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
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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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7
3
512
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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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OVNCFSPGJEFOUJDBMTRVBSFUJMFT.ST4NJUXBOUTUIFUJMFTUPCFBTMBSHFBTQPTTJCMF
(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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2 4JNQMJGZ
9
(a)
(e)
(i)
(m)
3
16
(b)
9 16 (c)
36
4
(f)
( 25 )
(g)
27 − 3 1 (j)
100 ÷ 4 (k)
(n)
1 +
4
×
2
( 13 ) 2
64 + 36 (d)
64 + 36 9 (h)
169 − 144
(l)
16 × 3 27
16
1+
9
16
(o) 3 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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(b) ¡$PS−¡$
(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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3 8SJUFFBDIOVNCFSBTBQSPEVDUPGJUTQSJNFGBDUPST
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EBZTBOEDZDMFFWFSZEBZT0OXIJDIEBUFTJO.BSDIXJMMTIFTXJNBOEDZDMFPOUIF
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
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(a) 5.4 × 12.2 4. 1
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(d) 3.8 × 12.6 4 35
(e)
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(c) 2 04 + 1.7 × 4.3
(
(f) 2.55 − 3.1 +
05
5
)
2
8 3PVOEFBDIOVNCFSUPUISFFTJHOJĕDBOUĕHVSFT
(a) (b) (c) (d) 9 "CVJMEJOHTVQQMZTUPSFJTTFMMJOHUJMFTXJUIBOBSFBPGDN
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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) UJNFTUIFTVNPGBOVNCFSBOE
(b) UJNFTUIFEJČFSFODFPGBOVNCFSBOE
(c) UXJDFUIFTVNPGBOEBOVNCFS
(d) BOVNCFSUJNFTUIFEJČFSFODFPGBOEo
(e) BEEFEUPUJNFTUIFTRVBSFPGBOVNCFS
(f) BOVNCFSTRVBSFEBEEFEUPUJNFTUIFEJČFSFODFPGBOE
(g) BOVNCFSTVCUSBDUFEGSPNUIFSFTVMUPGEJWJEFECZ
(h) BOVNCFSBEEFEUPUIFSFTVMUPGEJWJEFE
(i) UIFTVNPGUJNFT 12 BOEBOVNCFSUJNFT
(j) UIFEJČFSFODFPGBOVNCFSUJNFToBOEUJNFTo
2 "CPZJTpZFBSTPME
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3 ćSFFQFPQMFXJOBQSJ[FPGx
(a) *GUIFZTIBSFUIFQSJ[FFRVBMMZ IPXNVDIXJMMFBDIPGUIFNSFDFJWF
(b) * GUIFQSJ[FJTEJWJEFETPUIBUUIFĕSTUQFSTPOHFUTIBMGBTNVDINPOFZBTUIFTFDPOE
QFSTPOBOEUIFUIJSEQFSTPOHFUTUISFFUJNFTBTNVDIBTUIFTFDPOEQFSTPO IPXNVDI
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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) UIFCBTFJTDNBOEUIFIFJHIUJTDN
(b) UIFCBTFJTNBOEUIFIFJHIUJTN
(c) UIFCBTFJTDNBOEUIFIFJHIUJTIBMGBTMPOHBTUIFCBTF
(d) 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
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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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t 8IFOZPVSFNPWFCSBDLFUTJOQBSUPGBOFYQSFTTJPOZPVNBZFOEVQXJUIMJLFUFSNT"EEPSTVCUSBDUBOZMJLFUFSNT
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 )
(d) ( x + y ) − ( x
1
2
1
y)
(
(b) −3 ( − y ) − 2 x ( y − 2 x ) (c) −2 x
− −
(e) 5
(f) 2x (2x − 2) x ( x 2)
(3 − 2x ) )
(g) x (1 x ) x (2 x − 5) 2 x (1 3x )
2.5 Indices
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t x NFBOTx × xBOE y NFBOTy×y×y×y.
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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
7 y 3 x 2 5x 6 y 2
(h)
÷
5 y5 x 4 7x5 y3
(2 x y ) × ( x y )
(j)
( y x ) 3(x y ) 4
2
3
2
3
3
4
4
2
2
2
3
2
⎛ 2⎞ ⎛ 5⎞
(k) ⎜ x 4 ⎟ × ⎜ x 2 ⎟ ⎝y ⎠ ⎝y ⎠
(x y ) × (x y ) (i)
(x y )
(5x y ) ÷ ⎛ 2xy ⎞
(l)
2
5
3
3
3
2
4x7 y6
3
3
4
2
3
3
2
⎜⎝ 5x 2 y 4 ⎟⎠
Unit 1: Algebra
9
2 Making sense of algebra
Tip
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XJMMBDDFQUTJNQMJĕFE
FYQSFTTJPOTXJUIOFHBUJWF
JOEJDFT TVDIBTx−4
*G IPXFWFS UIFRVFTUJPO
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 x y )
(c)
(y x )
−33
x −4 y 3 x 7 y −5
(b) 2 −1 × −4 3
x y
x y 2
4
−10
y2
(e) x 2 ÷ ⎛⎜ 3 ⎞⎟
( y− ) ⎝ x ⎠
y −3
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
x2
1
x3
(x )
1
3
1
3
(a) x 4 × x 4
1
(b) x 3 × x 5
(e)
(64 x )
(f)
(8 x y )
1
3
(i)
( )
x2
x3
(j)
(
x 6 y 3 3 × x 8 y 10 2
1
(k) xy 3
(a)
(x ) × xx
(b)
(x ) × xx
( x y ) × (x y )
(d)
(x y )
(e)
⎛ x2 y3 ⎞ 2
1 ÷ ⎜
3 4⎟
x2 ⎝ x y ⎠
1
⎛ xy 4 ⎞ 4
4
(f) x 3 × ⎜ 3 2 ⎟
y2 ⎝ x y ⎠
(b)
−24
(−2)4
(c)
63
( 3)4
(d) 8 3
(g)
() (h) ( 18 ) 3
6
8
1
x2
1
2
×
9
(c)
(g)
) (
)
1
(d)
xy
( )
1
1
2
×
1
⎛ x6 ⎞ 2
(h) ⎜ 2 ⎟
⎝y ⎠
1
8
x2 y2
xy 4
4 4JNQMJGZ
Tip
"QQMZUIFJOEFYMBXTBOE
XPSLJOUIJTPSEFS
TJNQMJGZBOZUFSNTJO
CSBDLFUT
BQQMZUIFNVMUJQMJDBUJPO
MBXUPOVNFSBUPSTBOE
UIFOUPEFOPNJOBUPST
DBODFMOVNCFSTJG
ZPVDBO
BQQMZUIFEJWJTJPO
MBXJGUIFTBNF
MFUUFSBQQFBSTJO
UIFOVNFSBUPSBOE
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
1
2
2
1
2
3
4
3
4
1
3
1
Exercise 2.5 B
1 &WBMVBUF
(a) (
4
)(
)2 −1
(e) 256 4 (i)
( ) 1
8 −3
27
−4
(f) 125 3 (j)
5
1 −2
4
1
−2
( )
1
8 −2
18
2 $BMDVMBUF
(a) o o oŸ3
(b) o oo Ÿo
(c) o o Ÿ o 3
(d) o 4o o 6
3 4PMWFGPSx
(a) x=
(e) x=
10
3
5
1
2
x3 y3
×
xy 2
5
2
(b) 3x=
(f) ox= 251 (c) 3x= 811 (g) 3x−=
(d) x
(h) x+=
2 Making sense of algebra
Mixed exercise
1 8SJUFFBDIPGUIFGPMMPXJOHBTBOBMHFCSBJDFYQSFTTJPO6TFxUPSFQSFTFOUAUIFOVNCFS
(a) "OVNCFSJODSFBTFECZ
(b) "OVNCFSEFDSFBTFECZGPVS
(c) 'JWFUJNFTBOVNCFS
(d) "OVNCFSEJWJEFECZUISFF
(e) ćFQSPEVDUPGBOVNCFSBOEGPVS
(f) "RVBSUFSPGBOVNCFS
(g) "OVNCFSTVCUSBDUFEGSPN
(h) ćFEJČFSFODFCFUXFFOBOVNCFSBOEJUTDVCF
2 %FUFSNJOFUIFWBMVFPGxoxJG
(a) x=
(b) x=−
(c) x= 13
3 &WBMVBUFFBDIFYQSFTTJPOJGa=− b=BOEc=
b (c − a )
−2a + 3b
a b2
(b)
(c)
(a)
2ab
b a
c a2
3 2 ( − 1)
(d)
(e) a 3b2 − 2a 2 + a 4b2 ac 3
c − a (b − 1)
4 4JNQMJGZFBDIPGUIFGPMMPXJOHFYQSFTTJPOTBTGVMMZBTQPTTJCMF
(b) x 2 + 4x
4 x x 2
(a) 3a 4b + 6a 3b
(d) 2 (
(c) −2a 2b(2a 2 − 3b2 )
10 x 2 5xy
(f)
2x
3) − ( x 4 ) 2 x 2 (e) 16 x 2 y ÷ 4 y 2 x 5 &YQBOEBOETJNQMJGZJGQPTTJCMF
(a) 2 ( 4 x 3) 3( x + 1)
(c) x ( x + ) + 3x − 3( x − )
(b) 3 (2 x + 3) 2 ( 4 3x )
(d) x 2 ( x + 3) 2 x 3 ( x − 5)
6 4JNQMJGZ(JWFBMMBOTXFSTXJUIQPTJUJWFJOEJDFTPOMZ
7
(a) 15x 2
18 x 5
(b) 5x × 3x7 x
(d) (2 xy )
⎛ 4x3 ⎞
(e) ⎜ 5 ⎟
⎝ y ⎠
2 4
(g)
(x )
(c)
(x ) 3
2
(2xy ) (3x y )
−2
3
2
3
2
4
8
(x y )
(f) ( x y ) ×
(xy )
3
3
2
2
4
2
(x y ) ÷ ⎛ x y ⎞
(h)
⎜⎝ x y ⎟⎠
2 ( xy )
4
−
−3
−3
3
2
−1
3
3
2
7 4JNQMJGZFBDIPGUIFGPMMPXJOHFYQSFTTJPOT
(a)
(125x y )
3
2
1
3
(b)
( )
1
4
1
2
1
2
3
2
( x y ) × xx yy
6
3
1
2
(c)
(x y ) × (x y)
2
3
1
2
−4
5
2
1
2
x2 y
⎛ x3 y3 ⎞
÷ 3 5
(d) ⎜
1 ⎟
2x y
⎝ xy 3 ⎠
Unit 1: Algebra
11
3
Lines, angles and shapes
3.1 Lines and angles
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Exercise 3.1 A
1 -PPLBUUIFDMPDLGBDFPOUIFMFę$BMDVMBUFUIFGPMMPXJOH
10
11
12
1
9
2
3
8
4
7
6
5
(a) ćFTNBMMFTUBOHMFCFUXFFOUIFIBOETPGUIFDMPDLBU
(i) PDMPDL
(ii) IPVST (iii) BN
(b) ćSPVHIIPXNBOZEFHSFFTEPFTUIFIPVSIBOENPWFCFUXFFOQNBOEQN
(c) ćSPVHIIPXNBOZEFHSFFTEPFTUIFNJOVUFIBOEUVSOJO
1
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3 8JMMIBMWJOHBOPCUVTFBOHMFBMXBZTQSPEVDFBOBDVUFBOHMF &YQMBJOZPVSBOTXFS
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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VTFUIFSFMBUJPOTIJQT
CFUXFFOMJOFTBOEBOHMFT
UPDBMDVMBUFUIFWBMVFTPG
VOLOPXOBOHMFT
(c) x°
(f) +x °
Exercise 3.1 B
*OUIJTFYFSDJTF DBMDVMBUF EPOPUNFBTVSFGSPNUIFEJBHSBNT UIFWBMVFTPGUIFMFUUFSFEBOHMFT
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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)
I
(d)
A
K
E
Bx
F
108°
C 65°
G
y
D
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