVNFPGBDNIJHIDPOFXIJDIIBTBDNXJEFCBTF
BOEBTRVBSFCBTFEQZSBNJEUIBUJTDNXJEFBUJUTCBTFBOEDNIJHI
8 5FOOJTCBMMTPGEJBNFUFSDNBSFQBDLFEJOUPDZMJOESJDBMNFUBMUVCFTUIBUBSFTFBMFEBUCPUI
FOETćFJOTJEFPGUIFUVCFIBTBOJOUFSOBMEJBNFUFSPGDNBOEUIFUVCFJTDNMPOH
$BMDVMBUFUIFWPMVNFPGTQBDFMFęJOUIFUVCFJGUISFFUFOOJTCBMMTBSFQBDLFEJOUPJU
Unit 2: Shape, space and measures
47
8
Introduction to probability
8.1 Basic probability
t 1SPCBCJMJUZJTBNFBTVSFPGUIFDIBODFUIBUTPNFUIJOHXJMMIBQQFO*UJTNFBTVSFEPOBTDBMFPGUP
t
t
o PVUDPNFTXJUIBQSPCBCJMJUZPGBSFJNQPTTJCMF
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2
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5PDBMDVMBUFQSPCBCJMJUZGSPNUIFPVUDPNFTPGFYQFSJNFOUT VTFUIFGPSNVMB
t &YQFSJNFOUBMQSPCBCJMJUZJTBMTPDBMMFEUIFSFMBUJWFGSFRVFODZ
Exercise 8.1
1 4BMNBIBTBCBHDPOUBJOJOHPOFSFE POFXIJUFBOEPOFHSFFOCBMM4IFESBXTBCBMMBUSBOEPN
BOESFQMBDFTJUCFGPSFESBXJOHBHBJO4IFSFQFBUTUIJTUJNFT4IFVTFTBUBMMZUBCMFUPSFDPSE
UIFPVUDPNFTPGIFSFYQFSJNFOU
Red
//// //// ////
White
//// //// //// ///
Green
//// //// //// //
(a)
(b)
(c)
(d)
$BMDVMBUFUIFSFMBUJWFGSFRVFODZPGESBXJOHFBDIDPMPVS
&YQSFTTIFSDIBODFPGESBXJOHBSFECBMMBTBQFSDFOUBHF
8IBUJTUIFTVNPGUIFUISFFSFMBUJWFGSFRVFODJFT
8IBUTIPVMEZPVSDIBODFTCFJOUIFPSZPGESBXJOHFBDIDPMPVS
2 *UJT+PTITKPCUPDBMMDVTUPNFSTXIPIBWFIBEUIFJSDBSTFSWJDFEBUUIFEFBMFSUPDIFDL
XIFUIFSUIFZBSFIBQQZXJUIUIFTFSWJDFUIFZSFDFJWFE)FLFQUUIJTSFDPSEPGXIBUIBQQFOFE
GPSDBMMTNBEFPOFNPOUI
Result
4QPLFUPDVTUPNFS
Unit 2: Data handling
1IPOFOPUBOTXFSFE
44
-FęNFTTBHFPOBOTXFSJOHNBDIJOF
1IPOFFOHBHFEPSPVUPGPSEFS
8SPOHOVNCFS
48
Frequency
8 Introduction to probability
(a) $BMDVMBUFUIFSFMBUJWFGSFRVFODZPGFBDIFWFOUBTBEFDJNBMGSBDUJPO
(b) *TJUIJHIMZMJLFMZ MJLFMZ VOMJLFMZPSIJHIMZVOMJLFMZUIBUUIFGPMMPXJOHPVUDPNFTXJMM
PDDVSXIFO+PTINBLFTBDBMM
(i) ćFDBMMXJMMCFBOTXFSFECZUIFDVTUPNFS
(ii) ćFDBMMXJMMCFBOTXFSFECZBNBDIJOF
(iii) )FXJMMEJBMUIFXSPOHOVNCFS
8.2 Theoretical probability
t :PVDBODBMDVMBUFUIFUIFPSFUJDBMQSPCBCJMJUZPGBOFWFOUXJUIPVUEPJOHFYQFSJNFOUTJGUIFPVUDPNFTBSFFRVBMMZ
MJLFMZ6TFUIFGPSNVMB
number of favourable outcomes
number of possible outcomes
'PSFYBNQMF XIFOZPVUPTTBDPJOZPVDBOHFUIFBETPSUBJMT UXPQPTTJCMFPVUDPNFT ćFQSPCBCJMJUZPG
t
IFBETJTP(H) = 2
:PVOFFEUPXPSLPVUXIBUallUIFQPTTJCMFPVUDPNFTBSFCFGPSFZPVDBODBMDVMBUFUIFPSFUJDBMQSPCBCJMJUZ
P(outcome) =
1
Tip
Exercise 8.2
*UJTIFMQGVMUPMJTUUIF
QPTTJCMFPVUDPNFTTP
UIBUZPVLOPXXIBUUP
TVCTUJUVUFJOUIFGPSNVMB
1 4BMMZIBTUFOJEFOUJDBMDBSETOVNCFSFEPOFUPUFO4IFESBXTBDBSEBUSBOEPNBOESFDPSET
UIFOVNCFSPOJU
(a) 8IBUBSFUIFQPTTJCMFPVUDPNFTGPSUIJTFWFOU
(b) $BMDVMBUFUIFQSPCBCJMJUZUIBU4BMMZXJMMESBX
(i) UIFOVNCFSĕWF
(ii) BOZPOFPGUIFUFOOVNCFST
(iii) BNVMUJQMFPGUISFF
(iv) BOVNCFS<
(v) BOVNCFS<
(vi) BOVNCFS<
(vii) BTRVBSFOVNCFS
(viii) BOVNCFS<
(ix) BOVNCFS>
2 ćFSFBSFĕWFDVQTPGDPČFFPOBUSBZ5XPPGUIFNDPOUBJOTVHBS
(a) 8IBUBSFZPVSDIBODFTPGDIPPTJOHBDVQXJUITVHBSJOJU
(b) 8IJDIDIPJDFJTNPTUMJLFMZ 8IZ
3 .JLFIBTGPVSDBSETOVNCFSFEPOFUPGPVS)FESBXTPOFDBSEBOESFDPSETUIFOVNCFS
$BMDVMBUFUIFQSPCBCJMJUZUIBUUIFSFTVMUXJMMCF
(a) BNVMUJQMFPGUISFF
(b) BNVMUJQMFPGUXP
(c) BGBDUPSPGUISFF
4 'PSBĘZĕTIJOHDPNQFUJUJPO UIFPSHBOJTFSTQMBDFUSPVU TBMNPOBOEQJLFJOB
TNBMMEBN
(a) 8IBUJTBOBOHMFSTDIBODFPGDBUDIJOHBTBMNPOPOIFSĕSTUBUUFNQU
(b) WIBUJTUIFQSPCBCJMJUZUIFBOHMFSDBUDIFTBUSPVU
(c) 8IBUJTUIFQSPCBCJMJUZPGDBUDIJOHBQJLF
Unit 2: Data handling
49
8 Introduction to probability
5 "EBSUCPBSEJTEJWJEFEJOUPTFDUPSTOVNCFSFEGSPNPOFUP*GBEBSUJTFRVBMMZMJLFMZUP
MBOEJOBOZPGUIFTFTFDUPST DBMDVMBUF
(a) 1 < (d) 1 NVMUJQMFPG (b) 1 PEE (e) 1 NVMUJQMFPG (c) 1 QSJNF
6 "TDIPPMIBTGPSUZDMBTTSPPNTOVNCFSFEGSPNPOFUP8PSLPVUUIFQSPCBCJMJUZUIBUB
DMBTTSPPNOVNCFSIBTUIFOVNFSBMAJOJU
8.3 The probability that an event does not happen
t "OFWFOUNBZIBQQFOPSJUNBZOPUIBQQFO'PSFYBNQMF ZPVNBZUISPXBTJYXIFOZPVSPMMBEJF CVUZPVNBZOPU
t ćFQSPCBCJMJUZPGBOFWFOUIBQQFOJOHNBZCFEJČFSFOUGSPNUIFQSPCBCJMJUZPGUIFFWFOUOPUIBQQFOJOH CVUUIFUXP
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t *G"JTBOFWFOUIBQQFOJOH UIFO"′ PS A SFQSFTFOUTUIFFWFOU"OPUIBQQFOJOHBOE1 "′ = o1 " Exercise 8.3
1 ćFQSPCBCJMJUZUIBUBESJWFSJTTQFFEJOHPOBTUSFUDIPGSPBEJT8IBUJTUIFQSPCBCJMJUZ
UIBUBESJWFSJTOPUTQFFEJOH
2 ćFQSPCBCJMJUZPGESBXJOHBHSFFOCBMMJOBOFYQFSJNFOUJT 3 8IBUJTUIFQSPCBCJMJUZPGOPU
8
ESBXJOHBHSFFOCBMM
3 "DPOUBJOFSIPMETTXFFUTJOĕWFEJČFSFOUĘBWPVSTćFQSPCBCJMJUZPGDIPPTJOHB
QBSUJDVMBSĘBWPVSJTHJWFOJOUIFUBCMF
Flavour
P(flavour)
(a)
(b)
(c)
(d)
Strawberry
Lime
Lemon
Blackberry
Apple
$BMDVMBUF1 BQQMF 8IBUJT1 OPUBQQMF
$BMDVMBUFUIFQSPCBCJMJUZPGDIPPTJOH1 OFJUIFSMFNPOOPSMJNF
$BMDVMBUFUIFOVNCFSPGFBDIĘBWPVSFETXFFUJOUIFDPOUBJOFS
4 4UVEFOUTJOBTDIPPMIBWFĕWFBęFSTDIPPMDMVCTUPDIPPTFGSPNćFQSPCBCJMJUZUIBUB
TUVEFOUXJMMDIPPTFFBDIDMVCJTHJWFOJOUIFUBCMF
Club
P(Club)
(a)
(b)
(c)
(d)
50
Unit 2: Data handling
Computers
Sewing
Woodwork
Choir
Chess
$BMDVMBUF1 OPUTFXJOHOPSXPPEXPSL $BMDVMBUF1 OPUDIFTTOPSDIPJS * GTUVEFOUTIBWFUPDIPPTFBDMVC IPXNBOZXPVMEZPVFYQFDUUPDIPPTFTFXJOH
* GGPVSTUVEFOUTDIPTFDIPJS DBMDVMBUFIPXNBOZTUVEFOUTDIPTFDPNQVUFST
8 Introduction to probability
8.4 Possibility diagrams
t ćFTFUPGBMMQPTTJCMFPVUDPNFTJTDBMMFEUIFTBNQMFTQBDF PSQSPCBCJMJUZTQBDF PGBOFWFOU
t 1PTTJCJMJUZEJBHSBNTDBOCFVTFEUPTIPXBMMPVUDPNFTDMFBSMZ
t 8IFOZPVBSFEFBMJOHXJUIDPNCJOFEFWFOUT JUJTNVDIFBTJFSUPĕOEBQSPCBCJMJUZJGZPVSFQSFTFOUUIFTBNQMFTQBDF
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Tip
Exercise 8.4
ćJOLPGUIFQSPCBCJMJUZ
TQBDFEJBHSBNBTBNBQPG
BMMUIFQPTTJCMFPVUDPNFT
JOBOFYQFSJNFOU
1 %SBXBQPTTJCJMJUZEJBHSBNUPTIPXBMMQPTTJCMFPVUDPNFTXIFOZPVUPTTUXPDPJOTBUUIF
TBNFUJNF6TFZPVSEJBHSBNUPIFMQZPVBOTXFSUIFGPMMPXJOH
FAST FORWARD
Tree diagrams are also probability
space diagrams. These are dealt
with in detail in chapter 24. (a) 8IBUJT1 BUMFBTUPOFUBJM
(b) 8IBUJT1 OPUBJMT
2 +FTTIBTUISFFHSFFODBSETOVNCFSFEPOFUPUISFFBOEUISFFZFMMPXDBSETBMTPOVNCFSFEPOF
UPUISFF
(a) %
SBXBQPTTJCJMJUZEJBHSBNUPTIPXBMMQPTTJCMFPVUDPNFTXIFOPOFHSFFOBOEPOF
ZFMMPXDBSEJTDIPTFOBUSBOEPN
(b) )PXNBOZQPTTJCMFPVUDPNFTBSFUIFSF
(c) 8IBUJTUIFQSPCBCJMJUZUIBUUIFOVNCFSPOCPUIUIFDBSETXJMMCFUIFTBNF
(d) 8IBUJTUIFQSPCBCJMJUZPGHFUUJOHBUPUBMJGUIFTDPSFTPOUIFDBSETBSFBEEFE
3 0OBTDIPPMPVUJOH UIFTUVEFOUTBSFBMMPXFEUPDIPPTFPOFESJOLBOEPOFTOBDLGSPN
UIJTNFOV
Drinks: DPMB GSVJUKVJDF XBUFS
Snacks: CJTDVJU DBLF NVďO
(a) %SBXBQPTTJCJMJUZEJBHSBNUPTIPXUIFQPTTJCMFDIPJDFTUIBUBTUVEFOUDBONBLF
(b) 8
IBUJTUIFQSPCBCJMJUZBTUVEFOUXJMMDIPPTFDPMBBOEBCJTDVJU
(c) 8
IBUJTUIFQSPCBCJMJUZUIBUUIFESJOLDIPTFOJTOPUXBUFS
Unit 2: Data handling
51
8 Introduction to probability
8.5 Combining independent and mutually exclusive events
t 8IFOPOFPVUDPNFJOBUSJBMIBTOPFČFDUPOUIFOFYUPVUDPNFXFTBZUIFFWFOUTBSFJOEFQFOEFOU
t
t
t
t
– %SBXJOHBDPVOUFSBUSBOEPNGSPNBCBH SFQMBDJOHJUBOEUIFOESBXJOHBOPUIFSDPVOUFSJTBOFYBNQMFPG
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1 "BOE# 1 " ¨1 #
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– 'PSFYBNQMF ZPVDBOOPUUISPXBOPEEOVNCFSBOEBOFWFOOVNCFSBUUIFTBNFUJNFXIFOZPVSPMMBEJF
*G"BOE#areNVUVBMMZFYDMVTJWFFWFOUTUIFO1 "PS# 1 " 1 # 8IFOUIFPVUDPNFPGUIFĕSTUFWFOUBČFDUTUIFPVUDPNFPGUIFOFYUFWFOU UIFFWFOUTBSFTBJEUPCFEFQFOEFOU
– 'PSFYBNQMF JGZPVIBWFUXPSFEDPVOUFSTBOEUISFFXIJUFDPVOUFST ESBXBDPVOUFSXJUIPVUSFQMBDJOHJUBOEUIFO
ESBXBTFDPOEDPVOUFS UIFQSPCBCJMJUZPGESBXJOHBSFEPSBXIJUFPOUIFTFDPOEESBXEFQFOETPOXIBUZPVESFX
ĕSTUUJNFSPVOE:PVDBOĕOE1 "UIFO# CZDBMDVMBUJOH1 " ¨1 #HJWFOUIBU"IBTBMSFBEZIBQQFOFE Exercise 8.5
1 /JDPJTPOBCVTBOEIFJTCPSFE TPIFBNVTFTIJNTFMGCZDIPPTJOHBDPOTPOBOUBOEBWPXFM
BUSBOEPNGSPNUIFOBNFTPGUPXOTPOSPBETJHOTćFOFYUSPBETJHOJT$"-$655"
(a)
(b)
(c)
(d)
%
SBXVQBTBNQMFTQBDFEJBHSBNUPTIPXBMMUIFPQUJPOTUIBU/JDPIBT
$BMDVMBUF1 5BOE" $BMDVMBUF1 $PS-BOE6 $BMDVMBUF1 OPU-BOE6 2 "CBHDPOUBJOTUISFFSFEDPVOUFST GPVSHSFFODPVOUFST UXPZFMMPXDPVOUFSTBOEPOFXIJUF
DPVOUFS5XPDPVOUFSTBSFESBXOGSPNUIFCBHPOFBęFSUIFPUIFS XJUIPVUCFJOHSFQMBDFE
$BMDVMBUF
(a)
(b)
(c)
(d)
(e)
(f)
(g)
1 UXPSFEDPVOUFST
1 UXPHSFFODPVOUFST
1 UXPZFMMPXDPVOUFST
1 XIJUFBOEUIFOSFE
1 XIJUFPSZFMMPX JOFJUIFSPSEFSCVUOPUCPUI
1 XIJUFPSSFE JOFJUIFSPSEFSCVUOPUCPUI 8IBUJTUIFQSPCBCJMJUZPGESBXJOHBXIJUFPSZFMMPXDPVOUFSĕSTUBOEUIFOBOZ
DPMPVSTFDPOE
3 .BSJBIBTBCBHDPOUBJOJOHGSVJUESPQTXFFUTBSFBQQMFĘBWPVSFEBOEBSFCMBDLCFSSZ
ĘBWPVSFE4IFDIPPTFTBTXFFUBUSBOEPNBOEFBUTJUćFOTIFDIPPTFTBOPUIFSTXFFUBU
SBOEPN$BMDVMBUFUIFQSPCBCJMJUZUIBU
(a)
(b)
(c)
(d)
(e)
52
Unit 2: Data handling
CPUITXFFUTXFSFBQQMFĘBWPVSFE
CPUITXFFUTXFSFCMBDLCFSSZĘBWPVSFE
UIFĕSTUXBTBQQMFBOEUIFTFDPOEXBTCMBDLCFSSZ
U IFĕSTUXBTCMBDLCFSSZBOEUIFTFDPOEXBTBQQMF
:
PVSBOTXFSTUP B C D BOE E TIPVMEBEEVQUPPOF&YQMBJOXIZUIJTJTUIFDBTF
8 Introduction to probability
Mixed exercise
1 "DPJOJTUPTTFEBOVNCFSPGUJNFTHJWJOHUIFGPMMPXJOHSFTVMUT
)FBET
(a)
(b)
(c)
(d)
5BJMT
)PXNBOZUJNFTXBTUIFDPJOUPTTFE
$BMDVMBUFUIFSFMBUJWFGSFRVFODZPGFBDIPVUDPNF
8IBUJTUIFQSPCBCJMJUZUIBUUIFOFYUUPTTXJMMSFTVMUJOIFBET
+ FTTTBZTTIFUIJOLTUIFSFTVMUTTIPXUIBUUIFDPJOJTCJBTFE%PZPVBHSFF (JWFBSFBTPO
GPSZPVSBOTXFS
2 "CBHDPOUBJOTSFE FJHIUHSFFOBOEUXPXIJUFCBMMT&BDICBMMIBTBOFRVBMDIBODFPG
CFJOHDIPTFO$BMDVMBUFUIFQSPCBCJMJUZPG
(a)
(b)
(c)
(d)
(e)
(f)
(g)
DIPPTJOHBSFECBMM
DIPPTJOHBHSFFOCBMM
DIPPTJOHBXIJUFCBMM
DIPPTJOHBCMVFCBMM
DIPPTJOHBSFEPSBHSFFOCBMM
OPUDIPPTJOHBXIJUFCBMM
DIPPTJOHBCBMMUIBUJTOPUSFE
3 5XPOPSNBMVOCJBTFEEJDFBSFSPMMFEBOEUIFTVNPGUIFOVNCFSTPOUIFJSGBDFTJTSFDPSEFE
(a)
(b)
(c)
(d)
$BMDVMBUF1 8IJDITVNIBTUIFHSFBUFTUQSPCBCJMJUZ 8IBUJTUIFQSPCBCJMJUZPGSPMMJOHUIJTTVN
8IBUJT1 OPUFWFO
8IBUJT1 TVN
4 +PTIBOE$BSMPTFBDIUBLFBDPJOBUSBOEPNPVUPGUIFJSQPDLFUTBOEBEEUIFUPUBMTUPHFUIFS
UPHFUBOBNPVOU+PTIIBTUISFFDPJOT UXPDDPJOT BDPJOBOEUXPDDPJOTJOIJT
QPDLFU$BSMPTIBTGPVSDPJOT BDPJOBOEUXPDQJFDFT
(a) %SBXVQBQSPCBCJMJUZTQBDFEJBHSBNUPTIPXBMMUIFQPTTJCMFPVUDPNFTGPSUIFTVNPG
UIFUXPDPJOT
(b) 8IBUJTUIFQSPCBCJMJUZUIBUUIFDPJOTXJMMBEEVQUP
(c) 8IBUJTUIFQSPCBCJMJUZUIBUUIFDPJOTXJMMBEEVQUPMFTTUIBO
(d) 8IBUJTUIFQSPCBCJMJUZUIBUUIFDPJOTXJMMBEEVQUPPSNPSF
5 "GBJSEJFJTSPMMFEUISFFUJNFTBOEUIFOVNCFSSFWFBMFEXSJUUFOEPXO8IBUJTUIF
QSPCBCJMJUZUIBUBQSJNFOVNCFSXJMMCFXSJUUFOEPXOUISFFUJNFT
6 1BUSJDLLFFQTIJTTPDLTMPPTFJOBESBXFS)FIBTTJYEBSLPOFT GPVSXIJUFPOFTBOEUXP
TUSJQFEPOFT8IBUJTUIFQSPCBCJMJUZUIBUIFUBLFTPVUUXPTPDLTBOEUIFZBSFBQBJS
Unit 2: Data handling
53
9
Sequences and sets
9.1 Sequences
t "OVNCFSTFRVFODFJTBMJTUPGOVNCFSTUIBUGPMMPXTBTFUQBUUFSO&BDIOVNCFSJOUIFTFRVFODFJTDBMMFEB
UFSN5 JTUIFĕSTUUFSN 5 JTUIFUFOUIUFSNBOE5 JTUIFnUIUFSN PSHFOFSBMUFSN
t "MJOFBSTFRVFODFIBTBDPOTUBOUEJČFSFODF d CFUXFFOUIFUFSNTćFHFOFSBMSVMFGPSĕOEJOHUIFnUIUFSNPGBOZ
MJOFBSTFRVFODFJT5 =a+ no d XIFSFaJTUIFĕSTUWBMVFJOUIFTFRVFODF
t 8IFOZPVLOPXUIFSVMFGPSNBLJOHBTFRVFODF ZPVDBOĕOEUIFWBMVFPGBOZUFSN4VCTUJUVUFUIFUFSNOVNCFSJOUP
1
10
n
n
UIFSVMFBOETPMWFJU
You should recognise these
sequences of numbers:
square numbers: 1, 4, 9, 16 . . .
cube numbers: 1, 8, 27, 64 . . .
triangular numbers: 1, 3, 6, 10 . . .
Fibonacci numbers: 1, 1, 2, 3,
5, 8 . . .
Exercise 9.1
1 'JOEUIFOFYUUISFFUFSNTJOFBDITFRVFODFBOEEFTDSJCFUIFSVMFZPVVTFEUPĕOEUIFN
(a) (b) 1 1
(e) ¦ ¦ ¦ ¦(f)
4 2
(c) (g) (d) (h) 2 -JTUUIFĕSTUGPVSUFSNTPGUIFTFRVFODFTUIBUGPMMPXUIFTFSVMFT
(a)
(b)
(c)
(d)
(e)
4UBSUXJUITFWFOBOEBEEUXPFBDIUJNF
4UBSUXJUIBOETVCUSBDUĕWFFBDIUJNF
1
4UBSUXJUIPOFBOENVMUJQMZCZ 2 FBDIUJNF
4UBSUXJUIĕWFUIFONVMUJQMZCZUXPBOEBEEPOFFBDIUJNF
4UBSUXJUI EJWJEFCZUXPBOETVCUSBDUUISFFFBDIUJNF
3 8SJUFEPXOUIFĕSTUUISFFUFSNTPGFBDIPGUIFTFTFRVFODFTćFOĕOEUIFUIUFSN
(a)
(b)
(c)
(d)
(e)
(f)
5n=n+
5n=n
5n=n¦
5n=n¦
5n=n¦n
5n=¦n
4 $POTJEFSUIFTFRVFODF
(a)
(b)
(c)
(d)
54
Unit 3: Number
'JOEUIFnUIUFSNPGUIFTFRVFODF
'JOEUIFUIUFSN
8IJDIUFSNPGUIJTTFRVFODFIBTUIFWBMVF 4IPXGVMMXPSLJOH
4IPXUIBUJTOPUBUFSNJOUIFTFRVFODF
9 Sequences and sets
5 'PSFBDITFRVFODFCFMPXĕOEUIFHFOFSBMUFSNBOEUIFUIUFSN
(a)
(b)
(c)
(d)
(e)
− ¦ ¦ ¦
6 ćFEJBHSBNTIPXTBQBUUFSONBEFVTJOHNBUDITUJDLT
n=1
(a)
(b)
(c)
(d)
n=2
n=3
%SBXBTFRVFODFUBCMFGPSUIFOVNCFSPGNBUDITUJDLTJOUIFĕSTUTJYQBUUFSOT
'JOEBGPSNVMBGPSUIFnUIQBUUFSO
)PXNBOZNBUDIFTBSFOFFEFEGPSUIFUIQBUUFSO
8IJDIQBUUFSOXJMMOFFENBUDIFT
9.2 Rational and irrational numbers
t :PVDBOFYQSFTTBOZSBUJPOBMOVNCFSBTBGSBDUJPOJOUIFGPSNPGab XIFSFaBOEbBSFJOUFHFSTBOEbȶ
t 8IPMFOVNCFST JOUFHFST DPNNPOGSBDUJPOT NJYFEOVNCFST UFSNJOBUJOHEFDJNBMGSBDUJPOTBOESFDVSSJOHEFDJNBMT
BSFBMMSBUJPOBM
a
t :PVDBODPOWFSUSFDVSSJOHEFDJNBMGSBDUJPOTJOUPUIFGPSN
b
a
t *SSBUJPOBMOVNCFSTDBOOPUCFXSJUUFOJOUIFGPSNb *SSBUJPOBMOVNCFSTBSFBMMOPOSFDVSSJOH OPOUFSNJOBUJOH
EFDJNBMT
t ćFTFUPGSFBMOVNCFSTJTNBEFVQPGSBUJPOBMBOEJSSBUJPOBMOVNCFST
In 1.2 , the dot above the two
in the decimal part means it is
recurring (the ‘2’ repeats forever).
If a set of numbers recurs, e.g.
0.273273273..., there will be a dot
at the start and end of the recurring
.
set: 0.273
Exercise 9.2
1 8SJUFEPXOBMMUIFJSSBUJPOBMOVNCFSTJOFBDITFUPGSFBMOVNCFST
(a)
3
,
8
16 , 3 16 ,
(b) 23, 45 , 0.66 ,
22
,
7
3
4
12 , 0.090090009. . .,
31
,
3
0.020202. . .,
1
, 3 90 , π, 5 , 8 , 0.834
2
2 $POWFSUFBDIPGUIFGPMMPXJOHSFDVSSJOHEFDJNBMTUPBGSBDUJPOJOJUTTJNQMFTUGPSN
(a) 0 4 (d) 0.114 (b) 0 74
(e) 0.943
(c) 0 87
(f) 0.1857
Unit 3: Number
55
9 Sequences and sets
9.3 Sets
t "TFUJTBMJTUPSDPMMFDUJPOPGPCKFDUTUIBUTIBSFBDIBSBDUFSJTUJD
t "OFMFNFOU ∈ JTBNFNCFSPGBTFU
t "TFUUIBUDPOUBJOTOPFMFNFOUTJTDBMMFEUIFFNQUZTFU \^PS∅ t "VOJWFSTBMTFU ℰ DPOUBJOTBMMUIFQPTTJCMFFMFNFOUTBQQSPQSJBUFUPBQBSUJDVMBSQSPCMFN
t ćFFMFNFOUTPGBTVCTFU ⊂ BSFBMMDPOUBJOFEJOBMBSHFSTFU
t ćFFMFNFOUTPGUXPTFUTDBOCFDPNCJOFE XJUIPVUSFQFBUT UPGPSNUIFVOJPO ∪ PGUIFUXPTFUT
t ćFFMFNFOUTUIBUUXPTFUTIBWFJODPNNPOJTDBMMFEUIFJOUFSTFDUJPO PGUIFUXPTFUT
t ćFDPNQMFNFOUPGTFUA AҔ JTUIFFMFNFOUTUIBUBSFJOUIFVOJWFSTBMTFUGPSUIBUQSPCMFNCVUOPUJOTFUA
t "7FOOEJBHSBNJTBQJDUPSJBMNFUIPEPGTIPXJOHTFUT
t "TIPSUIBOEXBZPGEFTDSJCJOHUIFFMFNFOUTPGBTFUJTDBMMFETFUCVJMEFSOPUBUJPO'PSFYBNQMF\xxJTBOJOUFHFS x^
Exercise 9.3 A
Tip
.BLFTVSFZPVLOPXUIF
NFBOJOHPGUIFTZNCPMT
VTFEUPEFTDSJCFTFUTBOE
QBSUTPGTFUT
1 4BZXIFUIFSFBDIPGUIFGPMMPXJOHTUBUFNFOUTJTUSVFPSGBMTF
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
∈\PEEOVNCFST^
∈\DVCFEOVNCFST^
\ ^⊂\QSJNFOVNCFST^
\^ ⊂ \QSJNFOVNCFST^
\ ^∩\ ^\ ^
\ ^∪\ ^\ ^
A\ ^ B\ ^ TPAB
*Gℰ\MFUUFSTPGUIFBMQIBCFU^BOEA\DPOTPOBOUT^ UIFOAҔ\B F J P V^
2 AJTUIFTFU\ ^
(a)
(b)
(c)
(d)
(e)
(f)
Sometimes listing the elements
of each set will make it easier to
answer the questions.
3 ℰ\XIPMFOVNCFSTGSPNUP^ A\FWFOOVNCFSTGSPNUP^ B\PEEOVNCFST
GSPNUP^BOEC\NVMUJQMFTPGGSPNUP^
-JTUUIFFMFNFOUTPGUIFGPMMPXJOHTFUT
(a)
(b)
(c)
(d)
(e)
(f)
56
Unit 3: Number
%FTDSJCFTFUAJOXPSET
8IBUJTO A
-JTUTFUBXIJDIJTUIFQSJNFOVNCFSTJOA
-JTUTFUCXIJDIJTUIFTJOHMFEJHJUOVNCFSTJOA
-JTUB∩C
-JTUCҔ
A∩B
B∪C
AҔ∩B
B∩C)΄
A∩B΄
A∪B∪C
9 Sequences and sets
4 -JTUUIFFMFNFOUTPGUIFGPMMPXJOHTFUT
(a) \xx∈JOUFHFST oȳx^
(b) \xx∈OBUVSBMOVNCFST xȳ^
5 8SJUFJOTFUCVJMEFSOPUBUJPO
(a) \ ^
(b) \ ^
Exercise 9.3 B
Tip
:PVDBOVTFBOZTIBQFTUP
ESBXB7FOOEJBHSBNCVU
VTVBMMZUIFVOJWFSTBMTFUJT
ESBXOBTBSFDUBOHMFBOE
DJSDMFTXJUIJOJUTIPXUIF
TFUT
REWIND
Exam questions often combine
probability with Venn diagrams.
Revise chapter 8 if you’ve
forgotten how to work this out. 1 %SBXB7FOOEJBHSBNUPTIPXUIFGPMMPXJOHTFUTBOEXSJUFFBDIFMFNFOUJOJUTDPSSFDUTQBDF
ℰ\MFUUFSTJOUIFBMQIBCFU^
P\MFUUFSTJOUIFXPSEQIZTJDT^
C\MFUUFSTJOUIFXPSEDIFNJTUSZ^
2 6TFUIF7FOOEJBHSBNZPVESFXJORVFTUJPOUPĕOE
(a)
(b)
(c)
(d)
(e)
(f)
O C)
O P΄)
C∩P
P∪C
P∪C)΄
P⊂C
3 *OBTVSWFZPGTUVEFOUT TFWFOEJEOPUMJLFNBUITPSTDJFODF0GUIFSFTU TBJEUIFZMJLFE
NBUITBOETBJEUIFZMJLFETDJFODF
(a) %SBXB7FOOEJBHSBNUPTIPXUIJTJOGPSNBUJPO
(b) 'JOEUIFOVNCFSPGTUVEFOUTXIPMJLFECPUINBUITBOETDJFODF
(c) "
TUVEFOUJTDIPTFOBUSBOEPN'JOEUIFQSPCBCJMJUZUIBUUIFTUVEFOUMJLFTNBUITCVUOPU
TDJFODF
Mixed exercise
1 'PSFBDIPGUIFGPMMPXJOHTFRVFODFT ĕOEUIFnUIUFSNBOEUIFUIUFSN
(a) (b) (c) 2 5n no
(a) (JWFUIFĕSTUTJYUFSNTPGUIFTFRVFODF
(b) 8IBUJTUIFUIUFSN
(c) 8IJDIUFSNJTFRVBMUP
3 8IJDIPGUIFGPMMPXJOHOVNCFSTBSFJSSBUJPOBM
5
7
8
17
1 , 0.213231234. . ., 25 ,
, 0.1, − 0.654, 2 ,
22
5
, 4π
Unit 3: Number
57
9 Sequences and sets
4 8SJUFFBDISFDVSSJOHEFDJNBMBTBGSBDUJPOJOTJNQMFTUGPSN
(a)
3
(b) 0.286
5 *OBHSPVQPGTUVEFOUT TUVEZCJPMPHZBOETUVEZDIFNJTUSZTUVEFOUTTUVEZCPUI
TVCKFDUT
B
C
B = biology
C = chemistry
(a) .
BLFBDPQZPGUIF7FOOEJBHSBNBOEDPNQMFUFJUTIPXUIFJOGPSNBUJPOBCPVUUIF
TUVEFOUT
(b) )PXNBOZTUVEFOUTJOUIFHSPVQTUVEZOFJUIFSCJPMPHZOPSDIFNJTUSZ
(c) 8IBUJTn #∩$
(d) * GBTUVEFOUJTDIPTFOBUSBOEPNGSPNUIFHSPVQ XIBUJTUIFQSPCBCJMJUZUIBUIFPSTIF
TUVEJFT
(i) DIFNJTUSZ
(ii) CJPMPHZ
(iii) DIFNJTUSZBOECJPMPHZ
(iv) DIFNJTUSZPSCJPMPHZ PSCPUI
(v) OFJUIFSDIFNJTUSZOPSCJPMPHZ
58
Unit 3: Number
10
Straight lines and quadratic
equations
10.1 Straight lines
t ćFQPTJUJPOPGBQPJOUDBOCFVOJRVFMZEFTDSJCFEPOUIF$BSUFTJBOQMBOFVTJOHPSEFSFEQBJST x y PGDPPSEJOBUFT
t :PVDBOVTFFRVBUJPOTJOUFSNTPGxBOEyUPHFOFSBUFBUBCMFPGQBJSFEWBMVFTGPSxBOEy:PVDBOQMPUUIFTF
t
POUIF$BSUFTJBOQMBOFBOEKPJOUIFNUPESBXBHSBQI5PĕOEyWBMVFTJOBUBCMFPGWBMVFT TVCTUJUVUFUIFHJWFO
PSDIPTFO xWBMVFTJOUPUIFFRVBUJPOBOETPMWFGPSy
ćFHSBEJFOUPGBMJOFEFTDSJCFTJUTTMPQFPSTUFFQOFTT(SBEJFOUDBOCFEFĕOFEBT
t
t
t
m=
change in y
change in x
o MJOFTUIBUTMPQFVQUPUIFSJHIUIBWFBQPTJUJWFHSBEJFOU
o MJOFTUIBUTMPQFEPXOUPUIFSJHIUIBWFBOFHBUJWFHSBEJFOU
o MJOFTQBSBMMFMUPUIFxBYJT IPSJ[POUBMMJOFT IBWFBHSBEJFOUPG
o MJOFTQBSBMMFMUPUIFyBYJT WFSUJDBMMJOFT IBWFBOVOEFĕOFEHSBEJFOU
o MJOFTQBSBMMFMUPFBDIPUIFSIBWFUIFTBNFHSBEJFOUT
ćFFRVBUJPOPGBTUSBJHIUMJOFDBOCFXSJUUFOJOHFOFSBMUFSNTBT y
x + c XIFSFxBOEyBSFDPPSEJOBUFTPGQPJOUT
POUIFMJOF mJTUIFHSBEJFOUPGUIFMJOFBOEcJTUIFyJOUFSDFQU UIFQPJOUXIFSFUIFHSBQIDSPTTFTUIFyBYJT 5PĕOEUIFFRVBUJPOPGBHJWFOMJOFZPVOFFEUPĕOEUIFyJOUFSDFQUBOETVCTUJUVUFUIJTGPScćFOZPVOFFEUPĕOE
UIFHSBEJFOUPGUIFMJOFBOETVCTUJUVUFUIJTGPSm
:PVDBOĕOEUIFDPPSEJOBUFTPGUIFNJEQPJOUPGBMJOFTFHNFOUCZBEEJOHUIFxDPPSEJOBUFTPGJUTFOEQPJOUTBOE
EJWJEJOHCZUPHFUUIFxWBMVFPGUIFNJEQPJOUBOEUIFOEPJOHUIFTBNFXJUIUIFyDPPSEJOBUFTUPHFUUIFyWBMVFPG
UIFNJEQPJOU
Tip
Exercise 10.1
/PSNBMMZUIFxWBMVFTXJMM
CFHJWFO*GOPU DIPPTF
UISFFTNBMMWBMVFT GPS
FYBNQMF − BOE :PV
OFFEBNJOJNVNPGUISFF
QPJOUTUPESBXBHSBQI"MM
HSBQITTIPVMECFDMFBSMZ
MBCFMMFEXJUIUIFJSFRVBUJPO
1 'PSxWBMVFTPG− BOE ESBXBUBCMFPGWBMVFTGPSFBDIPGUIFGPMMPXJOHFRVBUJPOT
Remember, parallel lines have the
same gradient.
(a) y = x + 5
(b) y
(e) x = 4
(i) 0 x 2 y − 1
(f) y = −2
1
(j) x + y = − 2
2 x − 1
(c) y
(g) y
7 2x (d) y
1
2x − 2
x −2
(h) 4 2x
2 − 5y
2 %SBXBOEMBCFMHSBQIT a UP e JORVFTUJPOPOPOFTFUPGBYFTBOEHSBQIT f UP j POBOPUIFS
3 'JOEUIFFRVBUJPOPGBMJOFQBSBMMFMUPHSBQI a JORVFTUJPOBOEQBTTJOHUISPVHIQPJOU − 4 "SFUIFGPMMPXJOHQBJSTPGMJOFTQBSBMMFM
(a) y
(c) y
(e) 2 y
3x + 3 BOEy = x + 3 3x BOEy
3x + 7 3x + 2 BOEy =
(g) y = 8 BOEy = −9 3
x+2
2
(b) y
1
x − 4 BOEy
2
(d) y
0 8 x − 7 BOEy
(f) 2 y 3x = 2 BOEy
(h) x = −3 BOE x =
1
x −8
2
8x + 2
1 5x + 2
1
2
Unit 3: Algebra
59
10 Straight lines and quadratic equations
5 'JOEUIFHSBEJFOUPGUIFGPMMPXJOHMJOFT
(a)
(b)
y
6
(c)
y
(4, 6)
x
0
1
–1
4
2
4
(e)
0
–1
1
2
2
x
(1, –2)
0 1 2 3 4 5 6 7
4
(f)
x
4
–4
(h)
y
y
4
3
2
1
6
7
6
5
4
3
2
1
2
2
(g)
y
4
0
–2
x
–2
x
y
–2
–1
–2
(3, –3)
–3
0
–4
y
6
(–2, 1)
3
–2
2
(d)
y
3
–3
x
0
3
–8
–3
x
0
–4
–1
–2
–3
0
4
6 %FUFSNJOFUIFHSBEJFOU m BOEUIFyJOUFSDFQU c PGFBDIPGUIFGPMMPXJOHHSBQIT
Tip
:PVNBZOFFEUPSFXSJUF
UIFFRVBUJPOTJOUIFGPSN
y=mx+cCFGPSFZPVDBO
EPUIJT
(a)
3 x − 4
(b) y
(e) y =
x 1
+ 2 4
(f) y =
(i) x + 3 y = 14
y
(m) x + = −10
2
x −1
4x
− 2
5
(j) x + y + 4 = 0
(c) y
1
x + 5
2
(g) y = 7
(k) x 4
y
(d) y
x
(h) y
3x
(l) 2 x
5 y
7 %FUFSNJOFUIFFRVBUJPOPGFBDIPGUIFGPMMPXJOHHSBQIT
y
9
8
7
6
5
4
3
2
1
x
0
–9 –8 –7 –6–5 –4–3 –2–1 1 2 3 4 5 6
–2
–3
(j)
–4
–5
–6
–7
(l)
(k) –8
–9
(i)
y
(a)
(c)
(d)
(h)
3
2
1
0
–3 –2 –1
–2
(b)
–3
x
1 2 3
y
3
2
1
0
–3 –2 –1
–2
(e)
–3
(f)
x
1 2 3
(g)
8 'JOEUIFxBOEyJOUFSDFQUTPGUIFGPMMPXJOHMJOFT
(a) yxo
(b) yo 1 x (c) yox
2
(d) x + y (e) x y 2
60
Unit 3: Algebra
8
10 Straight lines and quadratic equations
9
8JUIPVUESBXJOHUIFHSBQIT DBMDVMBUFUIFHSBEJFOUPGUIFMJOFUIBUQBTTFTUISPVHIFBDIPG
UIFTFQBJSTPGQPJOUT
(a)
(b)
(c)
(d)
(e)
(f)
o o BOE o o BOE o BOE BOE o o BOE o
o BOE 10 6TJOHUIFTBNFTFUPGQPJOUTBTJORVFTUJPO ĕOEUIFDPPSEJOBUFTPGUIFNJEQPJOUPGFBDI
MJOFTFHNFOU
10.2 Quadratic expressions
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Exercise 10.2 A
Tip
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1 &YQBOEBOETJNQMJGZ
(e)
( x + 2))(( x + 3)
( x + 5))(( x + 7 )
( x 1))(( x − 3)
(g)
(y
(a)
(c)
(x
7 ))(( y − 2)
)(
(b) ( x + 2))(( x − 3)
(d)
(f)
)
)( x + 7)
( x 5)(
(2x 1)( x + 1)
(h) (2 x
y )(3x 2 y )
+ 1 2 x 2 − 3 (j)
(x
11) ( x + 12)
1
1 ⎞
(k) ⎛ x + 1⎞ ⎛ 1
x (l)
⎝2
⎠⎝ 2 ⎠
(x
3)(2 3x )
(i)
2
(m) (3x 2)(2 4 x )
Unit 3: Algebra
61
10 Straight lines and quadratic equations
Remember the expansions for the
square of a sum or a difference.
2 &YQBOEBOETJNQMJGZ
(a) ( x + 4 ) 2
(d) ( y − 2) 2
(g) (3 2) 2
(j) ( 4 6) 2
(m) (6 3 y )
2
(b)
(e)
(h)
(k)
( x − 3)2 (c) ( x + 5)
2
(f) (2 x y )
2
(i) (2 5)
2
(l) ( 4 2 x )
2
( x + y )2
(2x 3 y )2
(3 − x )2 3 8SJUFJOFYQBOEFEGPSN5SZUPEPUIJTCZJOTQFDUJPO
The difference of two squares
always gives you two brackets
that are identical except for the
signs, so you may be able to write
down the answer to an expansion
just by inspection.
(a)
(c)
(e)
5))(( x + 5)
(x
3x )(3x 4 ) (f)
(4
(j)
3
2z )( 4 x y 2z )
(
2
2y)
(2x 2 y )(2x
4 xy 5 y )
(5 y 4 xy )(4x
(8 x y 7 z )(8 x y 7 z )
(g) 4 x y
(i)
(
(x
7 y )(7 y 3) (d) x 2
(3
2
(h)
(b) (2 x 5)(2 x 5)
2
2
4
2
3
2
2
2
3
)
y2
)
4
2
3
2
)(
y )( x
y2 x2 + y2
2
2
2
2
Exercise 10.2 B
Tip
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QPXFS ĕSTU
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SFBMMZNBUUFS a b aob aob a b 62
Unit 3: Algebra
1 'BDUPSJTFGVMMZ
(a) x 2 + 4x
4 x 4
(c) x 2 + 6x
6x 9 (e) 15 + 8x
8 + x 2
2
(g) x 8x
8 x 15 (i) 26 − 27x
27 + x 2 2
(k) x + 3x
3 x 10 (m) 12 − 7x
7 + x2 (o) −54 + 3x + x 2
(b)
(d)
(f)
(h)
(j)
(l)
(n)
x 2 + 7x
7 x 12
4 5x
5 + x2
x 2 9x
9x 8
3 4x
4 + x2
x2 7x 8
x 2 4 x 32
−12 + x + x 2
(b)
(d)
(f)
(h)
(j)
3x 2 18 x 24
5x 2 15x 10
x 4 y + x 3 y − 2x 2 y
3x 2 15x 18
2 x 2 2 x − 112
(b)
(e)
(h)
(k)
(n)
(q)
16 − x 2 9x 2 4 y 2
121 y 2 144 x 2 200 − 2 x 2
x 2 y 2 − 100 1 81x 4 y 6
2 'BDUPSJTFGVMMZ
(a)
(c)
(e)
(g)
(i)
5x 2 15x 10
3x 3 122 2 9 x x3
x 2 + 20 x x 3 5x 2 14 x −2 x 2 + 4 x + 48 3 'BDUPSJTFGVMMZ
(a)
(d)
(g)
(j)
(m)
(p)
x 2 − 9
49 − x 2 x2 9 y2
2 x 2 18 25 − x16 25x10 − 1
(c)
(f)
(i)
(l)
(o)
x 2 − 25
81 − 4 x 2
16 x 2 49 y 2
x 4 y2
25x 2 64w 2
− 2
y4
z
10 Straight lines and quadratic equations
Exercise 10.2 C
Remember to make the righthand
side equal to zero and to factorise
before you try to solve the
equation.
Quadratic equations always have
two roots (although sometimes
the two roots are the same).
Mixed exercise
1 4PMWFUIFGPMMPXJOHRVBESBUJDFRVBUJPOT
(a)
(c)
(e)
(g)
(i)
(k)
(m)
(o)
(q)
x 2 3x 0 (b)
6 x 2 12 x (d)
x 2 −11 0
(f)
8 2 2
(h)
x 2 + 5x
5x 4 = 0 (j)
x 2 x − 20 = 0 (l)
x 2 + 15 8 x (n)
x 2 + 56 15x (p)
5x 2 20 x 20 0
8 2 32 0
−16 x − 24 x 2 = 0
49 − 4 2 0
x 2 + 6x
6x 8 = 0
2
x 4x 5 = 0
x 2 + 8x
8 x 20
−60 − 17 x = − x 2
x 2 − 20 x + 100 = 0
1 'PSFBDIFRVBUJPO DPQZBOEDPNQMFUFUIFUBCMFPGWBMVFTDSBXUIFHSBQITGPSBMMGPVS
FRVBUJPOTPOUIFTBNFTFUPGBYFT
(a) y
1
x
2
x
−1
−1
−1
y
(b) y
1
x+3
2
x
y
(c) y = 2
x
y
(d) y 2 x − 4 = 0
x
−1
y
2 %FUFSNJOFUIFHSBEJFOUBOEUIFyJOUFSDFQUPGFBDIHSBQI
(a) y
2 x − 1
(d) y = − 1 2
(b) y + 6 x (c) x
(e) 2 x 3y
3y = 6 (f) y
y = −8
x
3 8IBUFRVBUJPOEFĕOFTFBDIPGUIFTFMJOFT
(a)
(b)
(c)
(d)
BMJOFXJUIBHSBEJFOUPGBOEByJOUFSDFQUPG−
2
1
BMJOFXJUIByJOUFSDFQUPG 2 BOEBHSBEJFOUPG− 3
BMJOFQBSBMMFMUPy
x + 8 XJUIBy-JOUFSDFQUPG−
4
x XIJDIQBTTFTUISPVHIUIFQPJOU −
BMJOFQBSBMMFMUPy
5
Unit 3: Algebra
63
10 Straight lines and quadratic equations
(e)
(f)
(g)
(h)
BMJOFQBSBMMFMUP2 y 4x
4 x + 2 0 XJUIByJOUFSDFQUPG−
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4 'JOEUIFHSBEJFOUPGUIFGPMMPXJOHMJOFT
9
8
7
6
5
4
3
2
1
y
–9 –8 –7 –6–5 –4–3 –2–10 1 2 3 4 5 6 7 8 9
–2
–3
–4
–5
–6
–7
–8
–9
x
5 8IBUJTUIFFRVBUJPOPGFBDIPGUIFTFMJOFT
a
b
c
y
y
d)
y
4
2
–6 –3–3
–6
e
3 6
7
x
–6 –3 –2
–4 –2
2 4
f)
y
Tip
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64
Unit 3: Algebra
–5
6 x
–10 –2
10 20 x
–4
–6
y
3
5
–5 –3
–4
x
3
y
6
4
2
3 5 x
(5,–5)
–4 –2
2 4
x
–3
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(d) 8IBUJTUIFHSBEJFOUPGUIFMJOF
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(i) LN
(ii) LN
(iii) LN
10 Straight lines and quadratic equations
(f) 6TFZPVSHSBQIUPĕOEPVUIPXGBSTIFSVOTJO
1
(i) IPVST
(ii) 2 IPVST
(iii) 3 PGBOIPVS
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4
(g) $
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(i)
o BOE (ii) o BOE (iii) BOE (iv) o BOE (v)
o o BOE o 8
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(b) (2 x 2) ( x − 1)
(a)
( x − 8)2 (c)
(3x
(e)
(2
(g)
(3x
(i)
⎛ x + 1⎞ ⎛ x − 1⎞ ⎝
2⎠ ⎝
2⎠
(k)
(x
2 y ) 2
3x )(3x 2) 2
)
2
y 1 2)
( x − 7 )2 2
(d) (1 6 y )
2
(f)
5)
(2
2
1
(h) ⎛ x + y ⎞
⎝
2 ⎠
2
1
1
(j) ⎛ − 2⎞ ⎛ + 2⎞
⎝x ⎠⎝x
⎠
(l) −2 x ( x − 2) + 8 x 2
2
(m) 2 x ( x 1) − (7 2 x )( 2 x )
2
9
'BDUPSJTFGVMMZ
(a) a 3 4a (c) x 2 x − 2 (b) x 4 − 1
(d) x 2 22xx 1
(e) (2 x 3 y ) − 4 z 2 x2
(g) x 4 − 4
(i) 4 x 2 4 x − 48
(k) 5 20 x16 (f) x 2 + 16 x + 48
2
(h) x 2
5x 6
2
14xx 24
(j) 2 x 14
2
(l) 3x 15x 18
10 4PMWFGPSx
(a)
(b)
(c)
(d)
(e)
(f)
7x2
2 2
6x 2
3x 2
4x2
5x 2
42 x 35 0 8=0
18 x
12
6 x = −3
6x
16 x 20 0
20 x 20 0
Unit 3: Algebra
65
11
Pythagoras’ theorem and similar
shapes
11.1 Pythagoras’ theorem
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Exercise 11.1 A
Tip
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(a)
(b)
(c)
15 cm
x
3 cm
13 mm
y
8 cm
x
5 mm
4 cm
(d)
(e)
(f)
y
x
0.4 cm
24 cm
0.5 cm
1.2 cm
26 cm
(g)
(h)
x
11 cm
4 cm
y
7.3 cm
7 cm
66
Unit 3: Shape, space and measures
x
0.6 cm
11 Pythagoras’ theorem and similar shapes
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Tip
(a)
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(b)
70 mm
(c)
x
80 mm
6 cm
20.2 cm
z
y
4 cm
120 mm
8 cm
8.2 cm
(d)
(e)
5 mm
5
m
m
(f)
y
x
16 cm
13 mm
15 cm
9 cm
9 cm
z
5 cm
12 cm
16 cm
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DFOUJNFUSFT
(a)
(b)
A
8
7
B
D
10
15
(c)
(d) J
H
13
8
9
9
12
L
C
F
12
E
G
I
11
K
5
4 $BMDVMBUFUIFMFOHUIPGUIFMJOFTFHNFOUKPJOJOHFBDIPGUIFGPMMPXJOHQBJSTPGQPJOUT
(a) (−2, −2) BOE (2, 2)
(c) (−4, 5) BOE (0, 1)
(e) (−1, −3) BOE (2, −3)
Tip
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TJUVBUJPO ĕMMJOUIFLOPXO
MFOHUITBOENBSLUIF
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MFUUFST
(b) (−3, −1) BOE (0, 2)
(d) (2, 4) BOE (8, 16)
(f) (−2, 0) BOE (4, 3)
Exercise 11.1 B
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Unit 3: Shape, space and measures
67
11 Pythagoras’ theorem and similar shapes
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QMBDFUIFGPPUPGIJTMBEEFSGSPNUIFXBMM 11.2 Understanding similar triangles
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Tip
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Exercise 11.2
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FBDIDBTF
(a)
(b)
4.47 cm
x
2 cm
4 cm
2 mm
y
6 mm
β
(d)
5 mm
β
8 cm
β
y
y
x
y
z
4 mm
α
8 mm
α
4 mm
10 cm
x
8 cm
(f)
8 cm
y
β
α
10 cm
(g)
z
4 cm
β
α
43°
x
8.5 cm
α
α
β
x
27 cm
y
α
α
21 cm
40 cm
43°
y
α
5 cm
9.5 cm
(h)
β
7 cm
8 cm
6 cm
8 cm
Unit 3: Shape, space and measures
9 mm
6 mm
3 mm
α
68
β
1 cm
(c)
(e)
2 cm
12 mm
α
α
x
β
30 cm
y
25 cm
β
15 cm
11 Pythagoras’ theorem and similar shapes
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A
B
C
D
E
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building
3m
30 m
Nancy
4m
11.3 Understanding similar shapes
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2
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2
2
volume A ⎛ a ⎞ 3
=
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volume B ⎝ b ⎠
PGUIFSBUJPPGDPSSFTQPOEJOHMJOFBSNFBTVSFT*OPUIFSXPSET
surface area A ⎛ a ⎞ 2
=
surface area B ⎝ b ⎠
Exercise 11.3
1 'JOEUIFMFOHUIPGFBDITJEFNBSLFEXJUIBMFUUFSJOUIFTFQBJSTPGTJNJMBSTIBQFT"MM
EJNFOTJPOTBSFJODFOUJNFUSFT
(a)
15
(b)
25
45
x
24
30
30
y
18
x
Unit 3: Shape, space and measures
69
11 Pythagoras’ theorem and similar shapes
2 *GUIFBSFBTPGUXPTJNJMBSRVBESJMBUFSBMTBSFJOUIFSBUJP XIBUJTUIFSBUJPPGNBUDIJOH
TJEFT
3 ćFUXPTIBQFTJOFBDIQBJSCFMPXBSFTJNJMBSćFBSFBPGUIFTIBQFPOUIFMFęPGFBDIQBJSJT
HJWFO'JOEUIFBSFBPGUIFPUIFSTIBQF
(a)
(b)
18 cm
12 cm
69 mm
area = 113.10 cm2
23 mm
area = 4761 mm2
4 ćFUXPTIBQFTJOFBDIQBJSCFMPXBSFTJNJMBSćFBSFBPGCPUITIBQFTJTHJWFO6TFUIJTUP
DBMDVMBUFUIFMFOHUIPGUIFNJTTJOHTJEFJOFBDIĕHVSF
(a)
(b)
1 cm
area = 10 cm3
x
area = 40 cm3
x
3m
area = 16.2 m3
area = 405 m3
5 "TIJQQJOHDSBUFIBTBWPMVNFPGDN*GUIFEJNFOTJPOTPGUIFDSBUFBSFEPVCMFE XIBU
XJMMJUTOFXWPMVNFCF
6 5XPTJNJMBSDVCFT ABOEB IBWFTJEFTPGDNBOEDNSFTQFDUJWFMZ
(a) 8IBUJTUIFMJOFBSTDBMFGBDUPSPGAUPB (b) 8IBUJTUIFSBUJPPGUIFJSTVSGBDFBSFBT (c) 8IBUJTUIFSBUJPPGUIFJSWPMVNFT
11.4 Understanding congruence
t $POHSVFOUTIBQFTBSFJEFOUJDBMJOTIBQFBOETJ[F5XPTIBQFTBSFDPOHSVFOUJGUIFDPSSFTQPOEJOHTJEFTBSFFRVBMJO
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t 5SJBOHMFTBSFDPOHSVFOUJGBOZPGUIFGPMMPXJOHGPVSDPOEJUJPOTBSFNFU
o UISFFTJEFTPGPOFUSJBOHMFBSFFRVBMUPUISFFDPSSFTQPOEJOHTJEFTPGUIFPUIFS 444
o UXPTJEFTBOEUIFJODMVEFEBOHMFPGPOFUSJBOHMFBSFFRVBMUPUIFTBNFUXPTJEFTBOEJODMVEFEBOHMFPGUIFPUIFS 4"4
o UXPBOHMFTBOEUIFJODMVEFETJEF TJEFCFUXFFOUIFBOHMFT PGPOFUSJBOHMFBSFFRVBMUPUIFDPSSFTQPOEJOHBOHMFT
BOEJODMVEFETJEFPGUIFPUIFS "4"
o UIFIZQPUFOVTFBOEPOFPUIFSTJEFPGPOFSJHIUBOHMFEUSJBOHMFBSFFRVBMUPUIFIZQPUFOVTFBOEDPSSFTQPOEJOH
TJEFPGUIFPUIFSSJHIUBOHMFEUSJBOHMF 3)4 70
Unit 3: Shape, space and measures
11 Pythagoras’ theorem and similar shapes
Remember, the order in which you
name a triangle is important if you
are stating congruency. If Δ ABC is
congruent to Δ XYZ then:
AB = XY
BC = YZ
Exercise 11.4
1 State whether each pair of triangles is congruent or not. If they are congruent, state the
condition of congruency.
(a)
AC = XZ
(b)
X
B
x°
x°
C Y
A
Z
78°
(c) X
35°
35°
N
M
(d)
78°
67°
67°
P
D
x
Q
y
y
O
Z
Y
x
E
(e)
(f)
A
B
(g)
Q
N
O
(h)
Y
87°
P
M
P
D
C
R
F
D
A
C
87°
R
Z
E
X
B
Tip
2 MNOP is a trapezium. MP = NO and PN = MO. Show that Δ MPO is congruent to Δ NOP
giving reasons.
M
When you are asked
to show triangles are
congruent you must
always state the condition
that you are using.
N
Q
P
O
PN = MO
Unit 3: Shape, space and measures
71
11 Pythagoras’ theorem and similar shapes
3 *OUIFGPMMPXJOHĕHVSF TIPXUIBUABJTIBMGUIFMFOHUIPGACHJWJOHSFBTPOTGPSFBDI
TUBUFNFOUZPVNBLF
C
B
A
D
E
Mixed exercise
1 "TDIPPMDBSFUBLFSXBOUTUPNBSLPVUBTQPSUTĕFMENXJEFBOENMPOH5PNBLFTVSF
UIBUUIFĕFMEJTSFDUBOHVMBS IFOFFETUPLOPXIPXMPOHFBDIEJBHPOBMTIPVMECF
(a) %SBXBSPVHITLFUDIPGUIFĕFME
(b) $BMDVMBUFUIFSFRVJSFEMFOHUITPGUIFEJBHPOBMT
2 *OΔABC AB=DN BC=DNBOEAC=DN%FUFSNJOFXIFUIFSUIFUSJBOHMFJT
SJHIUBOHMFEPSOPUBOEHJWFSFBTPOTGPSZPVSBOTXFS
3 'JOEUIFMFOHUIPGUIFMJOFTFHNFOUKPJOJOHFBDIPGUIFGPMMPXJOHQBJSTPGQPJOUT
(a)
(c)
(e)
o BOE (b)
BOE (d)
o o BOE o o BOE o BOE 4 "USJBOHMFXJUITJEFTPGNN NNBOENNJTTJNJMBSUPBOPUIFSUSJBOHMFXJUIJUT
MPOHFTUTJEFNN$BMDVMBUFUIFQFSJNFUFSPGUIFMBSHFSUSJBOHMF
5 $BMDVMBUFUIFNJTTJOHEJNFOTJPOTPSBOHMFTJOFBDIPGUIFTFQBJSTPGTJNJMBSUSJBOHMFT
(a)
4.5 cm
α
1.5 cm
(b)
β
63°
x
13.5 cm
α
5 cm
Unit 3: Shape, space and measures
36 cm
15 cm
x
72
30°
10.5 cm
α
α
y
β
4.5 cm
(c)
30°
87°
x
11 Pythagoras’ theorem and similar shapes
6
$BMDVMBUFUIFSBUJPPGUIFBSFBTPGFBDIQBJSPGTJNJMBSTIBQFT
(a)
(b)
1.5 m
0.75 m
3.0 m
7
150 m
50 m
1.5 m
ćFUXPQBSBMMFMPHSBNTCFMPXBSFTJNJMBSćFBSFBPGUIFMBSHFSJTDN2'JOEUIFBSFBPG
UIFTNBMMFSQBSBMMFMPHSBN
18 cm
4.5 cm
area = 288 cm2
8 ćFUXPDVCPJET ABOEB BSFTJNJMBSćFMBSHFSIBTBTVSGBDFBSFBPGNN28IBUJT
UIFTVSGBDFBSFBPGUIFTNBMMFS
B
A
50 mm
9
80 mm
area = 60 800 mm2
"TRVBSFCBTFEQZSBNJEIBTBCBTFPGBSFBDN2BOEBWPMVNFPGDN$BMDVMBUF
(a) ćFQFSQFOEJDVMBSIFJHIUPGUIFQZSBNJE
(b) ćFIFJHIUBOEBSFBPGUIFCBTFPGBTJNJMBSQZSBNJEXJUIBWPMVNFPGDN
10 5XPPGUIFUSJBOHMFTJOFBDITFUPGUISFFBSFDPOHSVFOU4UBUFXIJDIUXPBSFDPOHSVFOUBOE
HJWFUIFDPOEJUJPOTUIBUZPVVTFEUPQSPWFUIFNDPOHSVFOU
(a)
H
D
A
(b)
D
A
G
y C
F
C
B
E
B
x
y
F
G
x E
y
x
I
H
I
(c)
(d)
A
6
D
6
7
C
5
G
x
F
y
B
y
x E
I
5
x
Unit 3: Shape, space and measures
H
7
73
11 Pythagoras’ theorem and similar shapes
11 "ONMPOHXJSFDBCMFJTVTFEUPTFDVSFBNBTUPGIFJHIUxNćFDBCMFJTBUUBDIFEUPUIF
UPQPGUIFNBTUBOETFDVSFEPOUIFHSPVOENBXBZGSPNUIFCBTFPGUIFNBTU)PXUBMM
JTUIFNBTU (JWFZPVSBOTXFSDPSSFDUUPUXPEFDJNBMQMBDFT
8.6 m
xm
6.5 m
12 /BEJBXBOUTUPIBWFBNFUBMOVNCFSNBEFGPSIFSHBUF4IFIBTGPVOEBTBNQMFCSBTT
OVNFSBMBOEOPUFEJUTEJNFOTJPOT4IFEFDJEFTUIBUIFSOVNFSBMTIPVMECFTJNJMBSUPUIJT
POF CVUUIBUJUTIPVMECFGPVSUJNFTMBSHFS
35 mm
17 mm
140 mm
105 mm
35 mm
(a) %
SBXBSPVHITLFUDIPGUIFOVNFSBMUIBU/BEJBXBOUTUPNBLFXJUIUIFDPSSFDU
EJNFOTJPOTXSJUUFOPOJUJONJMMJNFUSFT
(b) $
BMDVMBUFUIFMFOHUIPGUIFTMPQJOHFEHFBUUIFUPQPGUIFGVMMTJ[FOVNFSBMUPUIFOFBSFTU
XIPMFNJMMJNFUSF
74
Unit 3: Shape, space and measures
12
Averages and measures
of spread
12.1 Different types of average
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t "NFBTVSFPGTQSFBEJTUIFSBOHF MBSHFTUWBMVFNJOVTTNBMMFTUWBMVF t ćFNFBOJTUIFTVNPGUIFEBUBJUFNTEJWJEFECZUIFOVNCFSPGJUFNTJOUIFEBUBTFUćFNFBOEPFTOPUIBWF
t
t
UPCFPOFPGUIFOVNCFSTJOUIFEBUBTFU
o ćFNFBODBOCFBČFDUFECZFYUSFNFWBMVFTJOUIFEBUBTFU8IFOPOFWBMVFJTNVDIMPXFSPSIJHIFSUIBOUIF
SFTUPGUIFEBUBJUJTDBMMFEBOPVUMJFS0VUMJFSTTLFXUIFNFBOBOENBLFJUMFTTSFQSFTFOUBUJWFPGUIFEBUBTFU
ćFNFEJBOJTUIFNJEEMFWBMVFJOBTFUPGEBUBXIFOUIFEBUBJTBSSBOHFEJOJODSFBTJOHPSEFS
o 8IFOUIFSFJTBOFWFOOVNCFSPGEBUBJUFNT UIFNFEJBOJTUIFNFBOPGUIFUXPNJEEMFWBMVFT
ćFNPEFJTUIFOVNCFS PSJUFN UIBUBQQFBSTNPTUPęFOJOBEBUBTFU
o 8IFOUXPOVNCFSTBQQFBSNPTUPęFOUIFEBUBIBTUXPNPEFTBOEJTTBJEUPCFCJNPEBM8IFONPSFUIBO
UXPOVNCFSTBQQFBSFRVBMMZPęFOUIFNPEFIBTOPSFBMWBMVFBTBTUBUJTUJD
Exercise 12.1
1 %FUFSNJOFUIFNFBO NFEJBOBOENPEFPGUIFGPMMPXJOHTFUTPGEBUB
(a)
(b)
(c)
(d)
(e)
(f)
Tip
*GZPVNVMUJQMZUIFNFBO
CZUIFOVNCFSPGJUFNT
JOUIFEBUBTFU ZPVHFU
UIFUPUBMPGUIFTDPSFT
ćJTXJMMIFMQZPVTPMWF
QSPCMFNTMJLFRVFTUJPO
2 'JWFTUVEFOUTTDPSFEBNFBONBSLPGPVUPGGPSBNBUITUFTU
(a) 8IJDIPGUIFTFTFUTPGNBSLTĕUUIJTBWFSBHF
(i) (ii) (iii) (iv) (v) (vi) (b) $PNQBSFUIFTFUTPGOVNCFSTJOZPVSBOTXFSBCPWF&YQMBJOXIZZPVDBOHFUUIFTBNF
NFBOGSPNEJČFSFOUTFUTPGOVNCFST
3 ćFNFBOPGOVNCFSTJT8IBUJTUIFTVNPGUIFOVNCFST
4 ćFTVNPGOVNCFSTJT8IJDIPGUIFGPMMPXJOHOVNCFSTJTDMPTFTUUPUIFNFBOPG
UIFOVNCFST PS
Unit 3: Data handling
75
12 Averages and measures of spread
5 "OBHSJDVMUVSBMXPSLFSXBOUTUPLOPXXIJDIPGUXPEBJSZGBSNFSTIBWFUIFCFTUNJML
QSPEVDJOHDPXT'BSNFS4JOHITBZTIJTDPXTQSPEVDFMJUSFTPGNJMLQFSEBZ'BSNFS
/BJEPPTBZTIFSDPXTQSPEVDFMJUSFTPGNJMLQFSEBZ
ćFSFJTOPUFOPVHIJOGPSNBUJPOUPEFDJEFXIJDIDPXTBSFUIFCFUUFSQSPEVDFSTPGNJML
8IBUPUIFSJOGPSNBUJPOXPVMEZPVOFFEUPBOTXFSUIFRVFTUJPO
Tip
*UNBZIFMQUPESBXB
SPVHIGSFRVFODZUBCMFUP
TPMWFQSPCMFNTMJLFUIJT
POF
6 *OBHSPVQPGTUVEFOUT TJYIBEGPVSTJCMJOHT TFWFOIBEĕWFTJCMJOHT FJHIUIBEUISFFTJCMJOHT OJOFIBEUXPTJCMJOHTBOEUFOIBEPOFTJCMJOH 4JCMJOHTBSFCSPUIFSTBOETJTUFST
(a) 8IBUJTUIFNFBOOVNCFSPGTJCMJOHT
(b) 8IBUJTUIFNPEBMOVNCFSPGTJCMJOHT
7 ćFNBOBHFNFOUPGBGBDUPSZBOOPVODFETBMBSZJODSFBTFTBOETBJEUIBUXPSLFSTXPVME
SFDFJWFBOBWFSBHFJODSFBTFPGUP
ćFUBCMFTIPXTUIFPMEBOEOFXTBMBSJFTPGUIFXPSLFSTJOUIFGBDUPSZ
Previous salary
Salary with increase
'PVSXPSLFSTJO$BUFHPSZ"
5XPXPSLFSTJO$BUFHPSZ#
4JYXPSLFSTJO$BUFHPSZ$
&JHIUXPSLFSTJO$BUFHPSZ%
(a)
(b)
(c)
(d)
(e)
$BMDVMBUFUIFNFBOJODSFBTFGPSBMMXPSLFST
$BMDVMBUFUIFNPEBMJODSFBTF
8IBUJTUIFNFEJBOJODSFBTF
)PXNBOZXPSLFSTSFDFJWFEBOJODSFBTFPGCFUXFFOBOE
8BTUIFNBOBHFNFOUBOOPVODFNFOUUSVF 4BZXIZPSXIZOPU
12.2 Making comparisons using averages and ranges
t :PVDBOVTFBWFSBHFTUPDPNQBSFUXPPSNPSFTFUTPGEBUB)PXFWFS BWFSBHFTPOUIFJSPXONBZCFNJTMFBEJOH TPJUJT
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t ćFSBOHFJTBNFBTVSFPGIPXTQSFBEPVU EJTQFSTFE UIFEBUBJT3BOHFMBSHFTUWBMVFoTNBMMFTUWBMVF
t "MBSHFSBOHFNFBOTUIBUUIFEBUBJTTQSFBEPVU TPUIFNFBTVSFTPGDFOUSBMUFOEFODZ BWFSBHFT NBZOPUCF
SFQSFTFOUBUJWFPGUIFXIPMFEBUBTFU
Tip
8IFOUIFNFBOJT
BČFDUFECZFYUSFNF
WBMVFTUIFNFEJBOJTNPSF
SFQSFTFOUBUJWFPGUIFEBUB
76
Unit 3: Data handling
Exercise 12.2
1 'PSUIFGPMMPXJOHTFUTPGEBUB POFPGUIFUISFFBWFSBHFTJTOPUSFQSFTFOUBUJWF
4UBUFXIJDIPOFJTOPUSFQSFTFOUBUJWFJOFBDIDBTF
(a) (b) (c) 12 Averages and measures of spread
2 5XFOUZTUVEFOUTTDPSFEUIFGPMMPXJOHSFTVMUTJOBUFTU PVUPG Tip
ćFNPEFPOMZUFMMTZPV
UIFNPTUQPQVMBSWBMVF
BOEUIJTJTOPUOFDFTTBSJMZ
SFQSFTFOUBUJWFPGUIFXIPMF
EBUBTFU
(a) $BMDVMBUFUIFNFBO NFEJBO NPEFBOESBOHFPGUIFNBSLT
(b) 8IZJTUIFNFEJBOUIFCFTUTVNNBSZTUBUJTUJDGPSUIJTQBSUJDVMBSTFUPGEBUB
3 ćFUBCMFTIPXTUIFUJNFT JONJOVUFTBOETFDPOET UIBUUXPSVOOFSTBDIJFWFEPWFSN
EVSJOHPOFTFBTPO
Runner A
NT
NT
NT
NT
NT
NT
NT
Runner B
NT
NT
NT
NT
NT
NT
NT
(a) 8IJDISVOOFSJTUIFCFUUFSPGUIFUXP 8IZ
(b) 8IJDISVOOFSJTNPTUDPOTJTUFOU 8IZ
12.3 Calculating averages and ranges for frequency data
t ćFNFBODBOCFDBMDVMBUFEGSPNBGSFRVFODZUBCMF5PDBMDVMBUFUIFNFBOZPVBEEBDPMVNOUPUIFUBCMFBOE
total of (score frequency
f
) column
total of frequency
f
column
'JOEUIFNPEFJOBUBCMFCZMPPLJOHBUUIFGSFRVFODZDPMVNOćFEBUBJUFNXJUIUIFIJHIFTUGSFRVFODZJTUIFNPEF
*OBGSFRVFODZUBCMF UIFEBUBJTBMSFBEZPSEFSFECZTJ[F5PĕOEUIFNFEJBO XPSLPVUJUTQPTJUJPOJOUIFEBUBBOE
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DBMDVMBUFUIFTDPSF¨GSFRVFODZ fx Mean =
t
t
Exercise 12.3
1 $POTUSVDUBGSFRVFODZUBCMFGPSUIFEBUBCFMPXBOEUIFODBMDVMBUF
(a) UIFNFBO
(b) UIFNPEF
3
3
(c) UIFNFEJBO
3
2
(d) UIFSBOHF
2
2
2
3
3
3
2
2
2
5
3
2
3
3
3
2
3
2
2 'PSFBDIPGUIFGPMMPXJOHGSFRVFODZEJTUSJCVUJPOTDBMDVMBUF
(a) UIFNFBOTDPSF
%BUBTFU"
(b) UIFNFEJBOTDPSF
Score
Frequency
(c) UIFNPEBMTDPSF
2
3
5
%BUBTFU#
Score
Frequency
25
22
23
Score
Frequency
%BUBTFU$
Unit 3: Data handling
77
12 Averages and measures of spread
12.4 Calculating averages and ranges for grouped continuous data
t $POUJOVPVTEBUBDBOUBLFBOZWBMVFCFUXFFOUXPHJWFOWBMVFT
t 8IFOHJWFOBGSFRVFODZUBCMFDPOUBJOJOHHSPVQFEEBUB JODMBTTJOUFSWBMT ZPVEPOULOPXUIFFYBDUWBMVFTPGUIFEBUB
t
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– ćFNJEQPJOUJTUIFNFBOPGUIFTNBMMFTUBOEMBSHFTUTDPSFTJOUIFJOUFSWBM
– ćFMPXFTUBOEIJHIFTUTDPSFTJOBDMBTTJOUFSWBMBSFDBMMFEUIFMPXFSBOEVQQFSDMBTTMJNJUT
t
t
sum of (midpoint × frequency)
– 0ODFZPVIBWFGPVOEUIFNJEQPJOUTZPVDBOFTUJNBUFUIFNFBOEstimated mean =
sum off requencies
ćFNPEBMDMBTTJTUIFDMBTTJOUFSWBMXJUIUIFIJHIFTUGSFRVFODZ
ćFNFEJBODMBTTJTUIFDMBTTJOUFSWBMJOUPXIJDIUIFNJEEMFWBMVFJOUIFEBUBTFUGBMMT8PSLPVUUIFQPTJUJPOPGUIF
NFEJBO UPUBMGSFRVFODZŸUXP BOEUIFOBEEUIFUPUBMTJOUIFGSFRVFODZDPMVNOVOUJMZPVSFBDIUIBUQPTJUJPOUPĕOE
UIFNFEJBODMBTT
Exercise 12.4
Tip
$MBTTJOUFSWBMTBSFPęFO
HJWFOVTJOHJOFRVBMJUZ
TZNCPMTȳm NFBOTWBMVFTHSFBUFSPS
FRVBMUPVQUPWBMVFTUIBU
BSFMFTTUIBOćFOFYU
DMBTTJOUFSWBMIBTWBMVFT
FRVBMUPPSHSFBUFSUIBO
TPBWBMVFPGXPVME
HPJOUPUIFTFDPOEDMBTT
JOUFSWBM
Tip
*OTPNFUFYUT UIF
NJEQPJOUJTDBMMFEUIFDMBTT
DFOUSF
1 ćFUBCMFTIPXTUIFNBSLT m PCUBJOFECZBHSPVQPGTUVEFOUTGPSBOBTTJHONFOU
Marks (m)
Midpoint
Frequency (f)
≤m
2
≤m
5
≤m
≤m
≤m
≤m
Frequency × midpoint
5PUBM
(a)
(b)
(c)
(d)
$PQZBOEDPNQMFUFUIFUBCMF
$BMDVMBUFBOFTUJNBUFGPSUIFNFBONBSL
*OUPXIJDIDMBTTEPFTUIFNPEBMNBSLGBMM 8IBUJTUIFNFEJBODMBTT 2 ćFUBCMFTIPXTUIFOVNCFSPGXPSETQFSNJOVUFTUZQFECZBHSPVQPGDPNQVUFSQSPHSBNNFST
Words per minute (w)
Frequency
≤w
≤w
≤w
≤w
≤w
≤w
5PUBM
(a)
(b)
(c)
(d)
78
Unit 3: Data handling
%FUFSNJOFBOFTUJNBUFGPSUIFNFBOOVNCFSPGXPSETUZQFEQFSNJOVUF
)PXNBOZXPSETEPNPTUPGUIFQSPHSBNNFSTNBOBHFUPUZQFQFSNJOVUF 8IBUJTUIFNFEJBODMBTT 8IBUJTUIFSBOHFPGXPSETUZQFEQFSNJOVUF
12 Averages and measures of spread
12.5 Percentiles and quartiles
t 1FSDFOUJMFTBSFVTFEUPEJWJEFBEBUBTFUJOUPFRVBMHSPVQT*GZPVTDPSFJOUIFUIQFSDFOUJMFJOBUFTU JUNFBOT
UIBUPGUIFPUIFSNBSLTBSFMPXFSUIBOZPVST
t 2VBSUJMFTBSFVTFEUPEJWJEFBTFUPGEBUBJOUPGPVSFRVBMHSPVQT RVBSUFST t
– ćFMPXFSRVBSUJMF 2 JTUIFWBMVFCFMPXXIJDIPOFRVBSUFSPGUIFEBUBMJF
– ćFTFDPOERVBSUJMFJTBMTPUIFNFEJBOPGUIFEBUBTFU 22 – ćFVQQFSRVBSUJMF 23 JTUIFWBMVFCFMPXXIJDIUISFFRVBSUFSTPGUIFEBUBMJF PCWJPVTMZUIFPUIFSRVBSUFSMJF
BCPWFUIJT ćFJOUFSRVBSUJMFSBOHF *2323o2
Exercise 12.5
Tip
:PVDBOĕOEUIFQPTJUJPO
PGUIFRVBSUJMFTVTJOHUIFTF
SVMFT
1
Q ( 1) 4
1
Q2 (
2
3
23 (
4
1)
1)
8IFOUIFRVBSUJMFGBMMT
CFUXFFOUXPWBMVFT ĕOE
UIFNFBOPGUIFUXPTDPSFT
Mixed exercise
1 'PSFBDIPGUIFGPMMPXJOHTFUTPGEBUBDBMDVMBUFUIFNFEJBO VQQFSBOEMPXFSRVBSUJMFT*OFBDI
DBTFDBMDVMBUFUIFJOUFSRVBSUJMFSBOHF
(a) 55
(b) 3
(c) (d) 2
2
3
3
5
5
5
2
3
1 'JOEUIFNFBO NFEJBO NPEFBOESBOHFPGUIFGPMMPXJOHTFUTPGEBUB
(a) (b) (c) 2 ćFNFBOPGUXPDPOTFDVUJWFOVNCFSTJTćFNFBOPGFJHIUEJČFSFOUOVNCFSTJT
(a) $BMDVMBUFUIFUPUBMPGUIFĕSTUUXPOVNCFST
(b) 8IBUBSFUIFTFUXPOVNCFST
(c) $BMDVMBUFUIFNFBOPGUIFUFOOVNCFSTUPHFUIFS
3 ćSFFTVQQMJFSTTFMMTQFDJBMJTFESFNPUFDPOUSPMMFSTGPSBDDFTTTZTUFNT"TBNQMFPGSFNPUF
DPOUSPMMFSTJTUBLFOGSPNFBDITVQQMJFSBOEUIFXPSLJOHMJGFPGFBDIDPOUSPMMFSJTNFBTVSFEJO
XFFLTćFGPMMPXJOHUBCMFTIPXTUIFNFBOUJNFBOESBOHFGPSFBDITVQQMJFS
Supplier
Mean (weeks)
Range (weeks)
A
B
C
8IJDITVQQMJFSXPVMEZPVSFDPNNFOEUPTPNFPOFXIPJTMPPLJOHUPCVZBSFNPUF
DPOUSPMMFS 8IZ
Unit 3: Data handling
79
12 Averages and measures of spread
4 "CPYDPOUBJOTQMBTUJDCMPDLTPGEJČFSFOUWPMVNFBTTIPXOJOUIFGSFRVFODZUBCMF
Volume (cm3)
2
3
5
Frequency
(a) 'JOEUIFNFBOWPMVNFPGUIFCMPDLT
(b) 8IBUWPMVNFJTNPTUDPNNPO
(c) 8IBUJTUIFNFEJBOWPMVNF
5 ćFBHFTPGQFPQMFXIPWJTJUFEBOBSUFYIJCJUJPOBSFSFDPSEFEBOEPSHBOJTFEJOUIFHSPVQFE
GSFRVFODZUBCMFCFMPX
Age in years (a)
Frequency
≤a
≤a
≤a
≤a
≤a
≤a
≤a
5PUBM
(a)
(b)
(c)
(d)
&TUJNBUFUIFNFBOBHFPGQFPQMFBUUFOEJOHUIFFYIJCJUJPO
*OUPXIBUBHFHSPVQEJENPTUWJTJUPSTGBMM 8IBUJTUIFNFEJBOBHFPGWJTJUPSTUPUIFFYIJCJUJPO 8IZDBOZPVOPUDBMDVMBUFBOFYBDUNFBOGPSUIJTEBUBTFU 6 ćFOVNCFSPGTUVEFOUTBUUFOEJOHBDIFTTDMVCPOWBSJPVTEBZTXBTSFDPSEFE
(a)
(b)
(c)
(d)
'JOEUIFNFEJBOPGUIFEBUB
8IBUJTUIFSBOHFPGUIFEBUB 'JOE2BOE23BOEIFODFDBMDVMBUFUIF*23
$PNQBSFUIFSBOHFBOEUIF*238IBUEPFTUIJTUFMMZPVBCPVUUIFEBUB 7 "UFBDIFSBOOPVODFTUIBUBMMTUVEFOUTXIPTDPSFCFMPXUIFUIQFSDFOUJMFJOBUFTUXJMMIBWF
UPSFXSJUFJU8IFOUIFSFTVMUTBSFPVU PVUPGUIFTUVEFOUTIBWFUPSFXSJUFUIFUFTU
(a) 8IBUEPFTUIJTUFMMZPVBCPVUUIFTDPSFT (b) 8IBUEPFTUIJTUFMMZPVBCPVUUIFQFSGPSNBODFPGUIFDMBTTPWFSBMM 80
Unit 3: Data handling
13
Understanding measurement
13.1 Understanding units
t 6OJUTPGNFBTVSFJOUIFNFUSJDTZTUFNBSFNFUSFT N HSBNT H BOEMJUSFT M 4VCEJWJTJPOTIBWFQSFĕYFTTVDIBT
NJMMJBOEDFOUJUIFQSFĕYLJMPJTBNVMUJQMF
t 5PDPOWFSUGSPNBMBSHFSVOJUUPBTNBMMFSVOJUZPVNVMUJQMZUIFNFBTVSFNFOUCZUIFDPSSFDUNVMUJQMFPGUFO
t 5PDPOWFSUGSPNBTNBMMFSVOJUUPBMBSHFSVOJUZPVEJWJEFUIFNFBTVSFNFOUCZUIFDPSSFDUNVMUJQMFPGUFO
To change to a smaller unit, multiply by conversion factor
× 1000
× 100
× 10
km
m
cm
mm
kg
g
cg
mg
kl
l
cl
ml
.. 1000
.. 100
.. 10
To change to a larger unit, divide by conversion factor.
t "SFBJTBMXBZTNFBTVSFEJOTRVBSFVOJUT5PDPOWFSUBSFBTGSPNPOFVOJUUPBOPUIFSZPVOFFEUPTRVBSFUIF
BQQSPQSJBUFMFOHUIDPOWFSTJPOGBDUPS
t 7PMVNFJTNFBTVSFEJODVCJDVOJUT5PDPOWFSUWPMVNFTGSPNPOFVOJUUPBOPUIFSZPVOFFEUPDVCFUIF
BQQSPQSJBUFMFOHUIDPOWFSTJPOGBDUPS
Exercise 13.1
Tip
.FNPSJTFUIFTF
DPOWFSTJPOT
NN=DN
DN=N
N=LN
NH=H
H=LH
LH=U
NM=MJUSF
DN3=NM
1 6TFUIFDPOWFSTJPOEJBHSBNJOUIFCPYBCPWFBTBCBTJTUPESBXZPVSPXOEJBHSBNTUPTIPX
IPXUPDPOWFSU
(a) VOJUTPGBSFB
(b) VOJUTPGWPMVNF
2 $POWFSUUIFGPMMPXJOHMFOHUINFBTVSFNFOUTUPUIFVOJUTHJWFO
(a) LN=
(c) N=
(e) N=
N
DN
NN
(b) DN=
(d) N=
(f) DN=
NN
LN
N
3 $POWFSUUIFGPMMPXJOHNFBTVSFNFOUTPGNBTTUPUIFVOJUTHJWFO
(a) LH=
(c) LH=
(e) H=
H
H
LH
(b) LH=
(d) H=
(f) H=
H
LH
UPOOF
Unit 4: Number
81
13 Understanding measurement
4 *EFOUJGZUIFHSFBUFSMFOHUIJOFBDIPGUIFTFQBJSTPGMFOHUITćFODBMDVMBUFUIFEJČFSFODF
CFUXFFOUIFUXPMFOHUIT(JWFZPVSBOTXFSJOUIFNPTUBQQSPQSJBUFVOJUT
(a) LN
(c) DN
(e) N
(b) N
(d) DN
(f) LN
N
NN
LN
DN
N
DN
5 $POWFSUUIFGPMMPXJOHBSFBNFBTVSFNFOUTUPUIFVOJUTHJWFO
(a) DN2=
(c) DN=
(e) N2=
Cubic centimetres or cm3 is
sometimes written as cc. For
example, a scooter may have a
50 cc engine. That means the
total volume of all cylinders in the
engine is 50 cm3.
(b) DN=
(d) LN2=
(f) N=
NN2
NN2
LN2
NN2
N2
NN2
6 $POWFSUUIFGPMMPXJOHWPMVNFNFBTVSFNFOUTUPUIFVOJUTHJWFO
(a) DN3=
(c) DN3=
(e) NN3=
(b) DN3=
(d) N3=
(f) NN3=
NN3
NN3
DN3
NN3
DN3
N3
7 /BFFNMJWFTLNGSPNTDIPPMBOE4BEJRBMJWFTNGSPNTDIPPM)PXNVDIDMPTFSUP
UIFTDIPPMEPFT4BEJRBMJWF
8 "DPJOIBTBEJBNFUFSPGNN*GZPVQMBDFEDPJOTJOBSPX IPXMPOHXPVME
UIFSPXCFJODN
9 "TRVBSFPGGBCSJDIBTBOBSFBPGNN28IBUBSFUIFMFOHUITPGUIFTJEFTPG
UIFTRVBSFJODN
10 )PXNBOZDVCPJETIBQFECPYFT FBDIXJUIEJNFOTJPOTDN×DN×DN DBOZPV
ĕUJOUPBWPMVNFPGN3 13.2 Time
t 5JNFJTOPUEFDJNBMINFBOTPOFIPVSBOE1560 PS 14 PGBOIPVS OPUI
t 0OFIPVSBOENJOVUFTJTXSJUUFOBT
t 5JNFDBOCFXSJUUFOVTJOHBNBOEQNOPUBUJPOPSBTBIPVSUJNFVTJOHUIFOVNCFSTGSPNUPUP
HJWFUIFUJNFTGSPNNJEOJHIUPOPOFEBZ I UPPOFTFDPOECFGPSFNJEOJHIU &WFOJOUIFIPVSDMPDLTZTUFN UJNFJTOPUEFDJNBMćFUJNFPOFNJOVUFBęFSJT
Tip
:PVDBOFYQSFTTQBSUTPGBOIPVSBTBEFDJNBM%JWJEFUIFOVNCFSPGNJOVUFTCZ
'PSFYBNQMFNJOVUFT=
82
Unit 4: Number
12
60
1
5
= = 0 2IPVSTćJTDBONBLFZPVSDBMDVMBUJPOTFBTJFS
13 Understanding measurement
Exercise 13.2
1 'JWFQFPQMFSFDPSEUIFUJNFUIFZTUBSUXPSL UIFUJNFUIFZĕOJTIBOEUIFMFOHUIPGUIFJS
MVODICSFBL
(a) $
PQZBOEDPNQMFUFUIJTUBCMFUPTIPXIPXNVDIUJNFFBDIQFSTPOTQFOUBUXPSLPO
UIJTQBSUJDVMBSEBZ
Name
Time in
Time out
Lunch
%BXPPU
1
QBTU
4
)BMGQBTUĕWF
3
IPVS
4
/BEJSB
BN
QN
+PIO
1
2
Hours worked
IPVS
NJO
3PCZO
BN
QN
IPVS
.BSJ
NJO
(b) $BMDVMBUFFBDIQFSTPOTEBJMZFBSOJOHTUPUIFOFBSFTUXIPMFDFOUJGUIFZBSFQBJE
QFSIPVS
2 0OBQBSUJDVMBSEBZ UIFMPXUJEFJO)POH,POHIBSCPVSJTBUćFIJHIUJEFJTBU
UIFTBNFEBZ)PXNVDIUJNFQBTTFTCFUXFFOMPXUJEFBOEIJHIUJEF
3 4BSBITQMBOFXBTEVFUPMBOEBUQN)PXFWFS JUXBTEFMBZFEBOEJUMBOEFEBU)PX
NVDIMBUFSEJEUIFQMBOFBSSJWFUIBOJUXBTNFBOUUP
4 )PXNVDIUJNFQBTTFTCFUXFFO
(a)
(b)
(c)
(d)
QNBOEQNPOUIFTBNFEBZ
BNBOEQNPOUIFTBNFEBZ
QNBOEBNUIFOFYUEBZ
BNBOEPOUIFTBNFEBZ
5 6TFUIJTTFDUJPOGSPNBCVTUJNFUBCMFUPBOTXFSUIFRVFTUJPOTUIBUGPMMPX
Chavez Street
Castro Avenue
Peron Place
Marquez Lane
(a) 8IBUJTUIFFBSMJFTUCVTGSPN$IBWF[4USFFU
(b) )PXMPOHEPFTUIFKPVSOFZGSPN$IBWF[4USFFUUP.BSRVF[-BOFUBLF
(c) "
CVTBSSJWFTBU1FSPO1MBDFBURVBSUFSQBTUUFOćFCVTJTNJOVUFTMBUF"UXIBUUJNF
EJEJUMFBWF$IBWF[4USFFU
(d) 4BODIF[NJTTFTUIFCVTGSPN$BTUSP"WFOVF)PXMPOHXJMMIBWFUPXBJUCFGPSFUIF
OFYUTDIFEVMFECVTBSSJWFT
(e) ćFCVTGSPN$IBWF[4USFFUJTEFMBZFEJOSPBEXPSLTCFUXFFO$BTUSP"WFOVFBOE
1FSPO1MBDFGPSNJOVUFT)PXXJMMUIJTBČFDUUIFSFTUPGUIFUJNFUBCMF
Unit 4: Number
83
13 Understanding measurement
13.3 Upper and lower bounds
t "MMNFBTVSFNFOUTXFNBLFBSFSPVOEFEUPTPNFEFHSFFPGBDDVSBDZćFEFHSFFPGBDDVSBDZ GPSFYBNQMFUIFOFBSFTU
t
NFUSFPSUPUXPEFDJNBMQMBDFT BMMPXTZPVUPXPSLPVUUIFIJHIFTUBOEMPXFTUQPTTJCMFWBMVFPGUIFNFBTVSFNFOUT
ćFIJHIFTUQPTTJCMFWBMVFJTDBMMFEUIFVQQFSCPVOEBOEUIFMPXFTUQPTTJCMFWBMVFJTDBMMFEUIFMPXFSCPVOE
8IFOZPVXPSLXJUINPSFUIBOPOFSPVOEFEWBMVFZPVOFFEUPVTFUIFVQQFSBOEMPXFSCPVOETPGFBDI
dp means decimal places
sf means significant figures
Exercise 13.3
1 &BDIPGUIFOVNCFSTCFMPXIBTCFFOSPVOEFEUPUIFEFHSFFPGBDDVSBDZTIPXOJOUIF
CSBDLFUT'JOEUIFVQQFSBOEMPXFSCPVOETJOFBDIDBTF
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
OFBSFTUXIPMFOVNCFS
OFBSFTUXIPMFOVNCFS
TG
EQ
EQ
UPOFBSFTUUFOUI
OFBSFTUUFO
TG
2 "CVJMEJOHJTNUBMMNFBTVSFEUPUIFOFBSFTUNFUSF
(a) 8IBUBSFUIFVQQFSBOEMPXFSCPVOETPGUIFCVJMEJOHTIFJHIU
(b) * TNFUSFTBQPTTJCMFIFJHIUGPSUIFCVJMEJOH &YQMBJOXIZPSXIZOPU
3 ćFEJNFOTJPOTPGBSFDUBOHVMBSQJFDFPGMBOEBSFNCZNćFNFBTVSFNFOUTBSFFBDI
DPSSFDUUPPOFEFDJNBMQMBDF
(a) 'JOEUIFBSFBPGUIFQJFDFPGMBOE
(b) $BMDVMBUFUIFVQQFSBOEMPXFSCPVOETPGUIFBSFBPGUIFMBOE
4 6TBJO#PMUIPMETUIFXPSMESFDPSETGPSUIFNBOENTQSJOUT )FXBT
BMTPBNFNCFSPGUIF+BNBJDBOGPVSCZNSFMBZUFBNUIBUTFUBOFXXPSMESFDPSEPG
TFDPOETJO"VHVTU
(a) 6TBJO#PMUJTDNUBMM DPSSFDUUPUIFOFBSFTUDFOUJNFUSF BOEIJTNBTTJTLH DPSSFDU
UPUIFOFBSFTULJMPHSBN'JOEUIFVQQFSBOEMPXFSCPVOETPGIJTIFJHIUBOENBTT
(b) ćF+BNBJDBODPBDITBZTIJTUFBNDBOSVOUIFNSFMBZJOTFDPOET#PUIUIFTF
NFBTVSFNFOUTBSFHJWFOUPUXPTJHOJĕDBOUĕHVSFT8IBUJTUIFNBYJNVNTQFFE JO
NFUSFTQFSTFDPOE BUXIJDIUIFZDBOSVOUIFSFMBZ (JWFZPVSBOTXFSDPSSFDUUPUXP
EFDJNBMQMBDFT
5 ćFUXPTIPSUTJEFTPGBSJHIUBOHMFEUSJBOHMFBSFDN UPUIFOFBSFTUNN BOEDN UP
UIFOFBSFTUNN $BMDVMBUFVQQFSBOEMPXFSCPVOETGPS
(a) UIFBSFBPGUIFSFDUBOHMF
(b) UIFMFOHUIPGUIFIZQPUFOVTF
(JWFZPVSBOTXFSTJODFOUJNFUSFTUPGPVSEFDJNBMQMBDFT
84
Unit 4: Number
13 Understanding measurement
13.4 Conversion graphs
t $POWFSTJPOHSBQITBMMPXZPVUPDPOWFSUGSPNPOFVOJUPGNFBTVSFUPBOPUIFSCZQSPWJEJOHUIFWBMVFTPGCPUIVOJUT
POEJČFSFOUBYFT5PĕOEPOFWBMVF x XIFOUIFPUIFS y JTHJWFO ZPVOFFEUPĕOEUIFyWBMVFBHBJOTUUIFHSBQIBOE
UIFOSFBEPČUIFDPSSFTQPOEJOHWBMVFPOUIFPUIFSBYJT
Exercise 13.4
Tip
Exchange rate
600
Rupiah (in thousands)
.BLFTVSFZPVSFBEUIF
MBCFMTPOUIFBYJTTPUIBU
ZPVBSFSFBEJOHPČUIF
DPSSFDUWBMVFT
1 4IFJMB BO"VTUSBMJBO JTHPJOHPOIPMJEBZUPUIFJTMBOEPG#BMJJO*OEPOFTJB4IFĕOETUIJT
DPOWFSTJPOHSBQIUPTIPXUIFWBMVFPGSVQJBI UIFDVSSFODZPG*OEPOFTJB BHBJOTUUIF
"VTUSBMJBOEPMMBS
500
400
300
200
100
0
0
10 20 30 40 50 60 70 80 90 100
Australian dollars
(a) 8IBUJTUIFTDBMFPOUIFWFSUJDBMBYJT
(b) )PXNBOZSVQJBIXJMM4IFJMBHFUGPS
(i) "VT
(ii) "VT (iii) "VT
(c) ćFIPUFMTIFQMBOTUPTUBZBUDIBSHFTSVQJBIBOJHIU
(i) 8IBUJTUIJTBNPVOUJO"VTUSBMJBOEPMMBST
(ii) )PXNVDIXJMM4IFJMBQBZJO"VTUSBMJBOEPMMBSTGPSBOFJHIUOJHIUTUBZ
2 4UVEZUIFDPOWFSTJPOHSBQIBOEBOTXFSUIFRVFTUJPOT
Conversion graph,
Celsius to Fahrenheit
250
200
Temperature
in °F
Tip
ćF64"NBJOMZTUJMM
VTFTUIF'BISFOIFJU
TDBMFGPSUFNQFSBUVSF
"QQMJBODFT TVDIBTTUPWFT NBZIBWFUFNQFSBUVSFT
JO'BISFOIFJUPOUIFN QBSUJDVMBSMZJGUIFZBSFBO
"NFSJDBOCSBOE
150
100
50
–20
0
20
40
60
Temperature
in °C
80
100
(a) 8IBUJTTIPXOPOUIFHSBQI
(b) 8IBUJTUIFUFNQFSBUVSFJO'BISFOIFJUXIFOJUJT
(i) °$
(ii) °$
(iii) °$
Unit 4: Number
85
13 Understanding measurement
(c) 4 BSBIĕOETBSFDJQFGPSDIPDPMBUFCSPXOJFTUIBUTBZTTIFOFFETUPDPPLUIFNJYUVSFBU
°$GPSPOFIPVS"ęFSBOIPVSTIFĕOETUIBUJUIBTIBSEMZDPPLFEBUBMM8IBUDPVME
UIFQSPCMFNCF
(d) + FTTJT"NFSJDBO8IFOTIFDBMMTIFSGSJFOE/JDLJO&OHMBOETIFTBZT A*UTSFBMMZDPMEIFSF NVTUCFBCPVUEFHSFFTPVU8IBUUFNQFSBUVSFTDBMFJTTIFVTJOH )PXEPZPVLOPXUIJT
3 ćJTHSBQITIPXTUIFDPOWFSTJPOGBDUPSGPSQPVOET JNQFSJBMNFBTVSFNFOUPGNBTT BOELJMPHSBNT
Conversion graph, pounds to kilograms
60
40
Kilograms
20
0
40
80
Pounds
120
160
(a) /FUUJFTBZTTIFOFFETUPMPTFBCPVUQPVOET)PXNVDIJTUIJTJOLJMPHSBNT
(b) +PIOTBZTIFTBXFBLMJOH)FXFJHITQPVOET)PXNVDIEPFTIFXFJHIJOLJMPHSBNT
(c) 8IJDIJTUIFHSFBUFSNBTTJOFBDIPGUIFTFDBTFT
(i) QPVOETPSLJMPHSBNT
(ii) LJMPHSBNTPSQPVOET
(iii) LJMPHSBNTPSQPVOET
13.5 More money
t 8IFOZPVDIBOHFNPOFZGSPNPOFDVSSFODZUPBOPUIFSZPVEPTPBUBHJWFOSBUFPGFYDIBOHF$IBOHJOH
UPBOPUIFSDVSSFODZJTDBMMFECVZJOHGPSFJHODVSSFODZ
t &YDIBOHFSBUFTDBOCFXPSLFEPVUVTJOHDPOWFSTJPOHSBQIT BTJO CVUNPSFPęFO UIFZBSFXPSLFE
PVUCZEPJOHDBMDVMBUJPOT
t %PJOHDBMDVMBUJPOTXJUINPOFZJTKVTUMJLFEPJOHDBMDVMBUJPOTXJUIEFDJNBMTCVUZPVOFFEUPSFNFNCFSUPJODMVEF
UIFDVSSFODZTZNCPMTJOZPVSBOTXFST
Exercise 13.5
Tip
$VSSFODZSBUFTDIBOHFBMM
UIFUJNFćFTFSBUFTXFSF
DPSSFDUJO CVUUIFZ
NBZCFWFSZEJČFSFOUUPEBZ
The inverse rows show the
exchange rate of one unit of the
currency in the column to the
currency above the word inverse.
For example, using the inverse row
below US $ and the euro column,
€1 will buy $1.39.
86
Unit 4: Number
6TFUIFFYDIBOHFSBUFUBCMFCFMPXGPSUIFTFRVFTUJPOT
Currency exchange rates – October 2011
Currency
US
$
Euro
(€)
UK
£
Indian
rupee
Aus
$
Can
$
SA
rand
NZ
$
Yen
(¥)
1 US $
JOWFSTF
1 Euro
JOWFSTF
1 UK £
JOWFSTF
13 Understanding measurement
(a) 8IBUJTUIFFYDIBOHFSBUFGPS
(i) 64UPZFO
(ii) 6,bUP/;
(iii) FVSPUP*OEJBOSVQFF
(iv) $BOBEJBOEPMMBSUPFVSP
(v) ZFOUPQPVOE
(vi) 4PVUI"GSJDBOSBOEUP64
(b) )PXNBOZ*OEJBOSVQFFTXJMMZPVHFUGPS
(i) 64
(ii) FVSPT (iii) b
(c) )PXNBOZZFOXJMMZPVOFFEUPCVZ
(i) 64
(ii) FVSPT (iii) b
Mixed exercise
1 $POWFSUUIFGPMMPXJOHNFBTVSFNFOUTUPUIFVOJUTHJWFO
(a)
(d)
(g)
(j)
LNUPNFUSFT
HSBNTUPLJMPHSBNT
NMUPMJUSFT
N3UPDN
(b)
(e)
(h)
(k)
DNUPNN
HSBNTUPNJMMJHSBNT
DN2UPNN
LN3UPN
(c)
(f)
(i)
(l)
UPOOFTUPLJMPHSBNT
MJUSFTUPNJMMJMJUSFT
NUPLN2
NN3UPDN3
2 ćFBWFSBHFUJNFUBLFOUPXBMLBSPVOEBUSBDLJTPOFNJOVUFBOETFDPOET)PXMPOHXJMM
JUUBLFZPVUPXBMLBSPVOEUIFUSBDLUJNFTBUUIJTSBUF
3 "KPVSOFZUPPLINJOBOETUPDPNQMFUF0GUIJT INJOBOETXBTTQFOUIBWJOH
MVODIPSTUPQTGPSPUIFSSFBTPOTćFSFTUPGUIFUJNFXBTTQFOUUSBWFMMJOH)PXNVDIUJNF
XBTBDUVBMMZTQFOUUSBWFMMJOH
4 5BZPTIFJHIUJTN DPSSFDUUPUIFOFBSFTUDN$BMDVMBUFUIFMFBTUQPTTJCMFBOEHSFBUFTU
QPTTJCMFIFJHIUUIBUIFDPVMECF
5 ćFOVNCFSPGQFPQMFXIPBUUFOEFEBNFFUJOHXBTHJWFOBT DPSSFDUUPUIFOFBSFTU
(a) *TJUQPTTJCMFUIBUQFPQMFBUUFOEFE &YQMBJOXIZPSXIZOPU
(b) *TJUQPTTJCMFUIBUQFPQMFBUUFOEFE &YQMBJOXIZPSXIZOPU
6 ćFEJNFOTJPOTPGBSFDUBOHMFBSFDNBOEDN FBDIDPSSFDUUPUISFFTJHOJĕDBOU
ĕHVSFT
(a) 8SJUFEPXOUIFSBOHFPGQPTTJCMFWBMVFTPGFBDIEJNFOTJPO
(b) 'JOEUIFMPXFSBOEVQQFSCPVOETPGUIFBSFBPGUIFSFDUBOHMF
(c) 8SJUFEPXOUIFMPXFSBOEVQQFSCPVOETPGUIFBSFBDPSSFDUUPUISFFTJHOJĕDBOUĕHVSFT
Unit 4: Number
87
13 Understanding measurement
7 4UVEZUIFHSBQIBOEBOTXFSUIFRVFTUJPOT
(a) 8IBUEPFTUIFHSBQITIPX
(b) $POWFSUUPMJUSFT
(i) HBMMPOT (ii) HBMMPOT
(c) $POWFSUUPHBMMPOT
(i) MJUSFT
(ii) MJUSFT
(d) /BSFTITBZTIFHFUTNQHJOUIFDJUZBOENQHPOUIFIJHIXBZJOIJTDBS
(i) $POWFSUFBDISBUFUPLNQFSHBMMPO
(ii) (JWFOUIBUPOFHBMMPOJTFRVJWBMFOUUPMJUSFT DPOWFSUCPUISBUFT
UPLJMPNFUSFTQFSMJUSF
Conversion graph for
imperial gallons to litres
45
40
35
Litres
30
25
20
15
10
6TFUIFFYDIBOHFSBUFUBCMFCFMPX SFQFBUFEGSPN&YFSDJTF UPBOTXFSUIFGPMMPXJOH
RVFTUJPOT
Currency exchange rates – October 2011
5
0
0
4
6
8
2
Gallons (imperial)
10
1 mile = 1.61 km
The US gallon is different from the
imperial gallon with a conversion
factor of 1 US gallon to 3.785 litres.
$VSSFODZ
64
&VSP
ħ
6,b
*OEJBO
SVQFF
"VT
$BO
4"
SBOE
/;
:FO d
64
JOWFSTF
&VSP
*OWFSTF
6,b
JOWFSTF
8 +BOMJWFTJO4PVUI"GSJDBBOEJTHPJOHPOIPMJEBZUP*UBMZ)FIBT3UPFYDIBOHFGPS
FVSPT)PXNBOZ&VSPTXJMMIFHFU
9 1FUFJTBO"NFSJDBOXIPJTUSBWFMMJOHUP*OEJBGPSCVTJOFTT)FOFFETUPFYDIBOHF
GPSSVQFFT
(a) 8IBUJTUIFFYDIBOHFSBUF
(b) )PXNBOZSVQFFTXJMMIFHFUBUUIJTSBUF
(c) "
UUIFFOEPGUIFUSJQIFIBTSVQFFTMFęPWFS8IBUXJMMIFHFUJGIFDIBOHFTUIFTF
CBDLUPEPMMBSTBUUIFHJWFOSBUF
10 +JNNZJT#SJUJTIBOEIFJTHPJOHUP4QBJOPOBQBDLBHFIPMJEBZćFDPTUPGUIFIPMJEBZJT
FVSPT8IBUJTUIJTBNPVOUJO6,QPVOET
88
Unit 4: Number
14
Further solving of equations and
inequalities
14.1 Simultaneous linear equations
t 4JNVMUBOFPVTNFBOTABUUIFTBNFUJNF
t ćFSFBSFUXPNFUIPETGPSTPMWJOHTJNVMUBOFPVTFRVBUJPOTHSBQIJDBMMZBOEBMHFCSBJDBMMZ
t ćFHSBQIJDBMTPMVUJPOJTUIFQPJOUXIFSFUIFUXPMJOFTPGUIFFRVBUJPOTJOUFSTFDUćJTQPJOUIBTBOxBOEB
yDPPSEJOBUF
t ćFSFBSFUXPBMHFCSBJDNFUIPETCZTVCTUJUVUJPOBOECZFMJNJOBUJPO
o 4PNFUJNFTZPVOFFEUPNBOJQVMBUFPSSFBSSBOHFPOFPSCPUIPGUIFFRVBUJPOTCFGPSFZPVDBOTPMWFUIFN
BMHFCSBJDBMMZ
o 'PSUIFTVCTUJUVUJPONFUIPE POFFRVBUJPOJTTVCTUJUVUFEJOUPUIFPUIFS
o 'PSUIFFMJNJOBUJPONFUIPEZPVOFFEFJUIFSUIFTBNFDPFďDJFOUPGxPSUIFTBNFDPFďDJFOUPGyJOCPUI
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Exercise 14.1 A
1 %SBXUIFHSBQITGPSFBDIQBJSPGFRVBUJPOTHJWFOćFOVTFUIFQPJOUPGJOUFSTFDUJPOUPĕOE
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(a)
(b)
(c)
x 3 y
x y = 0
−<x<
x 2 y
x 2 y = 3
−<x<
2 x + 2
2 y 3x
3x − 1 0 −<x<
x = 4
x − 7
−<x<
(d)
y
(e)
x 33yy = 3
y
2 x + 3
−<x<
Unit 4: Algebra
89
14 Further solving of equations and inequalities
2 ćFHSBQICFMPXTIPXTMJOFTDPSSFTQPOEJOHUPTJYFRVBUJPOT
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(i) ABOEC
(ii) DBOEF
(iii) ABOEE
(c) /PXDIFDLZPVSTPMVUJPOTBMHFCSBJDBMMZ
A
C
10
y
D
B
8
6
F
4
2
x
–10 –8
–6
–4
–2 0
–2
2
4
6
8
10
E
–4
–6
–8
–10
3 4PMWFGPSxBOEyCZVTJOHUIFTVCTUJUVUJPONFUIPE$IFDLFBDITPMVUJPOCZTVCTUJUVUJOHUIF
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(a) y = 2
x+ y=6
(b) y x = 3
y 3x = 5
(c) x + y = 4
2 x + 3 y = 12
(d) 2 x
3x
y 7
y 8
4 4PMWFGPSxBOEyCZVTJOHUIFFMJNJOBUJPONFUIPE$IFDLFBDITPMVUJPO
Remember that you might
need to rearrange one or both
equations before solving.
(a) x + y = 5
x y=7
(b) 3
2x
(d) 2 x 33yy = 6
4 x 6 y = −4
(e) 2 x 5 y = 11
3x 2y
2y = 7
(f) y 2 x = 1
2 y 3x = 5
2x + y
=3
2
(b)
x
+ y=3
2
(c)
3x y
+ =4
(f) 4 2
3y
x+
=7
2
7 x 6 y = 17
(g) 2 x
1
+ y=6
3
3
3x
(a)
3x
y
y
7
5
2x 9 y
(i) x
= 6y 9
3
Unit 4: Algebra
(c) 2 x 33yy = 12
3x 3y
3 y = 30
5 4PMWFTJNVMUBOFPVTMZ
5x
+ y = 10
(e) 2
x
1
+ y=3
4
4
90
y 1
y 4
3x 22yy = 4
2x y 3
x y
+ =2
(d) 3 2
x + 4 y 11
3x + y 1
=
2
2
(h)
5x 7 y
=2
3
14 Further solving of equations and inequalities
6 4BMMZCPVHIUUXPDIPDPMBUFCBSTBOEPOFCPYPGHVNTGPSBOE&WBOQBJEGPSPOF
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8 &QISBIJNIBTDPJOTJOIJTQPDLFU DPOTJTUJOHPGRVBSUFSTBOEEJNFTPOMZ*GIFIBTJO
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Exercise 14.1 B
1 'PSNUXPFRVBUJPOTVTJOHUIFJOGPSNBUJPOPOFBDIEJBHSBNBOETPMWFUIFNTJNVMUBOFPVTMZ
UPĕOEUIFWBMVFTPGxBOEy
(a)
(c)
– 3x
(b)
20
6y
2
3x
y
+6
20
2x + 4
2x
9y
2y – 4x
(d)
–3y – 6
4x – 2y
(e)
(f)
2x + 7
x+y
1
2
5–
x
3x
9 + 3y
3y
2
5x – 2y
9–y
2x + 4
2 0OBCVTZEBZ BTIPQTFMMTBUPUBMPGEFTLTBOEDIBJSTGPSBUPUBMBNPVOUPGćFEFTLT
TFMMGPSFBDIBOEUIFDIBJSTTFMMGPSFBDI
(a) &
YQSFTTUIFJOGPSNBUJPOBTUXPFRVBUJPOT-FUdFRVBMUIFOVNCFSPGEFTLTBOE
MFUcFRVBMUIFOVNCFSPGDIBJST
(b) 4PMWFUIFFRVBUJPOTTJNVMUBOFPVTMZUPĕOEIPXNBOZPGFBDIXFSFTPME
14.2 Linear inequalities
t ćFTPMVUJPOUPBMJOFBSJOFRVBMJUZJTBSBOHFPGWBMVFT
t ćFTPMVUJPODBOCFSFQSFTFOUFEPOBOVNCFSMJOF
– "TPMJEDJSDMFPOUIFOVNCFSMJOFNFBOTUIFWBMVFJTJODMVEFE
– "OPQFODJSDMFPOUIFOVNCFSMJOFNFBOTUIFWBMVFJTOPUJODMVEFE
Unit 4: Algebra
91
14 Further solving of equations and inequalities
Tip
Exercise 14.2
"TXJUIFRVBUJPOT XIBU
ZPVEPUPPOFTJEFPGBO
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NVMUJQMZPSEJWJEFCPUI
TJEFTPGBOJOFRVBMJUZCZB
OFHBUJWFOVNCFS ZPVNVTU
SFNFNCFSUPSFWFSTFUIF
EJSFDUJPOPGUIFJOFRVBMJUZ
1 %SBXBOVNCFSMJOFUPSFQSFTFOUUIFQPTTJCMFWBMVFTPGUIFWBSJBCMFJOFBDIDBTF
(a) xȳ
(d) ȳaȳ
(g) oȳaȳo
(b) y ≥ o
(e) oȴn
(h) aȳ
(c) fo
(f) –m
(i) ȳa
2 8SJUFEPXOBMMJOUFHFSTUIBUTBUJTGZFBDIPGUIFGPMMPXJOHJOFRVBMJUJFT
(a) xȳ
(d) oȳsȳ
(b) oȳhȳ
1
(e) − ȳe
3
(c)
2 a
(f) ɀbȳɀ
3 4PMWFFBDIPGUIFGPMMPXJOHJOFRVBMJUJFT4PNFPGUIFBOTXFSTXJMMJOWPMWFGSBDUJPOT-FBWF
ZPVSBOTXFSTBTGSBDUJPOTJOUIFJSTJNQMFTUGPSNXIFSFBQQSPQSJBUF
x
(a) xȳo
(b) x¦
(c) x¦ȴ
(d)
ȳ
4
x
x −5
2
+ 8 ȴ
(e)
(f)
(g) x¦ (h) e ȳe 3
4
3
r 1
+ (i)
(j) x¦ ȴ x (k) g¦¦g
4 8
(l)
(o)
y+4
1
≥ y− 12
2
3⎛
8⎝
2x
1⎞
3⎠
2
9
(m)
(3 x
3
33) ≥
3
5
6
4
o (n)
3
+
n¦ ¦ ¦n n 1
3
14.3 Regions in a plane
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t 8IFOUIFJOFRVBMJUZJODMVEFTFRVBMUP ȳPSȴ UIFCPVOEBSZMJOFPOUIFHSBQINVTUCFJODMVEFEBTBTPMJEMJOFPO
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Tip
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ymx cJTBCPWFUIF
line
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t
92
Unit 4: Algebra
Exercise 14.3
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VOXBOUFESFHJPO
(a) yx (b) yȴox (c) y > –x 2 4IBEFUIFVOXBOUFESFHJPOUIBUSFQSFTFOUTFBDIJOFRVBMJUZPOTFQBSBUFBYFT
(a) yȳo
(b) x¦y
(c) x¦yȴ
(d) xy¦
(e) oyȳ
(f)
x
y
4
>
3
2
14 Further solving of equations and inequalities
3 'PSFBDIPGUIFGPMMPXJOHEJBHSBNT ĕOEUIFJOFRVBMJUZUIBUJTSFQSFTFOUFECZUIFVOTIBEFE
SFHJPO
5
y
5
4
4
3
3
2
2
1
1
x
–5 –4 –3 –2 –1 0
–1
1
2
3
4
5
–2
–3
–3
–4
–4
–5
–5
y
5
4
4
3
3
2
2
1
–5 –4 –3 –2 –1 0
–1
x
–5 –4 –3 –2 –1 0
–1
–2
5
y
1
2
3
4
5
1
2
3
4
5
y
1
x
1
2
3
4
5
–5 –4 –3 –2 –1 0
–1
–2
–2
–3
–3
–4
–4
–5
–5
x
4 #ZTIBEJOHUIFVOXBOUFESFHJPOT TIPXUIFSFHJPOEFĕOFECZUIFTFUPGJOFRVBMJUJFT
x y x¦yBOEyȴo
5 (a)#ZTIBEJOHUIFVOXBOUFESFHJPOT TIPXUIFSFHJPOUIBUTBUJTĕFTBMMUIFJOFRVBMJUJFT
x yȳ xoyȳoBOExȴo
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14.4 Linear programming
t "NBUIFNBUJDBMXBZUPFYQSFTTDPOTUSBJOUTJOCVTJOFTTBOEJOEVTUSZUPPCUBJOHSFBUFTUQSPĕU MFBTUDPTU FUD
t $POTUSBJOUTUBLFUIFGPSNPGMJOFBSJOFRVBMJUJFT
5
4
3
2
1
–5 –4 –3 –2 –1 0
–1
–2
–3
–4
–5
y
Exercise 14.4
x
1 2 3 4 5
1 *OUIFEJBHSBN UIFTIBEFESFHJPOSFQSFTFOUTUIFTFUPGJOFRVBMJUJFTyȳ xȴo yȴxBOE
x yȳ'JOEUIFHSFBUFTUBOEMFBTUQPTTJCMFWBMVFTPGx yTVCKFDUUPUIFTFJOFRVBMJUJFT
2 (a) 0OBHSJE TIBEFUPJOEJDBUFUIFSFHJPOTBUJTGZJOHBMMUIFJOFRVBMJUJFTyȴ ȳxȳ yȳx BOEyȳox (b) 8
IBUJTUIFHSFBUFTUQPTTJCMFWBMVFPGy xJGxBOEyTBUJTGZBMMUIFJOFRVBMJUJFTJO a
Unit 4: Algebra
93
14 Further solving of equations and inequalities
Tip
*EFOUJGZZPVSVOLOPXOT
BOEHJWFUIFNFBDIB
WBSJBCMF
1BZBUUFOUJPOUPXPSETMJLF
ABUMFBTU ANJOJNVN FUDUP
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JOFRVBMJUJFT
3 "UBTDIPPMCBLFTBMF DIPDPMBUFGVEHFJTTPMEGPSBQSPĕUPGBOEWBOJMMBGVEHFGPSBQSPĕU
PG4BMMZIBTJOHSFEJFOUTUPNBLFBUNPTUCBHTPGDIPDPMBUFGVEHFBOECBHTPGWBOJMMB
GVEHF4IFIBTFOPVHIUJNFUPNBLFBNBYJNVNPGCBHTBMUPHFUIFS)PXNBOZCBHTPG
FBDIUZQFTIPVMETIFNBLFUPNBYJNJTFIFSQSPĕUBOEXIBUJTUIJTNBYJNVNQSPĕU
4 "GBDUPSZNBLFTEJČFSFOUDPODFOUSBUFTGPSDPMEESJOLTćFQSPDFTTSFRVJSFTUIFQSPEVDUJPO
PGBUMFBTUUISFFMJUSFTPGPSBOHFGPSFBDIMJUSFPGMFNPODPODFOUSBUF'PSUIFTVNNFS BUMFBTU
MJUSFTCVUOPNPSFUIBOMJUSFTPGPSBOHFDPODFOUSBUFOFFETUPCFQSPEVDFEJOB
NPOUIćFEFNBOEGPSMFNPO POUIFPUIFSIBOE JTOPUNPSFUIBOMJUSFTBNPOUI
-FNPODPODFOUSBUFTFMMTGPSQFSMJUSFBOEPSBOHFDPODFOUSBUFTFMMTGPSQFSMJUSF)PX
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14.5 Completing the square
t $PNQMFUJOHUIFTRVBSFJTBNFUIPEVTFEGPSTPMWJOHRVBESBUJDFRVBUJPOTUIBUDBOOPUCFTPMWFECZGBDUPST
t &YQSFTTJPOTPGUIFGPSN x + x DBOCFXSJUUFOJOUIFGPSN ⎛⎝ x + a2 ⎞⎠ − ⎛⎝ a2 ⎞⎠ 2
2
2
Exercise 14.5
1 8SJUFUIFGPMMPXJOHFYQSFTTJPOTJOUIFGPSN( x +
Tip
x 2 + 6x
6x 4
x 2 − 12 x + 30
x 2 + 24 x + 121 x 2 2x
2 x 10 (b)
(e)
(h)
(k)
x 2 4x
4x 7 x 2 + 10 x + 17
x 2 − 16 x + 57 x 2 8x 5 (c)
(f)
(i)
(l)
x 2 + 14 x + 44
x 2 + 22 x + 141
x 2 + 18 x + 93
x 2 + 20 x + 83
*GUIFDPFďDJFOUPGx
JTOPU NBLFJUCZ
EJWJEJOHUIFFRVBUJPOCZ
UIFDPFďDJFOUPGx
2 4PMWFUIFGPMMPXJOHRVBESBUJDFRVBUJPOTCZUIFNFUIPEPGDPNQMFUJOHUIFTRVBSF HJWJOHZPVS
ĕOBMBOTXFSUPUXPEFDJNBMQMBDFTJGOFDFTTBSZ
Tip
3 4PMWFUIFGPMMPXJOHFRVBUJPOTCZDPNQMFUJOHUIFTRVBSF BOTXFSTUPUXPEFDJNBMQMBDFT
JGOFDFTTBSZ *GUIFFRVBUJPOJTOPUJO
UIFGPSN ax bx c UIFODIBOHFJUJOUPUIBU
GPSNCFGPSFCFHJOOJOHUP
TPMWFJU
94
(a)
(d)
(g)
(j)
)2 + b
Unit 4: Algebra
(a) x 2 5x 6 = 0 (b) x 2 x − 6 0 (e) x 2 − 16 x + 3 0 (f) x 2 + 7x
7x 1 = 0 2
(i) x 2 x − 100 = 0
(a) 2 x 2
(d) 2
(c) x 2 4x
4x 3 = 0 (g) x 2 + 9x
9x 1 = 0 3x − 2 0 (b) x ( x − 4 ) = −3 (c) 3x 2 2 (3x 2)
3
5= (e) ( x + 1))(( x − 7 ) = 4 (f) x + 2 x 2 = 8
x
(d) x 2 6 x 7 = 0
(h) x 2 + 11x + 27 = 0
14 Further solving of equations and inequalities
14.6 Quadratic formula
t ćFHFOFSBMGPSNPGUIFRVBESBUJDFRVBUJPOJTax bx c
t ćFRVBESBUJDGPSNVMBJT x = −b ± 2ba − 4ac t ćFRVBESBUJDGPSNVMBJTVTFEQSJNBSJMZXIFOUIFRVBESBUJDFYQSFTTJPODBOOPUCFGBDUPSJTFE
2
Tip
Exercise 14.6
ćFœJOUIFGPSNVMB
UFMMTZPVUPDBMDVMBUFUXP
WBMVFT
1 &BDIPGUIFGPMMPXJOHRVBESBUJDTXJMMGBDUPSJTF4PMWFFBDIPGUIFNCZGBDUPSJTBUJPOBOEUIFO
VTFUIFRVBESBUJDGPSNVMBUPTIPXUIBUZPVHFUUIFTBNFBOTXFSTJOCPUIDBTFT
Tip
5BLFDBSFXIFOUIF
DPFďDJFOUPGxJTOPU
FRVBMUP
Tip
:PVNVTUNBLFTVSF
UIBUZPVSFRVBUJPOUBLFT
UIFGPSNPGBRVBESBUJD
FYQSFTTJPOFRVBMUP[FSP
*GJUEPFTOPUUIFOZPVXJMM
OFFEUPDPMMFDUBMMUFSNTPO
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BQQFBSTPOUIFPUIFSTJEF
(a)
(d)
(g)
(j)
(b)
(e)
(h)
(k)
x 2 + 14 x − 120 = 0 x 2 3x 4 = 0 x 2 + 10 x + 25 = 0 x 2 9x
9 x 20 0 (c)
(f)
(i)
(l)
x 2 + 5x
5x
x 2 4x
4x
2
x + 2x
2x
x 2 + 3x
3x
6=0
4=0
8=0
40 0
2 4PMWFFBDIPGUIFGPMMPXJOHFRVBUJPOTCZVTJOHUIFRVBESBUJDGPSNVMB(JWFZPVSBOTXFST
DPSSFDUUPUXPEFDJNBMQMBDFTXIFSFOFDFTTBSZćFTFRVBESBUJDFYQSFTTJPOTEPOPUGBDUPSJTF
(a) 2 + 6x
6x 4 = 0 (d) x 2 6 x 8 = 0
(g) x 2 + 24 x + 121 = 0 (j) x 2 2 x 10 0 (b)
(e)
(h)
(k)
x2 + x − 4 0 x 2 + 10 x + 17 = 0
x 2 + 11x + 27 = 0 x 2 8x 5 = 0 (c)
(f)
(i)
(l)
x 2 + 14 x + 44 = 0
x 2 + 7x
7x 1 = 0
2
x + 18 x − 93 = 0
x 2 2 x − 100 = 0
3 4PMWFFBDIPGUIFGPMMPXJOHFRVBUJPOT(JWFZPVSBOTXFSTDPSSFDUUPUXPEFDJNBMQMBDFT
XIFSFOFDFTTBSZ
1
(a) 2 4 x 8 = 0 (b) 6 x 2 11x 35 0 (c) 2 x 2 3x
3x + = 0
2
2
2
2
(d) 2 x 5x = 25
(e) 4 x 133 9 = 0 (f) 8 x 4 x = 18
2
1 6x (h) x ( x − 16) + 57 = 0 (i) 9 (
2x 2
5x = 3 5x
(k) 6 x 2 13x 6 = 0 (l) 4
(g) 9
(j)
Consecutive numbers are one unit
apart.
x 2 − 14 x + 40 = 0 x 2 8x
8 x 15 0 2
x + 4x
4 x 12 0 x 2 4 x 12 0 1) x 2
2
3=0
4 5XPDPOTFDVUJWFOVNCFSTIBWFBQSPEVDUPG'PSNBRVBESBUJDFRVBUJPOUPĕOEXIBU
UIFUXPOVNCFSTBSF
5 ćFXJEUIPGBQPTUBHFTUBNQJTUXPUIJSETJUTIFJHIUćFQPTUBHFTUBNQJTUPCFFOMBSHFE
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QPTUFSCF
Unit 4: Algebra
95
14 Further solving of equations and inequalities
14.7 Factorising quadratics where the coefficient of x2 is not 1
t ćFDPFďDJFOUPGx JTOPUBMXBZT&YUSBDBSFNVTUCFUBLFOXIFOFYQSFTTJOHTVDIBRVBESBUJDBTBQSPEVDUPG
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Exercise 14.7
Tip
"MXBZTMPPLGPSDPNNPO
GBDUPSTBTUIFĕSTU
TUFQXIFOGBDUPSJTJOH
FYQSFTTJPOT
/PUF − x o x − 1 'BDUPSJTFFBDIPGUIFGPMMPXJOHFYQSFTTJPOT
(a) 2
2
(e) 4
2
(i)
2
3x
3
2
(b) 9
(c) 4 x 2 122
6x + 1 6x
2
3
(f) 14 x − 51x
51x + 7 10 x 25 (j)
3x
2
77xx − 66 (g) 3x
2
(k) 15x
9
11xx 20 11
2
(d) 6 x 2
(h) 6 x
16 x − 15 (l)
8x
7x − 5
2
11x 7
2
25x 3
2 'BDUPSJTFDPNQMFUFMZ:PVNBZOFFEUPSFNPWFBDPNNPOGBDUPSCFGPSFGBDUPSJTJOHUIF
USJOPNJBMT
(a) 4 x 2 122
(d) 6 x 2
2
(c) 50 x 2 + 40x
40 x + 8
2
7 xy 5 y 2 (g) 2 ( x 1) 4 x
(j)
1
(b) 2 x 2 − 9
(e) 4
4x 6 x 2 14x
1 x − 132 14
(h)
4
2
( x + 1)
2
(
(f) 12 x 2
3
+ 3 ( x + 1) + 2 )
(k) 3 x 2 1 + x ( x + 1) 2x 2
(i) 3 ( x 1)
2
(l) 8 x 2
10 ( x + 1) 25
25xy 3 y 2
14.8 Algebraic fractions
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t 'BDUPSJTF JGQPTTJCMF ĕSTU
Exercise 14.8
Tip
"MXBZTMPPLGPSDPNNPO
GBDUPSTĕSTU
1 4JNQMJGZUIFGPMMPXJOHGSBDUJPOT
2x
14
16z
(e)
6
(a)
(i)
3x
x
34 x
(f)
17 xy
(b)
x
7x
12 x 2
(g)
18 xy
(c)
12 x
30
5a
(h)
25a 2b
(d)
24ab2
36a 2b
2 4JNQMJGZUIFGPMMPXJOHGSBDUJPOT
(a)
(e)
(i)
96
Unit 4: Algebra
x2 + x − 6
x 2 2x
a2
5a 14
a 7a
2
35x 2 + 49 x
15x 2 + 21x
(b)
a 2 b2
a + 2abb b2
(c)
2 x 2 xy
x y2
2
x xy
x
(d)
(f)
a 2 b2
a + abb b2
(g)
2 x 2 77xx − 15
2 2
6
(h)
2
2
3x 2 10 x 3
x 2 2 x 15
x4
2x 2 3
x +1
2
14 Further solving of equations and inequalities
Tip
%JWJEJOHCZBGSBDUJPOJTUIF
TBNFBTNVMUJQMZJOHCZUIF
SFDJQSPDBMPGUIFGSBDUJPO
Tip
"OFHBUJWFTJHOJOGSPOU
PGBGSBDUJPOBČFDUTBMMUIF
UFSNTJOUIFOVNFSBUPS
3 8SJUFFBDIPGUIFGPMMPXJOHBTBTJOHMFGSBDUJPOJOJUTMPXFTUUFSNT
(a)
8 x 5x
×
15 16
(b)
3y 2y
×
4
9
(c)
5 7
×
a a
(d)
7 x 2 14 x
÷
5 y 25 y 2
(e)
a ⎛a c⎞
÷
×
b ⎝ b a⎠
(f)
4a 2 b 4 b2 2a
×
÷
2a
3
3b2
(g)
4a 2 b2 3ab
× ÷
7b 8a
1
(h)
66xx 8 x + 3
×
3x 9
2−x
(i)
4 9x 2
6
2
2
4 4JNQMJGZUIFGPMMPXJOH
(a)
3 2
+ x y
(b)
3 4− p 3
−
+
2
8p
4p
(c)
(e)
3
2
+
x + 2 x +1
(f)
2)
2m 3(
−
3
2
(g)
(h)
Mixed exercise
x2
x2
2
3x
−
3
(i)
x2 + 4x 2
1 4PMWFGPSxBOEyJG x
y
2
1
3
−
3
7
+
2p 5p
x2
(d)
5
4
+
p +1 2p 2
5
4
+
x − 6 x2 + x − 2
2
x2 − 1
BOEx 2 y = 12
2 4PMWFGPSxBOEyJG y 4x
4 x = 7BOE2 y 33xx − 4 0
p.a. stands for ‘per annum’, which
means ‘per year’.
3 .S)BCJCIBTUPJOWFTU)JTQPSUGPMJPIBTUXPQBSUT POFXIJDIZJFMETQBBOEUIF
PUIFSQBćFUPUBMJOUFSFTUPOUIFJOWFTUNFOUXBTBUUIFFOEPGUIFĕSTUZFBS)PX
NVDIEJEIFJOWFTUBUFBDISBUF
4 4JNQMJGZUIFGPMMPXJOHJOFRVBMJUJFT
(a) −3 < + 2 (b) 5 − 2x
2 ≥7
5 3FQSFTFOU −2 < ≤ 3 HSBQIJDBMMZ
6 ćFXIPMFOVNCFSTxBOEyTBUJTGZUIFGPMMPXJOHJOFRVBMJUJFT
y > 1 5 y ≤ 4 x ≥ −2 y > x + 1 BOE y ≥ x + 1 (SBQIJDBMMZĕOEUIFHSFBUFTUBOEMFBTUQPTTJCMFJOUFHFSWBMVFTPGxBOEyGPSUIFFYQSFTTJPO
x
+y
2
7 'BDUPSJTF
(a) x 2 2 xy
x 2
(d) 2 y 133 y 7
(b) a 4 b2 (e) −4 x 2 + 2 x + 6 (c) x 2 + 6x
6 x 55
2
(f) ( x + 1) 5 ( x + 1) − 14
Unit 4: Algebra
97
14 Further solving of equations and inequalities
8
4PMWFUIFGPMMPXJOHFRVBUJPOT
(a) 2a 2 22aa − 6 0 (d) 3x 2 5x
5x + 2 0 9
(b) 3 2
(e) 5x 2
4=0
3x = −33 2
5
(c) x 2 + 2x
2 x 15 0
(f) 3x 2 6 x = 6 2 3
6TFUIFRVBESBUJDGPSNVMBUPĕOEUIFWBMVF T PGx
(a) 5x 2
8x − 4 0 8x
(b)
p
(b)
16 − 4 x 2
4 8
(c)
1
1
+
2
5
p
2p
(e)
x ⎛ 2x
z ⎞
÷
×
2 y ⎝ yz 2 xy 3 ⎟⎠
(f)
3a 5 a a
−
+
2
3
5a
2
− qx + r = 0
10 4JNQMJGZUIFGPMMPXJOH
(a)
Unit 4: Algebra
y2
(x + y)
2
(d)
2
7 x 2 15 y
×
5 y 14 x
(g)
4
x2 + 2
(h)
98
x2
3x
6x
2
3x
8
÷
x2
4
2
−
x − 2 x2 + 4x
(i)
2x
2
1
111
5
−
2x
2
1
x 1
15
Scale drawings, bearings
and trigonometry
15.1 Scale drawings
t ćFTDBMFPGBEJBHSBN PSBNBQ DBOCFHJWFOJOUIFGPSNPGBGSBDUJPOPSBSBUJP1
t "TDBMFPGNFBOTUIBUFWFSZMJOFJOUIFEJBHSBNIBTBMFOHUIXIJDIJT 50000 PGUIFMFOHUIPGUIFMJOF
UIBUJUSFQSFTFOUTJOSFBMMJGF'PSFYBNQMFDNJOUIFEJBHSBNSFQSFTFOUTDN PSLN JOSFBMMJGF
Exercise 15.1
REWIND
Revise your metric conversions
from chapter 13. 1 (a) ć
FCBTJDQJUDITJ[FPGBSVHCZĕFMEJTNMPOHBOENXJEF"TDBMFESBXJOHPGB
ĕFMEJTNBEFXJUIBTDBMFPGDNUPN8IBUJTUIFMFOHUIBOEXJEUIPGUIFĕFMEJOUIF
ESBXJOH
(b) ć
FQJUDITJ[F JODMVEJOHUIFBSFBJOTJEFUIFHPBM JTNMPOHBOENXJEF8IBUBSF
UIFTFEJNFOTJPOTJOUIFESBXJOHPGUIJTQJUDI
2 (a) ć
FQJUDITJ[FPGBTUBOEBSEIPDLFZĕFMEJTNMPOHBOENXJEF"TDBMFESBXJOH
PGBIPDLFZĕFMEJTNBEFXJUIBTDBMFPG8IBUBSFUIFEJNFOTJPOTPGUIFIPDLFZ
ĕFMEJOUIFESBXJOH
(b) "
TDIPPMUIBUXBOUTUPIPMEB4FWFO"4JEFIPDLFZUPVSOBNFOUIBTUISFFTUBOEBSEIPDLFZ
ĕFMETBUUIFJS4QPSUT$FOUSF8PVMEJUCFQPTTJCMFUPIBWFĕWFNBUDIFTUBLJOHQMBDFBUUIF
TBNFUJNF JGUIFTJ[FPGUIFQJUDIVTFEGPS4FWFO"4JEFIPDLFZJTN×N
A5 is half A4 and has dimensions
14.8 cm × 21 cm.
3 (a) ć
FTJ[FPGBUFOOJTDPVSUJTN×N8IBUXPVMECFBHPPETDBMFGPSBESBXJOH
PGBUFOOJTDPVSUJGZPVDBOPOMZVTFIBMGPGBO"QBHF &YQSFTTUIJTTDBMFBTBGSBDUJPO
(b) (i) .BLFBOBDDVSBUFTDBMFESBXJOH VTJOHZPVSTDBMF*ODMVEFBMMUIFNBSLJOHTBT
TIPXOJOUIFEJBHSBNCFMPX
(ii) ć
FOFUQPTUTBSFQMBDFENPVUTJEFUIFEPVCMFTTJEFMJOFT.BSLFBDIOFUQPTUXJUI
BO×POZPVSTDBMFESBXJOH
5.5 m
Centre service line
Net
Service line
Base line
Singles side line
1.4 m
Doubles side line
Unit 4: Shape, space and measures
99
15 Scale drawings, bearings and trigonometry
4 (a) ć
FLBSBUFDPNCBUBSFBNFBTVSFTN×N6TJOHBTDBMFESBXJOHBOEBTDBMFPGZPVS
DIPJDF DBMDVMBUFUIFMFOHUIPGUIFEJBHPOBM
(b) 8
IBUXPVMECFBNPSFBDDVSBUFXBZUPEFUFSNJOFUIFMFOHUIPGUIFEJBHPOBM 15.2 Bearings
t "CFBSJOHJTBXBZPGEFTDSJCJOHEJSFDUJPO
t #FBSJOHTBSFNFBTVSFEDMPDLXJTFGSPNUIFOPSUIEJSFDUJPO
t #FBSJOHTBSFBMXBZTFYQSFTTFEVTJOHUISFFĕHVSFT
Exercise 15.2
1 (JWFUIFUISFFĕHVSFCFBSJOHDPSSFTQPOEJOHUP
(a) FBTU
(b) TPVUIXFTU
(c) OPSUIXFTU
2 8SJUFEPXOUIFUISFFĕHVSFCFBSJOHTPGXGSPNY
(a) N
(b)
N
(c)
N
Y
70°
X
45°
Y
130°
X
X
Y
3 7JMMBHF"JTLNFBTUBOELNOPSUIPGWJMMBHF#7JMMBHF$JTLNGSPNWJMMBHF#POB
CFBSJOHPG°6TJOHBTDBMFESBXJOHXJUIBTDBMFPGĕOE
(a)
(b)
(c)
(d)
100
Unit 4: Shape, space and measures
UIFCFBSJOHPGWJMMBHF#GSPNWJMMBHF"
UIFCFBSJOHPGWJMMBHF"GSPNWJMMBHF$
UIFEJSFDUEJTUBODFCFUXFFOWJMMBHF#BOEWJMMBHF"
UIFEJSFDUEJTUBODFCFUXFFOWJMMBHF$BOEWJMMBHF"
15 Scale drawings, bearings and trigonometry
15.3 Understanding the tangent, cosine and sine ratios
t ćFIZQPUFOVTFJTUIFMPOHFTUTJEFPGBSJHIUBOHMFEUSJBOHMF
t ćFPQQPTJUFTJEFJTUIFTJEFPQQPTJUFBTQFDJĕFEBOHMF
t ćFBEKBDFOUTJEFJTUIFTJEFUIBUGPSNTBTQFDJĕFEBOHMFXJUIUIFIZQPUFOVTF
the opposite side
t ćFUBOHFOUSBUJPJT the adjacent side PGBTQFDJĕFEBOHMF
opp( )
opp( )
opp( ) adj(
adj( ) × tanθ adj
d( )=
tanθ
adj
d( )
the opposite side
ćFTJOFSBUJPJT
PGBTQFDJĕFEBOHMF
the hypotenuse
opp( )
opp( )
opp
hyp sinθ hyp
y =
sinθ =
sinθ
hyp
y
the adjacent side
ćFDPTJOFSBUJPJT
PGBTQFDJĕFEBOHMF
the hypotenuse
adj
d( )
adj
d( )
d
hyp cosθ hyp
hyp
y =
cosθ =
adj
cosθ
hyp
y
tanθ =
t
t
hypotenuse
opposite side
θ
adjacent side
Exercise 15.3
1 $PQZBOEDPNQMFUFUIFGPMMPXJOHUBCMF
(a)
(b)
(c)
B
c
A
e
(d)
q
g
a
A
C
b
A
f
A
r
p
hypotenuse
opp(A)
adj(A)
2 $PQZBOEDPNQMFUFUIFTUBUFNFOU T BMPOHTJEFFBDIUSJBOHMF
Remember, when working with
right-angled triangles you may
still need to use Pythagoras.
(a)
60°
x cm
PQQ ° =
BEK ° =
=yDN
30°
y cm
(b)
° = qDN
° =qDN
=pDN
p cm
50°
q cm
40°
Unit 4: Shape, space and measures
101
15 Scale drawings, bearings and trigonometry
The memory aid,
SOHCAHTOA, or the triangle
diagrams
A
O
S
H
C
O
H
T
A
may help you remember the
trigonometric relationships.
3 $BMDVMBUFUIFWBMVFPGUIFGPMMPXJOHUBOHFOUSBUJPT VTJOHZPVSDBMDVMBUPS(JWFZPVSBOTXFST
UPUXPEFDJNBMQMBDFTXIFSFOFDFTTBSZ
(a) UBO°
(d) UBO°
(b) UBO°
(e) UBO°
(c) UBO°
4 $PQZBOEDPNQMFUFUIFTUBUFNFOUTGPSFBDIPGUIFGPMMPXJOHUSJBOHMFT HJWJOHZPVSBOTXFSBTB
GSBDUJPOJOJUTMPXFTUUFSNTXIFSFOFDFTTBSZ
(a)
(b)
(c)
B
A
y
55°
1.2 cm
x
3 cm
d cm
1.8 cm
C
UBOA=
UBOx=
UBOy=
(d)
(e)
B
1 cm
C
A
4 cm
x
UBO°= x=
UBOB=
A
Z 5m X
1.5 cm
12 m
y
C
B
2.5 cm
Y
AC=
UBOB=
UBOC=
UBOy=
∠X=
UBOX=
5 $BMDVMBUFUIFVOLOPXOMFOHUI UPUXPEFDJNBMQMBDFT JOFBDIDBTFQSFTFOUFECFMPX
(a)
(b)
(c)
ym
B
37° c cm
3 cm
C
A
X
72 m 15°
B
65°
A
x cm
Y
C
(d)
B
(e)
am
C
A
B
55°
13 m
45°
A
102
Unit 4: Shape, space and measures
x cm
25 cm
C
2.5 cm
15 Scale drawings, bearings and trigonometry
6 'JOE DPSSFDUUPPOFEFDJNBMQMBDF UIFBDVUFBOHMFTUIBUIBWFUIFGPMMPXJOHUBOHFOUSBUJPT
(a) (b) 1
4
(e)
(c) 13
15
(f)
(g)
(d) 1
5 2
(h)
61
63
7 'JOE DPSSFDUUPUIFOFBSFTUEFHSFF UIFWBMVFPGUIFMFUUFSFEBOHMFTJOUIFGPMMPXJOHEJBHSBNT
(a)
(b)
(c)
A
A
13.5 m
a
c
B
2.4 cm
15 cm
13 m
b
B
17 cm
C
0.7 cm
C
(d)
(e)
Z
5m
A
X
55 cm
12 m
d
e
C
70 cm
B
Y
8 'PSFBDIUSJBOHMFĕOE
(i) IZQ=
(a)
(ii) BEK θ =
(b)
(c)
B
y
θ
x
a
z
(iii) DPTθ=
(d)
c
b
a
θ
(e)
P
C
b
θ
Q
θ
X
q
r
c
y
Z
x
p
R
A
z
θ
Y
9 'PSFBDIPGUIFGPMMPXJOHUSJBOHMFTXSJUFEPXOUIFWBMVFGPS
(i) TJOF
(a)
(ii) DPTJOF
(b)
(c)
22
13
7
(iii) UBOHFOUPGUIFNBSLFEBOHMF
(d)
B
(e)
13 E
16
4x
5x
D
63
85
C
A
12
10
Unit 4: Shape, space and measures
103
15 Scale drawings, bearings and trigonometry
10
6TFZPVSDBMDVMBUPSUPĕOEUPUIFOFBSFTUEFHSFF
(a) BOBDVUFBOHMFXIPTFTJOFJT
(b) BOBDVUFBOHMFXIPTFDPTJOFJT
(c) BOBDVUFBOHMFXIPTFDPTJOFJT
3
2
1
BOBDVUFBOHMFXIPTFTJOFJT
2
(d) BOBDVUFBOHMFXIPTFTJOFJT
(e)
(f) BOBDVUFBOHMFXIPTFUBOHFOUJT
11 ćFEJBHSBNTIPXTUXPMBEEFSTQMBDFEJOBOBMMFZ CPUISFBDIJOHVQUPXJOEPXMFEHFTPO
PQQPTJUFTJEFTPOFJTUXJDFBTMPOHBTUIFPUIFSBOESFBDIFTIFJHIUH XIJMFUIFTIPSUFSPOF
SFBDIFTBIFJHIUhćFMFOHUIPGUIFMPOHFSMBEEFSJTN*GUIFBOHMFTPGJODMJOBUJPOPGUIF
MBEEFSTBSFBTTIPXO IPXNVDIIJHIFSJTUIFXJOEPXMFEHFPGUIFPOFXJOEPXUIBOUIBUPG
UIFPUIFS
7.5 m
H
h
61º
27º
Tip
12 'PSFBDIPGUIFGPMMPXJOHUSJBOHMFTĕOEUIFMFOHUIPGUIFVOLOPXOMFUUFSFETJEF DPSSFDUUP
UXPEFDJNBMQMBDFT PSUIFTJ[FPGUIFMFUUFSFEBOHMF DPSSFDUUPPOFEFDJNBMQMBDF 3FNFNCFS POMZSPVOE
ZPVSXPSLJOHBUUIFĕOBM
TUFQ
(a)
(b)
(c)
Q
A
A
12
R
5 cm
cm
3x m
52°
y cm
C
D
x°
E
xm
C
104
Unit 4: Shape, space and measures
y°
9m
O
B
6 cm
P 3 cm S
E
B
x°
D
75°
m
15 m
C
c
8.6
6 cm
B
(d)
D
cm
x°
A
45°
5.6 cm
E
15 Scale drawings, bearings and trigonometry
15.4 Solving problems using trigonometry
t *GOPEJBHSBNJTHJWFO ESBXPOFZPVSTFMG
t .BSLUIFSJHIUBOHMFTJOUIFEJBHSBN
t 4IPXBMMUIFHJWFONFBTVSFNFOUT
Tip
Exercise 15.4
.BLFTVSFZPVLOPXUIBU
1 "UBDFSUBJOUJNFPGUIFEBZ4BNVFMTTIBEPXJTNMPOHćFBOHMFPGFMFWBUJPOGSPNUIF
FOEPGIJTTIBEPXUPUIFUPQPGIJTIFBEJT¡)PXUBMMJT4BNVFM
horizontal
angle of depression
2 4BSBITMJOFPGTJHIUJTNBCPWFUIFHSPVOE4IFJTMPPLJOHBUUIFUPQPGBUSFFUIBUJTN
BXBZ
lin
eo
fs
igh
t
line
ight
of s
angle of
elevation
(a) *GUIFUSFFJTNUBMM XIBUJTUIFBOHMFPGFMFWBUJPOUISPVHIXIJDITIFJTMPPLJOH
DPSSFDUUPUIFOFBSFTUEFHSFF
(b) "CJSETJUTPOUPQPGUIFUSFF$BMDVMBUF DPSSFDUUPUIFOFBSFTUEFHSFF UIFBOHMFPG
EFQSFTTJPOGSPNUIFCJSEUP4BSBIhTGFFU
horizontal
15.5 Angles between 0° and 180°
t *GUXPBOHMFTTVN BEEVQ UP¡UIFZBSFTBJEUPCFTVQQMFNFOUBSZ
t "OBOHMFBOEJUTTVQQMFNFOUIBWFUIFTBNFTJOFWBMVF TJOθ=TJO ¡−θ t ćFDPTJOFPGBOBOHMFBOEJUTTVQQMFNFOUIBWFUIFTBNFWBMVFCVUEJČFSFOUTJHOT DPTθoDPT ¡oθ t ćFUBOHFOUPGBOBOHMFBOEJUTTVQQMFNFOUIBWFUIFTBNFWBMVFCVUEJČFSFOUTJHOT UBOθoUBO ¡oθ Exercise 15.5
1 &YQSFTTFBDIPGUIFGPMMPXJOHJOUFSNTPGUIFTVQQMFNFOUBSZBOHMFCFUXFFO¡BOE¡
Tip
(a) DPT¡
(e) −DPT¡
(i) DPT¡
TJOθTJO ¡oθ DPTθ –DPT ¡oθ
(b) TJO¡
(f) TJO¡
(j) DPT¡
(c) −DPT¡
(g) DPT¡
(d) TJO¡
(h) TJO¡
2 (JWFOUIBUθJTBOBOHMFPGBUSJBOHMF ĕOEBMMQPTTJCMFWBMVFTPGθCFUXFFO¡BOE¡ UPUIF
OFBSFTUEFHSFF JG
(a) TJOθ
(e) TJOθ
(i) TJOθ
(b) DPTθ−
(f) DPTθ
(j) TJOθ
(c) TJOθ
(g) DPTθ
15.6 The sine and cosine rules
t ćFTJOFBOEDPTJOFSVMFTDBOCFVTFEJOUSJBOHMFTUIBUEPOPUIBWFBSJHIUBOHMF
s B sin C
a
b
c
=
=
=
PS
t ćFTJOFSVMF sina A = sin
b
c
sin A sin
s B sin C
t ćFDPTJOFSVMF a b + c bc cos A PS b a + c 2ac cos B PS c = a + b
2
2
2
2
2
2
2
2
2
(d) DPTθ
(h) DPTθ−
ab cosC
Unit 4: Shape, space and measures
105
15 Scale drawings, bearings and trigonometry
Tip
Exercise 15.6
3FNFNCFS UIFTUBOEBSE
XBZPGMBCFMMJOHBUSJBOHMF
JTUPMBCFMUIFWFSUJDFTXJUI
VQQFSDBTF DBQJUBM MFUUFST
BOEUIFTJEFTPQQPTJUFXJUI
UIFTBNFMPXFSDBTFMFUUFST
1 5PPOFEFDJNBMQMBDFĕOEUIFWBMVFPGxJOFBDIPGUIFGPMMPXJOHFRVBUJPOT
Tip
ćFTJOFSVMFJTVTFEXIFO
EFBMJOHXJUIQBJSTPG
PQQPTJUFTJEFTBOEBOHMFT
(a)
sin
sin 45°
=
11
12
(b)
x
23
=
sin 50° sin 72°
(c)
sin x sin 38°
=
7.4
5.2
(d)
x
30
=
sin 35° sin 71°
(e)
x
8
=
(f)
sin 55° sin 68°
sin
sin 45°
=
4
6
(g)
sin
sin 35°
=
24
36
x
8.5
=
sin 59° sin 62°
(h)
sin
sin105°
=
4
16
2 'JOEUIFMFOHUIPGUIFNBSLFETJEFJOFBDIPGUIFGPMMPXJOHUSJBOHMFT
(i)
(a)
(b)
(c)
7.7 cm
7.7 cm
57.2°
42°
5.5 cm
x cm
114.5°
x cm
54.3°
72°
(d)
37°
x cm
(e)
(f)
4.9 cm
25.5°
81.7° 71.5°
45°
x cm
x cm
23.4 cm
9.7 cm
24°
x cm
60°
3 'JOEUIFTJ[FPGUIF BDVUF NBSLFEBOHMFJOUIFGPMMPXJOHUSJBOHMFT
(a)
(b)
(c)
θ
9.5
5.4 cm
4.8 cm
129°
42°
θ
cm
9.
9
θ
8 cm
cm
104°
5.6 cm
(d)
(e)
5.5
cm
124°
θ
10 cm
(f)
θ
18.2 cm
7.5 cm
78°
40°
5.2 cm
11.8 cm
θ
106
Unit 4: Shape, space and measures
15 Scale drawings, bearings and trigonometry
4 (JWFO b2
a2 + c 2
2ac cos B DPQZBOEDPNQMFUFUIFGPMMPXJOHFRVBUJPO
DPTByyyyyyy
5 'JOEUIFTJ[FPGUIFBOHMFNBSLFEθJOFBDIPGUIFUSJBOHMFTCFMPX DPSSFDUUPPOF
EFDJNBMQMBDF (a)
(b)
3 cm
3.5 cm
2.4 cm
3 cm
θ
θ
2.4 cm
2.4 cm
(c)
(d)
θ
4.7 cm
2.4 cm
5 cm
2.8 cm
θ
4.1 cm
Tip
ćFDPTJOFSVMFJTVTFE
XIFOBMMUISFFTJEFTBSF
LOPXOPSXIFOZPVLOPX
UXPTJEFTBOEUIFJODMVEFE
BOHMF
3.2 cm
6 'JOEUIFUIJSETJEFPGUIFGPMMPXJOHUSJBOHMFT DPSSFDUUPPOFEFDJNBMQMBDF (a)
(b)
A
5.6 m
X
25°
6.8 cm
z
6.7 m
31°
Y
B
a
Z
7 cm
C
(c)
(d)
O
P
4.7 cm
m
r
7 cm
131°
5.4 cm
N
M
Q
43.5°
3.8 cm
R
Unit 4: Shape, space and measures
107
15 Scale drawings, bearings and trigonometry
15.7 Area of a triangle
t :PVDBOĕOEUIFBSFBPGBUSJBOHMFFWFOJGZPVEPOPUIBWFBQFSQFOEJDVMBSIFJHIU BTMPOHBTZPVIBWFUXPTJEFTBOE
UIFJODMVEFEBOHMF
t "SFBPGUSJBOHMFABC area = 12 ab sin C PS area = 12 ac sin B PSarea = 12 bc sin A Exercise 15.7
Tip
5PĕOEUIFBSFBPGB
USJBOHMFZPVOFFEUXP
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1 Draw a rough sketch of each of these figures before calculating their area.
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x
2.5 cm
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7 cm
cm
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Tip
74°
65°
5 cm
6 cm
15.8 Trigonometry in three dimensions
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Exercise 15.8
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3 cm
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H
4 cm
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108
Unit 4: Shape, space and measures
9 cm
F
(a)
(c)
(e)
(g)
˘ DEF
DFE
F HF
CH
(b)
(d)
(f)
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15 Scale drawings, bearings and trigonometry
2 ćFEJBHSBNTIPXTBTRVBSFCBTFEQZSBNJE
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15 m
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Mixed exercise
1 'JOEUIFWBMVFPGx
(a)
sin x sin121°
=
5. 2
7. 3
(b)
3.5 cm
5 cm
x
4 cm
(c)
11.5 cm
7.7 cm
39.4°
x
Unit 4: Shape, space and measures
109
15 Scale drawings, bearings and trigonometry
Tip
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A
50°
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21°
C
530 m
D
4 BDDN'JOEAC
A
B
70°
C
50°
50 cm
D
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X
Y
1500 m
77°
R
110
Unit 4: Shape, space and measures
S
58°
T
16
Scatter diagrams
and correlation
16.1 Introduction to bivariate data
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Exercise16.1
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A
B
D
E
(a) XFBLOFHBUJWF
(d) TUSPOHOFHBUJWF
(b) OPDPSSFMBUJPO
(e) XFBLQPTJUJWF
C
(c) TUSPOHQPTJUJWF
Unit 4: Data handling
111
16
Scatter diagrams and correlation
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Student heights compared to distance jumped
Distance jumped (m)
6.0
5.5
5.0
4.5
4.0
150
155
160 165 170
Height (cm)
175
180
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(c) '
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Age (years)
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(b)
(c)
(d)
(e)
(f)
(g)
112
Unit 4: Data handling
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16
Accidents at a road junction
Number of accidents
50
40
30
20
10
0
0
35
70 105 140
Average speed (km/h)
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Comparison of car age with re-sale price
15
Price ($ 000)
Mixed exercise
Scatter diagrams and correlation
12
9
6
3
0
1
2
3
4
Age (years)
5
(a)
(b)
(c)
(d)
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Unit 4: Data handling
113
17
Managing money
17.1 Earning money
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Exercise 17.1
Casual workers are normally paid
an hourly rate for the hours they
work.
Tip
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Some workers are paid for each
piece of work they complete. This
is called piece work.
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114
Unit 5: Number
17 Managing money
17.2 Borrowing and investing money
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Exercise 17.2
‘per annum’ (p.a.) means
‘per year’
Tip
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I=
PRT
100
P=
100I
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R=
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T=
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(a)
(b)
(c)
(d)
(e)
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Unit 5: Number
115
17 Managing money
Using the multiplying factor for
compound interest gives the new
amount. Subtract the principal to
find the actual interest.
6 $BMDVMBUFUIFDPNQPVOEJOUFSFTUPO
(a)
(b)
(c)
(d)
(e)
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17.3 Buying and selling
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profit (or loss)
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×100%
cost price
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l selling
ll
price
Exercise 17.3
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(a) TFMMJOHQSJDF QSPĕU
(b) TFMMJOHQSJDF QSPĕU
(c) TFMMJOHQSJDF MPTT
1
(d) TFMMJOHQSJDF MPTT33 3
2 'JOEUIFDPTUQSJDFPGBOBSUJDMFTPMEBUXJUIBQSPĕUPG
3 *GBTIPQLFFQFSTFMMTBOBSUJDMFGPSBOEMPTFTPOUIFTBMF ĕOEIJTDPTUQSJDF
4 "EFOUJTUPČFSTBEJTDPVOUUPQBUJFOUTXIPQBZUIFJSBDDPVOUTJODBTIXJUIJOBXFFL
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116
Unit 5: Number
17 Managing money
5 $BMDVMBUFUIFOFXTFMMJOHQSJDFPGFBDIJUFNXJUIUIFGPMMPXJOHEJTDPVOUT
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(b) EJTDPVOU
1
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Years
1
2
Simple interest
Compound interest
3
4
5
6
7
8
(a) $PQZBOEDPNQMFUFUIFUBCMF
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ĕWFZFBST
(c) %
SBXBCBSDIBSUUPDPNQBSFUIFWBMVFPGUIFJOWFTUNFOUBęFSPOF ĕWFBOEZFBST
GPSCPUIUZQFTPGJOUFSFTU$PNNFOUPOXIBUZPVSHSBQITIPXTBCPVUUIFEJČFSFODF
CFUXFFOTJNQMFBOEDPNQPVOEJOUFSFTU
5 'JOEUIFTFMMJOHQSJDFPGBOBSUJDMFUIBUXBTCPVHIUGPSBOETPMEBUBQSPĕUPG
6 $BMDVMBUFUIFTFMMJOHQSJDFPGBOJUFNPGNFSDIBOEJTFCPVHIUGPSBOETPMEBUB
QSPĕU‫ڀ‬PG‫ڀ‬
Unit 5: Number
117
17 Managing money
7 "HBMMFSZPXOFSEJTQMBZTQBJOUJOHTGPSBSUJTUT4IFQVUTBNBSLVQPOUIFQSJDFBTLFE
CZUIFBSUJTUUPDPWFSIFSFYQFOTFTBOENBLFBQSPĕU"OBSUJTUTVQQMJFTUISFFQBJOUJOHTBU
UIFQSJDFTMJTUFECFMPX'PSFBDIPOF DBMDVMBUFUIFNBSLVQJOEPMMBST BOEUIFTFMMJOHQSJDF
UIFHBMMFSZPXOFSXPVMEDIBSHF
(a) 1BJOUJOH" (b) 1BJOUJOH# (c) 1BJOUJOH$ 8 "OBSUDPMMFDUPSXBOUTUPCVZQBJOUJOHT"BOE# GSPNRVFTUJPO )FBHSFFTUPQBZ
DBTIPODPOEJUJPOUIBUUIFHBMMFSZPXOFSHJWFTIJNBEJTDPVOUPOUIFTFMMJOHQSJDFPG
UIF‫ڀ‬QBJOUJOHT
(a) 8IBUQSJDFXJMMIFQBZ
(b) 8IBUQFSDFOUBHFQSPĕUEPFTUIFHBMMFSZPXOFSNBLFPOUIFTBMF
9 "CPZCPVHIUBCJDZDMFGPS"ęFSVTJOHJUGPSUXPZFBST IFTPMEJUBUBMPTTPG
$BMDVMBUFUIFTFMMJOHQSJDF
10 *UJTGPVOEUIBUBOBSUJDMFJTCFJOHTPMEBUBMPTTPGćFDPTUPGUIFBSUJDMFXBT
$BMDVMBUFUIFTFMMJOHQSJDF
11 "XPNBONBLFTESFTTFT)FSUPUBMDPTUTGPSUFOESFTTFTXFSF"UXIBUQSJDFTIPVMETIF
TFMMUIFESFTTFTUPNBLFQSPĕU
12 4BMXBOUTUPCVZBVTFETDPPUFSćFDBTIQSJDFJT5PCVZPODSFEJU TIFIBTUPQBZ
BEFQPTJUBOEUIFONPOUIMZJOTUBMNFOUTPGFBDI)PXNVDIXJMMTIFTBWFCZ
QBZJOHDBTI
118
Unit 5: Number
18
Curved graphs
18.1 Plotting quadratic graphs (the parabola)
t ćFIJHIFTUQPXFSPGBWBSJBCMFJOBRVBESBUJDFRVBUJPOJTUXP
t ćFHFOFSBMGPSNVMBGPSBRVBESBUJDHSBQIJTy=ax +bx+c
t ćFBYJTPGTZNNFUSZPGUIFHSBQIEJWJEFTUIFQBSBCPMBJOUPUXPTZNNFUSJDBMIBMWFT
t ćFUVSOJOHQPJOUJTUIFQPJOUBUXIJDIUIFHSBQIDIBOHFTEJSFDUJPOćJTQPJOUJTBMTPDBMMFEUIFWFSUFYPGUIFHSBQI
o *GUIFWBMVFPGaJOUIFHFOFSBMGPSNPGBRVBESBUJDFRVBUJPOJTQPTJUJWF UIFQBSBCPMBXJMMCFBAWBMMFZTIBQFBOEUIF
yWBMVFPGUIFUVSOJOHQPJOUBNJOJNVNWBMVF
o *GUIFWBMVFPGaJOUIFHFOFSBMGPSNPGBRVBESBUJDFRVBUJPOJTOFHBUJWF UIFQBSBCPMBXJMMCFBAIJMMTIBQFBOEUIF
yWBMVFPGUIFUVSOJOHQPJOUBNBYJNVNWBMVF
Exercise 18.1
Remember, the constant term
(c in the general formula) is the
y-intercept.
1 $PQZBOEDPNQMFUFUIFGPMMPXJOHUBCMFT1MPUBMMUIFHSBQITPOUPUIFTBNFTFUPGBYFT6TF
WBMVFTPG−UPPOUIFyBYJT
(a)
x
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
y = −x + 2
2
(b)
x
y = x2 − 3
(c)
x
y = −x − 2
2
(d)
x
y = −x2 − 3
(e)
x
y = x2 +
1
2
Unit 5: Algebra
119
18 Curved graphs
2 .BUDIFBDIPGUIFĕWFQBSBCPMBTTIPXOIFSFUPPOFPGUIFFRVBUJPOTHJWFO
13
12
11
10
9
8
7
6
5
4
3
2
1
–4
y
4 x2
–3
–2
(a)
(b)
(c)
x
–1 –10
–2
–3
–4
–5
–6
–7
–8
–9
–10
–11
–12
–13
x2 + 3
y
y
3
4
(d)
(e)
3 + x2
y
2
1
y = x2 + 2
y
x2
3 $PQZBOEDPNQMFUFUIFUBCMFPGWBMVFTGPSFBDIPGUIFFRVBUJPOTHJWFOCFMPX1MPUUIFQPJOUT
POTFQBSBUFQBJSTPGBYFTBOEKPJOUIFN XJUIBTNPPUIDVSWF UPESBXUIFHSBQIPGUIF
FRVBUJPO
(a)
x
−2
−1
1
0
2
3
4
5
y = x − 3x + 2
2
(b)
x
−3
−2
0
−1
1
2
3
y = x2 − 2x − 1
(c)
x
−2
y = −x + 4x + 1
2
120
Unit 5: Algebra
−1
0
1
2
3
4
5
6
18 Curved graphs
4 "UPZSPDLFUJTUISPXOVQJOUPUIFBJSćFHSBQICFMPXTIPXTJUTQBUI
Height in metres
10
8
6
4
2
0
(a)
(b)
(c)
(d)
(e)
0
1 2 3 4
Time in seconds
5
8IBUJTUIFHSFBUFTUIFJHIUUIFSPDLFUSFBDIFT
)PXMPOHEJEJUUBLFGPSUIFSPDLFUUPSFBDIUIJTIFJHIU
)PXIJHIEJEUIFSPDLFUSFBDIJOUIFĕSTUTFDPOE
'PSIPXMPOHXBTUIFSPDLFUJOUIFBJS
&TUJNBUFGPSIPXMPOHUIFSPDLFUXBTIJHIFSUIBONBCPWFHSPVOE
18.2 Plotting reciprocal graphs (the hyperbola)
t ćFHFOFSBMGPSNVMBGPSBIZQFSCPMBHSBQIJTy = ax PSxy = a
t xȶBOEyȶ
t ćFxBYJTBOEUIFyBYJTBSFBTZNQUPUFTćJTNFBOTUIFHSBQIBQQSPBDIFTUIFxBYJTBOEUIFyBYJTCVUOFWFS
JOUFSTFDUTXJUIUIFN
t ćFHSBQIJTTZNNFUSJDBMBCPVUUIFy=xBOEy=−xMJOF
Tip
Exercise 18.2
3FDJQSPDBMFRVBUJPOTIBWF
BDPOTUBOUQSPEVDUćJT
NFBOTJOxy=a xBOE
yBSFWBSJBCMFTCVUaJTB
DPOTUBOU
1 %SBXHSBQITGPSUIFGPMMPXJOHSFDJQSPDBMHSBQIT1MPUBUMFBTUUISFFQPJOUTJOFBDIPGUIFUXP
RVBESBOUTBOEKPJOUIFNVQXJUIBTNPPUIDVSWF
(a) xy = 5 (d) y = −
8
x
16
x 4
y=−
x
(b) y =
(e)
(c) xy = 9
2 ćFMFOHUIBOEXJEUIPGBDFSUBJOSFDUBOHMFDBOPOMZCFBXIPMFOVNCFSPGNFUSFTćFBSFB
PGUIFSFDUBOHMFJTN2
(a) %SBXBUBCMFUIBUTIPXTBMMUIFQPTTJCMFDPNCJOBUJPOTPGNFBTVSFNFOUTGPSUIFMFOHUI
BOEXJEUIPGUIFSFDUBOHMF
(b) 1MPUZPVSWBMVFTGSPN a BTQPJOUTPOBHSBQI
(c) +PJOUIFQPJOUTXJUIBTNPPUIDVSWF8IBUEPFTUIJTHSBQISFQSFTFOU
(d) "
TTVNJOH OPX UIBUUIFMFOHUIBOEXJEUIPGUIFSFDUBOHMFDBOUBLFBOZQPTJUJWFWBMVFT
UIBUHJWFBOBSFBPGN2 VTFZPVSHSBQIUPĕOEUIFXJEUIJGUIFMFOHUIJTN
Unit 5: Algebra
121
18 Curved graphs
18.3 Using graphs to solve quadratic equations
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XIFSFy=
t 5PTPMWFRVBESBUJDFRVBUJPOTHSBQIJDBMMZ SFBEPČUIFxDPPSEJOBUFTPGUIFQPJOUTGPSBHJWFOyWBMVF
Exercise 18.3
1 6TFUIJTHSBQIPGUIFSFMBUJPOTIJQ y
8
6
x 2 − x − 6 UPTPMWFUIFGPMMPXJOHFRVBUJPOT
y = x2 − x − 6
4
2
Tip
'PSQBSU(c)ZPVNJHIUĕOE
JUIFMQGVMUPSFBSSBOHFUIF
FRVBUJPOTPUIFMFęIBOE
TJEFNBUDIFTUIFFRVBUJPO
PGUIFHSBQI JFTVCUSBDU
GSPNCPUITJEFT
–4 –3 –2 –1 0
–2
1
2
3
4
5
–4
–6
–8
(a) x 2
x −6 0
(b) x 2
2 (a) %SBXUIFHSBQIPG y
x −6
4
(c) x 2
x = 12
2
x − x + 2 GPSWBMVFTPGxGSPN−UP
(b) 6TFZPVSHSBQIUPĕOEUIFBQQSPYJNBUFTPMVUJPOTUPUIFFRVBUJPOT
(i) − x 2 − x + 2 = 0
(ii) − x 2 − x + 2 = 1
(iii) − x 2 − x + 2 = −2
3 (a) 6TFBOJOUFSWBMPG¦ȳx ȳPOUIFxBYJTUPESBXUIFHSBQI
x 2 − x − 6
(b) 6TFUIFHSBQIUPTPMWFUIFGPMMPXJOHFRVBUJPOT
(i) −6 = 2 − − 6
(ii) x 2 x − 6 0
(iii) x 2 x = 12
18.4 Using graphs to solve simultaneous linear and non-linear equations
t ćFTPMVUJPOJTUIFQPJOUXIFSFUIFHSBQITJOUFSTFDU
Exercise 18.4
1 %SBXUIFTFQBJSTPGHSBQITBOEĕOEUIFQPJOUTXIFSFUIFZJOUFSTFDU
4
BOE y 2 x + 2
x
(b) y = x 2 + 2 x − 3 BOE y
3
(c) y
x 2 + 4 BOE y =
x
(a) y =
122
Unit 5: Algebra
x +1
18 Curved graphs
2 6TFBHSBQIJDBMNFUIPEUPTPMWFUIFGPMMPXJOHFRVBUJPOTTJNVMUBOFPVTMZ
(a) y 2 x 2 + 3x − 2 BOE y = x + 2
x+4
(b) y = x 2 + 2 x BOE y
2 x 2 + 2 x + 4 BOE y
2x − 4
(c) y
1
0 5x 2 + x + 1.5 BOE y
(d) y
x
2
18.5 Other non-linear graphs
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t
t
t
t
o *GxJTQPTJUJWF UIFOxJTQPTJUJWFBOEox3JTOFHBUJWF
o *GxJTOFHBUJWF UIFOxJTOFHBUJWFBOEox3JTQPTJUJWF
$VCJDFRVBUJPOTQSPEVDFHSBQITDBMMFEDVCJDDVSWFT
ćFHFOFSBMGPSNPGBDVCJDFRVBUJPOJTyax3 bx2 cx d
&YQPOFOUJBMHSPXUIJTGPVOEJONBOZSFBMMJGFTJUVBUJPOT
ćFHFOFSBMFRVBUJPOGPSBOFYQPOFOUJBMGVODUJPOJTyxa
Exercise 18.5
Tip
1 $POTUSVDUBUBCMFPGWBMVFTGSPNoȳxȳBOEQMPUUIFQPJOUTUPESBXHSBQITPGUIF
GPMMPXJOHFRVBUJPOT
#FGPSFESBXJOHZPVS
HSBQITDIFDLUIFSBOHFPG
yWBMVFTJOZPVSUBCMFPG
WBMVFT
(a) y
(d) y
(g) y
2
Tip
3FBSSBOHFUIFFRVBUJPO
TPUIBUUIF-)4PGUIF
FRVBUJPOJTFRVJWBMFOU
UPUIFHJWFOFRVBUJPOUP
IFMQZPVTPMWFUIFOFX
FRVBUJPO
Tip
8IFOZPVIBWFUPQMPU
HSBQITPGFRVBUJPOT
XJUIBDPNCJOBUJPOPG
MJOFBS RVBESBUJD DVCJD SFDJQSPDBMPSDPOTUBOU
UFSNTZPVOFFEUPESBX
VQBUBCMFPGWBMVFTXJUI
BUMFBTUFJHIUWBMVFTPGxUP
HFUBHPPEJOEJDBUJPOPG
UIFTIBQFPGUIFHSBQI
x
x3 − 4x2 x3 + 4x2 − 5 x 3 − 3x 2 + 6 y = x3 + 5 y = x 3 + 2 x − 10 y
3x 3 + 5x
(b)
(e)
(h)
(c)
(f)
B $PQZBOEDPNQMFUFUIFUBCMFPGWBMVFTGPSUIFFRVBUJPOy
o
–2
o
–1
o
0
1
2
y
y
2 x 3 + 5x 2 + 5
2x 3 + 4 x 2 − 7
x 3 − 5x 2 + 10
3
4
5
6
y
(b) 0OBTFUPGBYFT ESBXUIFHSBQIPGUIFFRVBUJPO y x 3 − 5x 2 + 10 GPS−≤x≤
(c) 6TFUIFHSBQIUPTPMWFUIFFRVBUJPOT
(i) x 3 5x 2 10 0 (ii) x 3 5x 2 10 10
3
2
(iii) x 5x 10 x 5
3 $POTUSVDUBUBCMFPGWBMVFTGPSoȳxȳGPSFBDIPGUIFGPMMPXJOHFRVBUJPOTBOEESBX
UIFHSBQIT
1
1
4
(a) y x − (b) y = x 3 + (c) y = x 2 + 2 −
x
x
x
2
1
3
(d) y 2 x + (e) y x 3 − (f) y x 2 − x +
x
x
x
4 0OUPUIFTBNFTFUPGBYFTESBXUIFHSBQITPG
(a) y = 2 x BOE y = 2 − x GPSoȳxȳ
(b)
= 10 x BOE y 2 x − 1 GPSȳxȳ
Unit 5: Algebra
123
18 Curved graphs
5 "TJOHMFDFMMFEBMHBFIBTCFFOEJTDPWFSFEUIBUTQMJUTJOUPUISFFTFQBSBUFDFMMTFWFSZIPVS
(a) 3FQSFTFOUUIFĕSTUĕWFIPVSTPGHSPXUIPOBHSBQI
(b) (i) 0OUPUIFTBNFTFUPGBYFTVTFEGPS a TLFUDIUIFHSBQIPG 12 x + 1 XIJDI
SFQSFTFOUTUIFHSPXUIPGBOPUIFSTJOHMFDFMMFEPSHBOJTNXJUIBĕYFESBUFPGHSPXUI
(ii) 8IBUJTUIFHSPXUISBUFPGUIFTFDPOEPSHBOJTN
(c) 6TFZPVSHSBQIUPBOTXFSUIFGPMMPXJOH
(i) AęFSIPXNBOZIPVSTBSFUIFSFFRVBMOVNCFSTPGFBDIPSHBOJTN
(ii) HPXNBOZDFMMTPGFBDIPSHBOJTNJTUIJT
18.6 Finding the gradient of a curve
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t ćFHSBEJFOUPGBDVSWFBUBQPJOUJTFRVBMUPUIFHSBEJFOUPGUIFUBOHFOUUPUIFDVSWFBUUIBUQPJOU
y change
vertical change
t (SBEJFOU = x change = horizontal change Exercise 18.6
1 %SBXUIFHSBQIPG y
x 2 − 2 x − 8'JOEUIFHSBEJFOUPGUIFHSBQI
(a) BUUIFQPJOUXIFSFUIFDVSWFJOUFSTFDUTXJUIUIFyBYJT
(b) BUFBDIPGUIFQPJOUTXIFSFUIFDVSWFJOUFSTFDUTXJUIUIFxBYJT
2 (a) %SBXUIFHSBQIPG y
x 3 − 1 GPSoȳxȳ
(b) 'JOEUIFHSBEJFOUPGUIFDVSWFBUUIFQPJOUA 3 ćFHSBQITIPXTIPXUIFSBUFPGHSPXUIPGBCFBOQMBOUJTBČFDUFECZTVOMJHIU'JOEUIFSBUF
PGHSPXUIXJUIBOEXJUIPVUMJHIUPOUIFUIJSUFFOUIEBZ
Growth rate of beans
30
y
Growth in cm
25
20
with light
15
without light
10
5
x
0
124
Unit 5: Algebra
5
Days
10
15
18 Curved graphs
Mixed exercise
1
(a) 8SJUFBOFRVBUJPOGPSFBDIPGUIFHSBQITBCPWF A BBOEC
(b) (i) 4PMWFFRVBUJPOTABOECTJNVMUBOFPVTMZ
(ii) )PXXPVMEZPVDIFDLZPVSTPMVUJPOHSBQIJDBMMZ
(c) 8IBUJTUIFNBYJNVNWBMVFPGB
2 (a) %SBXUIFHSBQITPGUIFGPMMPXJOHFRVBUJPOTPOUIFTBNFHSJE y
x 2BOE y
x 3
(b) 6TFZPVSHSBQIUPTPMWFUIFUXPFRVBUJPOTTJNVMUBOFPVTMZ
(c) #ZESBXJOHBTVJUBCMFTUSBJHIUMJOF PSMJOFT POUPUIFTBNFHSJE TPMWFUIFFRVBUJPOT
(i) 2 = 4
(ii) x 3 + 8 0
(d) 'JOEUIFSBUFPGDIBOHFGPSFBDIHSBQIBUUIFQPJOUTXIFSFx
3 ćFEPUUFEMJOFPOUIFHSJECFMPXJTUIFBYJTPGTZNNFUSZGPSUIFHJWFOIZQFSCPMB
y
8
7
6
5
4
3
2
1
x
0 1 2 3 4 5 6 7 8
–8 –7 –6–5–4–3 –2–1
–1
–2
–3
–4
–5
–6
–7
–8
(a) (JWFUIFFRVBUJPOGPSUIFIZQFSCPMB
(b) (JWFUIFFRVBUJPOGPSUIFHJWFOMJOFPGTZNNFUSZ
(c) $
PQZUIFEJBHSBNBOEESBXJOUIFPUIFSMJOFPGTZNNFUSZ HJWJOHUIFFRVBUJPOGPS
UIJTMJOF
Unit 5: Algebra
125
19
Symmetry and loci
19.1 Symmetry in two dimensions
t 5XPEJNFOTJPOBM ĘBU TIBQFTIBWFMJOFTZNNFUSZJGZPVBSFBCMFUPESBXBMJOFUISPVHIUIFTIBQFTPUIBU
t
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SPUBUJPOJTDBMMFEUIFPSEFSPGSPUBUJPOBMTZNNFUSZ
If a shape can only fit back into
itself after a full 360° rotation, it
has no rotational symmetry.
Exercise 19.1
1 'PSFBDIPGUIFGPMMPXJOHTIBQFT
(a) DPQZUIFTIBQFBOEESBXJOBOZMJOFTPGTZNNFUSZ
(b) EFUFSNJOFUIFPSEFSPGSPUBUJPOBMTZNNFUSZ
A
B
E
D
G
C
F
H
2 (a) )
PXNBOZMJOFTPGTZNNFUSZEPFTBSIPNCVTIBWF %SBXBEJBHSBNUPTIPX
ZPVS‫ڀ‬TPMVUJPO
(b) 8IBUJTUIFPSEFSPGSPUBUJPOBMTZNNFUSZPGBSIPNCVT
3 %SBXBRVBESJMBUFSBMUIBUIBTOPMJOFTPGTZNNFUSZBOEOPSPUBUJPOBMTZNNFUSZ
126
Unit 5: Shape, space and measures
19 Symmetry and loci
19.2 Symmetry in three dimensions
t ćSFFEJNFOTJPOBMTIBQFT TPMJET DBOBMTPCFTZNNFUSJDBM
t "QMBOFPGTZNNFUSZJTBTVSGBDF JNBHJOBSZ UIBUEJWJEFTUIFTIBQFJOUPUXPQBSUTUIBUBSFNJSSPSJNBHFTPGFBDI
PUIFS
t *GZPVSPUBUFBTPMJEBSPVOEBOBYJTBOEJUMPPLTUIFTBNFBUEJČFSFOUQPTJUJPOTPOJUTSPUBUJPO UIFOJUIBTSPUBUJPOBM
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Exercise 19.2
Tip
ćJOLPGB plane of
symmetryBTBTMJDFPSDVU
UISPVHIBTPMJEUPEJWJEF
JUJOUPUXPIBMWFTUIBUBSF
NJSSPSJNBHFTPGFBDI
PUIFS
Tip
ćJOLPGBOaxis of
symmetryBTBSPEPSBYMF
UISPVHIBTPMJE8IFOUIF
TPMJEUVSOTPOUIJTBYJT
BOESFBDIFTJUTPSJHJOBM
QPTJUJPOEVSJOHBUVSO JU
IBTSPUBUJPOBMTZNNFUSZ
1 'PSFBDIPGUIFGPMMPXJOHTPMJET TUBUFUIFOVNCFSPGQMBOFTPGTZNNFUSZ
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
2 8IBUJTUIFPSEFSPGSPUBUJPOBMTZNNFUSZBCPVUUIFHJWFOBYJTJOFBDIPGUIFTFTPMJET
(a)
(b)
(c)
(d)
(e)
(f)
Unit 5: Shape, space and measures
127
19 Symmetry and loci
19.3 Symmetry properties of circles
t "DJSDMFIBTMJOFTZNNFUSZBCPVUBOZEJBNFUFSBOEJUIBTSPUBUJPOBMTZNNFUSZBSPVOEJUTDFOUSF
t ćFGPMMPXJOHUIFPSFNTDBOCFVTFEUPTPMWFQSPCMFNTSFMBUFEUPDJSDMFT
o UIFQFSQFOEJDVMBSCJTFDUPSPGBDIPSEQBTTFTUISPVHIUIFDFOUSF
o FRVBMDIPSETBSFFRVJEJTUBOUGSPNUIFDFOUSFBOEDIPSETFRVJEJTUBOUGSPNUIFDFOUSFBSFFRVBMJOMFOHUI
o UXPUBOHFOUTESBXOUPBDJSDMFGSPNUIFTBNFQPJOUPVUTJEFUIFDJSDMFBSFFRVBMJOMFOHUI
Tip
Exercise 19.3
:PVOFFEUPMFBSOUIF
DJSDMFUIFPSFNT4UBUFUIF
UIFPSFNZPVBSFVTJOH
XIFOTPMWJOHBQSPCMFNJO
BOFYBN
1 'JOEUIFTJ[FPGUIFBOHMFTNBSLFExBOEy
(a)
(b)
A
O
x
230°
80° Z
X
x
B
y
Y
2 *OUIFEJBHSBN MNBOEPQBSFFRVBMDIPSETSJTUIFNJEQPJOUPGMNBOERJTUIF
NJEQPJOUPGPQMNDN'JOEUIFMFOHUIPGSODPSSFDUUPUXPTJHOJĕDBOUĕHVSFT
M
S
P
O
R
9 cm
N
Q
3 *OEJBHSBN(a) MNBOEPQBSFFRVBMDIPSETBOEMNDNSBOETBSFUIFNJEQPJOUTPG
MNBOEPQSFTQFDUJWFMZSODN
*OEJBHSBN(b) DIPSEABNN
'JOEUIFMFOHUIPGUIFEJBNFUFSPGFBDIDJSDMF BOEIFODFDBMDVMBUFJUTDJSDVNGFSFODF DPSSFDU
UPUXPEFDJNBMQMBDFT
(a)
(b)
M
P
C
A
S
O
T
O
E
N
128
Unit 5: Shape, space and measures
Q
B
D
19 Symmetry and loci
19.4 Angle relationships in circles
t 8IFOBUSJBOHMFJTESBXOJOBTFNJDJSDMF TPUIBUPOFTJEFJTUIFEJBNFUFSBOEUIFWFSUFYPQQPTJUFUIF
EJBNFUFSUPVDIFTUIFDJSDVNGFSFODF UIFBOHMFPGUIFWFSUFYPQQPTJUFUIFEJBNFUFSJTBSJHIUBOHMF ¡ t 8IFSFBUBOHFOUUPVDIFTBDJSDMF UIFSBEJVTESBXOUPUIFTBNFQPJOUNFFUTUIFUBOHFOUBU¡
t ćFBOHMFGPSNFEGSPNUIFFOETPGBDIPSEBOEUIFDFOUSFPGBDJSDMFJTUXJDFUIFBOHMFGPSNFECZUIFFOETPGUIF
DIPSEBOEBQPJOUBUUIFDJSDVNGFSFODF JOUIFTBNFTFHNFOU t "OHMFTJOUIFTBNFTFHNFOUBSFFRVBM
t ćFPQQPTJUFBOHMFTPGBDZDMJDRVBESJMBUFSBMBEEVQUP¡
t &BDIFYUFSJPSBOHMFPGBDZDMJDRVBESJMBUFSBMJTFRVBMUPUIFJOUFSJPSBOHMFPQQPTJUFUPJU
Tip
*GZPVMFBSOUIFDJSDMFUIFPSFNTXFMMZPVTIPVMECFBCMFUPTPMWFNPTU
DJSDMFQSPCMFNT
Exercise 19.4
The angle relationships for
triangles, quadrilaterals and
parallel lines (chapter 3), as well
as Pythagoras’ theorem (chapter
11), may be needed to solve circle
problems.
1 *OUIFEJBHSBN OJTUIFDFOUSFPGUIFDJSDMF ∠NPO¡BOE∠MOP¡
N
M
Q
O
70°
15°
P
'JOEUIFTJ[FPGUIFGPMMPXJOHBOHMFT HJWJOHSFBTPOT
(a) ∠PNO
(b) ∠PON
(c) ∠MPN
(d) ∠PMN
2 OJTUIFDFOUSFPGUIFDJSDMF
B
C
55°
25°
O
A
$BMDVMBUF
(a) ∠ACD
D
(b) ∠AOD
(c) ∠BDC
Unit 5: Shape, space and measures
129
19 Symmetry and loci
3 $BMDVMBUFUIFBOHMFTPGUIFDZDMJDRVBESJMBUFSBM
E
D
115°
A
C
B
4 ABJTUIFEJBNFUFSPGUIFDJSDMF$BMDVMBUFUIFTJ[FPGBOHMFABD
B
C
125°
D
A
5 SPTJTBUBOHFOUUPBDJSDMFXJUIDFOUSF O SR JTBTUSBJHIUMJOFXIJDIHPFTUISPVHIUIFDFOUSF
PGUIFDJSDMFBOE∠PSO¡'JOEUIFTJ[FPG∠TPR
S
29°
O
P
Q
R
T
6 (JWFOUIBUOJTUIFDFOUSFPGUIFDJSDMFCFMPXBOE∠BAC¡ DBMDVMBUFUIFTJ[FPG∠BOC
A
72°
O
B
130
Unit 5: Shape, space and measures
C
D
19 Symmetry and loci
7 *OUIFEJBHSBN AB]]DC ∠ADC ¡BOE∠DCA¡
D
A
64°
22°
B
C
$BMDVMBUFUIFTJ[FPG
(a) ∠BAC
(b) ∠ABC
(c) ∠ACB
8 SNTJTBUBOHFOUUPBDJSDMFXJUIDFOUSFO∠QMO¡BOE∠MPN¡
S
M
N
18°
O
Q
56°
T
P
'JOEUIFTJ[FPG
(a) ∠MNS
(b) ∠MOQ
(c) ∠PNT
19.5 Locus
t "MPDVTJTBTFUPGQPJOUT UIBUNBZPSNBZOPUCFDPOOFDUFE UIBUTBUJTGZBHJWFOSVMF"MPDVTDBOCFBTUSBJHIUMJOF B
DVSWFPSBDPNCJOBUJPOPGTUSBJHIUBOEDVSWFEMJOFT
t ćFMPDVTPGQPJOUTFRVJEJTUBOUGSPNBHJWFOQPJOUJTBDJSDMF
t ćFMPDVTPGQPJOUTFRVJEJTUBOUGSPNBĕYFEMJOFJTUXPMJOFTQBSBMMFMUPUIFHJWFOMJOF
t ćFMPDVTPGQPJOUTFRVJEJTUBOUGSPNBHJWFOMJOFTFHNFOUJTBASVOOJOHUSBDLTIBQFBSPVOEUIFMJOFTFHNFOU
t ćFMPDVTPGQPJOUTFRVJEJTUBOUGSPNUIFBSNTPGBOBOHMFJTUIFCJTFDUPSPGUIFBOHMF
t 5PĕOEUIFMPDVTPGQPJOUTUIBUBSFUIFTBNFEJTUBODFGSPNUXPPSNPSFHJWFOQPJOUTZPVIBWFUPVTFB
DPNCJOBUJPOPGUIFMPDJBCPWFUPĕOEUIFJOUFSTFDUJPOTPGUIFMPDJ
REWIND
Make sure you know to bisect an
angle as this is often required in
loci problems (see chapter 3). Exercise 19.5
1 $PQZUIFEJBHSBNTBOEESBXUIFMPDVTPGUIFQPJOUTUIBUBSFDNGSPNUIFQPJOUPSMJOFTJO
FBDIDBTF
(a)
A
(b) A
B
(c) A
B
C
Unit 5: Shape, space and measures
131
19 Symmetry and loci
2 $POTUSVDUBDJSDMFXJUIDFOUSFOBOEBSBEJVTPGDN
(a) %SBXUIFMPDVTPGQPJOUTUIBUBSFDNGSPNUIFDJSDVNGFSFODFPGUIFDJSDMF
(b) 4IBEFUIFMPDVTPGQPJOUTUIBUBSFMFTTUIBODNGSPNDJSDVNGFSFODFPGUIFDJSDMF
3 *OUIJTTDBMFEJBHSBN UIFTIBEFEBSFBSFQSFTFOUTBĕTIQPOEćFĕTIQPOEJTTVSSPVOEFECZ
BDPODSFUFXBMLXBZNXJEF$PQZUIFEJBHSBNBOEESBXUIFMPDVTPGQPJOUTUIBUBSFN
GSPNUIFFEHFTPGUIFĕTIQPOE4IBEFUIFBSFBDPWFSFECZUIFDPODSFUFXBMLXBZ
1m
4 /JDLMJWFTLNGSPNUIFSPBEBOELNGSPNUIFTDIPPMćFDMPTFTUEJTUBODFPGUIFTDIPPM
GSPNUIFSPBEJTLN6TJOHBTDBMFPGDNUPLN DPOTUSVDUUIFMPDJUPTIPXUXPQPTTJCMF
QPTJUJPOTGPS/JDLTIPNF
4 km
Road
HOOL
CH
SC
School
5 "OOBMJWFTBUABOE#FUUZMJWFTBUBćFZXBOUUPNFFUBUBQPJOUFRVJEJTUBOUGSPNCPUI
IPNFT#ZDPOTUSVDUJPO TIPXUIFMPDVTPGQPTTJCMFNFFUJOHQPJOUT
132
Unit 5: Shape, space and measures
19 Symmetry and loci
6 'PMMPXUIFJOTUSVDUJPOTDBSFGVMMZUPĕOEUIFMPDBUJPOPGBXBUFSNBJOPOBDPQZPGUIF
EJBHSBNCFMPX
Cell tower
A
Radio mast
C
30
m
15 m
B
Flagpole
Scale 1 cm = 5 m
(a) '
JOEUIFMPDVTPGQPJOUTUIBUBSFFRVJEJTUBOUGSPNUIFDFMMUPXFSBOEUIFĘBHQPMF UIFO
UIFTJOHMFMPDVTUIBUJTFRVJEJTUBOUGSPNABBOEBC-BCFMUIJTQPJOUDBOEESBXBMJOF
UPKPJOJUUPUIFCBTFPG‫ڀ‬UIFĘBHQPMF
(b) ćFXBUFSNBJOJTUFONFUSFTGSPNQPJOUDBOEPOUIFMJOFTFHNFOUKPJOJOHQPJOUDUP
UIFCBTFPGUIFĘBHQPMF.BSLUIJTBTQPJOUXPOZPVSEJBHSBN
Mixed exercise
1 )FSFBSFĕWFTIBQFT
(a)
(b)
(c)
(d)
(e)
'PSFBDIPOF
(i) JOEJDBUFUIFBYFTPGTZNNFUSZ JGBOZ
(ii) TUBUFUIFPSEFSPGSPUBUJPOBMTZNNFUSZ
2 4UVEZUIFEJBHSBN
(a)
(b)
(c)
(d)
8IBUUZQFPGTPMJEJTUIJT
8IBUJTUIFDPSSFDUNBUIFNBUJDBMOBNFGPSUIFSPEUISPVHIUIFTPMJE
8IBUJTUIFPSEFSPGSPUBUJPOBMTZNNFUSZPGUIJTTPMJE
)PXNBOZQMBOFTPGTZNNFUSZEPFTUIJTTPMJEIBWF
Unit 5: Shape, space and measures
133
19 Symmetry and loci
3 *OFBDIPGUIFGPMMPXJOHOJTUIFDFOUSFPGUIFDJSDMF'JOEUIFWBMVFPGUIFNBSLFEBOHMFT
(JWFSFBTPOTGPSZPVSTUBUFNFOUT
(a)
(b)
A
A
(c)
(d)
A
x
B
28°
B
O
O
D
O
160°
w
yz
A
x
C
C
D
O x
y
x
B
60°
29°
B
C
C
4 'JOEUIFMFOHUIPG xBOEyJOFBDIPGUIFTFEJBHSBNTOJTUIFDFOUSFPGUIFDJSDMFJOFBDIDBTF
(a)
(b)
18 cm
250 mm
y
y
x
120 mm
O
x 15 cm
O
5 +FTTJDBUFMMTBGSJFOEUIBUIFSIPVTFJTLNGSPNUIFQFUSPMTUBUJPOPO3PVUFBOELNGSPN
3PVUFJUTFMG6TFBTDBMFPGDNUPLNUPTIPXBMMQPTTJCMFMPDBUJPOTGPS+FTTJDBTIPVTF
Petrol station
Route 66
6 $POTUSVDUMJOFTFHNFOUABDNMPOH4IPXUIFMPDVTPGQPJOUTUIBUBSFFRVJEJTUBOUGSPN
CPUIABOEBBOEBMTPDNGSPNA
134
Unit 5: Shape, space and measures
20
Histograms and frequency
distribution diagrams
20.1 Histograms
t "IJTUPHSBNJTBTQFDJBMJTFEHSBQI*UJTVTFEUPTIPXHSPVQFEOVNFSJDBMEBUBVTJOHBDPOUJOVPVTTDBMF
t
t
t
POUIFIPSJ[POUBMBYJT ćJTNFBOTUIFSFBSFOPHBQTJOCFUXFFOUIFEBUBDBUFHPSJFToXIFSFPOFFOET UIF
PUIFSCFHJOT
#FDBVTFUIFTDBMFJTDPOUJOVPVT FBDIDPMVNOJTESBXOBCPWFBQBSUJDVMBSDMBTTJOUFSWBM
8IFOUIFDMBTTJOUFSWBMTBSFFRVBM UIFCBSTBSFBMMUIFTBNFXJEUIBOEJUJTDPNNPOQSBDUJDFUPMBCFMUIFWFSUJDBMTDBMF
BTGSFRVFODZ
8IFOUIFDMBTTJOUFSWBMTBSFOPUFRVBM UIFWFSUJDBMBYJTTIPXTUIFGSFRVFODZEFOTJUZ
frequency
ćFBSFBPGFBDIACBSSFQSFTFOUTUIFGSFRVFODZ
class interval
"HBQXJMMPOMZBQQFBSCFUXFFOCBSTJGBOJOUFSWBMIBTBGSFRVFODZGSFRVFODZEFOTJUZPG[FSP
Frequency density =
t
Exercise 20.1
1 ćFUBCMFTIPXTUIFNBSLTPCUBJOFECZBOVNCFSPGTUVEFOUTGPS&OHMJTIBOE.BUIFNBUJDTJO
BNPDLFYBN
Remember there can be no gaps
between the bars: use the upper
and lower bounds of each class
interval to prevent gaps.
REWIND
You met the mode in chapter 12. Marks class interval
English frequency
Mathematics frequency
o
o
3
o
7
5
o
6
o
o
51
o
o
6
o
3
o
1
(a) %
SBXUXPTFQBSBUFIJTUPHSBNTUPTIPXUIFEJTUSJCVUJPOPGNBSLTGPS&OHMJTIBOE
.BUIFNBUJDT
(b) 8IBUJTUIFNPEBMDMBTTGPS&OHMJTI
(c) 8IBUJTUIFNPEBMDMBTTGPS.BUIFNBUJDT
(d) 8
SJUFBGFXTFOUFODFTDPNQBSJOHUIFTUVEFOUTQFSGPSNBODFJO&OHMJTIBOE
.BUIFNBUJDT
Unit 5: Data handling
135
20 Histograms and frequency distribution diagrams
Frequency
2 4UVEZUIJTHSBQIBOEBOTXFSUIFRVFTUJPOTBCPVUJU
(a)
(b)
(c)
(d)
(e)
Tip
*UIFMQTUPSFNFNCFSUIBU
UIFGSFRVFODZEFOTJUZ
UFMMTZPVXIBUUIFDPMVNO
IFJHIUTIPVMECF*GPOF
DPMVNOJTUXJDFBTXJEF
BTBOPUIFS JUXJMMPOMZCF
IBMGBTIJHIGPSUIFTBNF
GSFRVFODZ
Ages of women in a clothing factory
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
19
21
23 25 27
Age (years)
29
31
)PXEPZPVLOPXUIJTJTBIJTUPHSBNBOEOPUBCBSHSBQI
)PXNBOZXPNFOBHFEoXPSLJOUIFDMPUIJOHGBDUPSZ
)PXNBOZXPNFOXPSLJOUIFGBDUPSZBMUPHFUIFS
8IBUJTUIFNPEBMDMBTTPGUIJTEBUB
&YQMBJOXIZUIFSFJTBCSPLFOMJOFPOUIFIPSJ[POUBMBYJT
3 4BMMZEJEBTVSWFZUPĕOEUIFBHFTPGQFPQMFVTJOHBOJOUFSOFUDBGÏćFTFBSFIFSSFTVMUT
Age (a)
ȳa
ȳa
ȳa
ȳa
ȳa
No. of people
%SBXBOBDDVSBUFIJTUPHSBNUPTIPXUIFTFEBUB6TFBTDBMFPGDNUPĕWFZFBSTPOUIF
IPSJ[POUBMBYJTBOEBOBSFBTDBMFPGPOFTRVBSFDFOUJNFUSFUPSFQSFTFOUPOFQFSTPO
4 ćJTIJTUPHSBNTIPXTUIFOVNCFSPGIPVTFTJOEJČFSFOUQSJDFSBOHFTUIBUBSFBEWFSUJTFEJOB
QSPQFSUZNBHB[JOF
House prices in property magazine
30
25
20
Frequency density 15
10
5
0
20
40
60
80
100
120
140
Price ($thousands)
160
180
(a) )PXNBOZIPVTFTXFSFJOUIFoQSJDFSBOHF (b) )PXNBOZIPVTFTXFSFJOUIFoQSJDFSBOHF (c) )PXNBOZIPVTFTBSFSFQSFTFOUFECZPOFTRVBSFDFOUJNFUSFPOUIJTHSBQI
136
Unit 5: Data handling
200
20 Histograms and frequency distribution diagrams
20.2 Cumulative frequency
t $VNVMBUJWFGSFRVFODZJTBASVOOJOHUPUBMPGUIFDMBTTGSFRVFODJFTVQUPFBDIVQQFSDMBTTCPVOEBSZ
t 8IFODVNVMBUJWFGSFRVFODJFTBSFQMPUUFEUIFZHJWFBDVNVMBUJWFGSFRVFODZDVSWFPSHSBQI
t :PVDBOVTFUIFDVSWFUPFTUJNBUFUIFNFEJBOWBMVFPGUIFEBUB
t :PVDBOEJWJEFUIFEBUBJOUPGPVSFRVBMHSPVQTDBMMFERVBSUJMFTćFJOUFSRVBSUJMFSBOHF *23 JTUIFEJČFSFODF
CFUXFFOUIFVQQFSBOEMPXFSRVBSUJMFT 2 ¦2 t %BUBDBOBMTPCFEJWJEFEJOUPFRVBMHSPVQTDBMMFEQFSDFOUJMFTćFUIQFSDFOUJMFJTFRVJWBMFOUUPUIFNFEJBO
3
1
Exercise 20.2
1 ćFUBCMFTIPXTUIFQFSDFOUBHFTDPSFECZBOVNCFSPGTUVEFOUTJOBOFYBNJOBUJPO
Percentage
:PVXJMMOPSNBMMZCFHJWFO
BTDBMFUPVTFXIFOZPV
IBWFUPESBXBDVNVMBUJWF
GSFRVFODZDVSWFJOBO
FYBNJOBUJPO
Tip
5PĕOEUIFQPTJUJPO
PGUIFRVBSUJMFTGSPNB
DVNVMBUJWFGSFRVFODZ
DVSWF VTFUIFGPSNVMBFn 4
n BOE 3n /PUFUIBUUIJT
2
4
JTEJČFSFOUUPUIFGPSNVMBF
VTFEGPSEJTDSFUFEBUB
ȳm
ȳm
3
ȳm
7
ȳm
ȳm
ȳm
51
ȳm
ȳm
ȳm
3
ȳm
1
(a) %SBXBDVNVMBUJWFGSFRVFODZDVSWFUPTIPXUIJTEBUB6TFBTDBMFPGDNQFSUFOQFSDFOU
POUIFIPSJ[POUBMBYJTBOEBTDBMFPGDNQFSUFOTUVEFOUTPOUIFWFSUJDBMBYJT
(b) 6
TFZPVSDVSWFUPFTUJNBUFUIFNFEJBO 21BOE23
(c) &TUJNBUFUIF*23
(d) ćFQBTTSBUFGPSUIJTUFTUJT8IBUQFSDFOUBHFPGUIFTUVEFOUTQBTTFEUIFUFTU (e) *OEJDBUFUIFUIBOEUIQFSDFOUJMFTPOZPVSHSBQI8IBUEPUIFTFQFSDFOUJMFT
JOEJDBUF
2 ćJTDVNVMBUJWFGSFRVFODZDVSWFTIPXTUIFIFJHIUJO
DFOUJNFUSFTPGQSPGFTTJPOBMCBTLFUCBMMQMBZFST
(a) &TUJNBUFUIFNFEJBOIFJHIUPGQMBZFSTJO
UIJTTBNQMF
(b) &TUJNBUF21BOE23
(c) &TUJNBUFUIF*23
(d) 8IBUQFSDFOUBHFPGCBTLFUCBMMQMBZFSTBSF
PWFSNUBMM Cumulative frequency
Tip
Number of students
200
Height of professional
basketball players
150
100
50
0
125 140 155 170 185 200
Height (cm)
Unit 5: Data handling
137
20 Histograms and frequency distribution diagrams
Mixed exercise
1 ćJTQBSUJBMMZDPNQMFUFEIJTUPHSBNTIPXTUIFIFJHIUTPGUSFFTJOBTFDUJPOPGUSPQJDBMGPSFTU
Heights of trees
10
Frequency
8
6
4
2
0
0
2
4
6
8 10
Height in metres
12
(a) "
TDJFOUJTUNFBTVSFEĕWFNPSFUSFFTBOEUIFJSIFJHIUTXFSFN N N NBOEN3FESBXUIFHSBQIUPJODMVEFUIJTEBUB
(b) )PXNBOZUSFFTJOUIJTTFDUPSPGGPSFTUBSFȴNUBMM
(c) 8IBUJTUIFNPEBMDMBTTPGUSFFIFJHIUT
2 "OVSTFNFBTVSFEUIFNBTTFTPGBTBNQMFPGTUVEFOUTJOBIJHITDIPPMBOEESFXUIFGPMMPXJOHUBCMF
Mass (kg)
Frequency
≤m<
≤m<
7
≤m<
13
≤m<
≤m<
11
(a)
(b)
(c)
(d)
%SBXBIJTUPHSBNUPTIPXUIFEJTUSJCVUJPOPGNBTTFT
8IBUJTUIFNPEBMNBTT
8IBUQFSDFOUBHFPGTUVEFOUTXFJHIFEMFTTUIBOLH
8IBUJTUIFNBYJNVNQPTTJCMFSBOHFPGUIFNBTTFT
3 ćFUBCMFTIPXTUIFBWFSBHFNJOVUFTPGBJSUJNFUIBUUFFOBHFSTCPVHIUGSPNBQSFQBJELJPTL
JOPOFXFFL
Minutes
≤m< ȳm< ȳm< ȳm< ȳm< ȳm<
No. of
teenagers
15
%SBXBOBDDVSBUFIJTUPHSBNUPEJTQMBZUIJTEBUB6TFBTDBMFPGDNUPSFQSFTFOUUFONJOVUFT
POUIFIPSJ[POUBMBYJTBOEBOBSFBTDBMFPGDNQFSĕWFQFSTPOT
4 ćJSUZTFFEMJOHTXFSFQMBOUFEGPSBCJPMPHZFYQFSJNFOUćFIFJHIUTPGUIFQMBOUTXFSF
NFBTVSFEBęFSUISFFXFFLTBOESFDPSEFEBTCFMPX
Heights (h cm)
Frequency
≤h
≤h<
≤h<
≤h<
3
15
(a) 'JOEBOFTUJNBUFGPSUIFNFBOIFJHIU
(b) %SBXBDVNVMBUJWFGSFRVFODZDVSWFBOEVTFJUUPĕOEUIFNFEJBOIFJHIU
(c) &TUJNBUF21BOE23BOEUIF*23
138
Unit 5: Data handling
21
Ratio, rate and proportion
21.1 Working with ratio
t "SBUJPJTBDPNQBSJTPOPGUXPPSNPSFRVBOUJUJFTNFBTVSFEJOUIFTBNFVOJUT*OHFOFSBM BSBUJPJT
XSJUUFOJOUIFGPSNab
t 3BUJPTTIPVMEBMXBZTCFHJWFOJOUIFJSTJNQMFTUGPSN5PFYQSFTTBSBUJPJOTJNQMFTUGPSN EJWJEFPS
NVMUJQMZCZUIFTBNFGBDUPS
t 2VBOUJUJFTDBOCFTIBSFEJOBHJWFOSBUJP5PEPUIJTZPVOFFEUPXPSLPVUUIFOVNCFSPGFRVBMQBSUT
JOUIFSBUJPBOEUIFOXPSLPVUUIFWBMVFPGFBDITIBSF'PSFYBNQMF BSBUJPPGNFBOTUIBUUIFSFBSF
3
5
2
5
FRVBMQBSUT0OFTIBSFJT PGUIFUPUBMBOEUIFPUIFSJT PGUIFUPUBM
Remember, simplest form is also
called ‘lowest terms’.
Exercise 21.1
1 &YQSFTTUIFGPMMPXJOHBTSBUJPTJOUIFJSTJNQMFTUGPSN
(a) 2
3
4
3
2
3
(c) DNUPN
(e) HUPH
Tip
:PVDBODSPTTNVMUJQMZ
UPNBLFBOFRVBUJPOBOE
TPMWFGPSx
1
2
(b) 1 IPVSTNJOVUFT
(d) HUPUISFFLJMPHSBNT
2 'JOEUIFWBMVFPGxJOFBDIPGUIFGPMMPXJOH
(a) x
(e) x
(i)
x 1
=
21 3 (b) x
2 x
(f)
=
7 4
5 3
(j)
=
x 8
(c) x
5 16
(g)
=
x 6
(d) x
x 10
(h)
=
4 15
3 "MFOHUIPGSPQFDNMPOHNVTUCFDVUJOUPUXPQBSUTTPUIBUUIFMFOHUITBSFJOUIFSBUJP
8IBUBSFUIFMFOHUITPGUIFQBSUT
4 5PNBLFTBMBEESFTTJOH ZPVNJYPJMBOEWJOFHBSJOUIFSBUJP$BMDVMBUFIPXNVDIPJMBOE
IPXNVDIWJOFHBSZPVXJMMOFFEUPNBLFUIFGPMMPXJOHBNPVOUTPGTBMBEESFTTJOH
(a) NM
(b) NM
(c) NM
5 ćFTJ[FTPGUISFFBOHMFTPGBUSJBOHMFBSFJOUIFSBUJPABC8IBUJTUIFTJ[FPG
FBDIBOHMF
6 "NFUBMEJTDDPOTJTUTPGUISFFQBSUTTJMWFSBOEUXPQBSUTDPQQFS CZNBTT *GUIFEJTDIBTB
NBTTPGNH IPXNVDITJMWFSEPFTJUDPOUBJO
Unit 6: Number
139
21
Ratio, rate and proportion
21.2 Ratio and scale
t 4DBMFJTBSBUJP*UDBOCFFYQSFTTFEBT
t
t
MFOHUIPOUIFESBXJOHSFBMMFOHUI
"MMSBUJPTDBMFTNVTUCFFYQSFTTFEJOUIFGPSNPGnPSn
5PDIBOHFBSBUJPTPUIBUPOFQBSU ZPVOFFEUPEJWJEFCPUIQBSUTCZUIFOVNCFSUIBUZPVXBOUUPCF
FYQSFTTFEBT'PSFYBNQMFXJUI JGZPVXBOUUIFUPCFFYQSFTTFEBT ZPVEJWJEFCPUIQBSUTCZ
ćFSFTVMUJT
Tip
Exercise 21.2 A
8JUISFEVDUJPOT TVDIBT
NBQT UIFTDBMFXJMMCFJO
UIFGPSNn XIFSFn
8JUIFOMBSHFNFOUTUIFTDBMF
XJMMCFJOUIFGPSNn XIFSFnnNBZOPUCF
BXIPMFOVNCFS
1 8SJUFUIFTFSBUJPTJOUIFGPSNPGn
(a) (b) NLN
1
2
(c) NJOVUFT1 IPVST
2 8SJUFUIFTFSBUJPTJOUIFGPSNPGn
(a) (b) NDN
(c) HUPNH
Exercise 21.2 B
1 ćFEJTUBODFCFUXFFOUXPQPJOUTPOBNBQXJUIBTDBMFPGJTNN8IBUJTUIF
EJTUBODFCFUXFFOUIFUXPQPJOUTJOSFBMJUZ (JWFZPVSBOTXFSJOLJMPNFUSFT
2 "QMBOJTESBXOVTJOHBTDBMFPG*GUIFMFOHUIPGBXBMMPOUIFQMBOJTDN IPXMPOHJT
UIFXBMMJOSFBMJUZ
3 .JHVFMNBLFTBTDBMFESBXJOHUPTPMWFBUSJHPOPNFUSZQSPCMFNDNPOIJTESBXJOH
SFQSFTFOUTNJOSFBMMJGF)FXBOUTUPTIPXBNMPOHMBEEFSQMBDFENGSPNUIFGPPU
PGBXBMM
(a) 8IBUMFOHUIXJMMUIFMBEEFSCFJOUIFEJBHSBN
(b) )PXGBSXJMMJUCFGSPNUIFGPPUPGUIFXBMMJOUIFEJBHSBN
4 "NBQIBTBTDBMFPG
(a) 8IBUEPFTBTDBMFPGNFBO
(b) $PQZBOEDPNQMFUFUIJTUBCMFVTJOHUIFNBQTDBMF
Map distance (mm)
Actual distance (km)
50
50
5 .BSZIBTBSFDUBOHVMBSQJDUVSFNNXJEFBOENNIJHI4IFFOMBSHFTJUPOUIF
QIPUPDPQJFSTPUIBUUIFFOMBSHFNFOUJTDNXJEF
(a) 8IBUJTUIFTDBMFGBDUPSPGUIFFOMBSHFNFOU
(b) 8IBUJTUIFIFJHIUPGIFSFOMBSHFEQJDUVSF
(c) *OUIFPSJHJOBMQJDUVSF BGFODFXBTNNMPOH)PXMPOHXJMMUIJTGFODFCFPOUIF
FOMBSHFEQJDUVSF
140
Unit 6: Number
21
Ratio, rate and proportion
21.3 Rates
t "SBUFDPNQBSFTUXPRVBOUJUJFTNFBTVSFEJOEJČFSFOUVOJUT'PSFYBNQMFTQFFEJTBSBUFUIBUDPNQBSFT
LJMPNFUSFTUSBWFMMFEQFSIPVS
t 3BUFTDBOCFTJNQMJĕFEKVTUMJLFSBUJPTćFZDBOBMTPCFFYQSFTTFEJOUIFGPSNPGn
t :PVTPMWFSBUFQSPCMFNTJOUIFTBNFXBZUIBUZPVTPMWFESBUJPBOEQSPQPSUJPOQSPCMFNT6TFUIFVOJUBSZ
PSSBUJPNFUIPET
Tip
Exercise 21.3
ćFXPSEAQFSJTPęFO
VTFEJOBSBUFT1FSDBO
NFBOAGPSFWFSZ AJOFBDI APVUPGFWFSZ PSAPVUPG EFQFOEJOHPOUIFDPOUFYU
1 "UBNBSLFU NJMLDPTUTQFSMJUSF)PXNVDINJMLDBOZPVCVZGPS
2 4BNUSBWFMTBEJTUBODFPGLNBOEVTFTMJUSFTPGQFUSPM&YQSFTTIJTQFUSPMDPOTVNQUJPO
BTBSBUFJOLNM
3 $BMDVMBUFUIFBWFSBHFTQFFEPGUIFGPMMPXJOHWFIJDMFT
Remember, speed is a very
important rate.
distan
t ce
speed =
time
4 )PXMPOHXPVMEJUUBLFUPUSBWFMUIFTFEJTUBODFT
distan
t ce = speed × time
time =
(a) "DBSUIBUUSBWFMTLNJOIPVST
(b) "QMBOFUIBUUSBWFMTLNJOPOFIPVSNJOVUFT
(c) "USBJOUIBUUSBWFMTLNJONJOVUFT
distan
t ce
speed
(a) LNBULNI
(c) LNBULNI
(b) LNBULNI
(d) NBULNI
1
2
5 )PXGBSXPVMEZPVUSBWFMJO2 IPVSTBUUIFTFTQFFET
(a) LNI
(c) NFUSFTQFSNJOVUF
(b) LNI
(d) UXPNFUSFTQFSTFDPOE
21.4 Kinematic graphs
t %JTUBODFoUJNFHSBQITTIPXUIFDPOOFDUJPOCFUXFFOUIFEJTUBODFBOPCKFDUIBTUSBWFMMFEBOEUIFUJNF
UBLFOUPUSBWFMUIBUEJTUBODFćFZBSFBMTPDBMMFEUSBWFMHSBQIT
t 5JNFJTOPSNBMMZTIPXOBMPOHUIFIPSJ[POUBMBYJTCFDBVTFJUJTUIFJOEFQFOEFOUWBSJBCMF%JTUBODFJT
TIPXOPOUIFWFSUJDBMBYJTCFDBVTFJUJTUIFEFQFOEFOUWBSJBCMF
t :PVDBOEFUFSNJOFTQFFEPOBEJTUBODFoUJNFHSBQICZMPPLJOHBUUIFTMPQF TUFFQOFTT PGUIFMJOFćFTUFFQFSUIF
MJOF UIFHSFBUFSUIFTQFFEBTUSBJHIUMJOFJOEJDBUFTDPOTUBOUTQFFEVQXBSEBOEEPXOXBSETMPQFTSFQSFTFOUNPWFNFOU
JOPQQPTJUFEJSFDUJPOTBOEBIPSJ[POUBMMJOFSFQSFTFOUTOPNPWFNFOU
o
t 4QFFEoUJNFHSBQITTIPXTQFFEPOUIFWFSUJDBMBYJTBOEUJNFPOUIFIPSJ[POUBMBYJT
t 'PSBTQFFEoUJNFHSBQI HSBEJFOUBDDFMFSBUJPO1PTJUJWFHSBEJFOU BDDFMFSBUJPO JTBOJODSFBTFJOTQFFE/FHBUJWF
HSBEJFOU EFDFMFSBUJPO JTBEFDSFBTFJOTQFFE
t %JTUBODFTQFFE¨UJNF:PVDBOĕOEUIFEJTUBODFDPWFSFEJOBDFSUBJOUJNFCZDBMDVMBUJOHUIFBSFBCFMPXFBDI
TFDUJPOPGUIFHSBQI"QQMZUIFBSFBGPSNVMBFGPSRVBESJMBUFSBMTBOEUSJBOHMFTUPEPUIJT
Unit 6: Number
141
21
Ratio, rate and proportion
Exercise 21.4
1 ćFHSBQICFMPXTIPXTUIFEJTUBODFDPWFSFECZBWFIJDMFJOBTJYIPVSQFSJPE
600
Distance (km)
500
400
300
200
100
0
1
2
3
4
5
Time (hours)
6
(a) 6TFUIFHSBQIUPĕOEUIFEJTUBODFDPWFSFEBęFS
(i) POFIPVS
(ii) UXPIPVST
(iii) UISFFIPVST
(b) $BMDVMBUFUIFBWFSBHFTQFFEPGUIFWFIJDMFEVSJOHUIFĕSTUUISFFIPVST
(c) %FTDSJCFXIBUUIFHSBQITIPXTCFUXFFOIPVSUISFFBOEGPVS
(d) 8IBUEJTUBODFEJEUIFWFIJDMFDPWFSEVSJOHUIFMBTUUXPIPVSTPGUIFKPVSOFZ
(e) 8IBUXBTJUTBWFSBHFTQFFEEVSJOHUIFMBTUUXPIPVSTPGUIFKPVSOFZ
2 %BCJMPBOE1BNMJWFLNBQBSUGSPNFBDIPUIFSćFZEFDJEFUPNFFUVQBUBTIPQQJOH
DFOUSFJOCFUXFFOUIFJSIPNFTPOB4BUVSEBZ1BNUSBWFMTCZCVTBOE%BCJMPDBUDIFTBUSBJO
ćFHSBQITIPXTCPUIKPVSOFZT
Da
0
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
142
Unit 6: Number
Pam
bi
100
Pam
lo
Distance (km)
200
60
120
Time (min)
180
Dabilo
240
)PXNVDIUJNFEJE%BCJMPTQFOEPOUIFUSBJO
)PXNVDIUJNFEJE1BNTQFOEPOUIFCVT
"UXIBUTQFFEEJEUIFUSBJOUSBWFMGPSUIFĕSTUIPVS
)PXGBSXBTUIFTIPQQJOHDFOUSFGSPN
(i) %BCJMPTIPNF (ii) 1BNTIPNF
8IBUXBTUIFBWFSBHFTQFFEPGUIFCVTGSPN1BNTIPNFUPUIFTIPQQJOHDFOUSF
)PXMPOHEJE%BCJMPIBWFUPXBJUCFGPSF1BNBSSJWFE
)PXMPOHEJEUIFUXPHJSMTTQFOEUPHFUIFS
)PXNVDIGBTUFSXBT1BNTKPVSOFZPOUIFXBZIPNF
*GUIFZMFęIPNFBUBN XIBUUJNFEJEFBDIHJSMSFUVSOIPNFBęFSUIFEBZTPVUJOH
21
Ratio, rate and proportion
3 ćJTTQFFEoUJNFHSBQITIPXTUIFTQFFEPGBDBSJOLNIBHBJOTUUIFUJNFJONJOVUFT
120
100
80
Speed (km/h) 60
40
20
0
1
2
3
4
Time (minutes)
5
6
(a) 8IBUJTUIFTQFFEPGUIFDBSBęFS
(i) UXPNJOVUFT
(ii) TJYNJOVUFT
(b) 8IFOJTUIFDBSUSBWFMMJOHBULNI
(c) $BMDVMBUFUIFBDDFMFSBUJPOPGUIFDBSJOLNI
(d) 8IBUEJTUBODFEJEUIFDBSDPWFSJOUIFĕSTUTJYNJOVUFT Tip
"DDFMFSBUJPODIBOHFJO
TQFFEŸUJNFUBLFO"T
UIFVOJUTPGTQFFEJOUIJT
FYBNQMFBSFNFUSFTQFS
TFDPOE UIFOUIFVOJUTPG
BDDFMFSBUJPOBSFNFUSFTQFS
TFDPOEQFSTFDPOEćJTJT
XSJUUFOBTm s −2 PSNT
4 ćJTTQFFEoUJNFHSBQITIPXTUIFTQFFEPGBUSBJOJONTBHBJOTUUIFUJNFJOTFDPOET
30
Speed
(m/s)
20
10
0
(a)
(b)
(c)
(d)
20
40
60
80
Time (seconds)
100
120
8IFOXBTUIFUSBJOBDDFMFSBUJOHBOEXIBUXBTUIFBDDFMFSBUJPO 8IFOEJEUIFUSBJOTUBSUEFDFMFSBUJOHBOEXIBUXBTUIFEFDFMFSBUJPO 8IFOUIFUSBJOXBTUSBWFMMJOHBUDPOTUBOUTQFFE XIBUXBTUIFTQFFEJOLNI 8IBUEJTUBODFEJEUIFUSBJOUSBWFMJOUXPNJOVUFT
Unit 6: Number
143
21
Ratio, rate and proportion
21.5 Proportion
t 1SPQPSUJPOJTBDPOTUBOUSBUJPCFUXFFOUIFDPSSFTQPOEJOHFMFNFOUTPGUXPTFUT
t 8IFORVBOUJUJFTBSFJOEJSFDUQSPQPSUJPOUIFZJODSFBTFPSEFDSFBTFBUUIFTBNFSBUFćFHSBQIPGBEJSFDUMZ
QSPQPSUJPOBUFSFMBUJPOTIJQJTBTUSBJHIUMJOFQBTTJOHUISPVHIUIFPSJHJO
t 8IFORVBOUJUJFTBSFJOWFSTFMZQSPQPSUJPOBM POFJODSFBTFTBTUIFPUIFSEFDSFBTFTćFHSBQIPGBOJOWFSTFMZ
QSPQPSUJPOBMSFMBUJPOTIJQJTBDVSWF
t ćFVOJUBSZNFUIPEJTVTFGVMGPSTPMWJOHSBUJPBOEQSPQPSUJPOQSPCMFNTćJTNFUIPEJOWPMWFTĕOEJOHUIF
WBMVFPGPOFVOJU PGXPSL UJNF FUD BOEVTJOHUIBUWBMVFUPĕOEUIFWBMVFPGBOVNCFSPGVOJUT
Tip
Exercise 21.5
*GxBOEyBSFEJSFDUMZ
QSPQPSUJPOBM UIFO x JTUIF
y
TBNFGPSWBSJPVTWBMVFTPG
xBOEy
1 %FUFSNJOFXIFUIFS"BOE#BSFEJSFDUMZQSPQPSUJPOBMJOFBDIDBTF
(a)
(b)
(c)
A
6
B
600
A
5
B
A
B
2 "UFYUCPPLDPTUT
(a) 8IBUJTUIFQSJDFPGTFWFOCPPLT
(b) 8IBUJTUIFQSJDFPGUFOCPPLT
3 'JOEUIFDPTUPGĕWFJEFOUJDBMMZQSJDFEJUFNTJGTFWFOJUFNTDPTU
4 *GBNUBMMQPMFDBTUTBNTIBEPX ĕOEUIFMFOHUIPGUIFTIBEPXDBTUCZBNUBMM
QPMFBUUIFTBNFUJNF
5 "DBSUSBWFMTBEJTUBODFPGLNJOUISFFIPVSTBUBDPOTUBOUTQFFE
(a) 8IBUEJTUBODFXJMMJUDPWFSJOPOFIPVSBUUIFTBNFTQFFE
(b) )PXGBSXJMMJUUSBWFMJOĕWFIPVSTBUUIFTBNFTQFFE
(c) )PXMPOHXJMMJUUBLFUPUSBWFMLNBUUIFTBNFTQFFE
6 "USVDLVTFTMJUSFTPGEJFTFMUPUSBWFMLJMPNFUSFT
(a) )PXNVDIEJFTFMXJMMJUVTFUPUSBWFMLNBUUIFTBNFSBUF
(b) )PXGBSDPVMEUIFUSVDLUSBWFMPOMJUSFTPGEJFTFMBUUIFTBNFSBUF
7 *UUBLFTPOFFNQMPZFFUFOEBZTUPDPNQMFUFBQSPKFDU*GBOPUIFSFNQMPZFFKPJOTIJN JUPOMZ
UBLFTĕWFEBZT'JWFFNQMPZFFTDBODPNQMFUFUIFKPCJOUXPEBZT
(a) %FTDSJCFUIJTSFMBUJPOTIJQ
(b) )PXMPOHXPVMEJUUBLFUPDPNQMFUFUIFQSPKFDUXJUI
(i) GPVSFNQMPZFFT
(ii) FNQMPZFFT
144
Unit 6: Number
21
Ratio, rate and proportion
8 "ęFSBUTVOBNJ UFOQFPQMFIBWFFOPVHIGSFTIXBUFSUPMBTUUIFNGPSTJYEBZTBUBTFUSBUF
QFSQFSTPO
(a) )PXMPOHXPVMEUIFXBUFSMBTU JGUIFSFXFSFPOMZĕWFQFPQMFESJOLJOHJUBUUIF
TBNFSBUF
(b) "OPUIFSUXPQFPQMFKPJOUIFHSPVQ)PXMPOHXJMMUIFXBUFSMBTUJGJUJTVTFEBUUIF
TBNFSBUF
9 /JDLUPPLGPVSIPVSTUPDPNQMFUFBKPVSOFZBULNI.BSJFEJEUIFTBNFKPVSOFZ
BULNI)PXMPOHEJEJUUBLFIFS
10 "QMBOFUSBWFMMJOHBUBOBWFSBHFTQFFEPGLNIUBLFTIPVSTUPDPNQMFUFBKPVSOFZ
)PXGBTUXPVMEJUOFFEUPUSBWFMUPDPWFSUIFTBNFEJTUBODFJOUFOIPVST
21.6 Direct and inverse proportion in algebraic terms
t 8IFOUXPRVBOUJUJFTBSFEJSFDUMZQSPQPSUJPOBMUIFOPkQ XIFSFkJTBDPOTUBOU
t 8IFOUXPRVBOUJUJFTBSFJOWFSTFMZQSPQPSUJPOBMUIFOPQk XIFSFkJTBDPOTUBOU
Exercise 21.6
Tip
ćFTZNCPM∝NFBOT
QSPQPSUJPOBMUP
1 (JWFOUIBUaWBSJFTEJSFDUMZXJUIbBOEUIBUaXIFOb (a) ĕOEUIFWBMVFPGUIFDPOTUBOUPGQSPQPSUJPOBMJUZ k
(b) ĕOEUIFWBMVFPGaXIFOb
2 ćFUBCMFTIPXTWBMVFTPGmBOET4IPXUIBUTJTEJSFDUMZQSPQPSUJPOBMUPm
m
T
3 FJTEJSFDUMZQSPQPSUJPOBMUPmBOEFXIFOm
(a) 'JOEUIFWBMVFPGFXIFOm
(b) 'JOEUIFWBMVFPGmXIFOF
4 *OUIFGPMMPXJOHSFMBUJPOTIJQ xyk'JOEUIFNJTTJOHWBMVFT
x
6
b
y
a
c
1
BOEyXIFOx
x
(a) 'JOEUIFWBMVFPGyXIFOx
(b) 'JOEUIFWBMVFPGxXIFOy
5 y∝
6 yJTEJSFDUMZQSPQPSUJPOBMUPx BOEyXIFOx
(a) 8SJUFUIFFRVBUJPOGPSUIJTSFMBUJPOTIJQ
(b) 'JOEyJGx
(c) 'JOExJGy
Unit 6: Number
145
21
Ratio, rate and proportion
7
yJTJOWFSTFMZQSPQPSUJPOBMUP x BOEyXIFOx
(a) 8SJUFUIFFRVBUJPOGPSUIJTSFMBUJPOTIJQ
(b) 'JOEyJGx
1
(c) 'JOExJGy 3 8
aWBSJFTJOWFSTFMZXJUIbBOEaXIFOb
(a) 'JOEUIFWBMVFPGbXIFOa
(b) 'JOEUIFWBMVFPGaXIFOb
9
ćFSFMBUJPOTIJQCFUXFFOxBOEyJTHJWFOBT y =
(a) 'JOEUIFWBMVFPGyXIFOx
(b) $BMDVMBUFxXIFOy
10
x
10 "MFOHUIPGXJSFNMPOHJTDVUJOUPBOVNCFS y PGFRVBMMFOHUIT xN (a) 4IPXUIBUyJTJOWFSTFMZQSPQPSUJPOBMUPx
(b) 8SJUFBOFRVBUJPOUPEFTDSJCFUIFSFMBUJPOTIJQCFUXFFOxBOEy
(c) 'JOEUIFWBMVFPGyXIFOxN
21.7 Increasing and decreasing amounts by a given ratio
t :PVDBOJODSFBTFPSEFDSFBTFBNPVOUTJOBHJWFOSBUJP
new amount
x × old amount
'PSBOJODSFBTF xy'PSBEFDSFBTFxy
t old amount = xy new amount =
y
Exercise 21.7
1 ćFQSJDFPGBOBSUJDMFDPTUJOHJTJODSFBTFEJOUIFSBUJP8IBUJTUIFOFXQSJDF 2 "OBQBSUNFOUQVSDIBTFEGPSIBTEFDSFBTFEJOWBMVFJOUIFSBUJP
8IBUJTJUXPSUIOPX 3 "DBSQVSDIBTFEGPSIBTEFDSFBTFEJOWBMVFJOUIFSBUJP
(a) 8IBUJTUIFOFXWBMVFPGUIFDBS (b) )PXNVDINPOFZXPVMEUIFPXOFSMPTFJGTIFTPMEUIFDBSGPSUIFEFDSFBTFEWBMVF 4 *ODSFBTFJOUIFSBUJP
Mixed exercise
1 "USJBOHMFPGQFSJNFUFSNNIBTTJEFMFOHUITJOUIFSBUJP
(a) 'JOEUIFMFOHUITPGUIFTJEFT
(b) *TUIFUSJBOHMFSJHIUBOHMFE (JWFBSFBTPOGPSZPVSBOTXFS
2 0OBĘPPSQMBOPGBTDIPPM DNSFQSFTFOUTNJOUIFSFBMTDIPPM
8IBUJTUIFTDBMFPGUIFQMBO
146
Unit 6: Number
21
Ratio, rate and proportion
3 "DBSUSBWFMTBUBOBWFSBHFTQFFEPGLNI
(a) 8IBUEJTUBODFXJMMUIFDBSUSBWFMJO
1
2
(i) IPVS
(ii) 4 IPVST
(iii) NJOVUFT
(b) )PXMPOHXJMMJUUBLFUIFDBSUPUSBWFM
(i) LN
(ii) LN
(iii) LN
4 ćJTUSBWFMHSBQITIPXTUIFKPVSOFZPGBQFUSPMUBOLFSEPJOHEFMJWFSJFT
Distance (km)
500
400
300
200
100
0
1
2
3
4
5
Time (hours)
6
7
(a) 8IBUEJTUBODFEJEUIFUBOLFSUSBWFMJOUIFĕSTUUXPIPVST
(b) 8IFOEJEUIFUBOLFSTUPQUPNBLFJUTĕSTUEFMJWFSZ 'PSIPXMPOHXBTJUTUPQQFE
(c) $BMDVMBUFUIFBWFSBHFTQFFEPGUIFUBOLFSCFUXFFOUIFĕSTUBOETFDPOETUPQ
POUIFSPVUF
(d) 8IBUXBTUIFBWFSBHFTQFFEPGUIFUBOLFSEVSJOHUIFMBTUUXPIPVSTPGUIFKPVSOFZ
(e) )PXGBSEJEUIFUBOLFSUSBWFMPOUIJTKPVSOFZ
5 ćJTTQFFEoUJNFHSBQITIPXTUIFTQFFEJONTGPSBDBSKPVSOFZ
20
Speed
(m/s) 10
0
(a)
(b)
(c)
(d)
10
20
30
40
Time (seconds)
50
60
'PSIPXMPOHXBTUIFDBSBDDFMFSBUJOH $BMDVMBUFUIFSBUFBUXIJDIJUEFDFMFSBUFEGSPNUPTFDPOET
8IBUEJTUBODFEJEUIFDBSDPWFSEVSJOHUIFĕSTUTFDPOET )PXNBOZNFUSFTEJEUIFDBSUBLFUPTUPQPODFJUTUBSUFEEFDFMFSBUJOH 6 /JOFTUVEFOUTDPNQMFUFBUBTLJOUISFFNJOVUFT)PXMPOHXPVMEJUUBLFTJYTUVEFOUTUP
DPNQMFUFUIFTBNFUBTLJGUIFZXPSLFEBUUIFTBNFSBUF
7 "DVCFXJUITJEFTPGDNIBTBNBTTPGHSBNT'JOEUIFNBTTPGBOPUIFSDVCFNBEF
PGUIFTBNFNBUFSJBMJGJUIBTTJEFTPGDN
Unit 6: Number
147
21
Ratio, rate and proportion
8 ćFQSFTTVSF P PGBHJWFOBNPVOUPGHBTJTJOWFSTFMZQSPQPSUJPOBMUPUIFWPMVNF V PG
UIFHBT
(a) &YQSFTTUIFSFMBUJPOTIJQCFUXFFOPBOEVJOBHFOFSBMJTFEFRVBUJPO
(b) *UJTGPVOEUIBUPXIFOV'JOEUIFWBMVFPGPXIFOV
9 4BMNBIBTBQIPUPHSBQINNMPOHBOENNXJEFćJTJTUPPCJHGPSIFSBMCVN
4IFSFEVDFTJUPOUIFDPNQVUFSTPUIBUJUJTNNMPOH
(a) *OXIBUSBUJPEJE4BMNBSFEVDFUIFQIPUPHSBQI (b) $BMDVMBUFUIFOFXXJEUIPGUIFQIPUPDPSSFDUUPUXPEFDJNBMQMBDFT
148
Unit 6: Number
22
More equations, formulae and
functions
22.1 Setting up equations to solve problems
t :PVDBOTFUVQZPVSPXOFRVBUJPOTBOEVTFUIFNUPTPMWFQSPCMFNTUIBUIBWFBOVNCFSBTBOBOTXFS
t ćFĕSTUTUFQJOTFUUJOHVQBOFRVBUJPOJTUPXPSLPVUXIBUOFFETUPCFDBMDVMBUFE3FQSFTFOUUIJTBNPVOU
VTJOHBWBSJBCMF VTVBMMZx :PVUIFODPOTUSVDUBOFRVBUJPOVTJOHUIFJOGPSNBUJPOZPVBSFHJWFOBOETPMWF
JUUPĕOEUIFBOTXFS
Tip
Exercise 22.1 A
8IFOZPVIBWFUPHJWF
BGPSNVMB ZPVTIPVME
FYQSFTTJUJOTJNQMFTUUFSNT
CZDPMMFDUJOHMJLFUFSNT
1 "SFDUBOHMFJTDNMPOHFSUIBOJUJTXJEF*GUIFSFDUBOHMFJTxDNMPOH XSJUFEPXO
(a) UIFXJEUI JOUFSNTPGx)
(b) BGPSNVMBGPSDBMDVMBUJOHUIFQFSJNFUFS P PGUIFSFDUBOHMF
(c) BGPSNVMBGPSĕOEJOHUIFBSFB A PGUIFSFDUBOHMFJOTJNQMFTUUFSNT
2 ćSFFOVNCFSTBSFSFQSFTFOUFECZx xBOEx (a) 8SJUFBGPSNVMBGPSĕOEJOHUIFTVN S PGUIFUISFFOVNCFST
(b) 8SJUFBGPSNVMBGPSĕOEJOHUIFNFBO M PGUIFUISFFOVNCFST
3 ćFTNBMMFTUPGUISFFDPOTFDVUJWFOVNCFSTJTx
(a) &YQSFTTUIFPUIFSUXPOVNCFSTJOUFSNTPGx
(b) 8SJUFBGPSNVMBGPSĕOEJOHUIFTVN S PGUIFOVNCFST
4 4BNNZJTUXPZFBSTPMEFSUIBO.BY5BZPJTUISFFZFBSTZPVOHFSUIBO.BY*G.BYJTxZFBST
PME XSJUFEPXO
(a) 4BNNZTBHFJOUFSNTPGx
(b) 5BZPTBHFJOUFSNPGx
(c) BGPSNVMBGPSĕOEJOHUIFDPNCJOFEBHFTPGUIFUISFFCPZT
Tip
Exercise 22.1 B
ćFSFJTVTVBMMZPOFXPSE
PSTIPSUTUBUFNFOUJOUIF
QSPCMFNUIBUNFBOTAFRVBM
UP&YBNQMFTBSFUPUBM HJWFT UIFBOTXFSJT UIF
SFTVMUJT QSPEVDUPGJT BOE
TVNPGJT
1 ćFSFBSFNPSFHJSMTUIBOCPZTJOBDMBTTPGTUVEFOUT)PXNBOZCPZTBSFUIFSF
2 "SFDUBOHMFIBTBOBSFBPGDNBOEUIFCSFBEUIJTDN'JOEUIFMFOHUIPGUIFSFDUBOHMF
3 ćFSFBSFUFOUJNFTBTNBOZTJMWFSDBSTUIBOSFEDBSTJOBQBSLJOHBSFB*GUIFSFTJMWFSBOE
SFEDBSTBMUPHFUIFS ĕOEUIFOVNCFSPGDBSTPGFBDIDPMPVS
4 /BEJSBJTZFBSTZPVOHFSUIBOIFSGBUIFS/BEJSBTNPUIFSJTUXPZFBSTZPVOHFSUIBOIFS
GBUIFS5PHFUIFS/BEJSB IFSNPUIFSBOEGBUIFSIBWFBDPNCJOFEBHFPG8PSLPVUUIFJSBHFT
Unit 6: Algebra
149
22
More equations, formulae and functions
5 ćFQFSJNFUFSPGBQBSBMMFMPHSBNJTDN*GUIFMFOHUIJTUISFFUJNFTUIFCSFBEUI DBMDVMBUF
UIFEJNFOTJPOTPGUIFQBSBMMFMPHSBN
6 ćSFFJUFNT X YBOEZBSFTVDIUIBUUIFQSJDFPGYJTDNPSFUIBOUIFQSJDFPGXBOEUIF
QSJDFPGZJTDDIFBQFSUIBOUIFQSJDFPGX5XPPGJUFNXQMVTUISFFPGJUFNYQMVTTJYPG
JUFNZDPTU8IBUJTUIFQSJDFPGFBDIJUFN
7 3VTIEJTGBUIFSJTQSFTFOUMZĕWFUJNFTIJTBHF*OUISFFZFBSTUJNF IFXJMMCFGPVSUJNFT
3VTIEJTBHF)PXPMEJT3VTIEJOPX 8 "DPODFSUUJDLFUDPTUT1FOTJPOFSTBOETUVEFOUTQBZ'PSPOFQFSGPSNBODF NPSF
EJTDPVOUUJDLFUTUIBOSFHVMBSQSJDFEUJDLFUTXFSFTPMEBUUIFEPPS*GXBTDPMMFDUFE IPX
NBOZSFHVMBSQSJDFEUJDLFUTXFSFTPME 22.2 Using and transforming formulae
t 5PDIBOHFUIFTVCKFDUPGBGPSNVMB
t
t
– FYQBOEUPHFUSJEPGBOZCSBDLFUT
– VTFJOWFSTFPQFSBUJPOTUPJTPMBUFUIFWBSJBCMFZPVSFRVJSF
8IFOBGPSNVMBDPOUBJOTTRVBSFEUFSNTPSTRVBSFSPPUT SFNFNCFSUIBUBTRVBSFEOVNCFSIBTCPUIBOFHBUJWFBOEB
QPTJUJWFSPPU
ćFWBSJBCMFUIBUJTUPCFNBEFUIFTVCKFDUNBZPDDVSNPSFUIBOPODFJOUIFGPSNVMB*GUIJTJTUIFDBTF HBUIFSUIFMJLF
UFSNTBOEGBDUPSJTFCFGPSFZPVFYQSFTTUIFGPSNVMBJOUFSNTPGUIFTVCKFDUWBSJBCMF
REWIND
You have met this topic in
chapter 6. This exercise uses the
same principles to transform
slightly more complicated
formulae. Exercise 22.2
1 ćFWBSJBCMFUIBUOFFETUPCFUIFTVCKFDUJTJOCSBDLFUT
$IBOHFUIFTVCKFDUPGFBDIGPSNVMBUPUIBUWBSJBCMF
(a) U TV W
(b) V W U¦T
C
(c) A = B
B
(d) A = C
(e) PQ
(f) PQ
(g) PQR
(h) Q
P
(V)
(V)
(i) Q
(P)
P PR
(j) Q
(P − R ) (k) PQR
(B)
(B)
(Q)
(Q)
(Q)
(P)
(P)
(Q)
2 0INTMBXJTBGPSNVMBUIBUMJOLTWPMUBHF V DVSSFOU I BOESFTJTUBODF R (JWFOUIBUVRI
(a) DIBOHFUIFTVCKFDUPGUIFGPSNVMBUPI
(b) ĕOEUIFDVSSFOU JOBNQT GPSBWPMUBHFPGWPMUTBOEBSFTJTUBODFPGPINT
150
Unit 6: Algebra
22
More equations, formulae and functions
3 ćFBSFBPGBDJSDMFDBOCFGPVOEVTJOHUIFGPSNVMBAɀr
(a) $IBOHFUIFTVCKFDUPGUIFGPSNVMBUPr
(b) '
JOEUIFMFOHUIPGUIFSBEJVTPGBDJSDMFXJUIBOBSFBPGNN(JWFZPVSBOTXFSDPSSFDU
UPUISFFTJHOJĕHBOUĕHVSFT
4 ćFGPSNVMBGPSĕOEJOHEFHSFFT$FMTJVTGSPNEFHSFFT'BISFOIFJUJTC
5
9
(F − 32) (a) .BLFFUIFTVCKFDUPGUIFGPSNVMB
(b) 'JOEUIFUFNQFSBUVSFJOEFHSFFT'BISFOIFJUXIFOJUJT¡$
(c) ć
FUFNQFSBUVSFJO,FMWJODBOCFGPVOEVTJOHUIFGPSNVMB KC XIFSFKJTUIF
UFNQFSBUVSFJO,FMWJOBOECJTUIFUFNQFSBUVSFJOEFHSFFT$FMTJVT6TFUIJTJOGPSNBUJPO
UPHFUIFSXJUIUIFQSFWJPVTGPSNVMB BTOFDFTTBSZ UPĕOEUIF,FMWJOFRVJWBMFOUPG¡'
22.3 Functions and function notation
t "GVODUJPOJTBSVMF PSTFUPGJOTUSVDUJPOT GPSDIBOHJOHBOJOQVUWBMVFJOUPBOPVUQVUWBMVF
t ćFTZNCPMG x JTVTFEUPEFOPUFBGVODUJPOPGxćJTJTDBMMFEGVODUJPOOPUBUJPO'PSFYBNQMFG x xo
t 5PĕOEUIFWBMVFPGBGVODUJPO TVDIBTG ZPVTVCTUJUVUFUIFHJWFOOVNCFSGPSxBOEDBMDVMBUFUIFWBMVFPGUIF
FYQSFTTJPO
"DPNQPTJUFGVODUJPOJTBDPNCJOBUJPOPGUXPPSNPSFGVODUJPOTćFPSEFSJOXIJDIUIFGVODUJPOTBSFXSJUUFOJT
t
JNQPSUBOUHGNFBOTĕSTUBQQMZGBOEUIFOBQQMZH
t ćFOPUBUJPOGPSUIFJOWFSTFPGBGVODUJPOPGGJTG ćJTJTSFBEBTGJOWFSTF:PVDBOUIJOLPGUIFJOWFSTFPGBGVODUJPO
o
BTJUTSFWFSTF"QQMZJOHUIFJOWFSTFNFBOTXPSLJOHCBDLXBSETBOEVOEPJOH EPJOHUIFJOWFSTF PGFBDIPQFSBUJPO
Tip
Exercise 22.3
'JOEJOHUIFJOWFSTFPGB
GVODUJPOJOQSBDUJDFNFBOT
NBLJOHxUIFTVCKFDUPGUIF
GPSNVMBJOTUFBEPGy
1 G x x DBMDVMBUF
(a) G (b) G o (c) G (d) G m 2 (JWFOGx→ x (a) 8SJUFEPXOUIFFYQSFTTJPOGPSG x (b) $BMDVMBUF
(i) G (ii) G (c) 4IPXXIFUIFSG G G (d) 'JOE
(iii) G (i) G a (ii) G b (e) 'JOEaJGG a (iii) G a b)
3 (JWFO h : x → 5 x (a) 8SJUFEPXOUIFFYQSFTTJPOGPSI x (b) 'JOE
(i) I (ii) I o
4 *GG x xBOEH x xo ĕOE
(a) GH
(b) HG
5 *GH x xBOEI x x ĕOEHI Unit 6: Algebra
151
22
More equations, formulae and functions
6 8SJUFEPXOUIFJOWFSTFPGFBDIPGUIFGPMMPXJOHGVODUJPOTVTJOHUIFDPSSFDUOPUBUJPO
(a) G x x (b) G x xo
x
7 (JWFOG x xoBOE g ( x ) = ĕOE
2
(a) GH x (b) HG x (d) HG o x (e) GoHo x Mixed exercise
(c) G x x
(d) f ( x ) =
x
−2
(c) GH o x)
(f) HoGo x)
1 "DFSUBJOOVNCFSJODSFBTFECZTJYJTFRVBMUPUXJDFUIFTBNFOVNCFSEFDSFBTFECZGPVS
8IBUJTUIFOVNCFS
2 ćFTVNPGUISFFDPOTFDVUJWFOVNCFSTJT'JOEUIFOVNCFST
3 ;PSJOBJTBRVBSUFSPGUIFBHFPGIFSNPUIFS XIPJTZFBSTPME*OIPXNBOZZFBSTUJNFXJMM
TIFCFBUIJSEPGIFSNPUIFSTBHF
4 $FESJDIBTNPSFUIBO/BUIJ5PHFUIFSUIFZIBWF)PXNVDIEPFTFBDI
QFSTPOIBWF
5 4JOEJ +POBTBOE.PQVUNPOFZUPHFUIFSUPDPOUSJCVUFUPDIBSJUZ+POBTQVUTJOIBMGUIF
BNPVOUUIBU4JOEJQVUTJOBOE.PQVUTJOMFTTUIBOUXJDFUIFBNPVOUUIBU4JOJEQVUTJO
8PSLPVUIPXNVDIFBDIQFSTPOQVUJO
6 "MUPHFUIFSBEVMUTBOEDIJMESFOUBLFQBSUJOBGVOIJLF"EVMUTQBZBOEDIJMESFOQBZ
*GXBTDPMMFDUFE IPXNBOZDIJMESFOUPPLQBSU
7 .BLFbUIFTVCKFDUPGFBDIGPSNVMB
5 (b ) (b − 3)
(a) a =
+
9
3
(b) a bo b)
8 (JWFO f ( x ) =
2x − 3
ĕOEGo x 5
9 G x x BOE g ( x ) =
(a) Go x (d) Č x 152
Unit 6: Algebra
x −2
ĕOE
5
(b) Go (e) HoG (c) UIFWBMVFPGaJGG a 23
Transformations and matrices
23.1 Simple plane transformations
t
t
t
t
t
t
t
"USBOTGPSNBUJPOJTBDIBOHFJOUIFQPTJUJPOPSTJ[FPGBQPJOUPSBTIBQF
ćFSFBSFGPVSCBTJDUSBOTGPSNBUJPOTSFĘFDUJPO SPUBUJPO USBOTMBUJPOBOEFOMBSHFNFOU
o ćFPSJHJOBMTIBQFJTDBMMFEUIFPCKFDU 0 BOEUIFUSBOTGPSNFETIBQFJTDBMMFEUIFJNBHF 0′ "SFĘFDUJPOĘJQTUIFTIBQFPWFS
o 6OEFSSFĘFDUJPO FWFSZQPJOUPOBTIBQFJTSFĘFDUFEJOBNJSSPSMJOFUPQSPEVDFBNJSSPSJNBHFPGUIFPCKFDU
1PJOUTPOUIFPCKFDUBOEDPSSFTQPOEJOHQPJOUTPOUIFJNBHFBSFUIFTBNFEJTUBODFGSPNUIFNJSSPSMJOF XIFO
ZPVNFBTVSFUIFEJTUBODFQFSQFOEJDVMBSUPUIFNJSSPSMJOF
o 5PEFTDSJCFBSFĘFDUJPOZPVOFFEUPHJWFUIFFRVBUJPOPGUIFNJSSPSMJOF
"SPUBUJPOUVSOTUIFTIBQFBSPVOEBQPJOU
o ćFQPJOUBCPVUXIJDIBTIBQFJTSPUBUFEJTDBMMFEUIFDFOUSFPGSPUBUJPOćFTIBQFNBZCFSPUBUFEDMPDLXJTFPS
BOUJDMPDLXJTF
o 5PEFTDSJCFBSPUBUJPOZPVOFFEUPHJWFUIFDFOUSFPGSPUBUJPOBOEUIFBOHMFBOEEJSFDUJPOPGUVSO
"USBOTMBUJPOJTBTMJEFNPWFNFOU
o 6OEFSUSBOTMBUJPO FWFSZQPJOUPOUIFPCKFDUNPWFTUIFTBNFEJTUBODFJOUIFTBNFEJSFDUJPOUPGPSNUIFJNBHF
5SBOTMBUJPOJOWPMWFTNPWJOHUIFTIBQFTJEFXBZTBOEPSVQBOEEPXOćFUSBOTMBUJPODBOUIFSFGPSFCF
⎛ x⎞
EFTDSJCFEVTJOHBDPMVNOWFDUPS⎜ ⎟ XIFSFxJTUIFNPWFNFOUUPUIFTJEF BMPOHUIFxBYJT BOEyJTUIF
⎝ y⎠
NPWFNFOUVQPSEPXOćFTJHOPGUIFxPSyWBMVFHJWFTZPVUIFEJSFDUJPOPGUIFUSBOTMBUJPO1PTJUJWFNFBOTUP
UIFSJHIUPSVQBOEOFHBUJWFNFBOTUPUIFMFęPSEPXO
"OFOMBSHFNFOUJOWPMWFTDIBOHJOHUIFTJ[FPGBOPCKFDUUPQSPEVDFBOJNBHFUIBUJTTJNJMBSJOTIBQFUPUIFPCKFDU
length of a side on the image
o The enlargement factor =
8IFOBOPCKFDUJTFOMBSHFEGSPNB
length of the corresponding side on the object
ĕYFEQPJOU JUIBTBDFOUSFPGFOMBSHFNFOUćFDFOUSFPGFOMBSHFNFOUEFUFSNJOFTUIFQPTJUJPOPGUIFJNBHF-JOFT
ESBXOUISPVHIDPSSFTQPOEJOHQPJOUTPOUIFPCKFDUBOEUIFJNBHFXJMMNFFUBUUIFDFOUSFPGFOMBSHFNFOU
ćFUSBOTGPSNBUJPOTBCPWFDBOCFDPNCJOFE
Tip
3FĘFDUJPOBOESPUBUJPO
DIBOHFUIFQPTJUJPOBOE
PSJFOUBUJPOPGUIFPCKFDU
XIJMFUSBOTMBUJPOPOMZ
DIBOHFTUIFQPTJUJPO
&OMBSHFNFOUDIBOHFT
UIFTJ[FPGUIFPCKFDUUP
QSPEVDFUIFJNBHFCVUJUT
PSJFOUBUJPOJTOPUDIBOHFE
Exercise 23.1 A
:PVXJMMOFFETRVBSFEQBQFSGPSUIJTFYFSDJTF
1 %SBXBOEMBCFMBOZSFDUBOHMFABCD
(a) 3PUBUFUIFSFDUBOHMFDMPDLXJTF°BCPVUQPJOUD-BCFMUIFJNBHFA′B′C′D′
(b) 3FĘFDUA′B′C′D′BCPVUB′D′
Unit 6: Shape, space and measures
153
23 Transformations and matrices
2 .BLFBDPQZPGUIFEJBHSBNCFMPXBOEDBSSZPVUUIFGPMMPXJOHUSBOTGPSNBUJPOT
y
8
A
6
4
2
B
–8
C
–6
–4
–2
0
–2
D
E
F
2
4
6
8
x
I
–4
–6
G
H
K
J
–8
(a) 5SBOTMBUFΔABCUISFFVOJUTUPUIFSJHIUBOEGPVSVOJUTVQ-BCFMUIFJNBHFDPSSFDUMZ
(b) 3FĘFDUSFDUBOHMFDEFGBCPVUUIFMJOFy=−-BCFMUIFJNBHFDPSSFDUMZ
(c) (i) 3PUBUFQBSBMMFMPHSBNHIJK°BOUJDMPDLXJTFBCPVUQPJOU − (ii) 5SBOTMBUFUIFJNBHFH ′I ′J ′K ′POFVOJUMFęBOEĕWFVOJUTVQ
3 .BLFBDPQZPGUIFEJBHSBNCFMPXBOEDBSSZPVUUIFGPMMPXJOHUSBOTGPSNBUJPOT
⎛ 10 ⎞
(a) ΔABCJTUSBOTMBUFEVTJOHUIFDPMVNOWFDUPS ⎜ ⎟ UPGPSNUIFJNBHFA′B ′C ′%SBXBOE
⎝ −9⎠
MBCFMUIFJNBHF
(b) 2VBESJMBUFSBMPQRSJTSFĘFDUFEJOUIFyBYJTBOEUIFOUSBOTMBUFEVTJOHUIFDPMVNO
⎛ 0⎞
WFDUPS ⎜ ⎟ %SBXUIFSFTVMUBOUJNBHFP′Q′R′S ′
⎝ −6⎠
154
Unit 6: Shape, space and measures
23 Transformations and matrices
4 'PSFBDIPGUIFSFĘFDUJPOTTIPXOJOUIFEJBHSBN HJWFUIFFRVBUJPOPGUIFNJSSPSMJOF
10
y
8
D
D'
A
6
4
A'
2
–10
C'
–8
–6
–4
–2
2
0
4
6
8
10
x
–2
C
–4
B'
B
–6
–8
–10
5 $PQZUIFEJBHSBNGPSRVFTUJPOBOEESBXUIFSFĘFDUJPOPGFBDITIBQF "o% JOUIFxBYJT
6 *OFBDIPGUIFGPMMPXJOH GVMMZEFTDSJCFBUMFBTUUXPEJČFSFOUUSBOTGPSNBUJPOTUIBUNBQUIF
PCKFDUPOUPJUTJNBHF
10
y
8
6
4
2
–10 –8
–6
–4
–2 0
–2
2
4
6
8
10
x
–4
–6
–8
–10
Unit 6: Shape, space and measures
155
23 Transformations and matrices
Exercise 23.1 B
REWIND
You dealt with enlargement and
scale factors in chapter 21. 1 'PSFBDIQBJSPGTIBQFT HJWFUIFDPPSEJOBUFTPGUIFDFOUSFPGFOMBSHFNFOUBOEUIFTDBMFGBDUPS
PGUIFFOMBSHFNFOU
y
8
You can find the centre of
enlargement by drawing lines
through the corresponding vertices
on the two shapes. The lines will
meet at the centre of enlargement.
6
B'
4
When the image is smaller than
the object, the scale factor of the
‘enlargement’ will be a fraction.
A'
–8
–6
–4
A
–2
2
0
B
–2
C'
4
2
6
D
–4
C
x
8
D'
–6
–8
2 $PQZFBDITIBQFPOUPTRVBSFEHSJEQBQFS6TJOHQPJOU9BTBDFOUSFPGFOMBSHFNFOUBOEB
TDBMFGBDUPSPGUXP ESBXUIFJNBHFPGFBDITIBQFVOEFSUIFHJWFOFOMBSHFNFOU
a
b
c
d
A
C
X
B
X
156
Unit 6: Shape, space and measures
X
D
X
23 Transformations and matrices
Exercise 23.1 C
1 5SJBOHMFABCJTUPCFSFĘFDUFEJOUIFyBYJTBOEJUTJNBHFΔA′B′C′JTUIFOUPCFSFĘFDUFEJO
UIFxBYJTUPGPSNΔA″ B″ C″
y
8
A
6
4
C
2
–1
–1
B
2
4
x
8
6
(a) %SBXBTFUPGBYFT FYUFOEJOHUIFNJOUPUIFOFHBUJWFEJSFDUJPO$PQZΔABCPOUPZPVS
HSJEBOEESBXUIFUSBOTGPSNBUJPOTEFTDSJCFE
(b) %FTDSJCFUIFTJOHMFUSBOTGPSNBUJPOUIBUNBQTΔABCEJSFDUMZPOUPΔA″ B″ C″ 2 4IBQF"JTUPCFFOMBSHFECZBTDBMFGBDUPSPGUXP VTJOHUIFPSJHJOBTUIFDFOUSFPG
⎛ −8⎞
FOMBSHFNFOU UPHFUTIBQF#4IBQF#JTUIFOUSBOTMBUFE VTJOHUIFDPMVNOWFDUPS ⎜ ⎟ UPHFU
⎝ 1⎠
TIBQF$
y
8
6
4
2
–1
–1
A
2
4
6
8
x
(a) %SBXBTFUPGBYFT FYUFOEJOHUIFxBYJTJOUPUIFOFHBUJWFEJSFDUJPO$PQZTIBQF"POUP
ZPVSHSJEBOEESBXUIFUXPUSBOTGPSNBUJPOTEFTDSJCFE
(b) 8IBUTJOHMFUSBOTGPSNBUJPOXPVMEIBWFUIFTBNFSFTVMUTBTUIFTFUXPUSBOTGPSNBUJPOT
3 "USBQF[JVNABCDXJUIJUTWFSUJDFTBUDPPSEJOBUFTA B C BOED JTUP
CFSFĘFDUFEJOUIFMJOFx=ćFJNBHFJTUPCFSFĘFDUFEJOUIFMJOFy=
(a) %SBXUIFTIBQFBOETIPXJUTQPTJUJPOBęFSFBDISFĘFDUJPO-BCFMUIFĕOBMJNBHF'′
(b) %FTDSJCFUIFTJOHMFUSBOTGPSNBUJPOZPVDPVMEVTFUPUSBOTGPSNABCDUP'′
Unit 6: Shape, space and measures
157
23 Transformations and matrices
23.2 Vectors
t
t
t
t
t
t
t
t
t
t
t
"WFDUPSJTBRVBOUJUZUIBUIBTCPUINBHOJUVEF TJ[F BOEEJSFDUJPO
7FDUPSTDBOCFSFQSFTFOUFECZMJOFTFHNFOUTćFMFOHUIPGUIFMJOFSFQSFTFOUTUIFNBHOJUVEFPGUIFWFDUPS
BOEUIFBSSPXPOUIFMJOFSFQSFTFOUTUIFEJSFDUJPOPGUIFWFDUPS"WFDUPSSFQSFTFOUFECZMJOFTFHNFOUABTUBSUT
BUABOEFYUFOETJOUIFEJSFDUJPOPGB
ćFOPUBUJPOa b c PSAB AB JTVTFEGPSWFDUPST
⎛ x⎞
7FDUPSTDBOBMTPCFXSJUUFOBTDPMVNOWFDUPSTJOUIFGPSN⎜ ⎟ ⎝ y⎠
"DPMVNOWFDUPSSFQSFTFOUTBUSBOTMBUJPO IPXUIFQPJOUBUPOFFOEPGUIFWFDUPSNPWFTUPHFUUPUIFPUIFS
FOEPGUIFMJOF ⎛ 1⎞
ćFDPMVNOWFDUPS⎜ ⎟ SFQSFTFOUTBUSBOTMBUJPOPOFVOJUJOUIFxEJSFDUJPOBOEUXPVOJUTJOUIFyEJSFDUJPO
⎝ 2⎠
7FDUPSTBSFFRVBMJGUIFZIBWFUIFTBNFNBHOJUVEFBOEUIFTBNFEJSFDUJPO
ćFOFHBUJWFPGBWFDUPSJTUIFWFDUPSXJUIUIFTBNFNBHOJUVEFCVUUIFPQQPTJUFEJSFDUJPO4P
UIF
OFHBUJWFPGa JT–aBOEUIFOFHBUJWFPG AB JT AB 7FDUPSTDBOOPUCFNVMUJQMJFECZFBDIPUIFS CVUUIFZDBOCFNVMUJQMJFECZBTDBMBS BOVNCFS ⎛ kx ⎞
⎛ x⎞
.VMUJQMZJOHBOZWFDUPS ⎜ ⎟ CZBTDBMBSk HJWFT ⎜ ⎟ y
⎝ ⎠
⎝ ky ⎠
7FDUPSTDBOCFBEEFEBOETVCUSBDUFEVTJOHUIFAOPTFUPUBJMNFUIPEPSUSJBOHMFSVMF TP
⎛ x1 ⎞ ⎛ x2 ⎞ ⎛ x1 + x2 ⎞
⎜⎝ y ⎟⎠ + ⎜⎝ y ⎟⎠ = ⎜⎝ y + y ⎟⎠ 1
2
1
2
5PTVCUSBDUWFDUPST ZPVOFFEUPSFNFNCFSUIBUTVCUSBDUJOHBWFDUPSJTUIFTBNFBTBEEJOHJUTOFHBUJWF
4P AB − CA = AB + AC
⎛ ⎞
:PVDBOVTF1ZUIBHPSBTUIFPSFNUPĕOEUIFNBHOJUVEF MFOHUI PGBWFDUPS*GUIFWFDUPSJT⎜⎝ y ⎟⎠ UIFOJUTNBHOJUVEFJT
x 2 + y 2 ćFOPUBUJPO]B]PS]AB ]JTVTFEUPSFQSFTFOUUIFNBHOJUVEFPGUIFWFDUPS
"WFDUPSUIBUTUBSUTGSPNUIFPSJHJO 0 JTDBMMFEBQPTJUJPOWFDUPS*GOAJTBQPTJUJPOWFDUPS UIFOUIFDPPSEJOBUFTPG
ONVTUCF BOEUIFDPPSEJOBUFTPGQPJOUAIBWFUPCFUIFTBNFBTUIFDPNQPOFOUTPGUIFDPMVNOWFDUPSOA
:PVDBOVTFQPTJUJPOWFDUPSTUPĕOEUIFNBHOJUVEFPGBOZSFMBUFEWFDUPSCFDBVTFZPVLOPXUIFDPPSEJOBUFTPGUIF
QPJOUT:PVDBOUIFOVTF1ZUIBHPSBTUPXPSLPVUUIFMFOHUITPGUIFTJEFTPGBSFMBUFESJHIUBOHMFEUSJBOHMF
Exercise 23.2 A
10
Remember, a scalar is a quantity
without direction, basically just a
number or measurement.
y
8
6
A
B
4
2
Remember, equal vectors have the
same magnitude and direction;
negative vectors have the same
magnitude and opposite directions.
158
–6
Unit 6: Shape, space and measures
–4
–2
E
2
0
–2
D
4
6
8
1 6TJOHUIFQPJOUTPOUIFHSJE FYQSFTT
FBDIPGUIFGPMMPXJOHBTBDPMVNO
WFDUPS
(a)
(b) BC (c) AE (d) BD
(e) DB (f) EC (g) CD (h) BE
(i) 8IBUJTUIFSFMBUJPOTIJQCFUXFFO
BE BOECD
(j) 8IBUJTAB + BC
x
(k)
8IBUJT AE − AB 10
(l) *TBC = ED
23 Transformations and matrices
2 3FQSFTFOUUIFGPMMPXJOHWFDUPSTPOTRVBSFEQBQFS
Tip
⎛ 2⎞
(a) AB = ⎜ ⎟
⎝ 3⎠
8IFOZPVESBXBWFDUPS
POTRVBSFEQBQFSZPVDBO
DIPPTFBOZTUBSUJOHQPJOU
⎛ −1⎞
(b) CD = ⎜ ⎟ ⎝ 4⎠
⎛ 4⎞
(c) EF = ⎜ ⎟
⎝ 5⎠
⎛ 3⎞
(d) GH = ⎜ ⎟
⎝ −4⎠
3 'JOEUIFDPMVNOWFDUPSUIBUEFTDSJCFTUIFUSBOTMBUJPOGSPNUIFPCKFDUUPJUTJNBHFJOFBDIPG
UIFGPMMPXJOHFYBNQMFT
Remember, A is the object and A’
is the image.
10
y
8
B'
6
C
B
4
C'
–10 –8
–6
–4
E
A
2
–2 0
–2
E'
2
D
–4
D'
–6
4
6
8
10
x
A'
–8
–10
Exercise 23.2 B
⎛ 1⎞
⎛ −2⎞
⎛ 0⎞
1 (JWFOUIBUa =⎜ ⎟ b =⎜ ⎟ BOEc=⎜ ⎟ ĕOE
4
3
⎝ ⎠
⎝ ⎠
⎝ −3⎠
(a) b
(f) a+b+c
(b) a
(g) a+b
(c) −c
(h) b¦c
(d) a+b
(i) −a¦b
(e) b+c
(j) a¦b+c
2 ćFEJBHSBNTIPXTUIFQBUUFSOPOBĘPPSUJMFćFUJMFTBSFTRVBSFTEJWJEFEJOUPGPVS
DPOHSVFOUUSJBOHMFTCZUIFJOUFSTFDUJOHEJBHPOBMTPGFBDITRVBSF
A
B
G
a
D
F
c
O
FWFDUPST = a OC =bBOEOG =c BSFTIPXO6TFUIJT
ć
JOGPSNBUJPOUPXSJUFFBDIPGUIFGPMMPXJOHJOUFSNTPGa bBOEc
(a) DE (b) AD
(c) AG (d) OB
(e) OE (f) CD (g) BF + FD
(h) DE + EF
1 (i) 4 BF 3EF (j) OC + 3GB
2
b
C
E
Unit 6: Shape, space and measures
159
23 Transformations and matrices
3 %SBXBOZQPTJUJPOWFDUPSOABPOBTFUPGBYFTBOEUIFOJOEJDBUF
(a) –a
Tip
*GZPVDBOOPUSFNFNCFS
UIFSVMFGPSĕOEJOHUIF
NBHOJUVEFPGBWFDUPS UIFOESBXUIFWFDUPSPOB
HSJEBTUIFIZQPUFOVTFPGB
SJHIUBOHMFEUSJBOHMF:PV
XJMMRVJDLMZTFFIPXUP
XPSLPVUJUTMFOHUIVTJOH
1ZUIBHPSBTUIFPSFN
(b) a
(c)
1
2
a
(d) oa
(e) − 1 a
2
4 ćFGPMMPXJOHWFDUPSTBSFESBXOPOBDNTRVBSFEHSJE$BMDVMBUFUIFNBHOJUVEFPGFBDI
WFDUPSJODFOUJNFUSFT(JWFZPVSBOTXFSTDPSSFDUUPUXPEFDJNBMQMBDFT
⎛ −2⎞
⎛ 4⎞
(a) MD = ⎜ ⎟ (b) PQ = ⎜ ⎟
⎝ 7⎠
⎝ 5⎠
⎛ 9 ⎞
⎛ −13⎞
(c) XY = ⎜
(d) ST = ⎜
⎟
⎝ −12⎠
⎝ −12⎟⎠
5 1PJOUOJTUIFPSJHJO1PJOUXJT QPJOUYJT o BOEQPJOUZJT o o 'JOEUIFWBMVF
PG
(a) ]OX ]
(b) ]OY ]
(d) ]XY ]
(c) ]OZ ]
⎛ −6⎞ ⎛ −2⎞
⎛ 5⎞
6 (JWFOUIFQPTJUJPOWFDUPSTOA = ⎜ ⎟ OB = ⎜ ⎟ BOEOC = ⎜ ⎟ ⎝ −4⎠
⎝ 2⎠
⎝ 1⎠
(a) HJWFUIFDPPSEJOBUFTPGQPJOUTA BBOEC
(b) FYQSFTTWFDUPSTAB BC BOECABTDPMVNOWFDUPST
7 *O6XYZ xBOEy
xBOEYZ yMJTUIFNJEQPJOUPGXY &YQSFTTXZ ZX BOEMZ JOUFSNTPG
Y
x
y
M
Tip
8IFOUXPWFDUPSTBSF
FRVBMUIFZIBWFUIFTBNF
NBHOJUVEFBOEEJSFDUJPO IFODFUIFMJOFTBTTPDJBUFE
XJUIUIFNBSFFRVBMBOE
QBSBMMFM
7FDUPSTXJUIUIFTBNF
EJSFDUJPOBSFQBSBMMFM FWFO
JGUIFZEPOPUIBWFUIF
TBNFNBHOJUVEF
X
Z
⎛1⎞
⎛ −2⎞
⎛ 3⎞
8 a⎜ 2⎟ b⎜ ⎟ BOEc ⎜ ⎟ ⎝ ⎠
⎝ 5⎠
⎝ 0⎠
(a) 8SJUFBDPMVNOWFDUPSGPSx yBOEz JG
(i) xa b c
(ii) yaoboc
(iii) za boc
(b) $BMDVMBUFBOEHJWFZPVSBOTXFSTUPUISFFTJHOJĕDBOUĕHVSFT
(i) ]x]
Tip
%SBXBEJBHSBNUP
SFQSFTFOUUIJTTJUVBUJPO
BOEUIFOBQQMZUIFWFDUPS
DBMDVMBUJPOT
(ii) ]y]
9 *OUIJTUSJBOHMF XBOEYBSFUIFNJEQPJOUTPGABBOEACSFTQFDUJWFMZDJTUIFNJEQPJOUPG
XY AX aBOEAY b
B
X
a
A
D
b
Y
C
160
(iii) ]z]
Unit 6: Shape, space and measures
23 Transformations and matrices
(a) &YQSFTTUIFTFWFDUPSTJOUFSNTPGaBOEb HJWJOHZPVSBOTXFSTJOUIFJSTJNQMFTUGPSN
(i) XY
(ii) AD
(iii) BC
(b) 4IPXUIBUXY ]]BCBOEJTIBMGJUTMFOHUI
10 $BSMPTKPHHFEGPSNJOVUFTBUBTUFBEZQBDFPGLNIJOBOPSUIFBTUFSMZEJSFDUJPO)F
UIFOXBMLFEXFTUXBSETVOUJMIFXBTEVFOPSUIPGIJTTUBSUJOHQPJOU XIFSFIFTUPQQFEGPS
MVODI*G$BSMPTKPHHFEIPNFUSBWFMMJOHEVFTPVUIBUUIFTBNFQBDFBTIFTFUPČ IPXNBOZ
NJOVUFTEJEUIFMBTUMFHPGIJTKPVSOFZUBLFIJN (JWFZPVSBOTXFSUPPOFEFDJNBMQMBDF
23.3 Further transformations
t
t
t
"TIFBSJTBUSBOTGPSNBUJPOUIBULFFQTPOFMJOFPGBTIBQFĕYFE JOWBSJBOU BOENPWFTBMMPUIFSQPJOUTPOUIFTIBQF
QBSBMMFMUPUIFJOWBSJBOUMJOF
o ćFEJTUBODFUIBUUIFQPJOUTNPWFJTQSPQPSUJPOBMUPUIFEJTUBODFPGUIFQPJOUTGSPNUIFJOWBSJBOUMJOF UIF
DPOTUBOUPGQSPQPSUJPOBMJUZJTDBMMFEUIFTIFBSGBDUPS
o 5PEFTDSJCFBTIFBSZPVOFFEUPHJWFUIFTIFBSGBDUPSBOETUBUFXIJDIMJOFJTJOWBSJBOU
"POFXBZTUSFUDIJTBUSBOTGPSNBUJPOUIBUFOMBSHFTUIFTIBQFJOPOFEJSFDUJPO QFSQFOEJDVMBSUPBOJOWBSJBOUMJOF
o ćFEJTUBODFUIBUUIFQPJOUTNPWFGSPNUIFJOWBSJBOUMJOFJTQSPQPSUJPOBMUPUIFJSPSJHJOBMEJTUBODFGSPNUIF
JOWBSJBOUMJOF UIFDPOTUBOUPGQSPQPSUJPOBMJUZJTDBMMFEUIFTDBMFGBDUPS
o 5PEFTDSJCFBPOFXBZTUSFUDIZPVOFFEUPHJWFUIFTDBMFGBDUPSBOETUBUFXIJDIMJOFJTJOWBSJBOU
"UXPXBZTUSFUDIJTBDPNCJOBUJPOPGUXPPOFXBZTUSFUDIFTBU¡UPFBDIPUIFS
o ćFPOMZJOWBSJBOUQPJOUPOUIFPCKFDUJTUIFQPJOUXIFSFUIFUXPQFSQFOEJDVMBSJOWBSJBOUMJOFTDSPTT UIJOLPGUIJT
BTUIFPSJHJOPGUIFTUSFUDIoPOFTUSFUDIJTUIFOJOyEJSFDUJPOBOEUIFPUIFSJTJOxEJSFDUJPO o "UXPXBZTUSFUDIBČFDUTCPUIUIFxBOEyDPPSEJOBUFTPGFBDIQPJOUCZUIFSFTQFDUJWFTDBMFGBDUPST
Exercise 23.3
1 %SBXUIFJNBHFPGFBDITIBQFBęFSUIFHJWFOUSBOTGPSNBUJPO
(a)
4IFBSGBDUPS xJTJOWBSJBOU (b) 4IFBSGBDUPS yJTJOWBSJBOU
y
y
10
9
8
7
6
5
4
3
2
1
–1 0
–1
x
1 2 3 4 5 6 7 8 9 10
10
9
8
7
6
5
4
3
2
1
–1 0
–1
x
1 2 3 4 5 6 7 8 9 10
Unit 6: Shape, space and measures
161
23 Transformations and matrices
Tip
(c)
t:PVDBOUIJOLPGBPOF
4IFBSGBDUPS yBYJTJTJOWBSJBOU
(d) 4USFUDI TDBMFGBDUPS yBYJTJTJOWBSJBOU
y
y
10
9
8
7
6
5
4
3
2
1
XBZTUSFUDIBTBTIFBS
QFSQFOEJDVMBSUPUIF
JOWBSJBOUMJOF
t"TDBMFGBDUPSkM
SFEVDFTUIFTJ[FPGUIF
TIBQFJOBTUSFUDIBOE
UIFJNBHFXJMMCFDMPTFS
UP SBUIFSUIBOGVSUIFS
GSPN UIFJOWBSJBOUMJOF
–1 0
–1
x
1 2 3 4 5 6 7 8 9 10
10
9
8
7
6
5
4
3
2
1
–1 0
–1
x
1 2 3 4 5 6 7 8 9 10
2 %FTDSJCFUIFTJOHMFUSBOTGPSNBUJPOUIBUNBQTFBDIPCKFDUPOUPJUTJNBHF
162
(a)
(b)
(c)
(d)
Unit 6: Shape, space and measures
23 Transformations and matrices
23.4 Matrices and matrix transformation
t
t
t
t
t
t
t
t
t
t
"NBUSJYJTBOBSSBZPGOVNCFSTćFOVNCFSTJOUIFNBUSJYBSFDBMMFEUIFFMFNFOUTPGUIFNBUSJY
ćFPSEFSPGNBUSJYJTUIFOVNCFSPGSPXTNVMUJQMJFECZUIFOVNCFSPGDPMVNOT4PBNBUSJYXJUIUISFFSPXTBOE
GPVSDPMVNOTIBTBOPSEFSPG¨
.BUSJDFTPGUIFTBNFPSEFSDBOCFBEEFEBOETVCUSBDUFE5PEPUIJT BEEPSTVCUSBDUUIFDPSSFTQPOEJOHFMFNFOUTPG
UIFUXPNBUSJDFTUPĕOEUIFTPMVUJPO XIJDIXJMMCFBNBUSJYPGUIFTBNFPSEFSBTUIFPSJHJOBMT .BUSJDFTDBOCFNVMUJQMJFECZTDBMBST OVNCFST .VMUJQMZFBDIFMFNFOUPGUIFNBUSJYCZUIFTDBMBS:PVSBOTXFSXJMM
CFBNBUSJYPGUIFTBNFPSEFSBTUIFPSJHJOBM
:PVDBONVMUJQMZUXPNBUSJDFT A¨B POMZJGUIFOVNCFSPGDPMVNOTJOAJTFRVBMUPUIFOVNCFSPGSPXTJOBćF
FMFNFOUTPGBSPXJOABSFNVMUJQMJFECZUIFDPSSFTQPOEJOHFMFNFOUTPGBDPMVNOJOBBOEUIFQSPEVDUTBEEFEoUIF
SFTVMUHPFTJOUPBQPTJUJPOFRVJWBMFOUUPUIFSPXOVNCFSPGABOEUIFDPMVNOOVNCFSPGBćJTJTSFQFBUFEGPSBMM
DPNCJOBUJPOTPGSPXTGSPNAXJUIDPMVNOTGSPNB
⎛ 1 0⎞
ćFTRVBSF¨NBUSJYI⎜
JTDBMMFEUIFidentityNBUSJY8IFOZPVNVMUJQMZB¨NBUSJYACZ*PS*CZB
⎝ 0 1⎟⎠
¨NBUSJYAUIFOAIABOEIAA
⎛ 0 0⎞
ćFTRVBSF¨NBUSJYO⎜
JTDBMMFEUIFzeroNBUSJYGPS¨NBUSJDFT.VMUJQMJDBUJPOCZUIF[FSPNBUSJYJT
⎝ 0 0⎟⎠
FRVBMUPUIF[FSPNBUSJY FBDIFMFNFOUJTNVMUJQMJFECZ HJWJOHBMMSPVOE ⎛ a b⎞
'PSBOZ¨NBUSJYA
UIFDBMDVMBUJPO adobc JTDBMMFEUIFdeterminantPGUIFNBUSJYćFTZNCPM]A]
⎜⎝ c d ⎟⎠
EFOPUFTUIFEFUFSNJOBOUPGA
:PVDBOOPUEJWJEFNBUSJDFT CVUZPVDBOĕOEUIFJOWFSTFPGBOPOTJOHVMBSNBUSJY+VTUBTBOVNCFS¨JUTJOWFSTF BNBUSJY¨JUTJOWFSTFI UIFidentityNBUSJY ćFJOWFSTFPGNBUSJYAJTEFOPUFECZA–1 TPA¨A–1I XIFSFUIF
JOWFSTFFYJTUT b⎞
⎛d
1
⎛
⎞ –1
1
'PSA⎜
A ⎜
⎟
(ad − bc ) ⎝ −c a ⎟⎠ ćFNVMUJQMJFS (ad − bc ) JTUIFSFDJQSPDBMPGUIFEFUFSNJOBOUPGUIFNBUSJY
⎝ c d⎠
BOEUIFFMFNFOUTPGUIFNBUSJYJOUIFNVMUJQMJDBUJPOBSFGPVOECZTXBQQJOHFMFNFOUTaBOEdBOEDIBOHJOHUIFTJHO
PGFMFNFOUTbBOEcPGUIFPSJHJOBMNBUSJYA
t
8IFOadbc UIFO
1
(ad − bc )
1
1
= #FDBVTF JTVOEFĕOFE ZPVDBOOPUEJWJEFCZ UIFNBUSJYIBTOPJOWFSTFBOEJU
0
0
JTDBMMFEBTJOHVMBSNBUSJY
Exercise 23.4
⎛1
⎛ −1 2 3 ⎞
1 (JWFOA⎜
BOEB
⎜⎝ 3
⎝ 4 3 −3⎟⎠
(a) A B
(b) AoB
4 2⎞
ĕOE
1 0⎟⎠
(c) BoA
⎛ 1 4⎞
2 D⎜
⎝ 5 8⎟⎠
(a) 'JOEUIFEFUFSNJOBOUPGD
(b) 'JOED–1
Unit 6: Shape, space and measures
163
23 Transformations and matrices
⎛3
3 (JWFOA⎜
⎝1
0⎞
⎛ 1 3⎞
BOEB⎜
2⎟⎠
⎝ −4 2⎟⎠
(a) 4IPXCZDBMDVMBUJPOUIBUABȶBA
(b) 'JOE
(i) ]A]
(ii) ]AB]
⎛1
4 (JWFOA ⎜
⎝4
3 0⎞
⎛6
BOEB
⎜⎝ 7
2 −1⎟⎠
(a) A B
(b) 3A
3 −3⎞
ĕOE
2 5 ⎟⎠
(c) oB
⎛ 2 x⎞
5 (JWFOUIBUC ⎜
BOE]C]
⎝ 1 4 ⎟⎠
(a) 'JOEUIFWBMVFPGx
(b) 'JOEC–1
23.5 Matrices and transformations
t
$PMVNOWFDUPSTDBOCFVTFEUPSFQSFTFOUUSBOTMBUJPOTPOBHSJEBOE¨NBUSJDFTDBOSFQSFTFOUSFĘFDUJPO SPUBUJPO FOMBSHFNFOU TUSFUDIBOETIFBSUSBOTGPSNBUJPOT
t
5PĕOEUIFJNBHFPGBOZQPJOUZPVNVMUJQMZUIFNBUSJYCZUIFQPTJUJPOWFDUPSPGUIFQPJOU 3FNFNCFSUIFQPTJUJPO
⎛ ⎞
WFDUPSPGBQPJOU⎜ ⎟ HJWFTZPVUIFDPPSEJOBUFT x y PGUIFQPJOU
⎝ y⎠
t
5PĕOEXIBU¨NBUSJYSFQSFTFOUTBHJWFOUSBOTGPSNBUJPO
o TFMFDUUIFQPJOUT BOE o ĕOEUIFJNBHFTPGUIFTFQPJOUTVOEFSUIFHJWFOUSBOTGPSNBUJPO
o XSJUFEPXOUIFQPTJUJPOWFDUPSTPGUIFTFJNBHFT
o UIFQPTJUJPOWFDUPSPGUIFJNBHFPGUIFĕSTUQPJOUCFDPNFTUIFĕSTUDPMVNOBOEUIFQPTJUJPOWFDUPSPGUIFJNBHFPG
UIFTFDPOEQPJOUJTUIFTFDPOEDPMVNOPGUIFSFRVJSFENBUSJY
'PSDPNCJOFEUSBOTGPSNBUJPOTSFQSFTFOUFECZUXPNBUSJDFT USBOTGPSNBUJPOCZABOEUIFOCZB UIFNBUSJYQSPEVDU
BAEFTDSJCFTUIFDPNCJOFEUSBOTGPSNBUJPO/PUFUIBUUIFPSEFSPGNVMUJQMJDBUJPOJTWFSZJNQPSUBOUJONBUSJDFT
CFDBVTFABȶBA
t
Exercise 23.5
Tip
:PVEPOUOFFEUPLOPX
UIFNBUSJDFTUIBUSFQSFTFOU
EJČFSFOUUSBOTGPSNBUJPOT
GSPNNFNPSZCVUZPV
NVTUCFBCMFUPXPSL
UIFNPVUBOEVTFUIFNUP
EFTDSJCFUSBOTGPSNBUJPOT
1 'PSCPUIQBJSTPGTIBQFT
(i) EFTDSJCFGVMMZUIFUSBOTGPSNBUJPOUIBUNBQTTIBQF"POUPTIBQF#
(ii) ĕOEUIF¨NBUSJYUIBUSFQSFTFOUTFBDIUSBOTGPSNBUJPO
y
(a)
(b) y
A
10
9
8
7
6
5
4
3
2
1
–4 –3 –2 –1 0
–1
164
Unit 6: Shape, space and measures
B
x
1 2 3 4 5 6
10
9
8
7
6
5
4
3
2
1
A
B
–1 0 1 2 3 4 5 6 7 8 9 10
–1
x
23 Transformations and matrices
2 "TRVBSFXJUIWFSUJDFTBUP Q R BOES JTUSBOTGPSNFECZUIF
⎛ 1 0⎞
NBUSJY ⎜
⎝ 0 4⎟⎠
(a) 'JOEUIFDPPSEJOBUFTPGWFSUFY3hVOEFSUIJTUSBOTGPSNBUJPO
(b) %FTDSJCFUIFUSBOTGPSNBUJPOJOXPSET
⎛ 3 0⎞
3 ćFUSBOTGPSNBUJPO5JTEFĕOFECZUIFNBUSJYAXIFSFA ⎜ 0 3⎟
⎝
⎠
(a) %FTDSJCFUIFUSBOTGPSNBUJPO5JOXPSET
⎛ −1 0⎞
(b) "TFDPOEUSBOTGPSNBUJPO 5 JTSFQSFTFOUFECZUIFNBUSJYBXIFSFB⎜
'JOE
⎝ 0 1⎟⎠
UIFNBUSJYUIBUSFQSFTFOUTUIFDPNCJOFEUSBOTGPSNBUJPOBA
4 'JOEUIFNBUSJYUIBUSFQSFTFOUTUIFDPNCJOFEUSBOTGPSNBUJPOSFĘFDUJPOJOUIFxBYJTGPMMPXFE
CZBOFOMBSHFNFOUPGTDBMFGBDUPSBCPVUUIFPSJHJO
Mixed exercise
1
10
y
8
6
B
H
A
4
2
E
–10 –8 –6 –4 –2 0
–2
2
4
6
C
8
x
10
–4
G
–6
F
–8
D
–10
(a) %FTDSJCFBTJOHMFUSBOTGPSNBUJPOUIBUNBQTUSJBOHMF"POUP
(i) USJBOHMF#
(ii) USJBOHMF$
(iii) USJBOHMF%
(b) %FTDSJCFUIFQBJSPGUSBOTGPSNBUJPOTUIBUZPVDPVMEVTFUPNBQUSJBOHMF"POUP
(i) USJBOHMF&
(iii) USJBOHMF(
(iv) USJBOHMF)
⎛ −4⎞
2 4BMMZUSBOTMBUFEQBSBMMFMPHSBNDEFGBMPOHUIFDPMVNOWFDUPS ⎜ ⎟ BOEUIFOSPUBUFEJU°
⎝ 5⎠
y
DMPDLXJTFBCPVUUIFPSJHJOUPHFUUIFJNBHFD′E′F′G′TIPXOPOUIFHSJE
8
(a) %SBXBEJBHSBNBOESFWFSTFUIFUSBOTGPSNBUJPOT4BMMZQFSGPSNFEPOUIFTIBQFUPTIPX
UIFPSJHJOBMQPTJUJPOPGDEFG
(b) &OMBSHFDEFGCZBTDBMFGBDUPSPGVTJOHUIFPSJHJOBTUIFDFOUSFPGFOMBSHFNFOU-BCFM
ZPVSFOMBSHFNFOUD″ E″ F″ G″ 6
4
(ii) USJBOHMF'
D'
G'
2
E'
F'
2
4
6
8
x
Unit 6: Shape, space and measures
165
23 Transformations and matrices
3 4RVBSFABCDJTTIPXOPOUIFHSJE
10
y
8
6
A
B
4
2
–10 –8
–6
–4
–2 D
–2
C
2
4
6
8
10
x
–4
–6
–8
–10
0OUPBDPQZPGUIFEJBHSBN ESBXUIFGPMMPXJOHUSBOTGPSNBUJPOTBOE JOFBDIDBTF HJWFUIF
DPPSEJOBUFTPGUIFOFXQPTJUJPOPGWFSUFYB
(a) 3FĘFDUABCDBCPVUUIFMJOFx=−
(b) 3PUBUFABCD°DMPDLXJTFBCPVUUIFPSJHJO
⎛ −3⎞
(c) 5SBOTMBUFABCDBMPOHUIFDPMVNOWFDUPS⎜ ⎟ ⎝ 2⎠
(d) &OMBSHFABCDCZBTDBMFGBDUPSPGVTJOHUIFPSJHJOBTUIFDFOUSFPGFOMBSHFNFOU
x
A
⎛ 3⎞
⎛ 2⎞
⎛ −2⎞
4 (JWFOUIBUa= ⎜ ⎟ b= ⎜ ⎟ BOEc= ⎜ ⎟
6
−
4
⎝ ⎠
⎝ ⎠
⎝ −4⎠
(a) ĕOE
(i) a
(ii) b+c
(iii) a¦b
(iv) a+b
(b) %SBXGPVSTFQBSBUFWFDUPSEJBHSBNTPOTRVBSFEQBQFSUPSFQSFTFOU
(i) a a
(ii) b c b+c (iii) a b aob (iv) a b a+b
5 ABCDEFJTBSFHVMBSIFYBHPOXJUIDFOUSFOFA xBOEAB y
y
B
F
C
E
D
166
(a) &YQSFTTUIFGPMMPXJOHWFDUPSTJOUFSNTPGxBOEy
(i) ED (ii) DE (iii) FB
(iv) EF (v) FD
(b) * GUIFDPPSEJOBUFTPGQPJOUEBSF BOEUIFDPPSEJOBUFTPGQPJOUBBSF $BMDVMBUF] EB ]DPSSFDUUPUISFFTJHOJĕDBOUĕHVSFT
Unit 6: Shape, space and measures
23 Transformations and matrices
6 %FTDSJCFUIFGPMMPXJOHUSBOTGPSNBUJPOT
(a)
(b) y
B′
B
A
(c)
C
x
(d)
1
2
7 %SBXUIFJNBHFPGABCDVOEFSBTUSFUDIPGTDBMFGBDUPS XJUIUIFxBYJTJOWBSJBOU
Unit 6: Shape, space and measures
167
23 Transformations and matrices
%SBXUIFJNBHFPGABCDVOEFSBTIFBSPGGBDUPSXJUIyJOWBSJBOU
8
y
10
9
8
7
6
5
4
3
2
1 D
–1 0
–1
A
B
x
1 2 C 3 4 5 6 7 8 9 10
⎛ 1 3⎞
⎛ 1 2⎞
P ⎜
BOEQ
⎜⎝ 2 2⎟⎠ $BMDVMBUF
⎝ 2 1⎟⎠
9
(a)
(b)
(c)
(d)
(e)
(f)
P Q
PoQ
PQ
QP
]P]
P–I
⎛ 2 5 a ⎞ ⎛ a⎞ ⎛ 45⎞
10 (JWFO ⎜ 3 4 6 ⎟ ⎜ b ⎟ = ⎜ 47⎟ ĕOEUIFWBMVFTPGa bBOEc
⎜
⎟⎜ ⎟ ⎜ ⎟
⎜⎝ −2 4 −1⎟⎠ ⎜⎝ 3⎟⎠ ⎜⎝ c ⎟⎠
11 $PQZUIFEJBHSBNPOUPBTFUPGBYFT
y
10
9
8
7
6
A
5
4
3
2
1
B
x
C
–1 0 1 2 3 4 5 6 7 8 9 10
–1
(a)
(b)
(c)
(d)
(e)
168
Unit 6: Shape, space and measures
%SBXUIFJNBHFPGABCBęFSSFĘFDUJPOJOUIFxBYJT
8SJUFEPXOUIF¨NBUSJYAUIBUSFQSFTFOUTUIJTUSBOTGPSNBUJPO
&OMBSHFUIFJNBHFA'B'C'CZBTDBMFGBDUPSPGXJUIDFOUSFPGFOMBSHFNFOUBU 8SJUFEPXOUIF¨NBUSJYBUIBUSFQSFTFOUTUIJTUSBOTGPSNBUJPO
'JOEUIFNBUSJYPGUIFDPNCJOFEUSBOTGPSNBUJPOAGPMMPXFECZB
24
Probability using tree
diagrams
24.1 Using tree diagrams to show outcomes
t "QSPCBCJMJUZUSFFTIPXTBMMUIFQPTTJCMFPVUDPNFTGPSTJNQMFDPNCJOFEFWFOUT
t &BDIMJOFTFHNFOUPSCSBODISFQSFTFOUTPOFPVUDPNFćFFOEPGFBDICSBODITFHNFOUJTMBCFMMFEXJUIUIFPVUDPNF
BOEUIFQSPCBCJMJUZPGFBDIPVUDPNFJTXSJUUFOPOUIFCSBODI
Tip
Exercise 24.1
3FNFNCFS GPS
JOEFQFOEFOUFWFOUT
1 "BOEUIFO# 1 " ¨1 # BOEGPSNVUVBMMZ
FYDMVTJWFFWFOUT
1 "PS# 1 " 1 # 1 "OJUBIBTGPVSDBSETćFZBSFZFMMPX SFE HSFFOBOECMVF4IFESBXTBDBSEBUSBOEPNBOE
UIFOUPTTFTBDPJO%SBXBUSFFEJBHSBNUPTIPXBMMQPTTJCMFPVUDPNFT
2 ćFTQJOOFSTIPXOIBTOVNCFSTPOBOJOOFSDJSDMFBOEMFUUFSTPOBOPVUFSSJOH8IFOTQVO JUHJWFTBSFTVMUDPOTJTUJOHPGBOVNCFSBOEBMFUUFS%SBXBUSFFEJBHSBNUPTIPXBMMQPTTJCMF
PVUDPNFTXIFOZPVTQJOJU
H
A
G
1
2
B
F
4
3
C
E
D
3 "UJODPOUBJOTFJHIUHSFFODPVOUFSTBOEGPVSZFMMPXDPVOUFST0OFDPVOUFSJTESBXOBOEOPU
SFQMBDFE BOEUIFOBOPUIFSJTESBXOGSPNUIFCBH
(a) %SBXBUSFFEJBHSBNUPTIPXUIFQPTTJCMFPVUDPNFTGPSESBXJOHUXPDPVOUFST
(b) -BCFMUIFCSBODIFTUPTIPXUIFQSPCBCJMJUZPGFBDIFWFOU
24.2 Calculating probability from tree diagrams
t 5PEFUFSNJOFUIFQSPCBCJMJUZPGBDPNCJOBUJPOPGPVUDPNFT NVMUJQMZBMPOHFBDIPGUIFDPOTFDVUJWFCSBODIFT*G
TFWFSBMDPNCJOBUJPOTTBUJTGZUIFTBNFPVUDPNFDPOEJUJPOT UIFOBEEUIFQSPCBCJMJUJFTPGUIFEJČFSFOUQBUIT
t ćFTVNPGBMMUIFQSPCBCJMJUJFTPOBTFUPGCSBODIFTNVTUFRVBMPOF
Exercise 24.2
1 8IFOBDPMPVSFECBMMJTESBXOGSPNBCBH BDPJOJTUPTTFEPODFPSUXJDF EFQFOEJOHPOUIF
DPMPVSPGUIFCBMMESBXOćFSFBSFUISFFCMVFCBMMT UXPZFMMPXCBMMTBOEBCMBDLCBMMJOUIF
CBHćFUSFFEJBHSBNTIPXTUIFQPTTJCMFPVUDPNFT
Unit 6: Data handling
169
24 Probability using tree diagrams
H
Blue
T
H
Yellow
T
H
Black
H
T
H
T
T
(a) $PQZBOEMBCFMUIFEJBHSBNUPTIPXUIFQSPCBCJMJUZPGFBDIFWFOU"TTVNFUIFESBXPG
UIFCBMMTJTSBOEPNBOEUIFDPJOJTGBJS
(b) $BMDVMBUFUIFQSPCBCJMJUZPGBCMVFCBMMBOEBIFBE
(c) $BMDVMBUFUIFQSPCBCJMJUZPGBZFMMPXCBMMBOEUXPIFBET
(d) $BMDVMBUFUIFQSPCBCJMJUZUIBUZPVXJMMOPUHFUIFBETBUBMM
2 ćFUSFFEJBHSBNCFMPXTIPXTUIFQPTTJCMFPVUDPNFTXIFOUXPDPJOTBSFUPTTFE
First
toss
1
2
1
2
(a)
(b)
(c)
(d)
(e)
Mixed exercise
Second
toss
1
2
H
1
2
T
1
2
H
1
2
T
H
T
$PQZBOEUIFUSFFEJBHSBNUPTIPXUIFQPTTJCMFPVUDPNFTXIFOBUIJSEDPJOJTUPTTFE
$BMDVMBUFUIFQSPCBCJMJUZPGUPTTJOHUISFFIFBET
$BMDVMBUFUIFQSPCBCJMJUZPGHFUUJOHBUMFBTUUXPUBJMT
$BMDVMBUFUIFQSPCBCJMJUZPGHFUUJOHGFXFSIFBETUIBOUBJMT
$BMDVMBUFUIFQSPCBCJMJUZPGHFUUJOHBOFRVBMOVNCFSPGIFBETBOEUBJMT
1 %JOFPJTQMBZJOHBHBNFXIFSFTIFSPMMTBOPSNBMTJYTJEFEEJFBOEUPTTFTBDPJO*GUIFTDPSF
POUIFEJFJTFWFO TIFUPTTFTUIFDPJOPODF*GUIFTDPSFPOUIFEJFJTPEE TIFUPTTFTUIFDPJO
UXJDF
(a) %SBXBUSFFEJBHSBNUPTIPXBMMQPTTJCMFPVUDPNFT
(b) "TTVNJOHUIFEJFBOEUIFDPJOBSFGBJSBOEBMMPVUDPNFTBSFFRVBMMZMJLFMZ MBCFMUIF
CSBODIFTXJUIUIFDPSSFDUQSPCBCJMJUJFT
(c) 8IBUJTUIFQSPCBCJMJUZPGPCUBJOJOHUXPUBJMT
(d) 8IBUJTUIFQSPCBCJMJUZPGPCUBJOJOHBĕWF BIFBEBOEBUBJM JOBOZPSEFS
Tip
ćJTJTBOFYBNQMFPG
DPOEJUJPOBMQSPCBCJMJUZ
ćFĕSTUDPJOJTSFNPWFE
BOEOPUSFQMBDFE MFBWJOH
GFXFSQPTTJCMFPVUDPNFT
GPSUIFTFDPOEDPJOJO
FBDIDBTF
170
Unit 6: Data handling
2 "CBHDPOUBJOTUXPDBOEĕWFDDPJOT:PVBSFBTLFEUPESBXBDPJOGSPNUIFCBHBU
SBOEPNXJUIPVUSFQMBDJOHBOZVOUJMZPVHFUBDDPJO
(a)
(b)
(c)
(d)
(e)
%SBXBQSPCBCJMJUZUSFFUPTIPXUIFQPTTJCMFPVUDPNFT
-BCFMUIFCSBODIFTUPTIPXUIFQSPCBCJMJUZPGFBDIFWFOU
$BMDVMBUFUIFQSPCBCJMJUZPGHFUUJOHBDPOZPVSĕSTUESBX
8IBUJTUIFQSPCBCJMJUZPGESBXJOHUIFUXPDDPJOTCFGPSFZPVESBXBDDPJO
* GZPVESBXUXPDDPJOTPOZPVSĕSTUUXPESBXT XIBUJTUIFQSPCBCJMJUZPGHFUUJOHBD
DPJOPOZPVSUIJSEESBX 8IZ
Answers
1 (a) 2, 3, 5, 7
(b) 53, 59
(c) 97, 101, 103
Chapter 1
Exercise 1.1
1
Number Natural Integer Prime Fraction
−0.2
✓
−57
✓
3.142
✓
0
0.3̇
✓
1
✓
✓
51
✓
✓
10 270
✓
✓
1
−
4
✓
2
7
✓
9
11
3
512
(s) 4.03
(u) 6.61
(w) −19.10
Exercise 1.3
Exercise 1.7
2 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
2×2×3×3
5 × 13
2×2×2×2×2×2
2×2×3×7
2×2×2×2×5
2×2×2×5×5×5
2 × 5 × 127
13 × 151
3 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
LCM = 378, HCF = 1
LCM = 255, HCF = 5
LCM = 864, HCF = 3
LCM = 848, HCF = 1
LCM = 24 264, HCF = 2
LCM = 2574, HCF = 6
LCM = 35 200, HCF = 2
LCM = 17 325, HCF = 5
✓
✓
Exercise 1.4
✓
✓
✓
✓
✓
1 square: 121, 144, 169, 196, 225, 256, 289
cube: 125, 216
2 (a) 7
(d) 10
(b) 5
(e) 3
(c) 14
(f) 25
(g)
(h) 5
(i) 2
3
4
(j) 5
(k)
(m) –5
(n)
3 (a) 23 cm
Exercise 1.2 A
1 (a) 18
(d) 24
(g) 72
(b) 36
(e) 36
(h) 96
(c) 90
(f) 24
2 (a) 6
(d) 3
(g) 12
(b) 18
(e) 10
(h) 50
(c) 9
(f) 1
5 (a) 1024 cm2
(b) 200 tiles
(b) 529 cm
2
(i)
5.65
9.88
12.87
0.01
10.10
45.44
14.00
26.00
(ii)
5.7
9.9
12.9
0.0
10.1
45.4
14.0
26.0
2 (a) 53 200
(c) 17.4
3 (a)
(c)
(e)
(g)
(iii)
6
10
13
0
10
45
14
26
(b) 713 000
(d) 0.00728
36
12 000
430 000
0.0046
5.2
0.0088
120
10
(b)
(d)
(f)
(h)
1 natural: 24, 17
3
1
4
2
rational: − , 0.65, 3 , 0 66
integer: 24, –12, 0, 17
prime: 17
2 (a) two are prime: 2 and 3
(b) 2 × 2 × 3 × 3
(c) 2 and 36
(d) 36
3 (a) 2 × 2 × 7 × 7
(b) 3 × 3 × 5 × 41
(c) 2 × 2 × 3 × 3 × 5 × 7 × 7
4 14th and 28th March
(b) −9 °C
(c) −12 °C
3 (a) 4
(d) −2
(b) 7
(e) −3
(c) −1
1 (a) 26
(c) 23.2
(e) 3.39
(g) 3.83
(i) 1.76
(k) 7.82
(m) 8.04
(o) 8.78
(q) 94.78
4 after 420 seconds; 21 laps (Francesca),
5 laps (Ayuba) and 4 laps (Claire)
(o) 6
6
2 (a) −2 °C
1 18 m
3 20 students
(l) 12
1 −3 °C
Exercise 1.6
2 120 shoppers
3
1
4
5
Exercise 1.5
Exercise 1.2 B
1 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Mixed exercise
✓
2 (a) 121, 144, 169, 196, . . .
1 1 2 2
(b) , , , , etc.
4 6 7 9
(c) 83, 89, 97, 101, . . .
(d) 2, 3, 5, 7
(t) 6.87
(v) 3.90
(x) 20.19
(b)
(d)
(f)
(h)
(j)
(l)
(n)
(p)
(r)
66
15.66
2.44
2.15
2.79
0.21
1.09
304.82
0.63
5 (a) true
(c) false
(b) true
(d) false
6 (a) 5
(c) 64
(b) 5
(d) 145
7 (a) 16.07
(d) 11.01
(b) 9.79
(e) 0.12
8 (a) 1240
(c) 0.0238
(c) 13.51
(f) −7.74
(b) 0.765
(d) 31.5
9 (a) It is not strictly possible – to have
a tile of 790 cm2, the builders
must be using rounded values.
(b) 28.1 cm
(c) 110 tiles
Answers
171
(e) –2x + 8x
1 (a) 3(x + 2)
(c) 2(11 + x)
(e) 3x2 + 4
1
(g) − x
5
(i) 4 + 3x
3 (a)
6(x − 1)
18x
x2 + 8
1
x+
(b)
(d)
(f)
(h)
3
(j) 12 – 5x
(b) p − 4
(c) 4p
x
$
3
x
9
1 (a)
(b) 1.875 m2
(d) 8 cm2
2 –104
(k)
2 (a)
(c)
4 17.75
(b) 91
(e)
Exercise 2.3
3x 2 − 2x + 3
4x 2y − 2xy
5ab − 4ac
4x 2 + 5x − y − 5
−30mn
(f) 6x 2y
3
6xy
(h) −4x 3y
(i) 4b
(j)
1
4y
(k) 3b
(l)
9m
4
(n)
3x
y
(p)
y
2
(r)
−14 y
5
(o)
(q)
(s)
20 y
3x
2y
2
x
2
15a
4
x
6y
(t)
2x 2 − 4x
−2x − 2
−2x 2 + 6x
x 3 − 2x 2 − x
2
2 (a) x +
172
x
2
Answers
2
2x
3y
5x
9
2y
3
4
16
(l)
x
8
5
y
2
(b)
8
16
x
22
3 (a) x
(c) x
7
1
2
1
6
(e) 8x
3
1
(g) x 2 y 4
2
7
2
(d)
(f)
x
(c) y 3
3
4
(e) x y 2
2
x
9
y
22
2x
(b) x
4
(f) 2 x y
(h) x 3 y
1
3
1
x3
y
or
2 4
(b)
(d)
(f)
(h)
xy − 3x
−3x + 2
3x + 1
x2 + x + 2
(b) x + xy
2
(d) 2
(g) 32
(j)
3
2
(h) x 3 − x
2 (a) −6
(b) 24
3 (a) −2
(b)
or
1
29
x 4 y 16
4x
y
4
(h) 4
(f)
11x − 3
6x 2 + 15x − 8
−2x 2 + 5x + 12
−x 3 + 3x 2 − x + 5
6 (a)
5x
6
5
(b) 15
1
x
(d) 16x4y8
4
64 x
y
9
(f) x 9y 8
15
27 x
4y
4
(h)
3
(i)
6
1
(c) x −9 y or
y
x
9
5
5
2y3
1
x3
3
1
Chapter 3
625
Exercise 3.1 A
2
xy
2
(b) x 2
8
3
(c) 5
5 (a)
(b)
(c)
(d)
−1
1
2
3
−14
9
5y
2
(d) 2 x 3 y 3 or
(c)
(c)
7
(e) −2
9a + b
x 2 + 3x − 2
−4a 4b + 6a 2b3
−7x + 4
1
(b) –1
(e)
(g) 12 − x
7 (a) 5xy 3
1
Exercise 2.5 B
1 (a) –1296
x
4
(g)
(d) x2
(b) x − 4
(f)
(c)
x y
(b) x2
(f) x y
3
2
(e) 4x
(e)
29
− 16
(f) 2
x
3
1
1
y2
− 14
(e) 2
(h)
(f) 5x −
1
9
3
(c) –4
(d)
(e)
8
15
(d) x
4
(b) 4
(c) 5x
(d)
4 (a)
(b)
(c)
(d)
4
y
1
(j) x −2 y −4 or
4 (a) x 3
27 x
10
3125 x y
16
y
y
3
8x y
3
16
x y
3
(j)
x
5
49
25 x y
10
4
3
1 (a) x + 12
(f) x y
(h)
1
Mixed exercise
7 3
50 x
27 y
(b) 65
(d) –163
(g) 2
(d) xy 10
(k) y −2 or
2
Exercise 2.4
1 (a)
(c)
(e)
(g)
y
(i) x3
2
2
(d)
(b) 3x 4y
2
(c)
3 17
(m)
6
(i) x 7y
1 (a) 54 cm2
(c) 110.25 cm2
1 (a)
(b)
(c)
(d)
(e)
(g)
x
2 (a) 17
(c) 15
3 (a)
Exercise 2.5 A
(g)
Exercise 2.2
5 (a) 6
2
3
2x
2x
, and $
9
3
3y
2
(g) –5x2 – 6x
(e)
(b) $ , $
+
(f) 3x – 6x
2
Exercise 2.1
2 (a) p + 5
x
2
(c) −8x 3 + 4x 2 + 2x (d)
Chapter 2
1 (a) (i) 150°
(b) 45°
(ii) 180° (iii) 135°
(f) x = 58° (base ∠s isosceles and
∠s in triangle); y = 26°
(ext ∠s equals sum int opps)
(g) x = 33° (base ∠s isosceles then ext
∠s equals sum int opps)
(h) x = 45° (co-int ∠s then ∠s in
triangle)
(i) x = 45° (base ∠s isosceles);
y = 75° (base ∠s isosceles)
(c) (i) 810°
(ii) 72°
(d) quarter to one or 12:45
2 No. If the acute angle is ≤ 45° it will
produce an acute or right angle.
3 Yes. The smallest obtuse angle is 91°
and the largest is 179°. Half of those
will range from 45.5° to 89.5°, all of
which are acute.
4 (a) 45°
(b) (90 − x)°
(c) x°
5 (a) 135°
(c) (180 − x)°
(e) (90 + x)°
(b) 90°
(d) x°
(f) (90 − x)°
Exercise 3.1 B
1 ∠QON = 48°, so a = 48° (vertically
opposite)
2 (a) ∠EOD = 41° (∠s on line), so
x = 41° (vertically opposite)
(b) x = 20° (∠s round point)
2 (a) x = 36°; so A = 36° and B = 72°
(b) x = 40°; so A = 80°; B = 40° and
∠ACD = 120°
(c) x = 60°
(d) x = 72°
3 ∠B = 34°; ∠C = 68°
Exercise 3.3
1 (a)
(b)
(c)
(d)
(e)
(f)
Exercise 3.1 C
1 (a) x = 85° (co-int ∠s);
y = 72° (alt ∠s)
(b) x = 99° (co-int ∠s); y = 123°
(∠ABF = 123°, co-int ∠s then
vertically opposite)
(c) x = 72° (∠BFE = 72°, then alt ∠s);
y = 43° (∠s in triangle BCJ)
(d) x = 45° (∠s round a point);
y = 90° (co-int ∠s)
2 (a) x = 112°
(b) x = 45° ( STQ corr ∠s then
vertically opposite)
(c) x = 90° (∠ECD and ∠ACD co-int
∠s then ∠s round a point)
(d) x = 18° (∠DFE co-int with ∠CDF
then ∠BFE co-int with ∠ABF)
(e) x = 85°
Exercise 3.2
1 (a)
(b)
(c)
(d)
(e)
103° ( s in triangle)
51° (ext ∠ equals sum int opps)
68° (ext ∠ equals sum int opps)
53° (base ∠s isosceles)
60° (equilateral triangle)
(g)
(h)
(i)
2
square, rhombus
rectangle, square
square, rectangle
square, rectangle, rhombus,
parallelogram
square, rectangle
square, rectangle, parallelogram,
rhombus
square, rhombus, kite
rhombus, square, kite
rhombus, square, kite
45°
45°
45°
45°
45°
45°
45°
45°
27°
27°
3 (a)
(b)
(c)
(d)
(e)
(f)
1 (a) (i) 1080°
(b) (i) 1440°
(c) (i) 2340°
2
900
7
(ii) 135°
(ii) 144°
(ii) 156°
= 128.57°
3 20 sides
4 (a) 165.6°
(b)
360
14.4
= 25 sides
5 (a) x = 156°
(b) x = 85°
(c) x = 113°; y = 104°
Exercise 3.5
1
M
(d) tangent
D
(b) chord
( ) major
arc
sector
50° (a)
O
diameter
E
P
N
( )
1, 2 student’s own diagrams
3 2.1 cm
4 (a) scalene
(b) yes
5 6.6 cm
27°
27°
x = 69°
x = 64°
x = 52°
x = 115°
x = 30°; 2x = 60°; 3x = 90°
a = 44°; b = 68°; c = 44°; d = e = 68°
4 (a) ∠Q + ∠R = 210°
(b) ∠R = 140°
(c) ∠Q = 70°
Exercise 3.4
Exercise 3.6
63°
63°
63°
63°
5 (a) ∠MNP = 42°
(b) ∠MNO = 104°
(c) ∠PON = 56°
Mixed exercise
1 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
x = 113°
x = 41°
x = 66°
x = 74°; y = 106°; z = 37°
x = 46°; y = 104°
x = 110°; y = 124°
x = 40°; y = 70°; z = 70°
x = 35°; y = 55°
2 (a) x = 60 + 60 + 120 = 240°
(b) x = 90 + 90 + 135 = 315°
(c) x = 80°
Answers
173
4 (a)
Eye colour
Brown
Blue
Green
Male
4
0
1
Female
2
1
2
(b)
Hair colour Brown Black
Blonde
Male
2
2
1
Female
1
4
0
Exercise 4.2
1
Phone Tally
calls
Frequency
1
/
1
2
//
2
3
4
//
////
2
5
5
//// ////
9
6
//// //
7
7
6
8
//// /
///
9
///
3
10
//
2
1
2
3
4
Male
0
1
1
2
1
Female
2
1
1
1
0
Exercise 4.3
1 (a) pictogram
(b) number of students in each year
group in a school
(c) 30 students
(d) half a stick figure
(e) 225
(f) Year 11; 285
(g) rounded; unlikely the year groups
will all be multiples of 15
3
No. of
mosquitoes
0
1
2
3
4
5
6
Frequency
7
6
9
7
8
7
6
(b) It is impossible to say; frequency
is very similar for all numbers of
mosquitoes.
Answers
0
(c) student’s own sentences
2 (a)
174
No. of brothers/
sisters
2 student’s own pictogram
3 (a) number of boys and girls in
Class 10A
(b) 18
(c) 30
(d) the favourite sports of students in
Class 10A, separated by gender
(e) athletics
(f) athletics
(g) 9
(d) C
26° C
May–Nov
northern hemisphere
No
35 mm
April
none
Mixed exercise
1 (a) survey or questionnaire
(b) discrete; you cannot have half a
child
(c) quantitative; it can be counted
(d)
No. of
children in
family
0
4
5
6
Frequency
7 10 11 12 5
2
1
1
2
3
(e) student’s own chart
(f) student’s own chart
2 student’s own pictogram
3 (a) compound bar chart
(b) It shows how many people, out of
every 100, have a mobile phone
and how many have a land line
phone.
(c) No. The figures are percentages.
(d) Canada, USA and Denmark
(e) Germany, UK, Sweden and Italy
(f) Denmark
(g) own opinion with reason
4 (a) Student’s own chart
(b) Student’s own chart
(c) Student’s own chart
5 (a)
15
Value of car over time
10
5
0
10
20
11
20
12
20
13
1 (a) gender, eye colour, hair colour
(b) height, shoe size, mass, number of
brothers/sisters
(c) shoe size, number of brothers/
sisters
(d) height, mass
(e) possible answers include: gender,
eye colour, hair colour − collected
by observation; height,
mass − collected by measuring;
shoe size, number of siblings −
collected by survey, questionnaire
(b) 10
(c) 2
(d) 26
(e) There are very few marks at the
low and high end of the scale.
7 (a)
(b)
(c)
(d)
(e)
(f)
(g)
09
Exercise 4.1
6 (a) student’s own chart
(b) 6
(c) 50
20
Chapter 4
5 (a) cars
(b) 17%
(c) 20
(d) handcarts and bicycles
08
5 student’s own diagram
4 (a) student’s own chart
(b) student’s own chart
Frequency
1
1
7
19
12
6
4
20
4 student’s own diagram
Score
0−29
30−39
40−49
50−59
60−69
70−79
80−100
20
3 (a)
Value ($ 000)
3 (a) (i) radius
(ii) chord
(iii) diameter
(b) AO, DO, OC, OB
(c) 24.8 cm
(d) student’s own diagram
Time (years)
(b) 49.6%
(c) $3900
3 (a) 83%
(d) 37.5%
Chapter 5
Exercise 5.1
1 (a) x = 65 (b) x = 168 (c) x = 55
(d) x = 117 (e) x = 48 (f) x = 104
Exercise 5.2
1 (a)
25
8
(b)
17
11
(c)
59
5
(d)
15
4
(e)
59
4
(f)
25
9
2 (a)
(d)
108
5
28
5
(b)
63
13
(e) 3
(c) 14
(f)
6
19
(g) 120
(h)
3
14
(i) 72
(j) 3
(k)
233
50
(l)
7
4
(b)
19
60
(c)
19
21
(f)
161
20
3 (a)
13
24
(d)
35
6
(e)
183
56
(g)
18
65
−5
6
(h)
41
40
(i)
(k)
−10
3
(l)
(j)
4 (a) 24
(b)
10
27
(e)
(d)
(g) 2
5 (a)
(d)
38
9
19
4
(g) 0
6 (a) $525
(h)
(b)
(e)
(h)
96
7
10
9
3
5
4
5
5
12
11
170
(c)
(f)
29
21
−26
9
7
96
9
14
(i) 4
(c)
(f)
(i)
39
7
215
72
187
9
(b) $375
(b) 62.5%
(e) 4%
(h) 207%
2 (a)
1
8
(b)
1
2
(d)
3
5
(e)
11
50
60 kg
150 litres
$64
18 km
$2.08
5 (a)
(c)
(e)
(g)
+20%
+53.3%
−28.3%
+2566.7%
(b)
(d)
(f)
(h)
(j)
(c) 7%
(f) 250%
$24
55 ml
£19.50
0.2 grams
475 cubic metres
(c) 29.8%
(f) 47%
(c)
49
50
(g) 0.000028
(i) 0.00245
(h) 94 000 000
Exercise 5.4 B
1 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
6.56 × 10−17
1.28 × 10−14
1.44 × 1013
1.58 × 10−20
5.04 × 1018
1.98 × 1012
1.52 × 1017
2.29 × 108
4.50 × 10−3
1 28 595 tickets
2 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
12 × 1030
4.5 × 1011
3.375 × 1036
1.32 × 10−11
2 × 1026
2.67 × 105
1.2 × 102
2 × 10−3
2.09 × 10−8
2 1800 shares
3 (a) the Sun
3 $129 375
4 (a) 500 seconds = 5 × 102 seconds
(b) 19 166.67 seconds = 1.92 × 104
seconds
(b) −10%
(d) +3.3%
(f) +33.3%
6 (a) $54.72
(c) $32.28
(e) $98.55
(b) $945
(d) $40 236
(f) $99.68
7 (a) $58.48
(c) $83.16
(e) $76.93
(b) $520
(d) $19 882
(f) $45.24
Exercise 5.3 B
4 21.95%
5 $15 696
7 2.5 grams
8
7
25
= 28% increase, so $7 more is better
Exercise 5.3 C
4 × 5 = 20
70 × 5 = 350
1000 × 7 = 7000
42 ÷ 6 = 7
(b) 3
(d) 243
(b) 960
Mixed exercise
(b) 27 750 (c) $114 885
1 (a) 40
(c) 22
3 $150
1 (a)
(c)
(e)
(g)
(i)
(k)
1 (a)
(b)
(c)
(d)
2 (a) 20
(c) 12
1 $50
2 (a) 1200
(b) 6.051 × 106
Exercise 5.5
6 $6228
Exercise 5.4 A
Exercise 5.3 A
1 (a) 16.7%
(d) 30%
(g) 112%
4 (a)
(c)
(e)
(g)
(i)
4 (a) $12
7 (a) 300
(b) 6 hours 56 min
(b) 60%
(e) 125%
4.5 × 104
8 × 10
4.19 × 106
6.5 × 10−3
4.5 × 10−4
6.75 × 10−3
2 (a) 2500
(c) 426 500
(e) 0.00000915
(b)
(d)
(f)
(h)
(j)
(l)
8 × 105
2.345 × 106
3.2 × 1010
9 × 10−3
8 × 10−7
4.5 × 10−10
(b) 39 000
(d) 0.00001045
(f) 0.000000001
(b) 6
(d) 72
2 (a)
4
5
(b)
3 (a)
1
6
(d)
3
44
(g)
361
16
(j)
2
3
(c)
2
3
(b) 63
(c)
5
3
(e)
31
48
(f)
71
6
(h)
334
45
(i)
68
15
14
9
4 (a) $760
(b) $40 000
Answers
175
5 $10 000
6 (a) 720
(b) 11 779
7 67.9%
(f) x =
(g) x = −1
(h) x = = 4
(i) x = −
8 (a) 5.9 × 10 km
(b) 5.753 × 109 km
9
(k) x =
9 (a) 9.4637 × 10 km
(b) 1.6 × 10−5 light years
(c) 3.975 × 1013 km
x(x + 8)
x(9x + 4)
2b(3ab + 4)
3x(2 − 3x)
3abc2(3c –ab)
b2(3a − 4c)
(b)
(d)
(f)
(h)
(j)
(l)
a(12 − a)
2x(11 − 8x)
18xy(1 − 2x)
2xy 2(7x − 3)
x(4x − 7y)
7ab(2a − 3b)
(h) x = −5
3
1
(j) x = − = −1
4 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(3 + y)(x + 4)
(y − 3)(x + 5)
(a + 2b)(3 − 2a)
(2a − b)(4a − 3)
(2 − y)(x + 1)
(x − 3)(x + 4)
(2 + y)(9 − x)
(2b − c)(4a + 1)
(x − 6)(3x − 5)
(x − y)(x − 2)
(2x + 3)(3x + y)
(x − y)(4 − 3x)
(l) x = 3
Exercise 6.4 A
(b) x = −2
1 m=
(d) x = 4
=3
(g) x = 2
(i) x = 4
(k) x =
3
5
(f) x = 5
2
11
2
1
=5
2
2 (a) x = 10
8
3
(c) x = − = −2
2
3
(d) x =
4
3
1
4
2
=1
(e) x = 8
(f) x =
(g) x = −4
(h) x = −9
(i) x = −10
(j) x = −13
(k) x = −34
(l) x =
3 (a) x = 18
(c) x = 24
176
Answers
−11
2
3 (a)
(c)
(e)
(g)
(i)
(k)
−5a + 5b
8x − 4xy
−9x + 9
3 − 4x − y
−3x − 7
−3x2 + 6xy
(b) x = 4
18
5
(n) x =
2(1 + 4y)
x(3 − y)
3(x − 5y)
3b(3a − 4c)
2x(7 − 13y)
(b)
(d)
(f)
(h)
(j)
(l)
Exercise 6.2
=
(l) x = 10
(b)
(d)
(f)
(h)
(j)
14x − 2y − 9x
−5xy + 10x
6x − 6y − 2xy
−2x − 6y − xy
12xy − 14 − y + 3x
4x2 − 2x2y − 3y
−2x2 + 2x + 5
6x2 + 4y − 8xy
–5x – 3
36
10
3
13
12(x + 4)
4(a − 4)
a(b + 5)
8xz(3y − 1)
2y(3x − 2z)
2 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(e) x =
=1
2 (a)
(c)
(e)
(g)
(i)
−2x −2y
6x − 3y
−2x2 − 6xy
12 − 6a
3
2x2 − 2xy
(c) x = =
(j) x = 9
3
a
4xy
xy2z
1 (a)
(c)
(e)
(g)
(i)
(k)
1
4
2
1
2
1 (a)
(d)
(g)
(j)
Exercise 6.1
9
2
5
6
Exercise 6.3
Chapter 6
1 (a) x = 3
1
3
16
13
=3
9
2
(m) x = 42
12
23
6
(e) x = 17
20
13
1
3
=1
(b) x = 27
(d) x = −44
7
13
(b)
(e)
(h)
(k)
8
3y
pq
ab3
(c)
(f)
(i)
(l)
5
5ab
7ab
3xy
P+c
a
4 b=
a c
x
5 (a) a = c − b
(c) a =
c d
b
de − c
b
(i)
(j) a =
ef − d
bc
(k) a =
(l) a =
d (e c )
b
(m) a
(n) a
d
c
(b) a = 2c + 3b
(d) a =
(e) a = bc − d (or a = −d + bc)
d
c
b
a=
cd − b
2
e d
bc
c( f
d
c
b
de )
b
b
Exercise 6.4 B
P
2
1 (a) b
l
C
2 (a) r =
2A
h
3 use
(b) b = 35.5 cm
(b) 9 cm
2π
(c)
T 70P
12
(c) 46 cm
; b = 3.8 cm
4 (a) (i) 70 kg
(b) 11 656 kg
(ii) 12 kg
=B
(d) 960 kg
5 (a) t =
m
5
(b) 6 seconds
Mixed exercise
x = −3
x=9
x=2
x = 1.5
2 (a) x =
2 c = y − mx
(g) a =
(h) a =
1 (a)
(c)
(e)
(g)
D
k
3 b=
(f) a = d + bc
m+r
np
(b)
(d)
(f)
(h)
x = −6
x = −6
x = −13
x=5
(b) x =
mq − p
n
3 (a)
(c)
(e)
(g)
3x + 2
−8x + 4y − 6
11x + 5
−2x 2 + 11x
4 (a)
(c)
(e)
(g)
(h)
4(x − 2)
(b) 3(4x − y)
−2(x + 2)
(d) 3x(y − 8)
7xy(2xy + 1)
(f) (x − y)(2 + x)
(4 + 3x)(x − 3)
4x(x + y)(x − 2)
5 (a)
(b)
(c)
(d)
4(x − 7) = 4x − 28
2x(x + 9) = 2x 2 + 18x
4x(4x + 3y) = 16x 2 + 12xy
19x(x + 2y) = 19x 2 + 38xy
6 (a) x = 15°
(b) x = 26°
(c) x = 30°
(b)
(d)
(f)
(h)
−12x 2 + 8x
−16y + 2y2
5x 2 + 7x − 4
10x 2 + 24x
Chapter 7
Exercise 7.2
Exercise 7.1 A
1 (a)
(b)
(c)
(d)
1 (a) 120 mm
(c) 128 mm
(e) 36.2 cm
(b) 45 cm
(d) 98 mm
(f) 233 mm
15.7 m
54.0 mm
18.8 m
24.4 cm
(b) 44.0 cm
(d) 21.6 m
(f) 151 mm
2 (a)
(c)
(e)
(g)
3 90 m
cube
cuboid
square-based pyramid
octahedron
2 (a) cuboid
(b) triangular prism
(c) cylinder
3 The following are examples; there are
other possible nets.
(a)
Exercise 7.3 A
1 (a) 2.56 mm2
(c) 13.5 cm2
(b) 523.2 m2
(d) 401.92 mm2
2 (a) 384 cm2
(b) 8 cm
3 (a) 340 cm2
(c) 4 tins
(b) 153 000 cm2
4 (a)
(c)
(e)
(g)
90 000 mm3
20 410 mm3
960 cm3
1800 cm3
4 164 × 45.50 = $7462
5 332.5 cm3
5 9 cm each
6 (a) 224 m3
6 about 88 cm
7 211.95 m3
7 (a) 197.82 cm
332.5 cm2
399 cm2
59.5 cm2
2296 mm2
7261.92 cm2
2 (a) 7850 mm2
(c) 7693 mm2
(e) 167.47 cm2
3 (a)
(c)
(e)
(g)
(i)
288 cm
373.5 cm2
366 cm2
272.93 cm2
5639.63 cm2
2
4 (a) 30 cm2
(c) 33.6 cm2
(e) 720 cm2
(b)
(d)
(f)
(h)
1.53 m2
150 cm2
71.5 cm2
243 cm2
(b)
82 cm
581.5 cm2
39 cm2
4000 cm2
Volume (mm3)
64 000
64 000
64 000
Length (mm)
80
50
100
50
Breadth (mm)
40
64
80
80
Height (mm)
20
20
8
16
1 (a) 5.28 cm3
(b) 33 493.33 m3
3
(c) 25.2 cm
(d) 169.56 cm3
3
(e) 65 111.04 mm
2 (a) (i) 1.08 × 1012 km3
(ii) 5.10 × 108 km2
(b) 1.48 × 108 km2
2
Mixed exercise
(c)
(b) 90 cm2
(d) 61.2 cm2
(f) 2562.5 cm2
1 (a) 346.19 cm2
2000 mm2
40 cm2
106 cm2
175.84 cm2
6 70 mm = 7 cm
Exercise 7.1 C
4 15 m
5 11.1 m2
(b) 47.1 cm
(d)
2 6668.31 km
3 (a) 75.36 cm2
(c) 92.34 mm2
4 61.33 cm2
(b) 732.67 cm2
(b) 65.94 cm
2 4.55 cm
3 (a)
(c)
(e)
(g)
1 (a) 43.96 mm
(c) 8.37 mm
64 000
Exercise 7.3 B
(b) 153.86 mm2
(d) 17.44 cm2
(b)
(d)
(f)
(h)
(b) 44 people
8 various answers − for example:
(b) 219.8 cm
Exercise 7.1 B
1 (a)
(c)
(e)
(g)
(i)
60 cm3
1120 cm3
5.76 m3
1.95 m3
(b)
(d)
(f)
(h)
5 (a)
(b)
(c)
(d)
(b) 33 000 mm2
(d) 80 cm2
(f) 35 cm2
cuboid is smaller
14 240 mm3
student’s own diagram
cylinder 7536 mm2, cuboid
9000 mm2
6 42
7 volume cone = 23.55 cm3
volume pyramid = 30 cm3
difference = 6.45 cm3
Answers
177
8 volume 3 balls = 1144.53 cm3
volume tube = 1860.39 cm3
space = 715.86 cm3
Exercise 8.4
1
Chapter 8
Exercise 8.1
1 (a) red =
3
9
17
, white = , green =
10
25
50
(b) 30%
(c) 1
(d)
= 0.1 (ii)
1
(v)
(c)
4 (a)
(c)
5 (a)
(d)
6
13
40
(viii)
2
5
(vi)
(b)
1
2
(ix) 0
3
5
or 0.33
1
or 0.16
6
7
20
3
10
(b)
1
2
(b)
(e)
1
2
1
5
(c)
2
5
(d)
3
1
3
cola,
cake
cola,
muffin
water,
biscuit
water,
cake
water,
muffin
2
3
C
L
C
T
T
3 (a) 0.16
(b) 0.84
(c) 0.6
(d) strawberry 63, lime 66, lemon 54,
blackberry 69, apple 48
(b) 0.97
A
CA
LA
CA
TA
TA
(c) 11
2 (a)
(d)
(g)
4
15
1
15
1
30
3
10
(c)
(b)
(e)
U
CU
LU
CU
TU
TU
1
5
2
15
7
15
28
153
(c)
40
153
40
153
1 (a) 10 000 (b) heads 0.4083;
tails 0.5917
1
(c)
2
(d) could be − probability of the tails
outcome is higher than the heads
outcome for a great many tosses
1
2
(b)
(d) 0
(e)
2 (a)
(g)
1 (a)
(b)
Answers
1
3
1, 3
2, 3
3, 3
Exercise 8.5
5
8
178
Yellow
2
1, 2
2, 2
3, 2
fruit
juice,
muffin
(c)
(b)
Mixed exercise
fruit
juice,
cake
9
or 0.5
1 0.73
4 (a) 0.6
(d) 114
4
fruit
juice,
biscuit
1
5
17
(e) The four situations represent all
the possible outcomes, so they
must add up to one.
1
cola,
biscuit
(b)
1
3
(c)
(d)
Snack
Drink
1
4
1
2
T
HT
TT
1
1, 1
2, 1
3, 1
1
2
3
3
10
1
2
Exercise 8.3
2
4
2 (a)
3 (a)
(iii)
2
5
9
10
(b) no sugar; probability =
3 (a)
(b)
(b) 9
1 (a) 1, 2, 3, 4, 5, 6, 7, 8, 9 or 10
2 (a)
3
(a)
Green
Exercise 8.2
(b)
H
T
H
HH
HT
1
3
2 (a) 0.61, 0.22, 0.11, 0.05, 0.01
(b) (i) highly likely
(ii) unlikely
(iii) highly unlikely
1
(i)
10
3
(iv)
10
3
(vii)
10
3 (a)
A
CA
LA
CA
TA
TA
(d)
(c)
(f)
14
15
1
45
2
5
3 (a)
(c)
1
2
1
(b)
5
6
(b) 7,
36
1
(d)
2
4 (a)
$1
$5 6
$5 6
$5 6
$5 6
$2 3
50c 1.5
50c 1.5
$1
6
6
6
6
3
1.5
1.5
2
5
9
10
$1
6
6
6
6
3
1.5
1.5
3
14
(f)
1
6
6
1
4
50c
5.5
5.5
5.5
5.5
2.5
1
1
$5
10
10
10
10
7
5.5
5.5
20c
5.2
5.2
5.2
5.2
2.2
0.7
0.7
(d)
1
8
1
3
Chapter 9
Exercise 9.1
1 (a)
(b)
(c)
(d)
(e)
(f)
1
10
9
10
1
50c
5.5
5.5
5.5
5.5
2.5
1
1
(c)
(c)
17, 19, 21 (add 2)
121, 132, 143 (add 11)
8, 4, 2 (divide by 2)
40, 48, 56 (add 8)
−10, −12, −14 (subtract 2)
2, 4, 8 (multiply by 2)
20c
5.2
5.2
5.2
5.2
2.2
0.7
0.7
35
56
(g) 11, 16, 22 (add one more each
time than added to previous
term)
(h) 21, 26, 31 (add 5)
2 (a)
(b)
(c)
(d)
(e)
7, 9, 11, 13
37, 32, 27, 22
1 1 1
1, , ,
2 4 8
5, 11, 23, 47
100, 47, 20.5, 7.25
3 (a)
(b)
(c)
(d)
(e)
(f)
5, 7, 9
1, 4, 9
5, 11, 17
0, 7, 26
0, 2, 6
1, −1, −3
{}
{1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 18}
{1, 3, 5, 7, 9, 11, 13, 15}
{1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13,
14, 16, 17, 18, 19, 20}
(e) {2, 4, 6, 8, 10, 12}
(f) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 15, 18}
3 (a)
(b)
(c)
(d)
4 (a) {–2, –1, 0, 1, 2}
(b) {1, 2, 3, 4, 5}
T35 = 73
T35 = 1225
T35 = 209
T35 = 42 874
T35 = 1190
T35 = −67
a
2n + 5
3 − 8n
6n − 4
n2
1.2n + 1.1
T50 = 105
T50 = −397
T50 = 296
T50 = 2500
T50 = 61.1
6 (a)
n
1
2
3
4
Tn 6 11 6 21
(b) Tn = 5n + 1
(c) 496
(d) 55th
6
31
1 (a)
(d)
o
f
g
q
u
j
t e
m
r
k
(d) (i)
v
w
M
4
9
(b)
74
99
(c)
79
90
103
900
(e)
943
999
(f)
928
4995
x
36 – x
(c) false
(f) true
(c) 0.57
2 (a) The set of even number from two
to twelve.
(b) 6
(c) {2}
(d) {2, 4, 6, 8}
(e) {2}
(f) {10, 12}
Mixed exercise
1 (a) 5n − 4
(b) 26 − 6n
(c) 3n − 1
80
T120 = 596
T120 = −694
T120 = 359
2 (a) –4, –2, 0, 2, 4, 8
(b) 174
(c) T46
3 0.213231234. . ., 2 , 4 π
(ii)
(v)
41
1
(iii)
80
21
5
80
x
1
2
3
y
4
5
6
7
8
x
−1
0
1
2
3
y
1
−1
−3
−5
−7
−1
0
1
2
3
9
7
5
3
1
(d) x
−1
0
1
2
3
y
−1
−2
−3
−4
−5
x
4
4
4
4
4
y
−1
0
1
2
3
z
(b)
(c) x
y
(in fact, any five values of y are correct)
(f) x
−1
0
1
2
3
y
−2
−2
−2
−2
−2
7
Exercise 9.3 A
(b) true
(e) false
(h) true
40
59
(c) 16
0
S
(b) 21
17
−1
(e)
78 – x
18
Exercise 10.1
9
20
{c, h, i, s, y}
{c, e, h, i, m, p, r, s, t, y}
{a, b, d, f, g, j, k, l, n, o, q, u, v, w,
x, z}
(f) {c, h, i, s, y}
12 , 0.090090009. . .
16
Chapter 10
C
2 (a)
(b)
(c)
(d)
(e)
45 , 90 , π, 8
1 (a) false
(d) true
(g) false
(b) 21
1 (a) x
3
(b)
2 (a)
16 ,
n
C
25
(iv)
h
i
s
c
y
p
286
999
B
3 (a)
Exercise 9.2
3
d
P
l
5
26
b
(b)
5 (a)
Exercise 9.3 B
1
23
99
21
5 (a) {x: x is even, x ≤ 10}
(b) {x: x is square numbers, x ≤ 25}
4 (a) 8n − 6
(b) 1594 (c) 30th
(d) T18 = 138 and T19 = 146, so 139 is
not a term
5 (a)
(b)
(c)
(d)
(e)
4 (a)
(g) x
y
1
2
3
−1 0
1.5 −0.5 −2.5 −4.5 −6.5
(h) x
y
0
1 2 3
−1
−1.2 −0.8 −0.4 0 0.4
(i)
x −1
0
y −1 −0.5
(j)
1
2
3
0
0.5
1
x −1
0
1
2
3
y 0.5 −0.5 −1.5 −2.5 −3.5
2 student’s graphs of values above
3 y=x−2
Answers
179
4 (a)
(d)
(g)
(h)
no
(b) yes
(c) yes
no
(e) no
(f) no
yes (horizontal lines)
yes (vertical lines)
5 (a) m = 1
(b) m = −1
(c) m = −1 (d) m =
6
7
1
1
2
4
4
, c = −2
5
(f) m =
(g) m is undefined, c = 7
(h) m = −3, c = 0
14
3
(j) m = −1, c = −4
(k) m = 1, c = −4
(l) m = −2, c = 5
(m) m = –2, c = −20
2
(g) x = 2
(i) y = −2x
(k) y = 3x − 2
8 (a)
(b)
(c)
(d)
1
x
2
(b) y
(d) y
(f) y
2x − 1
2x + 1
1
(h) y
x+2
3
(j) y = x + 4
(l) y = x − 3
x = 2, y = –6
x = 6, y = 3
x = –4, y = 6
x = 10, y = 10
(e) x =
−5
2 , y = –5
9 (a) 1
(d) 2
(b) 1
(e) 0
(c) –1
(f) 12
10 (a) (0, 0)
(b) (–1.5, 0.5)
(c) (–2, 3) (d) (5, 10)
(e) (0.5, –3) (f) (1, 1.5)
Exercise 10.2 A
1 (a) x 2 + 5x
5x 6
(b) x 2
x −6
(c) x 2 + 12 x + 35
(d) x 2 + 2x
2 x 35
(e) x 2
4x 3
4x
(f) 2 x 2
(g) y 2
9 y 14
9y
180
Answers
3
2
(j) x + x − 132
1
(k) 1 − x 2
2 (a) x 2 + 8x
8 x 16
(b) x 2 6x
6x 9
2
(c) x + 10 x + 25
(d) y 2 4 y 4
(e) x 2 + 2 xy
x + y2
2
(f) 4 x 4 xy
xy + y 2
(g) 9 2 12
12xx 4
2
(h) 4 x 122 y 9 y 2
(i) 4 x 2 200 x 25
(j) 16 x 2 − 48x
48 x + 36
(k) 9 6x
6 + x2
(l) 16 − 16x
16 + 4 x 2
(m) 36 − 36y
36 + 9 y 2
(e) m = , c =
7 (a) y
x
(c) y = 2 5
1
(e) y
x −1
(i) 2
2
(m) −12 x 2 + 14 x − 4
m = 3, c = −4
m = −1, c = −1
1
m=− ,c=5
2
m = 1, c = 0
1
3
4
(l) −3x 2 + 11x − 6
16
(i) m = − , c = +
7 xy + 2 y 2
4
(e) m = 2 (f) m = 0
(g) undefined
1
(h) m =
6 (a)
(b)
(c)
(d)
(h) x 2
x 1
3 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
x 2 − 25
4 x 2 25
49 y 2 − 9
x4 y4
9 2 16
x6
y4
4 4
16 x y − 4 z 4
4x8 4 y2
16 x 2 y 4 − 25 y 2
64 x 6 y 4 − 49z 4
Exercise 10.2 B
1 (a) ( x + 2))(( x + 2)
(b) ( x + 4 ))(( x + 3)
(c) ( x + 3))(( x + 3)
(d) ( x + 1))(( x + 4 )
(e) ( x + 3))(( x + 5)
(f) ( x 1))(( x − 8)
(g) ( x 5))(( x − 3)
(h) ( x 1))(( x − 3)
(i) ( x 26) ( x − 1)
(j) ( x 8))(( x + 1)
(k) ( x + 5))(( x − 2)
(l) ( x + 4 ))(( x − 8)
(m) ( x 3))(( x − 4 )
(n) ( x + 4 ))(( x − 3)
(o) ( x + 9))(( x − 6)
2 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
5 ( x 2) ( x + 1)
3 ( x 4 ) ( x − 2)
3 ( 3) ( x − 1)
5 ( x 2) ( x − 1)
x ( x + 10) ( x + 2)
x 2 y ( x + 2) ( x 1)
x ( x + 7 ))(( x − 2)
3 ( x 3) ( x − 2 )
−2 ( x + 4 )(
) ( x − 6)
2 ( x 7 ) ( x − 8)
3 (a) ( x + 3))(( x − 3)
(b) ( 4 x ))(( 4 − x )
(c) ( x + 5))(( x − 5)
(d) (7
)(7 − x )
(e) (3x 2 y )(3x 2 y )
(f) (9 2 x )(9 2 x )
(g) ( x + 3 y )( x 3 y )
(h) (11 y + 12 x )(11 y 12 x )
(i) ( 4 x 7 y )( 4 x 7 y )
(j) 2 ( x 3) ( x − 3)
(k) 2 (10 )(10 − x )
(l) x 2 + y x 2 y
(m) 5 8 5 − 8
(n) ( xy + 10)( xy − 10)
(
(
)(
)(
)
)
⎛5
8w ⎞ ⎛ 5
8w ⎞
− ⎟
(o) ⎜ 2 +
2
⎟
⎜
z ⎠⎝ y
z ⎠
⎝y
(
(1
)(
)
(p) 5x 5 1 55x 5 1
(q)
9
2
3
)(1
9
Exercise 10.2 C
1 (a)
(b)
(c)
(d)
x = 0 or x = 3
x = –2 or x = 2
x = 0 or x = 2
2
x = 0 or x = −
3
(e) x = –1 or x = 1
(f) x = −
(g) x = −
7
2
1
2
or x =
or x =
7
2
1
2
(h) x = –4 or x = –2
(i) x = –4 or x = –1
2
3
)
(j) x = 5 or x = –1
(k) x = 5 or x = –4
(l) x = –10 or x = 2
(m) x = 5 or x = 3
(n) x = 20 or x = –3
(o) x = 7 or x = 8
(p) x = 10
(q) x = 2
(b)
40
x
y
Distance
−1 0 2 3
−0.5 0 1 1.5
x
y
−1 0
3.5 3
−1
2
−1
2
0
4
2
2
3
2
2
8
3
10
(e)
m = −2, c = −1
m = 1, c = −6
m = 1, c = 8
1
m = 0, c = −
2
2
m=− ,c=2
3
(f) m = −1, c = 0
3 (a) y
x −3
(c) y
2
1
x+
3
2
4
x −3
5
(b) y
x −2
(d) y
(e) y 2 x − 3
(g) y = 2
(f) y
x+2
(h) x = −4
4 A 0, B 1, C 2, D 1, E 4
5 (a) y
4
3
2x − 6
(b) y = 7
(c) y = x + 4
(d) x = −10
(e) y
(f) y = −3
6 (a) t
D
x
0
0
2
14
4
28
2
4 6
Time
7
All four plotted on the same graph.
2 (a)
(b)
(c)
(d)
x
(c)
7
(d) 7
(e) (i) 3 hours (ii) 1 h 24 min
(iii) 48 min
(f) (i) 21 km
(ii) 17.5 km
(iii) 5.25 km
(g) 7 hours
(d) y 2 x − 4 = 0
x
y
15
0
2 3
2 1.5
0
2
20
5
(c) y = 2
x
y
25
10
1
x+3
2
(b) y
a (a + 2))((a − 2)
x 2 + 1 ( x 1)( x 1)
( x 2))(( x + 1)
( x 1))(( x − 1)
(2x 33yy + 2z )(2x − 3 y 2z )
( x + 12)( x + 4 )
x⎞ ⎛ 2 x⎞
⎛
(g) x 2 +
x −
⎝
2⎠ ⎝
2⎠
30
1
x
2
6
42
(a)
(b)
(i)
1
(0.5, 6.5)
(ii)
2
(0, 5)
(iii)
–1
(1, 3)
(iv)
−
(v)
4
3
undefined
x 2 − 16 x + 64
2x 2 2
9 2 12
1 xy 4 y 2
12x
1 −12y
12 36 y 2
9 2 4
4 x 2 200 x 25
9 4 y 2 6x 2 y + 1
1
(h) x 2 + xy
x + y2
4
1
2
(i) x −
4
1
−4
(j)
x2
8 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(k) 10 x − 45
(l) −2 x 3 + 16 x 2 − 8 x
(m) 2 x 3 8x
8 x 2 + 16 x
9 (a)
(b)
(c)
(d)
(e)
(f)
35
Mixed exercise
1 (a) y
Caroline’s distance at 7 km/h
y
45
(–0.5, 3)
(–1.5, 0.25)
(
)
( x + 1))(( x − 6)
4 ( x 3) ( x − 4 )
2 ( x 3) ( x − 4 )
(h)
(i)
(j)
(k)
(l)
5 1+ 2 8 1− 2
3 ( x 3) ( x + 2 )
10 (a)
(b)
(c)
(d)
(e)
(f)
x = –5 or x = –1
x = –2 or x = 2
x = 2 or x = 1
x = –1
x = 5 or x = –1
x=2
(
)()(
)
8
Chapter 11
Exercise 11.1 A
1 (a)
(c)
(e)
(g)
5 cm
12 mm
1.09 cm
8.49 cm
(b)
(d)
(f)
(h)
2 (a) 55.7 mm
(c) 5.29 cm
(e) 9.85 cm
3 (a) no
(d) yes
17 cm
10 cm
0.45 cm
6.11 cm
(b) 14.4 cm
(d) 10.9 mm
(f) 9.33 cm
(b) yes
4 (a)
32 = 5.66
(b)
18 = 4.24
(c)
32 = 5.66
(c) no
(d) 180 = 13.4
(e) 3
(f) 45 = 6.71
Exercise 11.1 B
1 20 mm
2 44 cm
Answers
181
3 height = 86.6 mm, area = 4330 mm2
4 13 m and 15 m
5 0.7 m
Exercise 11.2
1 (a)
(c)
(e)
(f)
(g)
(h)
2.24 cm
(b) 6 mm
7.5 mm
(d) 6.4 cm
y = 6.67 cm, z = 4.8 cm
x = 5.59 cm, y = 13.6 cm
x = 9 cm, y = 24 cm
x = 50 cm, y = 20 cm
2 ∠ABC = ∠ADE (corr ∠ are equal)
∠ACB = ∠AED (corr ∠ are equal)
∠A = ∠A (common)
∴ 6 ABC is similar to 6 ADE
3 25.5 m
Exercise 11.3
1 (a) x = 18 cm
(b) x = 27 cm, y = 16 cm
2 9:4
3 (a) 254.48 cm
(b) 529 mm2
2
4 (a) x = 2 cm
1
2
∴ AB = AC
Mixed exercise
1 (a) sketch
(b) 130 m
2 10 = 6 + 8 ∴ 6 ABC is right-angled
(converse Pythagoras)
2
2
2
3 (a) 18 = 4.24
(b) 20 = 4.47
(c) 8 2.83
(d) 5
(e) 3.5
(b) x = 15 cm
5 28 000 cm
Exercise 11.4
1 (a) 6 ABC is congruent to 6 ZXY (SAS)
(b) although the two triangles look
to be the same size, there are no
marks to confirm this to be the
case so we cannot be certain they
are congruent
(c) 6 XYZ is congruent to 6 ONM (ASA)
(d) 6 DEF is congruent to 6 RQP (ASA)
(e) 6 ACB is congruent to 6 ACD (RHS)
(f) 6 PMN is congruent to 6 NOP (SSS
or SAS)
(g) 6 PRQ is congruent to 6 ZYX (SAS)
(h) 6 ABC is congruent to 6 EDC (ASA)
2 PO = PO (common)
MP = NO (given)
MO = NP (diagonals of isosceles
trapezium are equal)
Δ MPO is congruent to Δ NOP (SSS)
5 (a) x = 3.5 cm
(b) x = 63°, y = 87°
(c) x = 12 cm
182
Answers
(a)
(b)
(c)
(d)
(e) (f)
mean
6.14 27.44 13.08
5
4.89 5.22
median 6
27
13
5
5
5
mode
6
27 and 12
no
4
6
38
mode
2 (a) (iii) and (vi)
(b) sensible answer from student,
e.g. different sets can still add up
to the same total as another set. If
divided by the same number they
will have the same mean.
6 (a) 2.78
(b) 1
7 (a) $20.40
(b) $6
(c) $10
(d) 2 (only the category B workers)
(e) The mean is between $20 and $40
so the statement is true.
7 18 cm2
9 (a) 3 cm
(b) height = 12 cm, area of base =
256 cm2
10 (a) 6 ABC is congruent to 6 HGI (RHS)
(b) 6 ABC is congruent to 6 DEF
(ASA)
(c) 6 ACB is congruent to 6 EDF (SAS)
(d) 6 CAB is congruent to 6 GIH (SAS)
11 5.63 m
140 mm
68 mm
560 mm
420 mm
140 mm
(b) 156 mm
1
5 Need to know how many cows there
are to work out mean litres of milk
produced per cow.
4 P = 2250 mm
12 (a)
Exercise 12.1
4 15
8 23 750 mm2
(b) 25 : 1
Chapter 12
3 255
6 (a) 4 : 1
(b) 1 : 9
3
6 (a) 5 : 1
(c) 125 : 1
3 ∠CDB = ∠DBE = ∠BEA (alt ∠ equal)
∠BAE = ∠CBD (corr ∠ equal)
AE = BD (given)
Δ BAE is congruent to Δ CBD (ASA)
AB = BC
Exercise 12.2
1 (a) mean = 4.3, median = 5,
mode = 2 and 5.
The data is bimodal and the lower
mode (2) is not representative of
the data.
(b) mean = 3.15, median = 2,
mode = 2.
The mean is not representative
of the data because it is too high.
This is because there are some
values in the data set that are
much higher than the others.
(This gives a big range, and when
the range is big, the mean is
generally not representative.)
(c) mean = 17.67, median = 17,
no mode.
There is no mode, so this cannot be
representative of the data. The mean
and median are similar, so they are
both representative of the data.
2 (a) mean = 12.8, median = 15,
mode = 17, range = 19
(b) mode too high, mean not reliable
as range is large
(b) 36.74 ≈ 37
(c) 30 ≤ m < 40
(d) 30 ≤ m < 40
6 (a)
(b)
(c)
(d)
2
3 (a) Runner B has the faster mean
time; he or she also achieved the
faster time, so would technically
be beating Runner A.
(b) A is more consistent with a range
of only 2 seconds (B has a range
of 3.8 seconds).
31 ≤ w < 36 33.5
40
1340
36 ≤ w < 41 38.5
70
2695
41 ≤ w < 46 43.5
80
3480
Exercise 12.3
46 ≤ w < 51 48.5
90
4365
51 ≤ w < 55 53.5
60
3210
55 ≤ w < 60 58.5
20
1170
Total
360
16 260
1
Score
Frequency
0
6
0
1
6
6
2
10
20
3
11
33
4
5
20
5
1
5
6
1
6
Total
40
90
(b) 3
(c) 2
(a) 2.25
(d) 6
2
Score ×
frequency
(fx)
Data set
mean
A
B
3.5
46.14
7 (a) 15 scores fell into the bottom 60%
of marks
(b) Less than half students scored
above the 60th percentile, so you
can assume the marks overall
were pretty low.
Chapter 13
Exercise 13.1
1 student’s own diagrams
Exercise 12.5
3 (a) 9080 g
(c) 500 g
(e) 0.0152 kg
(b) 49 340 g
(d) 0.068 kg
(f) 2.3 tonne
1 (a) Q1 = 47, Q2 = 55.5, Q3 = 63,
IQR = 16
(b) Q1 = 8, Q2 = 15, Q3 = 17, IQR = 9
(c) Q1 = 0.7, Q2 = 1.05, Q3 = 1.4,
IQR = 0.7
(d) Q1 = 1, Q2 = 2.5, Q3 = 4, IQR = 3
4 (a)
(b)
(c)
(d)
(e)
(f)
4.12
Mixed exercise
1 (a) mean 6.4, median 6,
mode 6, range 6
(b) mean 2.6, median 2,
mode 2, range 5
(c) mean 13.8, median 12.8,
no mode, range 11.9
5 (a) 1200 mm2
(c) 16 420 mm2
(e) 0.009441 km2
45.17
46 ≤ w < 51
41 ≤ w < 46
29
C
3
40
4.5
3 and 5
40
6.5
Exercise 12.4
1 (a)
Frequency Frequency ×
(f )
midpoint
0 ≤ m < 10
5
2
10
10 ≤ m < 20
15
5
75
20 ≤ m < 30
25
13
325
30 ≤ m < 40
35
16
560
40 ≤ m < 50
45
14
630
50 ≤ m < 60
55
13
715
63
2315
Total
Frequency f × midpoint
(b) 230 mm
(d) 2450.809 km
(f) 0.157 m
median
Mid
point
Mid
point
2 (a) 2600 m
(c) 820 cm
(e) 20 mm
(a)
(b)
(c)
(d)
mode
Marks
(m)
Words per
minute (w)
19
5
Q1 = 18, Q3 = 23, IQR = 5
fairly consistent, so data not
spread out
2 (a) 19
(b) 9 and 10
(c) 5.66
3 C − although B’s mean is bigger it
has a larger range. C’s smaller range
suggests that its mean is probably
more representative.
4 (a) 4.82 cm3 (b)
(c) 5 cm3
5 (a)
(b)
(c)
(d)
5 cm3
6 (a)
(b)
(c)
(d)
(e)
(f)
19 km
9015 cm
435 mm
492 cm
635 m
580 500 cm
100 m
15 cm
2 mm
63 cm
35 m
500 cm
(b) 900 mm2
(d) 370 000 m2
(f) 423 000 mm2
69 000 mm3
19 000 mm3
30 040 mm3
4 815 000 cm3
0.103 cm3
0.0000469 m3
7 220 m
8 110 cm
9 42 cm
10 88 (round down as you cannot have
part of a box)
36.47 years
40 ≤ a < 50
30 ≤ a < 40
don’t know the actual ages
Answers
183
Half
past 3 hour 7 1
$55.88
Dawoot 1
past 9
hours
4
2
five 4
8:17
Nadira
a.m.
8h
5:30 1
p.m. 2 hour 43 min
John
08:23
Robyn
7:22
a.m.
08:08
17:50 45 min 8 h
42 min
4:30 1 hour 8 h
p.m.
8 min
18:30 45 min 9 h
37 min
Mari
$64.94
$64.82
$60.59
$71.64
4 (a) 195.5 cm ≤ h < 196.5 cm
93.5 kg ≤ m < 94.5 kg
(b) maximum speed =
greatest distance
shortest time
=
405
33.5
= 12.09 m/s
5 (a) upper bound of area: 15.5563 cm2
lower bound of area: 14.9963 cm2
(b) upper bound of hypotenuse:
8.0910 cm
lower bound of hypotenuse:
7.9514 cm
2 6 h 25 min
Exercise 13.4
3 20 min
1 (a) 1 unit = 100 000 rupiah
(b) (i) 250 000
(ii) 500 000
(iii) 2 500 000
(c) (i) Aus $80
(ii) Aus $640
4 (a) 5 h 47 min
(c) 12 h 12 min
(b) 10 h 26 min
(d) 14 h 30 min
5 (a) 09:00 (b) 1 hour (c) 09:30
(d) 30 minutes
(e) It would arrive late at Peron Place
at 10:54 and at Marquez Lane
at 11:19.
Exercise 13.3
1 The upper bound is ‘inexact’ so 42.5
in table means < 42.5
Upper bound Lower bound
(a)
42.5
41.5
(b)
13325.5
13324.5
(c)
450
350
(d)
12.245
12.235
(e)
11.495
11.485
(f)
2.55
(g)
395
(h)
1.1325
2.45
385
1.1315
2 (a) 71.5 ≤ h < 72.5
(b) Yes, it is less than 72.5 (although
it would be impossible to measure
to that accuracy).
184
Answers
2 (a) temperature in degrees F against
temperature in degrees C
(b) (i) 32 °F
(ii) 50 °F
(iii) 210 °F
(c) oven could be marked in
Fahrenheit, but of course she
could also have experienced a
power failure or other practical
problem
(d) Fahrenheit scale as 50 °C is hot,
not cold
3 (a) 9 kg
(b) 45 kg
(c) (i) 20 kg
(ii) 35 kg
(iii) 145 lb
2 23 min 45 s
3 2 h 19 min 55 s
4 1.615 m ≤ h < 1.625 m
5 (a) No, that is lower than the lower
bound of 45
(b) Yes, that is within the bounds
6 (a) 3.605 cm to 3.615 cm
2.565 cm to 2.575 cm
(b) lower bound of area: 9.246825 cm2
upper bound of area:
9.308625 cm2
(c) lower: 9.25 cm2, upper: 9.31 cm2
7 (a) conversion graph showing litres
against gallons (conversion factor)
(b) (i) 45l
(ii) 112.5l
(c) (i) 3.33 gallons
(ii) 26.67 gallons
(d) (i) 48.3 km/gal and 67.62 km/gal
(ii) 10.62 km/l and 14.87 km/l
8 €892.06
9 (a) US$1 = IR49.81
(c) US$249.95
10 £4239.13
Chapter 14
Exercise 14.1 A
5
1 (a)
US$1 = ¥76.16
£1 = NZ$1.99
€1 = IR69.10
Can$1 = €0.71
¥1 = £0.01
R1 = US$0.12
2490.50
(ii) 41 460
7540.15
9139.20
(ii) 52 820
145 632
y
4
Exercise 13.5
1 (a) (i)
(ii)
(iii)
(iv)
(v)
(vi)
(b) (i)
(iii)
(c) (i)
(iii)
(b) 99 620
0
(b)
Daily
earnings
3
–5 –4 –3
2
1
–2 –1
0
–1
–2
(–3,–3)
–3
–4
–5
x = −3, y = −3
y=
Time Lunch (a)
out
Hours
worked
(b) 690 mm
(d) 0.0235 kg
(f) 29 250 ml
(h) 1000 mm2
(j) 7 900 000 cm3
(l) 0.168 cm3
y
Time
in
2700 m
6000 kg
263 000 mg
0.24 l
0.006428 km2
29 000 000 m3
3=
Name
1 (a)
(c)
(e)
(g)
(i)
(k)
2x
+
1
Mixed exercise
x–
3 (a) 27.52 m2
(b) upper bound: 28.0575 m2
lower bound: 26.9875 m2
Exercise 13.2
1
2
3
4
5
x
(b)
5
4x+2
=y
4
3
2
2 (a) (A) y
x+4
3
(B) y = x + 6
y
1
x
–5 –4 –3 –2 –1–10 1 2 3 4
3
(–1, –2) –2 – 2y =
x
–3
5
–4
–5
x = −1, y = −2
(c)
y
y=
+
–2x
2
5
4
3
2
1
2y
x
–5–4 –3–2–1 0 1 2 3 4 5 6 7 8 9 10
–2
–3
(3,–4)
–4
–5
–6
–7
–8
–9
–10
1=
x–
+3
0
x = 3, y = −4
(d)
5
4
4 3
=
x
2
y–
1
y
–x
–7
x = −5.5, y = −1.5
(e)
5
y
y
4x − 2
y x
y x−4
y = −2
(−2, 6) (ii)
(4, 0)
3x
+
3 (a)
(b)
(c)
(d)
x = 4, y = 2
x = −1, y = 2
x = 0, y = 4
x = 3, y = 1
4 (a)
(b)
(c)
(d)
(e)
(f)
x = 6, y = −1
x = 1, y = 2
x = 18, y = −8
1
x = 1, y = 1
3
x = 3, y = −1
x = 3, y = 7
5 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
x = 2, y = 1
x = 2, y = 2
x = 2, y = −1
x = 3, y = 2
x = 3, y = 2.5
x = 4, y = 2
x = 5, y = 3
x = 0.5, y = −0.5
x = −9, y = −2
3
+3
1
–2x
2
–5 –4 –3 –2 –1–10 1 2 3 4 5
(2, –1)
–2
–3
–4
–5
x = 2, y = −1
x
0
1
(d)
(e)
(−2, −2)
–7 –6 –5 –4 –3 –2 –1 2
0.25
0
0.75
0.5
–2 –1 0
2 (a)
(b)
(c)
(d)
(e)
(f)
1.25
1
1
1.75
1
2
3
2
4
5
6
7
3, 4
–2, –1, 0, 1, 2, 3
2
–1, 0, 1, 2, 3
0, 1, 2, 3, 4
4, 5, 6, 7, 8, 9
(k) g < 9
(b) x > 11
(d) x ≤ 52
(f) x > 73
1
(h) e ≥ –3
4
1
(j) x ≥ 45
2
1
(l) y ≤
4
(m) n > 31
(o) x ≤ −6
x = 3, y = –4
x = 15, y = 30
x = 4, y = 2
12
x = 15 , y =
10
11
(n) n > –2
1
28
Exercise 14.3
1 (a)
7
2 (a) c + d = 15
(b) 3 desks and 12 chairs
Exercise 14.2
1 (a)
3
4
5
(b)
2
5
0
1.5
3 (a) x ≤ –2
(c) x ≥ 4
(e) x ≥ 9
1 (a) x = –3, y = 4
1
(b) x = –1, y =
1
4
(i)
Exercise 14.1 B
–5 –4 –3 –2 –1 0
3
(h)
8 6 quarters and 6 dimes
2
2
–1
(i) r < 11
1
1
–2
7 number of students is 7
–5 –4 –3 –2 –1 0
1
(g)
–4 –3.5 –3
(g) x > 9
7
0
(f)
–5 –4 –3 –2 –1 0
6 a chocolate bar costs $1.20 and a box
of gums $0.75
(c)
(d)
(e)
(f)
y=
=
3y
3
–1
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
3
4
–2
2
y=
x
–10–9–8 –7–6–5–4–3–2–1 0 1 2 3 4 5
(–5.5, 1.5) –2
–3
–4
–5
–6
–7
–8
–9
–10
(C)
(D)
(E)
(F)
(b) (i)
(iii)
(c)
–4 –3.5 –3
3
4
5
5
4
3
2
1
0
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
y
x
1 2 3 4 5
y < 4x + 1
Answers
185
(b)
3 (a) y ≥ 2x − 1
(b) y < –4x + 3
(c)
5
4
3
2
1
y
5
4
3
2
1
y ≥ –2x + 5
x
0
–5 –4 –3 –2 –1
–1
–2
y
(c) y >
–3
–4
–5
2
x
(d) y ≥ –x – 3
4
x
0
–5 –4 –3 –2 –1
–1
–2
1 2 3 4 5
1
1 2 3 4 5
–3
–4
–5
3x – y ≥ 3
(c)
5
4
3
2
1
y
(d)
2x < 3y – 6
x
0
–5 –4 –3 –2 –1
–1
–2
1 2 3 4 5
y
(e)
x
0
–5 –4 –3 –2 –1
–1
–2
1 2 3 4 5
–2 < y ≤ 4
186
Answers
(b) (–2, 2), (–2, 1), (–1, 2), (–1, –1),
(0, 1)
x
1 2 3 4 5
y
5
4
3
2
1
(f)
1
x
1 2 3 4 5
x–y>7
–5 –4 –3 –2 –1 0
–1
x–y 3
–2
<
2
4
–3
–4
–5
–6
–7
–8
–9
–10
y
x
1 2
Exercise 14.4
1
–3
–4
–5
(b)
–3
–4
–5
y
5
4
3
2
1
0
–5 –4 –3 –2 –1
–1
–2
–3
2y ≤ –4
–4
–5
0
–5 –4 –3 –2 –1
–1
–2
x
1 2 3 4 5
–3
–4
–5
2 (a)
5
4
3
2
1
5 (a)
0
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
5
4
3
2
1
y
5
4
3
2
1
y > –x + 1
3 4
5 6
7 8 9 10
0
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
y
x
1 2 3 4 5
Exercise 14.5
2
1 (a) ( x + 3) − 5
2
(b) ( x − 2) + 3
2
(c) ( x + 7 ) − 5
2
(d) ( x − 6) − 6
2
(e) ( x + 5) − 8
2
(f) ( x + 11) + 20
2
(g) ( x + 12) − 23
2
(h) ( x − 8) − 7
2
(i) ( x + 9) + 12
2
(j) ( x − 1) + 9
2
(k) ( x − 4 ) − 21
2
(l) ( x + 10) − 17
2 (a)
y
8
7
6
5
4
3
2
1
x
–1 0
–1
1
2 3 4 5
6 7
8
(b) greatest posible value is at the
point (2, 5) so 2y + x = 12
3 let x = number of chocolate fudge
cakes and y = number of vanilla fudge
cakes
60
x ≤ 30
of
pr
50
y
x+
it 3
40
y ≤ 20
2y
30
x+
20
y
10
≤
40
x
10 20 30 40 50 60
–1 0
–1
to maximise profit x = 30 and y = 10
maximum profit = $110
4 let x = number of litres of orange and
y = number of litres of lemon
2000
1900
1800
1700
1600
1500
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
200
100
0
y
orange = x
pro
fit
x+
x ≤ 1800
1.9
y
y ≤ 600
1x
y≤ 3
200 400 600 800 1000 1200
x
1500
1800
maximum income ($2940) when
x = 1800 and y = 600
x = 6, x = –1
x = 3, x = –2
x = 3, x = 1
x = 7, x = –1
x = 15.81, x = 0.19
x = –6.85, x = –0.15
x = 0.11, x = –9.11
x = –3.70, x = –7.30
x = 11.05, x = –9.05
3 (a)
(b)
(c)
(d)
(e)
(f)
x = 2, x = –0.5
x = 3, x = 1
x = 2.53, x = –0.53
x = 3, x = –0.5
x = 7.47, x = –1.47
x = –2.27, x = 1.77
Exercise 14.6
lemon = y
x ≥ 1000
2 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
1 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
x = 10 or 4
x = 6 or –20
x = 1 or –6
x = 3 or 5
x = 4 or –1
x=2
x = 2 or –6
x = –5
x = 2 or –4
x = 6 or –2
x = 4 or 5
x = 5 or –8
2 (a)
(b)
(c)
(d)
(e)
(f)
x = –0.76 or – 5.24
x = 1.56 or –2.56
x = –4.76 or –9.24
x = 7.12 or –1.12
x = –2.17 or –7.83
x = –0.15 or –6.85
(g)
(h)
(i)
(j)
(k)
(l)
3 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
x = –7.20 or –16.80
x = –3.70 or –7.30
x = 4.19 or –22.19
x = 4.32 or –2.32
x = 8.58 or –0.58
x = 11.05 or –9.05
x = 5.46 or –1.46
x = 1.67 or –3.5
x = 1.31 or 0.19
x = 5 or –2.5
x = 2.25 or 1
x = 1.77 or –1.27
x = 0.80 or –0.14
x = 10.65 or 5.35
x = 9.91 or –0.91
x = 0.5 or –3
x = –0.67 or –1.5
x = 1 or –0.75
4 numbers are 57 and 58 or –58 and –57
5 dimensions are 12 cm × 18 cm
Exercise 14.7
1 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(2x − 3)(x + 1)
(3x +1)2
(2x − 3)2
(2x + 1)(3x − 5)
(4x − 3)(x + 1)
(7x − 1)(2x − 7)
(x + 5)(3x − 4)
(2x − 1)(3x + 7)
(3x + 5)(x − 5)
(3x − 11)(x + 6)
(5x + 3)(3x − 5)
(8x + 1)(x + 3)
2 (a) (2x + 3)2
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
1
2
(2x 1)(2x 1)
or
1⎞
⎛
⎜⎝ x + 2 ⎠ ( x + )
2(5x + 2)2
(2x + y)(3x − 5y)
(4x2 − 3)(x2 + 1)
2(2x − 1)(3x + 1)
2(1 − 2x)(1 + x)
(x + 3)(x + 2)
(3x + 8)(x − 4)
2(3x − 11)(x + 6)
(4x − 3)(x + 1)
(8x + y)(x + 3y)
Answers
187
1 (a)
(c)
x
7
7
(e) 8z
3
2x
(g)
3y
2b
(i)
3a
x+3
2 (a)
x
2x + y
(c)
x
a+2
(e)
a
x+5
(g)
x+2
(i)
(c)
(e)
(g)
(i)
2 x = −2, y = 5
2x
(d)
5
2
(f)
y
1
(h)
5ab
4 (a) x > –5
a b
a b
3 1
(d)
x −5
a b
(f)
a b
(b)
5
4
3
2
1
(d)
y
0
–5 –4 –3 –2 –1
–1
–2
x = –2
–3
–4
–5
(e)
(f)
x
1 2 3 4 5
x=3
(g)
(h)
(i)
5 2p
10 p2
3xy
2
xy 3 z
4
45a 2 + 16a
16a − 50
30a
( x + 2)(( x − 1)
x (x + 4)
1
2x 1
−6
)( x + 5))(( − 1)
(2x + 1)(
(h) x 2 − 3
6
Chapter 15
3
Exercise 15.1
x
6
35
a2
a
c
1
42
−2 − 3x
2x + 1
2
y
6
5xy
(d)
2
1 (a) length = 10 cm, width = 7 cm
(b) length = 14.4 cm, width = 7 cm
(b)
(f)
(b )(b − 1)
(h)
4−x
3
the integer solutions are:
(c)
29
10p
(d)
(e)
5 7
18 − 5m
(f)
x
+
2
x
1
6
)( )
(
(g)
9
x
3
( )(
(h)
17 − x
x ( x − 3)(
x
+ y = 1.5
2
x
greatest, x = 2, y = 4, giving + y = 5
2
7 (a) x(x − 2y)
(b) (a2 + b)(a2 − b)
(c) (x − 5)(x + 11)
(d) (2y – 1)(y + 7)
(e) –2(x + 1)(2x − 3)
(f) (x − 6)(x + 3)
least, x = –1, y = 2, giving
(b)
13 p + 2
8p
17
2) ( x − 1)
4)
5 3x
(i)
)( x + 1)( x 1)
(2x 3)(
7
p +1
8 (a)
(b)
(c)
(d)
(e)
(f)
a = 1.30 or –2.30
x = 1.33 or –1
x = 3 or –5
x = 0.67 or 1
x = 1 or –0.625
x=1
9 (a) x = –2 or 0.4
(b) x =
188
(b) x ≤ –1
5
3x 2 y
xy
4 (a)
(c)
3 $5000 at 5% and $10 000 at 8%
7
2
3 (a)
10 (a)
1 x = 2, y = −5
(b) 3
1
x y
x+y
(b) 2 – x
Mixed exercise
Exercise 14.8
Answers
q ± q 2 − 4 pr
2p
2 (a) length = 9.14 cm, width = 5.5 cm
(b) Yes. The length of the mini
pitch = width of standard pitch
and 2 × width of mini pitch <
length of standard pitch. It is
possible to have two mini pitches
on a standard pitch so, with
three standard pitches, up to six
matches could take place at the
same time.
3 (a)
1
200
or
1
150
(b) (i) and (ii) diagram drawn using
student’s scale including × for net
posts
4 (a) student’s scale drawing − diagonal
distance = 11.3 m
(b) using Pythagoras’ theorem
Exercise 15.2
1 (a) 090°
(b) 225°
(c) 315°
2 (a) 250°
(b) 310°
(c) 135°
65
85
(a) (b) (c) (d)
hypotenuse
c
z
f
q
opp(A)
a
y
g
p
adj(A)
b
x
e
r
(b) 1.43
(e) 0
4 (a) tan A =
3
4
2
3
(b) tan x = 3 , tan y = 2
1
5
12
12
5
3
4
(e) AC = 2 cm, tan B = 3 , tan C = 4
x = 1.40 cm
y = 19.29 m
c = 3.32 cm
a = 13 m
x = 35.70 cm
6 (a) 26.6°
(d) 85.2°
(g) 79.7°
(b) 40.9°
(e) 14.0°
(h) 44.1°
(c) 51.3°
(f) 40.9°
7 (a) 16°
(d) 23°
(b) 46°
(e) 38°
(c) 49°
8 (a) hyp = y, adj(θ) = z, cos θ =
(b) hyp = c, adj(θ) = b, cos θ =
(c) hyp = c, adj(θ) = a, cos θ =
(d) hyp = p, adj(θ) = r, cos θ =
z
y
b
c
c
r
p
z
22
, tan B =
4
3
5
4
(c) sin C = 3, cos C = , tan C =
5
(c) 57°
(f) 27°
11 4.86 m
x = 30° and y = 4.69 cm
x = 3 m and y = 53.1°
x = 48.2°
x = 22.9° and y = 8.90 cm
(b) 33°
–cos 68°
cos 105°
cos 135°
–cos 60°
–cos 125°
2 (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
θ = 15° or 165°
θ = 124°
θ = 52° or 128°
θ = 70°
θ = 20° or 160°
θ = 48°
θ =39°
θ =110°
θ =7° or 173°
θ =84° or 96°
(b)
(d)
(f)
(h)
(j)
sin 24°
sin 55°
sin 35°
sin 82°
–cos 50°
x = 40.4°
x = 61.2°
x = 7.1
x = 22.5°
x = 14.0°
10
19.6
4
(b) θ = 43.2°
(d) θ = 31.1°
6 (a) a = 2.9 m (b) z = 3.7 cm
(c) m = 9.2 m (d) r = 5.0 cm
Exercise 15.7
1 (a) 11.2 cm2
(c) 17.0 cm2
(e) 3.8 cm2
(b) 25.4 cm2
(d) 70.4 cm2
(f) 24.6 cm2
2 44.9 cm2
3 (a) x = 115° and area = 31.7 cm2
(b) x = 108° and area = 43.0 cm2
(c) x = 122.2° and area = 16.3 cm2
(b)
(d)
(f)
(h)
1 (a)
(c)
(e)
(g)
90°
36.9°
9.85 cm
10.3 cm
(b)
(d)
(f)
(h)
5 cm
4 cm
66.0°
16.9°
2 (a) 21.2 m
(c) 21.2 m
(b) 10.6 m
(d) 18.4 m
3 (a) 9.80 cm
(b) 24.1°
Mixed exercise
1 (a) x = 37.6°
(b) x = 44.0°
(c) x = 71.4°
2 35.3°
3 AB = 300 m
Exercise 15.6
1 (a)
(c)
(e)
(g)
(i)
5 (a) θ = 66.4°
(c) θ = 60.7°
Exercise 15.8
1 (a)
(c)
(e)
(g)
(i)
3 (a) θ = 82.4°
(c) θ = 33.3°
(e) θ = 38.4°
7 , cos A = 12 tan A = 7
13 ,
12
13
11
13
x = 18.5
x = 18.2
x = 28.1°
x = 8.3
4 AC ≈ 41.6 cm
5 ≈ 16 m
lookout
(L)
50°
25 m
2 (a) x = 6.3 cm (b) x = 6.8 cm
(c) x = 4.4 cm (d) x = 13.2 cm
(e) x = 8.7 cm (f) x = 10.8 cm
a
(e) hyp = x, adj(θ) = z, cos θ = x
(b) sin B = 5 , cos B =
84
85
Exercise 15.5
(c) 5.14
19.6
13
(b) 64°
(e) 30°
1 1.68 m
2 (a) 30°
(d) tan y = , ∠X = (90 − y)°, tan X =
9 (a) sin A =
16
Exercise 15.4
(c) tan 55° = d , tan B = d
5 (a)
(b)
(c)
(d)
(e)
10 (a) 45°
(d) 60°
12 (a)
(b)
(c)
(d)
2 (a) opp(30°) = x cm
adj(60°) = x cm
opp(60°) = adj(30°) = y cm
(b) adj(40°) = q cm
opp(50°) = q cm
opp(40°) = adj(50°) = p cm
3 (a) 0.65
(d) 0.41
63
65
(e) sin E = 84 , cos E = , tan E =
Exercise 15.3
1
16
(d) sin D = 63, cos D = , tan D =
3 student’s drawing
(a) 223°
(b) 065°
(c) 11 km (d) 13 km
(b) θ = 23.1°
(d) θ = 28.9°
(f) θ = 82.5°
a 2 + c 2 − b2
cos B =
2ac
base of lookout
(B)
5m
swimmer
(W)
shark
(S)
6 RS = 591 m
Chapter 16
Exercise 16.1
1 (a) E
(d) D
(b) C
(e) B
(c) A
Answers
189
2 (a) student’s own line (line should go
close to (160, 4.2) and (175, 5.55));
answers (b) and (c) depend on
student’s best fit line
(b) ≈ 4.7 m
(c) between 175 cm and 185 cm
(d) fairly strongly positive
(e) taller athletes can jump further
3 (a) distance (m)
(b)
600
(d)
Distance (m)
500
400
1 (a) $7.50
(d) $448
(b) $160
(c) $210
(e) $343.75
2 5 years
0
6
8 10 12 14
Age (years)
(c) weak positive
(e) 12 years old
(f) Not very reliable because
correlation is very weak
(g) 600 m
Mixed exercise
1 (a) the number of accidents for
different speeds
(b) average speed
(c) answers to (c) depend on
student’s best fit line
(i) ≈ 35 accidents
(ii) < 40 km/h
(d) strong positive
(e) there are more accidents when
vehicles are travelling at a higher
average speed
2 (a) there a strong negative correlation
at first, but this becomes weaker as
the cars get older
(b) 0−2 years
(c) it stabilises around the $6000 level
(d) 2−3 years
(e) $5000−$8000
Chapter 17
Exercise 17.1
1 $19.26
2 $25 560
Answers
(b) $92.74
(c) student’s own graph; a comment
such as, the amount of compound
interest increases faster than the
simple interest
5 $862.50
6 $3360
7 (a) $1335, $2225
(b) $1950, $3250
(c) $18 000, $30 000
8 (a) $4818
3 2.8%
5 $2281 more
200
100
190
Exercise 17.2
4 $2800 more
300
0
3 (a) $930.75
(b) $1083.75
(c) $765
(d) $1179.38
4 $1203.40
5 $542.75
6 (a) $625 (b) $25 (c) $506.50
6 (a) $7.50
(c) $225.75
(e) $346.08
(b) $187.73
(d) $574.55
(b) 120%
9 $425
10 $211.20
11 $43.36 (each)
12 $204
Chapter 18
7 $562.75
Exercise 18.1
8 $27 085.85
1
Exercise 17.3
1 (a) $100
(d) $900
x
(b) $200
(c) $340
2 $300
3 $500
4 $64.41
5 (a) $179.10
(b) $40.04
(c) $963.90
2
(a)
(b)
y
(c)
+2
2
2 (a) $1190
(b) $1386
(c) $1232
1
2
3
−7 −2
1
2
1
−2
−7
1
6
−3 −2
y
x − 2 −11 −6
2
−3
−2 −3
−6 −11
(d) y
x − 3 −12 −7
2
−4
−3 −4
−7 −12
2
y=x +
1
2
6
9.5 4.5 1.5 0.5 1.5 4.5 9.5
12
1
25 h
2
(c)
0
−2
x −3
Mixed exercise
(b) 40 h
−1
1
(e)
1 (a) 12 h
−3 −2
y
10
(e)
8
3 (a) $62 808 (b) $4149.02
6
4 (a)
4
(b)
2
Years
Simple
interest
Compound
interest
1
300
300
2
600
609
3
900
927.27
4
1200
1125.09
5
1500
1592.74
6
1800
1940.52
–8
7
2100
2298.74
–10
8
2400
2667.70
–12
–4
–2
0
–2
x
2
4
–4
–6
(a)
(c)
(d)
2 (a) y = 3 + x2
(c) y = x2
(e) y = −x2 − 4
(c) 6 m
(d) just short of 4 seconds
(e) 3 seconds
(b) y = x2 + 2
(d) y = −x2 + 3
3 (a)
−2 −1 0 1 2 3 4 5
1 (a)
2
x − 3 x + 2 12 6 2 0 0 2 6 12
y
y = x2 − 3x + 2
12
10
8
6
4
2
x
–3 –2 –1 0
–2
1
2
3
4
5
6
10
9
8
7
6
5
4
3
2
1
–8 –6 –4 –2 0
–2
–3
–4
–5
–6
xy = 5
–7
–8
–9
(b)
(b)
x
−3 −2 −1
2
x − 2 x − 1 14
7
0
1
2 3
2 −1 −2 −1 2
y = x2 − 2x + 1
10
–4 –3 –2 –1 0
–5
1
2
3
4
(c)
−2 −1 0 1 2 3 4 5
2
6
x + 4 x + 1 −11 −4 1 4 5 4 1 −4 −11
y
4
2
x
–4 –2 0
–2
2
4
6
8
–4
–6
–8
–10
–12
y = x2 + 4x + 1
4 (a) 8 m
(b) 2 seconds
4
6
8
10
(e)
y
8
7
y = − 4x 6
5
4
3
2
1
0
–8 –7 –6 –5 –4 –3 –2 –1
1 2 3 4 5 6 7 8x
–2
–3
–4
–5
y = − 4x
–6
–7
–8
y
16
y= x
x
2 (a)
–4
–6
–8
–10
–12
16 –14
y = x –16
x
6
x
2
0
–16–14–12–10 –8–6 –4 –2
–2 2 4 6 8 10 12 14 16
5
x
xy = 5
16
14
12
10
8
6
4
2
y
15
y
y
(c)
10
9
8
7
6
5
4
3
2
1
–10 –8 –6 –4 –2 0
–2
–3
–4
–5
–6
–7
–8
xy = 9 –9
Length
1
Width
24 12 8 6 4 3
(b)
y
Width (m)
14
y
y
8
7
6
5
4
3
2
1
0
–8 –7 –6 –5 –4 –3 –2 –1
1 2 3 4 5 6 7 8x
–2
–3
–4
–5
y = − 8x
–6
–7
–8
y = − 8x
Exercise 18.2
x
y
(d)
xy = 9
x
2
4
6
8
10
24
22
20
18
16
14
12
10
8
6
4
2
0
2
3 4 6 8 12
24
2
1
y
x
0 2 4 6 8 10 12 14 16 18 20 22 24
Length (m)
(c) the curve represents all the
possible measurements for the
rectangle with an area of 24 m2
(d) ≈ 3.4 m
Answers
191
Exercise 18.3
Exercise 18.4
1 (a) −2 or 3
(c) −3 or 4
(b) −1 or 2
2 (a)
3
1 (a)
y
2
–4 –3 –2 –1 0
–1
x
1
2
3
–2
–3
–4
(b) (i) −2 or 1
(ii) ≈ −1.6 or 0.6
(iii) ≈ −2.6 or 1.6
3 (a)
y
14
y = x2 − x − 6
13
12
11
10
9
8
7
6
5
4
3
2
1
x
0
–5 –4 –3 –2 –1
1 2 3 4 5
–2
–3
–4
–5
–6
–7
(b) (i) 0 or 1
(ii) −2 or 3
(iii) −3 or 4
–4
192
y
14 y = x2 − x − 6
13
12
11
10
9
8
7 (iii) y = 6
6
5
4
3
2
1 (ii) y = 0 x
0
–2 –1 1 2 3 4 5
–2
–3
–4
–5
(i) y = –6
–6
–7
–8
Answers
(1, 4) and (−2, −2)
(−2, 0) and (1, 3)
(b)
(b)
y
12
11
10
9
8
7
6
5
y = x2 + 2x − 3
4
3
2
1
x
–8–7–6–5–4–3–2–10 1 2 3 4 5 6 7 8
–2
–3
–4
–5 y = −x + 1
–6
–7
–8
y = −x2 + 4
6
5
4
3
2
1
–6 –5 –4 –3 –2 –10
–2
–3
–4
–5
–6
y
y=
3
x
y
8
7
6
y = −2x2 + 2x + 4
5
4
3
2
1
x
–3 –2 –1 0
x
1 2 3 4
–2
–3
y = −2x + 4
–4
1 2 3 4 5 6
≈ (−2.3, −1.3), (1, 3) and
(1.3, 2.3)
y
8
7
6
5
y = x2 + 2x
4
3
y = −x + 4
2
1
x
–8–7–6–5–4–3–2–10 1 2 3 4 5 6 7 8
–2
–3
–4
–5
–6
–7
–8
(−4, 8) and (1, 3)
(c)
(−4, 5) and (1, 0)
(c)
y
8
7
y=
+ 3x − 2
6
y=x+2
5
4
3
2
1
x
–8–7–6–5–4–3–2–10 1 2 3 4 5 6 7 8
–2
–3
–4
–5
–6
–7
–8
2x2
8
y = 2x + 2
7
6
5
4
3
2
y = 4x
1
x
–8–7–6–5–4–3–2–10 1 2 3 4 5 6 7 8
–2
–3
–4
–5
–6
–7
–8
y
1
2 (a)
(0, 4) and (2, 0)
(d)
y
4
3
y =1 x
2
2
2
y = −0.5x + 1x + 1.5
1
x
0
–8–7–6–5–4–3–2–1 1 2 3 4 5 6 7 8
–2
–3
–4
–5
–6
–7
–8
(−3, −6) and (1, 2)
(e)
Exercise 18.5
50
40
30
1
x
(a) y
–5
x3 − 4x2
3
(b) y = x + 5
–4
–3
–2
–1
0
1
2
3
4
5
–225 –128 –63 –24 –5
0
–3
–8
–9
0
25
–120 –59 –22 –3
4
5
6
13
32
69
130
(c) y
2 x 3 + 5x 2 + 5
380
213 104 41
12
5
8
9
–4
–43 –120
(d) y
x3 + 4x2 − 5
220
123
0
–5
–2
3
4
–5
–30
–145 –82 –43 –22 –13 –10 –7
2
23
62
125
–157 –71 –25 –7
83
185
3
(e) y = x + 2 x − 10
(f) y
2x 3 + 4 x 2 − 7
(g) y
x 3 − 3x 2 + 6
(h) y
56
3
3x + 5x
350
22
58
6
172
66
(a)
50
40
30
20
10
x
2 4 6 8 10
(b)
50 y
40
30
20
10
–10 –8 –6 –4 –2 0
–10
–20
–30
3
y = x + 5 –40
–50
2
14
6
x
8 10
–10 –8 –6 –4 –2 0
–10
–20
–30
–40
y = x3 + 2x − 10
–50
–7
–1
25
343
4
6
2
–14 –48 –106 –194
20
10
–2
0
2
–14 –66 –172 –350
50
40
30
20
10
y
–10 –8 –6 –4 –2 0
–10
–20
–30
–40
y = − x3 + 4x2 − 5
–50
6
x
8 10
2 4
6
x
8 10
2 4
6
x
8 10
2 4
6
x
8 10
Answers
193
y
–10 –8 –6 –4 –2 0
–10
–20
–30
–40
y = 2x3 + 4x2 − 7
–50
(g)
–10 –8 –6 –4 –2 0
–10
–20
–30
–40
y = − 2x3 + 5x2 + 5
–50
50
40
30
20
10
2 4
(f)
–5
2 4
50
40
30
x
6 8 10
y
20
10
–10 –8 –6 –4 –2 0
–10
–20
–30
–40
y = −x3 − 3x2 + 6
–50
(d)
2 4
20
10
50
40
30
(c)
y
–10 –8 –6 –4 –2 0
–10
–20
–30
–40
y = x3 − 4x3
–50
19
y
y
(h)
2 4
6
x
8 10
50
40
30
y
20
10
–10 –8 –6 –4 –2 0
–10
–20
–30
–40
y = −3x3 + 5x
–50
(d)
2 (a)
x
−2.5
y
−2
−1.5
−1 −0.5
0
0.5
1
1.5
2
2.5
3
4
5
50
6
20
y
50
40
30
10
(b)
50 y
y = x3 +
40
20
10
–4
1
x
–2
2 4
6
–2
–50
x
0
–10
2
4
(e)
–20
50
–30
(c) (i) x = –1.3, 1.8 or 4.5
(ii) x = 0 or 5
(iii) x = –1.6, 2.1 or 4.5
–40
1
(a) y x −
x
1
3
(b) y = x +
x
20
–2 –1 –0.5
1.5
–0.2
–0.1
4.8
9.9
0
0.1
0.2
0.5
1 2
–9.9
–4.8
–1.5
0 1.5 2.67
–2
3
x
0
–10
2
4
–20
–30
–40
4
3
x
2
x3 −
x
2x +
–27.33 –8.5 –2 –2.125 –5.008 –10.001
10.001 5.008 2.125 2 8.5 27.33
12.33
–37.99 –17.96 –5.75 –1 4 9. 67
8
–7
x2 − x +
7 10.25 22.04 42.01
–5.5 –5
–26.33 –7
(a)
–7
–15.4 –30.2
30.2
1 3.875 9.992 19.99
9.91
10
50
40
y=x−
1
x
0
–5
–10
–15
–20
–25
Answers
5 5.5
7
4.84
1.75
1 2.5 6.33
y
50
4
y = x2 + 2 −
x
40
30
30
20
20
10
5
–2
7
(f)
y
20
15
15.4
–50
–19.99 –9.992 –3.875 –1 7 26.33
1
11. 67 5.5 1 –1.25 –4.76 –9.89
x
(c)
25
194
2
x
10
–2.67 –1.5 0
2
(c) y = x + 2 − x
–4
y = x3 −
30
–50
–4
–3
y
40
3
x
4
–40
10
–4
2
–30
20
x
8 10
x
0
–10
–20
30
–10 –8 –6 –4 –2 0
–10
y=x−5
–20
–30
–40
y = x3 − 5x2 + 10
–50
(f) y
3
x
30
(b)
(e) y
y = 2x +
40
–36.875 –18 –4.625 4 8.625 10 8.875 6 2.125 –2 –5.625 –8 –6 10 46
(d) y
y
x
2
4
–4
–2
0
–10
y
y = x2 − x +
1
x
10
x
2
4
x
–4
–2
0
–10
–20
–20
–30
–30
–40
–40
–50
–50
2
4
2 (a)
4 (a)
x
–4
–1 0
1
2
3
4
y = 2x 0.0625 0.125 0.25 0.5 1
2
4
8
16
y = 2–x
–3
16
–2
8
18
y = 2–
4
2
1 0.5 0.25 0.125 0.0625
y
5 (a) & (b) (i)
y=2
16
300
Number of organisms
14
12
10
8
6
y
y = 3x
250
200
150
100
50
y = 12x + 1
4
–1 0
2
–6
–4
–2
0
2
4
6
y = 10
y
x
0
0.2 0.4 0.6 0.8
1
1
1.58 2.51 3.98 6.31 10
2 x − 1 –1 –0.6 –0.2 0.2 0.6
12
10
8
1
y
y = 10 x
6
4
2
y = 2x − 1
x
0
–2
0.2 0.4 0.6 0.8
1
1
2
3
4
Time (hours)
(b) (ii) 12 per hour
(c) (i) ≈ 3.4 hours
(b)
x
60 y
3
55 y = x − 1
50
45
40
35
30
25
20
15
10
5
1.2
5
x
6
(ii) ≈ 42
Exercise 18.6
0
–6 –5 –4 –3 –2 –1
–5
–10
–15
–20
–25
–30
–35
–40
–45
–50
–55
–60
x
1 2 3 4 5 6
(b) gradient = 12
1
10 y
y = x2 − 2x − 8
9
8
7
6
5
4
3
2
1
0
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
1 2 3 4 5 6 7
3 answers should be close to: with light
≈ 1.4 cm per day, without light ≈ 1 cm
per day
Mixed exercise
1 (a) A: y
B: y
x
3x − 2
x2 + 3
x
1
4
4
C: y = − − 5
(b) (i) (–1, –5)
(ii) answer should be the point of
intersection of graphs A and C.
(c) maximum value y = 3
(a) –2
(b) 6 and –6
Answers
195
2 (a)
C
y
10
Exercise 19.4
D
1 (a) 15° (isosceles 6)
(b) 150° (angles in a 6)
(c) 35° (∠MON = 80°, and 6MNO in
isosceles, so ∠NMO = ∠NOM =
50°, so ∠MPN = 35°)
(d) 105° (∠PON = 210° so ∠PMN =
105° – half the angle at the centre)
8
y = x2
6
E
4
F
2
x
–3
–1 0
–2
–2
y = x3
1
2
2 (a) 55° (angles in same segment)
(b) 110° (angle at centre twice angle
at circumference)
(c) 25° (∠ABD = ∠ACD, opposite
angles of intersecting lines AC
and BD, so third angle same)
3
G
–4
–6
H has no line symmetry
–8
(b) A = 0, B = 3, C = 4, D = 4, E = 5,
F = 2, G = 2, H = 2
–10
(b) (0, 0) and (1, 1)
(c) (i) x = 2 or x = –2
(ii) x = –2
(d) gradient for y x 2 when x = 2 is 4
and gradient for y x 3 when
x = 2 is 12
4
3 (a) y = − x
(c)
(b) y
2 (a) 2, student’s diagram
(b) 2
3 student’s own diagrams but as an
example:
1 (a)
B
A
(c) 42°
8 (a) 56°
(b) 68°
(c) 52°
(b)
A
A 3 cm
Exercise 19.2
1 (a) 3
(b) 4
(c) infinite number corresponding to
the number of diameters of the
circle face
(d) as per part (c)
(e) 2
(f) 3 (5 if face is a square)
(g) 1
(h) infinite number corresponding to
the number of diameters of the
sphere
2 (a) 4
(b) 3
(d) infinite (e) 4
A
B
C
2
O
3
2 6.5 cm
3 (a) 49.47 cm
(c)
4 cm
(c) 1
(f) 8
1 (a) x = 25° (b) x = 160°, y = 20°
Answers
(b) 116°
1
(a)
Exercise 19.3
196
7 (a) 22°
DIAGRAMS ARE NOT TO SCALE BUT
STUDENTS’ SHOULD BE WHERE
REQUESTED
y = −x
Exercise 19.1
6 144°
Exercise 19.5
y
Chapter 19
4 35°
5 59.5°
x
8
7
6
5
4
3
y = − 4x
2
1
0
–8 –7 –6 –5 –4 –3 –2 –1
1 2 3 4 5 6 7 8x
–2
–3
–4
–5
–6
–7
–8
3 ∠DAB = 65°, ∠ADC = 115°,
∠DCB = 115°, ∠CBA = 65°
(b) 177.72 cm
1m
B
4
(ii) eight
(d) (i)
Chapter 20
Exercise 20.1
Road
3 km
OOL
OL
SCHO
SCH
SC
1.5 km
School
5
A
B
6 NOT TO SCALE – note that on
student drawing point X should be
2 cm (10 m) from B.
D
A
Cell
tower
X
C
Radio
mast
B Flagpole
2 (a)
(b)
(c)
(d)
a hexagonal prism
the axis of rotational symmetry
6
7
3 (a) w = 72° (base angle isosceles
ΔOCB), x = 90° (angle in semicircle), y = 62° (angles in a Δ)
z = 18° (base angle isosceles
ΔODC)
(b) x = 100° (reflex ∠ADB = 200°,
angle at circumference = half
angle at centre)
(c) x = 29° (∠ADB is angle in a semicircle so ∠BDC = 90°, then angles
in a Δ)
(d) x = 120° (angle at centre), y = 30°
(base angle isosceles Δ)
4 (a) x = 7.5 cm, y = 19.5 cm
(b) x = 277.3 mm, y = 250 mm
5 NOT TO SCALE
Mixed exercise
1
(a) (i)
(ii) none
Petrol station
6 NOT TO SCALE
X
(b) (i)
(ii) none
A
(c) (i)
(ii) four
5 cm
B
Route 66
1 (a)
60
50
40
30
20
10
0
English exam marks
Frequency
(ii) none
(e) (i)
Frequency
4 km
60
50
40
30
20
10
0
0 10 20 30 40 50 60 70 80 90 100
Mock exam score
Maths exam marks
0 10 20 30 40 50 60 70 80 90 100
Mock exam score
(b) 51−60
(c) 51−60
(d) students to compare based
on histograms, but possible
comments are: The modal value
for both subjects is the same
but the number in the English
mode is higher than the Maths.
Maths has more students scoring
between 40 and 70 marks. Maths
has more students scoring more
than 90 marks and fewer scoring
less than 20.
2 (a) bars are touching, scale on
horizontal axis is continuous,
vertical axis shows frequency
(b) 55
(c) 315
(d) 29−31
(e) scale does not start from 0
X
Answers
197
3
Histogram of ages
Mixed exercise
Chapter 21
1 (a)
Exercise 21.1
one person
1 (a) 3 : 4
(c) 7 : 8
Frequency density
Frequency
12
10
8
6
4
2
0
Heights of trees
0
2
2 (a)
(c)
(e)
(g)
(i)
4 6 8 10 12
Height in metres
(b) 19
(c) 6−8 metres
2 (a)
4 (a) 300
(b) 480
(c) 100
Exercise 20.2
P80
Q3
P60
x
0 10 20 30 40 50 60 70 80 90 100
Percentage
(b) Median = 57%, Q1 = 49% and
Q3 = 65%
(c) IQR = 16
(d) 91%
(e) 60% of students scored at least
59%; 80% of the students scored
at least 67%
56
2 (a)
(b)
(c)
(d)
166 cm
Q1 = 158, Q3 = 176
18
12.5%
198
Answers
58 60 62
Mass (kg)
64
Frequency density
5 teenagers
20 30 40 50 60 70 80 90 100 110 120130140150
Time (minutes)
4 (a) 6.5 cm
(b)
Cumulative frequency of plant heights
y
20
10
Q1
0
4 (a) 20 ml oil and 30 ml vinegar
(b) 240 ml oil and 360 ml vinegar
(c) 300 ml oil and 450 ml vinegar
1
2
3
1 (a) 1 : 2.25
(b) 1 : 3.25
2 (a) 1.5 : 1
(c) 5 : 1
(b) 5 : 1
(c) 1 : 1.8
1 240 km
Histogram of airtime minutes
0
3 60 cm and 100 cm
Exercise 21.2 B
3
30
x=4
x=3
x = 1.14
x = 2.67
x = 13.33
Exercise 21.2 A
Q2
Q1
(b)
(d)
(f)
(h)
(j)
6 810 mg
(b) 60−62 kg
(c) 7.4%
(d) 10 kg
Cumulative
frequency
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
Number of students
1 (a)
x=9
x = 16
x=4
x = 1.875
x=7
(e) 1 : 4
5 60°, 30° and 90°
Frequency
15 20 25 30 35 40 45 50 55
Age (years)
Masses of students
20
18
16
14
12
10
8
6
4
2
0
54
(b) 6 : 1
(d) 1 : 5
4
Q2
Q3
5 6 7 8
Height (cm)
median height = 6.8 cm
(c) IQR = 8.3 – 4.7 = 3.6
x
9 10 11 12
2 30 m
3 (a) 5 cm
(b) 3.5 cm
4 (a) it means one unit on the map is
equivalent to 700 000 of the same
units in reality
(b)
Map
distance 10 71 50 80 1714 2143
(mm)
Actual
distance 7
(km)
50 35 56 1200 1500
5 (a) 4 : 1
(b) 14.8 cm
(c) 120 mm or 12 cm
Exercise 21.3
1 25.6 l
2 11.5 km/l
3 (a) 78.4 km/h
(b) 520 km/h
(c) 240 km/h
4 (a) 5 h
(c) 40 h
(b) 9 h 28 min
(d) 4.29 min
5 (a) 150 km
(c) 3.75 km
(b) 300 km
(d) 18 km
1 (a) (i) 100 km
(ii) 200 km
(iii) 300 km
(b) 100 km/h
(c) vehicle stopped
(d) 250 km
(e) 125 km/h
(e)
(f)
(g)
(h)
(i)
2 hours
190 min = 3 h 10 min
120 km/h
(i) 120 km
(ii) 80 km
48 km/h
40 min
50 min
53.3 − 48 = 5.3 km/h
Pam 12 noon, Dabilo 11:30 a.m.
3 (a) (i) 40 km/h
(ii) 120 km/h
(b) 3.5 min
(c) 1200 km/h2
(d) 6 km
5
4 (a) 0–30 s, m/s
2
6
(b) after 70 s, 0.5 m/s2
(c) 90 km/h
(d) 2 km
Exercise 21.5
1 (a)
(ii)
1
2
(b) (i) 0.35 h
(ii) 4.7 h
(iii) 1.18 h
day
8 (a) 12 days (b) 5 days
9 5 h 30 min
Exercise 21.4
2 (a)
(b)
(c)
(d)
1
(b) (i) 2 2 days
A
1
Yes, B = 150
8
1
No, 15 is not = 2
10 1200 km/h
Exercise 21.6
1 (a) k = 7
(b) a = 84
m
= 0.4587 ,
T
so m varies directly with T
3 (a) F = 40
(b) m = 4.5
2 ratio of m to T is constant,
4 a = 2, b = 8, c = 1
1
3
5 (a) y = 2
(b) x = 0.5
6 (a) y = 2x2
(c) x = 9
(b) y = 1250
4 (a)
(b)
(c)
(d)
(e)
150 km
after 2 hours; stopped for 1 hour
100 km/h
100 km/h
500 km
5 (a)
(b)
(c)
(d)
20 seconds
2 m/s2
200 m
100 m
6 4.5 min
7 187.5 g
k
or PV = k
V
(b) P = 80
8 (a) P =
9 (a) 7 : 4
(b) 54.86 mm
Chapter 22
7 (a) y x = 80
(b) y = 8
(c) x = 15.49
Exercise 22.1 A
17 7
9
8 (a) b = 40
(b) a =
9 (a) y = 2.5
(b) x = 2
10 (a) xy = 18 for all cases, so
relationship is inversely
proportional
18
(b) xy = 18 or y =
x
(c) y = 36
Exercise 21.7
1 (a) x − 4
(b) P = 4x − 8
(c) A = x2 − 4x
2 (a) S = 5x + 2
5x + 2
(b) M =
3
3 (a) x + 1, x + 2
(b) S = 3x + 3
4 (a) x + 2
(b) x − 3
(c) S = 3x − 1
1 $300
Exercise 22.1 B
2 $72 000
1 14
10
1
3 (a) $51 000 (b) $34 000
2 9 cm
(b) $250
4 14 350
3 80 silver cars, 8 red cars
3 $12.50
Mixed exercise
4 60 m
1 (a) 90 mm, 150 mm and 120 mm
(b) Yes, (150)2 = (90)2 + (120)2
4 father = 35, mother = 33 and
Nadira = 10
(b)
(c) Yes,
2 (a) $175
A
B
=
5 (a) 75 km (b) 375 km
(c) 3 h 20 min
6 (a) 15 litres (b) 540 km
7 (a) inversely proportional
5 breadth = 13 cm, length = 39 cm
2 1 : 50
6 X cost 90c, Y cost $1.80 and Z cost 30c
3 (a) (i) 85 km
(ii) 382.5 km
(iii) 21.25 km
8 97 tickets
7 9 years
Answers
199
Exercise 22.2
1 (a) V = U + T − W
U T2
(b) V =
− W or
3
V
(U − T
1
3
W
)
P
P
2
P
Q=±
R
Q2
P=
2
Q2
P=
R
P = Q2 + R
R2
Q=
P
V
I=
R
20 amps
A
r=
(note, radius cannot be
π
negative)
r = 5.64 mm
(f) Q = ±
(g)
(h)
(i)
(j)
(k)
2 (a)
(b)
3 (a)
(b)
4 (a) F
9
5
C + 32
(b) 80.6 °F
(c) 323 K
Exercise 22.3
1 (a) 11
(c) 5
(b) –1
(d) 2m + 5
2 (a) f(x) = 3x + 5
(b) (i) 17
(ii) 53
(iii) 113
(c) f(2) + f(4) = 17 + 53 = 70 ≠ f(6)
which is = 113
(d) (i) 3a2 + 5
(ii) 3b2 + 5
(iii) 3(a + b)2 + 5
(e) a = ±3
2
200
Answers
4 (a) 4(x – 5)
(b) 4x – 5
Chapter 23
Exercise 23.1 A
1
5 18
C
A
(d) B = AC
(c) B =
(e) Q
3 (a) h ( x )
5 x
(b) (i) h(1) = ±2
(ii) h(–4) = ±3
6 (a) f–1(x) = x – 4
(b) f–1(x) = x + 9
x
(c) f −1 ( x ) =
5
(d) f–1(x) = –2x
x
7 (a)
−3
2
x −3
(b)
2
(c) 2(x + 3)
(d) 2x + 3
(e) 2x + 3
(f) 2(x + 3)
2
Mixed exercise
1 10
2 41, 42, 43
3 4 years
4 Nathi has $67 and Cedric has $83
5 Sindi puts in $40, Jonas $20 and
Mo $70
6 44 children
9a − 26
7 (a) b =
8
a2 − 4
(b) b =
17
8
5 +3
2
x−4
f −1 ( x ) =
3
3
a=6
9x + 16
37
3
A
P
QQ
C
S
f −1 ( x ) =
9 (a)
(b)
(c)
(d)
(e)
P
B
S
P' R
R
A'
Q'
S'
B'
R'
C'
4 A: y = 5
B: x = 0
C: y = −1.5
D: x = −6
5
10
8
D
D'
3 (a)
A
x
B''
4
C'
–10 –8 –6 –4 –2 0
–2
C
–4
B'
–6
D''
–8
2
4
6
8
10
X
2
y
–10 –8 –6 –4 –2 0
–2
A''
D'
D
1 (a)
y
A
7
6
5
4
3
B'
2
1
C'
y
B
–5 –4 –3 –2 –1 0
4
2
4
x
2 (a)
(a)
B
A
6
C
B
4
2
–4
2
8
4
X
⎛ −5⎞
(k) ⎜ ⎟
⎝ −6⎠
10
y
–4 –2
B
–2
u r ⎛ 9⎞
(f) EC = ⎜ ⎟
⎝ 6⎠
y
X
0
u r ⎛ 1⎞
(e) DB = ⎜ ⎟
⎝ 6⎠
(l) Yes
(b) rotation 180° about (0, 0)
–4
u r ⎛ −1⎞
(d) BD = ⎜ ⎟
⎝ −6⎠
⎛ 9⎞
(j) ⎜ ⎟
⎝ 0⎠
–4
–5
–6
A'' –7
2
A
x
1 2 3 4 5
u r ⎛ 0⎞
(c) AE = ⎜ ⎟
⎝ −6⎠
u r ⎛ −5⎞
u r ⎛ −5⎞
(g) CD = ⎜ ⎟
(h) BE = ⎜ ⎟
⎝ −6⎠
⎝ −6⎠
(i) they are equal
C
–2
B'' –3
C''
u r ⎛ 4⎞
(b) BC = ⎜ ⎟
⎝ 0⎠
u r ⎛ 5⎞
1 (a) AB = ⎜ ⎟
⎝ 0⎠
A'
2
B'
10
Exercise 23.2 A
1 A: centre (0, 2), scale factor 2
B: centre (1, 0), scale factor 2
C: centre (−4, −7), scale factor 2
1
D: centre (9, −5), scale factor 4
(b)
8
–10
Exercise 23.1 C
–2
6
(b) rotation 180° about (4, 5)
Exercise 23.1 B
–2
C
4
–8
x
for MNOP: rotate 90° clockwise about
(2, 1) then reflect in the line x = 5.5
–4
2
–6
for ABC: rotate 90° clockwise about B
then reflect in the line x = −6
0
D
–4
6 possible answers are:
A'
B
A
4
x
–10
(a)
F'
6
(d)
B
y
8
C
A'
2
10
C'
6
C''
(b) enlargement scale factor 2, using
(8, −1) as centre
y
(c)
y
2
4
A
2
x
–4
–2
0
2
4
6
8
x
–2
–4
Answers
201
x
(j)
D
(c)
F
G
H
⎛ 4⎞
C⎜ ⎟
⎝ −3⎠
⎛ 2⎞
B⎜ ⎟
⎝ 3⎠
⎛ 2⎞
(b) ⎜ ⎟
⎝ 6⎠
⎛ 0⎞
(c) ⎜ ⎟
⎝ 12⎠
⎛ −1⎞
(d) ⎜ ⎟
⎝ 7⎠
⎛ −2⎞
(e) ⎜ ⎟
⎝ 1⎠
⎛ −1⎞
(f) ⎜ ⎟
⎝ 4⎠
⎛ −4⎞
(g) ⎜ ⎟
⎝ 18 ⎠
(h)
⎛ −8⎞
⎜⎝ 22 ⎟⎠
⎛ 0 ⎞
(i) ⎜
⎝ −20⎟⎠
⎛ 10 ⎞
(j) ⎜
⎝ −16⎟⎠
2 (a) –a
(d) 2c
Answers
4 (a)
(b)
(c)
(d)
6.40 cm
7.28 cm
15 cm
17.69 cm
5 (a)
(b)
(c)
(d)
5.10
5
8.06
9.22
⎛ −3⎞
(ii) y = ⎜ ⎟
⎝ −3⎠
⎛ 10 ⎞
(iii) z = ⎜ ⎟
⎝ −4⎠
⎛ 9⎞
E⎜ ⎟
⎝ 3⎠
Exercise 23.2 B
⎛ −8⎞
1 (a) ⎜ ⎟
⎝ 16 ⎠
+ 3c
u r ⎛ 7⎞
BC = ⎜ ⎟
⎝ 5⎠
u r ⎛ −11⎞
CA = ⎜
⎝ 1 ⎟⎠
u r
7 (a) XZ = x + y
u r
(b) ZX = –x – y
u r
1
(c) MZ = x + y
2
⎛ 2⎞
8 (a) (i) x = ⎜ ⎟
⎝ 7⎠
(d)
⎛ −3⎞
D⎜ ⎟
⎝ −3⎠
(i) −7a + 7c
6 (a) A(–6, 2), B (–2, –4), C (5, 1)
u r ⎛ 4⎞
(b) AB = ⎜ ⎟
⎝ −6⎠
E
⎛ 8⎞
3 A⎜ ⎟
⎝ 1⎠
b
2
3 (a) – (e) student’s own diagrams
C
202
(h) –c
(g) b
(b)
(b) (i) 7.28
(ii) 4.24
(iii) 21.5
u r
9 (a) (i) XY = b – a
u r
1
(ii) AD = (a + b)
u r 2
(iii) BC = 2(b – a)
u r
u r
(b) XY = b – a and, BC = 2(b – a)
so they are both multiples of
(b
u r– a), and hence
u r parallel, and
is double XY
10 28.2(3sf)
(b) 2b
(e) 2b
(c) −a + c
(f) 2c
Exercise 23.3
1 (a)
y
10
9
8
7
6
5
4
3
2
1
x
–1 0
–1
1 2 3 4 5 6 7 8 9 10
(b)
y
10
9
8
7
6
5
4
3
2
1
x
–1 0
–1
1 2 3 4 5 6 7 8 9 10
(c)
y
10
9
8
7
6
5
4
3
2
1
–1 0
–1
x
1 2 3 4 5 6 7 8 9 10
(d)
⎛3
(b) ⎜
⎝ 12
y
10
9
8
7
6
5
4
3
2
1
–1 0
–1
9
6
0⎞
3⎟⎠
2 (a) & (b)
y
7
6
5
4
3
2
1
⎛ −12 −6 6 ⎞
(c) ⎜
⎝ −14 4 −10⎟⎠
5 (a) x = 3
(b)
4 −1⎞
⎜
⎟
−
5 ⎝ 3 2⎠
1⎛
–5 –4 –3 –2 –1
Exercise 23.5
x
1 2 3 4 5 6 7 8 9 10
2 (a) shear, x-axis invariant, shear
factor 1.5
(b) 2-way stretch: stretch of scale
factor 4, invariant line x = 1 and
stretch of scale factor 2, invariant
line y = 2
(c) shear, x = 0 (y-axis) invariant,
scale factor 0.75
(d) shear, line y = 4 invariant, shear
factor 1
1 (a) (i) reflection in the y-axis
(ii) shear with x-axis invariant
and shear factor 2
⎛ −1 0⎞
(b) (i) ⎜
⎝ 0 1⎟⎠
⎛ 1 2⎞
(ii) ⎜
⎝ 0 1⎟⎠
⎛0 6 5 ⎞
1 (a) ⎜
⎝ 7 2 −3⎟⎠
⎛3
⎜⎝ 0
reflect in the line x = −1
⎛2
(c) ⎜
⎝ −11
(iii) reflect in the line y = −1
3 (a) AB =
⎛3
⎜⎝ 9
8 −5⎞
⎜
⎟
12 ⎝ −4 1⎠
−1 ⎛
9⎞
and BA =
1⎟⎠
6⎞
⎛ 6
⎜⎝ −10 −4⎟⎠ ; so AB ≠ BA
(b) (i) –6
(ii) –84
⎛ 7 0 −3⎞
4 (a) ⎜⎝ 11 0 4 ⎟⎠
4
(a)
2
x
–10 –8 –6 –4 –2
–2
(ii)
2
(b)
4
6
8
10
–4
–8
(ii) rotate 90° clockwise about
the origin
(b)
(d)
6
(c)
Mixed exercise
(b) (i)
y
8
–10
⎛ −2 −2 1 ⎞
(b) ⎜
4 −3⎟⎠
⎝ 1
2 (a) –12
10
–6
1 (a) (i)
2 −1⎞
4 3 ⎟⎠
3
0⎞
3⎟⎠
4
Exercise 23.4
–2
–3
–4
–5
–6
–7
2 (a) R' (2, 8)
(b) a stretch parallel to the y-axis
(x-axis invariant), scale factor 4
3 (a) enlargement, scale factor 3, centre
of enlargement the origin
⎛ −3 0⎞
(b) ⎜
⎝ 0 3⎟⎠
x
1 2 3 4 5
rotate 90° anticlockwise
about (0, 0) then translate
⎛ 2⎞
⎜⎝ −1⎟⎠
reflect in the line y = −1
⎛ −8⎞
then translate ⎜ ⎟
⎝ 0⎠
(a) B′ (−6, −6)
(c) B′ (−1, 8)
(b) B′ (6, −2)
(d) B′ (3, 9)
⎛ ⎞
⎜⎝ 12⎟⎠
⎛ 0⎞
(ii) ⎜ ⎟
⎝ −8⎠
⎛ 1⎞
(iii) ⎜ ⎟
⎝ 10⎠
⎛ 12⎞
(iv) ⎜ ⎟
⎝ 0⎠
4 (a) (i)
(b) (i)
a
2a
(iii) rotate 180° about origin
⎛ 6⎞
then translate ⎜ ⎟
⎝ 0⎠
(iv) reflect in the line x = 0
(y-axis) then translate
⎛ 0⎞
⎜⎝ −2⎟⎠
Answers
203
(ii)
7
c
b
10
8
4
A
2
C
–10 –8 –6 –4 –2 0
–2
B
2
(iii)
a
b
(iv)
a
b
8
2a + 3b
u r
5 (a) (i)
=y
u r
(ii) DE = –y
ur
(iii) FB = x + y
ur
(iv) EF = x – y
ur
(v) FD = 2y – x
(b) 4. 47
6 (a) one-way stretch with y-axis
invariant and scale factor of 2
(b) one-way stretch with x-axis
invariant and scale factor of 2
(c) a shear with x-axis invariant and
shear factor = 3
(d) a shear with y-axis invariant,
shear factor = 2
Answers
B
4 6
C
2
2
4
6
C 8
–8
A
–10
10
9
8
7
6
5
4
3
2
1
⎛1
(b) ⎜
⎝0
0⎞
1⎟⎠
⎛ 2 0⎞
(d) ⎜
⎝ 0 2⎟⎠
⎛2
(e) ⎜
⎝0
0⎞
2⎟⎠
x
1 2 3 4 5 6 7 8 9 10
⎛ 2 5⎞
9 (a) ⎜
⎝ 4 3⎟⎠
1⎞
⎛0
(b) ⎜
1⎟⎠
⎝0
⎛ 5 7⎞
(c) ⎜
⎝ 4 8⎟⎠
Chapter 24
Exercise 24.1
Card
1
H
T
H
T
H
T
H
T
Y
G
⎛ 7 5⎞
(d) ⎜
⎝ 6 6⎟⎠
(e) |P| = –3
⎛ 1 2 ⎞
−
⎜
⎟
(f) P–1 = ⎜ 3 3 ⎟
2
1
⎜
− ⎟
⎝ 3
3⎠
10 a = 7, b = 2 and c = –9
Coin
R
B
2
1
2
3
4
G
H
A
B
C
D
E
F
3 (a) & (b)
2
3
1
3
7
11
Green
4
11
Yellow
8
11
Green
3
11
Yellow
Green
Yellow
B x
8 10
A
–6
x
y
–1 0
–1
A
–4
D
–2
–2
a–b
y
8
6
4
6
b+c
204
11 (a) & (c)
y
Exercise 24.2
Mixed exercise
1 (a) & (b)
1 (a)
Blue
1
2
1
3
Yellow
1
6
(b)
Black
1
4
H
1
2
1
2
1
2
T
H
1
2
1
2
T
H
1
2
(c)
H
T
H
T
1
2
1
2
1
2
1
2
1
6
T
1
12
(d)
5
12
1
6
2
1
6
3
1
6
2 (a)
1
2
H
1
2
1
2
1
2
1
2
T
1
2
(b)
(d)
1
8
(c)
H
T
H
T
1
2
H
1
6
1
2
1
2
T
H
1
6
1
2
1
2
T
H
1
2
1
2
T
H
1
2
T
4
5
6
(c)
1
8
(d)
H
1
2
1
2
T
H
1
2
1
2
T
H
1
2
1
2
T
H
1
2
1
2
T
H
1
2
1
2
T
H
1
2
T
H
T
H
T
1
2
1
2
1
2
1
2
H
T
H
T
1
2
1
2
1
2
1
2
H
T
H
T
1
2
1
2
1
2
1
2
1
12
2 (a) & (b)
1
6
2
7
1
2
5
7
(c)
5
7
5c
5c
1
5
6
1
2
(e) 0, not possible on three coin
tosses
1
1
2
10c
10c
5
5
10c
(d)
1
21
(e) 1 (there are no 5c coins left)
Answers
205