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
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Information on this title: www.cambridge.org/9781107672727
© Cambridge University Press 2013
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permission of Cambridge University Press.
First published 2013
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ISBN-13 978-1-107-67272-7 Paperback
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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
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
139
139
140
141
141
144
145
146
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
149
149
150
151
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
a
b
Tip
a
b
b≠
Exercise 1.1
1
Number
Natural
Integer
Prime
Fraction
−57
03
− 14
2
7
3
512
2
(a)
(b)
(c)
(d)
>
1
3
<
Unit 1: Number
1
1 Reviewing number concepts
1.2 Multiples and factors
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
1
(a)
(e)
(b)
(f)
(c)
(g)
(d)
(h)
(a)
(e)
(b)
(f)
(c)
(g)
(d)
(h)
2
Exercise 1.2 B
1
2
3
4
×
5
(a)
(b)
2
Unit 1: Number
×
1 Reviewing number concepts
1.3 Prime numbers
not
Exercise 1.3
You can use a tree diagram or
division to find the prime factors
of a composite whole number.
1
(a)
(b)
(c)
2
(a)
(e)
(b)
(f)
(c)
(g)
(d)
(h)
(a)
(e)
(b)
(f)
(c)
(g)
(d)
(h)
3
1.4 Powers and roots
n×n
n
n
n×n×n
n3
3
FAST FORWARD
Powers greater than 3 are dealt
with in chapter 2. See topic 2.5
indices. !
n
Exercise 1.4
1
2
(a)
9
36
4
(e)
(i)
(m)
3
16
3
(b)
(f)
( 25 )
2
(g)
27 − 1
(j)
100 ÷ 4
×
(n)
1
4
3
(c)
9 16
2
(o)
(d)
64 + 36
9
(h)
169 − 144
(l)
16 × 3 27
16
(k)
( 13 )
+
64 + 36
1+
3
9
16
1 − 3 −125
3
(a)
(b)
Unit 1: Number
3
1 Reviewing number concepts
1.5 Working with directed numbers
−
+
Exercise 1.5
1
2
(a)
Draw a number line to help you.
(b)
−
(c) −
−
− −
3
(a)
(b)
(c)
(d)
(e)
−
−
−
−
1.6 Order of operations
×
Remember the order of operations
using BODMAS:
Exercise 1.6
Brackets
Of
Divide
Multiply
Add
Subtract
Tip
FAST FORWARD
The next section will remind you
of the rules for rounding
numbers. !
4
Unit 1: Number
1
(a)
+ ×
(d)
+
(b)
×
(e)
+
×
(c)
×
−
(f)
× − ÷
÷
+
1 Reviewing number concepts
(g) 1.453 +
5.34 3.315
4 03
11.5
(k)
2 9 − 1.43
12.6 1 98
−
(n)
8 3 4 62
76
32
(h)
5 27
1.4 × 1.35
89
(m) 8 9 −
10.4
(j)
(p)
−
(s)
4 072
8.2 − 4.09
(q)
3
(t) 6 8 +
( )
(v) 6 1 + 2.1
28 16
2
−
1. 4 1. 2
−
69 93
(w) 6.4 (1.2
.22 + 1 92 )2
(i)
(l)
(o)
(r)
6 54
− 1 08
23
0.23 4.26
1.32 3.43
(
−
)
2
16.8
− 1 01
93
(
(u) 4.33 + 1.2 + 1 6
5
(x)
)
2
(4 8 − 916 ) × 4 3
1.7 Rounding numbers
≥
≤
Exercise 1.7
FAST FORWARD
1
(i)
(ii)
(iii)
Rounding is very useful when you
have to estimate an answer. You
will deal with this in more detail in
chapter 5. !
(a)
(d)
(g)
(b)
(e)
(h)
(c)
(f)
(a)
(b)
(c)
(d)
(a)
(e)
(b)
(f)
(c)
(g)
(d)
(h)
2
3
Unit 1: Number
5
1 Reviewing number concepts
Mixed exercise
1
−
3
−
4
3
1
2
2
(a)
(b)
(c)
(e)
3
(a)
(b)
(c)
4
5
(a)
(b)
(c) 30 + 10
30
(d)
6
(a)
100 ÷ 4
(b)
100 ÷ 4
(c)
( 64 )
3
3
(d)
7
(a) 5.4 × 12.2
4. 1
2
(b) 12.2
3 92
(d) 3.8 × 12.6
4 35
(e)
(a)
(b)
2. 8 × 4. 2 2
3.32 × 6.22
12.65
(c) 2 04 + 1.7 × 4.3
(
(f) 2.55 − 3.1 +
05
5
)
2
8
9
(a)
(b)
(c)
6
Unit 1: Number
(c)
(d)
3
+
2
Making sense of algebra
2.1 Using letters to represent unknown values
x
y
+
−
Exercise 2.1
x,
1
Tip
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
x
x
1
2
p
2
(a)
(b)
(c)
H
x
3
(a)
(b)
Unit 1: Algebra
7
2 Making sense of algebra
2.2 Substitution
x=−
x
× −
=−
Exercise 2.2
1
A = bh
A
1
(a)
(b)
(c)
(d)
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.
b
2
h
2
xy
3
a
x
x
b
m
4
y
y
c
n
a
b
c
m3
n3
m2
mn
n
x
5
1
2
x
1
2
x
(a)
(b)
2.3 Simplifying expressions
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
(a) 3x 2 + 6 x 8
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
3 (b) x 2 y + 3x
ac + 3ba
3x 2 y 2 yx (c) 2ab − 44ac
(d) x + 2xx 4 + 3x
3x 2
y 3x − 1
(g) −2 xy × −3 y
(h) −2 xy × 2 x
2
(k) 33abc
11ca
y 2y
(o)
×
x x
x 2
(s)
×
4 3y
2
(l) 45mn
20n
xy y
(p)
×
2 x
3
x
9x
(t)
×
5
2
2
(e) −6m × 5n
(f) 3xy 2 x
(i) 12ab ÷ 3a
(j) 12 x ÷ 48 xy
80 xy 2
(m)
12 x 2 y
(q) 5a × 3a
4
3
(n) −36 x
−12 xy
−2 y
(r) 7 ×
5
2 Making sense of algebra
2.4 Working with brackets
a b + c = ab + ac
Exercise 2.4
Remember the rules for multiplying
integers:
1
(a)
(e)
(h)
+×+=+
−×−=+
+×−=−
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.
(b) ( y 3) x
x)
(f) 2 ( x 1) (1 x )
− x ( − ) + 2( + ) − 4
2x ( x 2)
(x
)(
)(
(c) ( x ) 3x
(g) x ( x 2 − 2x
2 x 1)
(d) −2 x − ( − )
2
(a) 2 x ( 12 x + 14 )
(b) −3
(d) ( x + y ) − ( 12 x
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
x×x
x
1
index laws:
x m x n = x m+ n
xm xn = xm n
( )n x mn
x0 = 1
x −m =
m
y × y × y × y.
4
Exercise 2.5 A
Tip
xn
y
(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
(g)
10 x 5 y 2 3x 3 y
÷
9 x 6 y 6 5x 7 y 4
(h)
7 y 3 x 2 5x 6 y 2
÷
5 y5 x 4 7x5 y3
(i)
(j)
(2 x y ) × ( x y )
( y x ) 3(x y )
1
x
m
(x ) = ( x )
1
n
m
n
m
4
3
2
2
3
3
4
4
2
2
2
3
 2  5
(k)  x 4  ×  x 2 
y  y 
2
(l)
(x y ) × (x y )
(x y )
(5x y ) ÷  2xy
2
5
3
3
3
7
2
4x y
6
3
3
4
2
3
3

 5x 2 y 4 
Unit 1: Algebra
2
9
2 Making sense of algebra
2
Tip
(a)
x−4
x −m =
1
xm
5x −4 =
y
= x2 y
x −2
x5 y 4
x −33 y 2
5
x4
(x )
÷
−1
 
(d)  x3 
y 
2
4
−10
y2
(e) x 2 ÷  3 
−
x


(y )
y −3
(2 x y )
(y x )
−33
x −4 y 3 x 7 y −5
×
x 2 y −1 x −4 y 3
(b)
(c)
2
1
1
1
(e)
(64 x )
(i)
( x ) × xx
(a)
(x ) × xx
(d)
(x y )
6
1
2
8
1
2
2
3
(b) x 3 × x 5
(c)
(f)
(8 x y )
(g)
(j)
(x y ) × (x
(b)
(x ) × xx
(e)
 x2 y3  2
1 ÷ 
3 4
x2  x y 
(b)
−24
(−2)4
9
6
3
1
3
1
3
8
y
10
)
1
2
(
2
1
1
2
2
)
( )
2
x −2 y 6
 x4 y 1 
(f)  5 3  ×
 x y  2 xy 3 −2
−4
3
(a) x 4 × x 4
3
1
x2
1
x3
(d)
( )
1
1
2
×
1
3
1
3
 x6 
(h)  2 
y 
xy 8
(k) xy 3
(x )
1
x2 y2
xy 4
1
2
4
Tip
1
2
2
3
2
3
2
3
1
3
4
1
3
5
5
2
1
2
x3 y3
×
xy 2
(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
(a) (
4
(e) 256
(i)
)2
− 14
( )
8
27
)(
− 13
(f) 125
− 43
( )
− 12
(j)
8
18
(c)
63
( 3)4
(d) 8 3
(g)
()
(h)
5
1 −2
4
1
()
2
1 −3
8
2
(a)
(b)
(c)
(d)
3
3
4
x
3
(a)
(e)
10
Unit 1: Algebra
6
=
x
=
x
(b) 3x =
x
(f)
=
1
25
(c) 3x = 811
(g) 3 x − =
(d)
(h)
x
x+
=
2 Making sense of algebra
Mixed exercise
x
1
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
x
2
(a) x =
x
(b) x = −
(c) x =
a=− b=
b (c − a )
(b)
b a
3
−2a + 3b
(a)
2ab
3 2 ( − 1)
(d)
c − a (b − 1)
1
3
c=
a b2
c a2
(c)
(e) a 3b2 − 2a 2 + a 4b2 ac 3
4
(a) 3a 4b + 6a 3b
(b) x 2 + 4x
4x x 2
(d) 2 (
(e) 16 x 2 y ÷ 4 y 2 x
3) − ( x 4 ) 2 x 2
(c) −2a 2b(2a 2 − 3b2 )
10 x 2 5xy
(f)
2x
5
(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
(b) 5x 2 × 3x7
x
(d) (2 xy 2 )
 4x3 
(e)  5 
 y 
4
(2xy ) (3x y )
(h)
(x y )
2 ( xy )
(a)
(125x y )
(b)
(x y )
−2
2
3
(x )
(x )
(f)
(x y )
×
(c)
(x
) × (x y)
3
(g)
3
(c)
3
5
7
(a) 15x 2
18 x
−
4
−3
 x −3 y 3 
÷  2 −1 
x y 
2
4
8
2
3
(x y )
(xy )
2
4
2
3
3
2
7
3
2
1
3
( )
1
4
6
3
1
2
1
×
1
x2 y2
x3 y2
2
y
3
1
2
−4
5
2
1
2
x2 y
 x3 y3 
÷ 3 5
(d) 
1 
3
x y
2
 xy 
Unit 1: Algebra
11
3
Lines, angles and shapes
3.1 Lines and angles
<
°
°
>
>
°
°
<
<
°
°.
°
°
°.
°.
°.
°,
Exercise 3.1 A
1
10
11
12
1
9
(a)
(i)
2
3
8
7
(i) 2
4
6
(ii)
(iii)
(b)
(c)
5
1
4
(ii)
(d)
°
2
3
4
(a)
12
Unit 1: Shape, space and measures
°
(b) x°
(c)
x°
3 Lines, angles and shapes
5
(a)
(d)
Tip
(b)
(e)
°
x°
(c) x°
(f)
+x °
°
x°
Exercise 3.1 B
MN
1
PQ
a.
Q
M
Remember, give reasons
for statements. Use these
abbreviations:
a
O
N
37°
P
Supp ∠s
∠s on line
ˆ = 85°
QOR
R
Comp ∠s
x
2
(a)
∠s round point
(b)
A
Vertically opposite ∠s
E
27°
D
3x + 25°
O
x 112°
x
6x
C
5x + 35°
P
B
Exercise 3.1 C
x
1
Remember, give reasons
for statements. Use these
abbreviations to refer to types of
angles:
Alt ∠s
M
x
Corr ∠s
N
O
y
P
R
M
E
y
B
57°
72°
S
95°
Q
Co-int ∠s
y
(b)
(a)
P
T
C x
F
81° G
N
Q
H
(c)
A
K
Bx
I
C 65°
D
(d)
E
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
x
2
(a)
(b)
E
A
F
112°
B
M
x
G
C
S
N
57°
Q
T
D
H
(c) A
30°
(d) A
B
C
x
C
D
D
E 60°
(e)
F
A
x
C
B
14
Unit 1: Shape, space and measures
D
50°
G
45°
E
F
B
E
126°
108°
x
F
3 Lines, angles and shapes
3.2 Triangles
°
°.
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
(a)
29°
48°
(b)
(c)
113°
37°
31°
a
b
62°
(d)
c
d
53°
(e)
e
(f)
32°
x
61°
(g)
21°
54°
(h)
(i)
x
x
y
x
30°
102°
57°
y
Unit 1: Shape, space and measures
15
3 Lines, angles and shapes
x
2
(a)
(b)
A
A
x
B
B 2x
C
C
D
(c)
(d)
M
O
M
O
N x
x N
18°
P
P
Tip
3
ABC ∠A
∠B
x
∠C
B
C
x.
3.3 Quadrilaterals
°
°
°
°
°.
16
Unit 1: Shape, space and measures
3 Lines, angles and shapes
Exercise 3.3
1
REWIND
The angle relationships for parallel
lines will apply when aquadrilateral
has parallel sides. !
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
°
2
a
f
e
d
g
f
b
63° a
b
c
ed
c
3
(a)
A 79°
(b) A
B
(c)
B
32°
38°
x
D
D
D 58°
(d)
M
B
C
E
C
N
x
A
O
(e)
(f)
3x
F
C
N
M
c
d
68°
O
x
Q 65°
R e
2x
P
PQRS
4
(a) ∠R
5
∠P
∠S
∠Q
(b) ∠R
PMNO
PN
(a) ∠MNP
(b) ∠MNO
∠R
Q
a
b P
∠Q.
(c) ∠Q.
MO
Q ∠QMN
∠PNO
(c) ∠PON.
Unit 1: Shape, space and measures
17
3 Lines, angles and shapes
3.4 Polygons
n−
×
°
n
°
Exercise 3.4
1
Tip
(i)
(ii)
(a)
(b)
(c)
°
2
°
3
°.
4
(a)
(b)
°
5
(a)
(b)
120° 125°
C
50
°
98°
100°
x – 10°
B
H
56° 100° y
17°
D
x–
x
130°
84°
x
161°
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
Exercise 3.5
O
1
(a)
(b)
DE
(c) MON
(d)
(e)
M
MP.
3.6 Construction
A
How to bisect an angle.
Tip
B
How to draw the perpendicular bisector of a line.
Exercise 3.6
1
! =
ABC
2
ΔABC
3
ΔMNO
°
AC =
MN =
NO
CB =
NO =
NO X
AB =
MO =
MO
Y
XY
Unit 1: Shape, space and measures
19
3 Lines, angles and shapes
ΔDEF
4
DE =
(a)
(b)
FE =
DF =
DEF
5
Mixed exercise
(a)
1
(b)
G
A
23°
C
x
E
F
A
x
(c)
B
C
D
71° C
D
x
D
(e)
y
104°
32°
(f)
y
110°
x
30°
40°
68°
B
(d)
(g)
E
A
111°
98°
16°
(h)
y
x
y
110°
x
z
x
2
(a)
(b)
x
(c)
A
x
x
50°
C 70°
120°
E
60°
20
Unit 1: Shape, space and measures
B
50° D
110°
B
3 Lines, angles and shapes
O
3
A
(a)
(i) DO
(b)
(c)
(d)
O
(ii) AB
OB
(iii) AC
AC
B.
B
AB
4
D
AB
C
5
ΔABC
X
AB
XY
BX̂Y.
AB = BC = AC =
AB
Unit 1: Shape, space and measures
21
4
Collecting, organising
and displaying data
4.1 Collecting and classifying data
2 12
Exercise 4.1
Student
1
2
Gender
3
4
5
M
M
M
6
7
8
M
7
7
8
7
Mass (kg)
Eye colour
Hair colour
No. of brothers/
sisters
1
(a)
(b)
(c)
(d)
(e)
22
Unit 1: Data handling
10
M
Height (m)
Shoe size
9
1
4 Collecting, organising and displaying data
4.2 Organising data
Exercise 4.2
In data handling, the word
frequency means the number
of times a score or observation
occurs.
1
Phone calls
Tally
Frequency
1
2
(a)
Number of
mosquitoes
0
1
2
3
4
5
6
Frequency
(b)
Unit 1: Data handling
23
4 Collecting, organising and displaying data
Score
Frequency
3
(a)
(b)
(c)
(d)
(e)
4
Student
1
Gender
2
3
4
5
M
M
M
6
7
8
M
Hair colour
1
(a)
Eye colour
Male
Female
(b)
(c)
4.3 Using charts to display data
24
Unit 1: Data handling
Brown
Blue
Green
10
M
Eye colour
No. of siblings
(brothers/sisters)
9
1
4 Collecting, organising and displaying data
Exercise 4.3
1
Number of students in each year
Year 8
Key
Year 9
Year 10
(a)
(b)
= 30 students (c)
(d)
(e)
(f)
Year 11
(g)
Year 12
Tip
2
City
Tokyo
Seoul
Mexico City New York
Mumbai
Population (millions)
3
(a)
(b)
(c)
(e)
(f)
(g)
(d)
A. Number of students in 10A
20
B. Favourite sport of students in 10A
18
Frequency
16
14
Frequency
Tip
12
10
8
6
4
14
12
10
8
6
4
2
0
Key
boys
girls
soccer
basketball athletics
Sport
2
0
boys girls
Gender
Unit 1: Data handling
25
4 Collecting, organising and displaying data
4
Cereal
Girls
Hot porridge
Bread
8
Boys
(a)
(b)
5
Tip
Traffic passing my home
n
× 100
360
(a)
(b)
(c)
(d)
handcarts
bicycles
taxis
motorcycles
n
cars
trucks
buses
tuk-tuks
6
(a)
(b)
(c)
(d)
7
20
140
10
120
Rainfall (mm)
Temperature (°C)
30
0
–10
–20
–30
–40
Unit 1: Data handling
80
60
40
J
F
(a)
(b)
26
100
M
A
M
J J
Month
A
S
O
N
D
20
J
F
M
A
M
J J
Month
A
S
O
N
D
4 Collecting, organising and displaying data
(c)
(d)
(e)
(f)
(g)
Mixed exercise
1
(a)
(b)
(c)
(d)
(e)
(f)
2
3
Mobile phones and land lines, per 100 people
Number of phones per 100 people
100
90
80
70
(a)
(b)
(c)
(d)
60
50
(e)
40
30
(f)
20
(g)
10
0
Canada USA Germany Denmark UK
Key
Mobile phones
Sweden Italy
Land lines
Unit 1: Data handling
27
4 Collecting, organising and displaying data
4
Student
1
Gender
7
M
M
M
8
M
M
Height (m)
Shoe size
7
7
8
7
Mass (kg)
Eye colour
Hair colour
No. of brothers/sisters
(a)
(b)
(c)
5
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
(a)
(b)
(c)
Value of car
1
1
5
Fractions
5.1 Equivalent fractions
You can cross multiply to make
an equation and then solve it. For
example:
Exercise 5.1
1
1
x
=
2
28
2 x = 28
x = 14
(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
÷
Tip
×
Exercise 5.2
1
(a) 3 5
40
(e) 14
(b) 1 12
22
3
(f) 2
4
(c) 11 24
30
(d) 3
75
100
35
45
2
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
(c) 3
(f)
1 12
1
× ×2
5 19
2
2
1
(g)
(j)
3
of 4
2
(k)
1
3
1
2
1
2
(d) 2
4
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
Tip
3
1
6
+
(e) 2
1
8
1
1
7
(f) 4
(i)
5
7
1
1
3
(j) 1 −
(a)
2
3
8
(b)
9
10
7
12
−
3
10
(c)
3
1
2
3
4
4 1
+
7 3
(g) 1
7
3
1
13
−
1
3
(k) 2 −
4
5
17
3
(d) 2
1
2
(h) 3
9
10
2
(l)
1 −
4
9
13
3
2
3
5×
3
1
3
7
8
4
(a) 8 ÷
(d)
(b) 12 ÷
3
2 18
÷
9 30
(g) 1
REWIND
1
14
26
÷
(e)
10
13
7
8
(c)
8 4
÷
9 5
(h) 3
6
15
7
8
÷ 12
3
7
2
(f) 1
5
2
3
(i) 5
1
5
2
9
1
3
10
5
The order of operations rules
(BODMAS) that were covered in
chapter 1 apply here too. !
2
3
(a) 4 + ×
(d) 2
7
8
1
3
(b) 2
 8 1 − 6 3
 4
8
(e)
5 15
5 1
(g)  ÷  −  × 
 8 4   6 5
1
8
5 1
×
6 4
2 1 − 7
 5 8
5
8
+ ×
(c)
3 2
× +6
7 3
3
5
1
(f)  5 ÷ −  ×
 11 12  6
1
3
2
3
3
(h)  2
4−  ×
 3
10  17
(i)
2 1 2

7÷ − ×

9 3 3
7
12
6
(a)
(b)
3
4
7
(a)
4
1
2
(b)
5.3 Percentages
Exercise 5.3 A
1
(a)
(e)
30
Unit 2: Number
1
6
(b)
(f)
5
8
(c)
(g)
93
312
(d)
(h)
2
7
5 Fractions
2
(a)
(b)
(c)
(d)
(e)
3
Tip
(a)
(e)
(b)
(f)
(c)
(d)
(a)
(e)
(i)
(b)
(f)
(j)
(c)
(g)
(d)
(h)
4
5
Original amount
New amount
Percentage increase or
decrease
(a)
(b)
(c)
(d)
(e)
(f)
(g)
6
(a)
(c)
(e)
(b)
(d)
(f)
(a)
(c)
(e)
(b)
(d)
(f)
7
Exercise 5.3 B
1
2
3
3
1
2
4
Unit 2: Number
31
5 Fractions
5
6
7
8
Tip
Exercise 5.3 C
1
2
(a)
(b)
3
4
(a)
(b)
(c)
5.4 Standard form
a×
looks
Tip
Exercise 5.4 A
1
(a)
(e)
(i)
32
Unit 2: Number
(b)
(f)
(j)
(c)
(g)
(k)
(d)
(h)
(l)
k
5 Fractions
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. !
2
(a)
(e)
(i)
×
×
×
(b)
(f)
−
(c)
(g)
×
×
−
(d)
(h)
×
×
−
×
−
×
−
Exercise 5.4 B
1
(a)
(b)
(d)
9500
0.00054
(g)
(h)
(c)
÷
(f) 4525 × 8760
0.00002
÷
(e)
×
×
5 25 108
(i)
3
9 1 × 10 −8
2
(a)
(d)
(g)
×
×
×
×
−
(b)
(e)
(h)
×
×
÷
×
×
−
×
×
×
−
(c)
(f)
(i)
×
×
÷
3
9 1 × 10 −8
×
×
S
3
× ×
×
×
÷ × −
×
(a)
(b)
×
4
×
×
(a)
(b)
5.5 Estimation
Exercise 5.5
Remember, the symbol ≈ means
‘is approximately equal to’.
1
(a)
(c)
×
×
(a)
+
(b)
(d)
≈
≈
×
≈
÷
≈
2
(c) 9 3 × 7.6
5 9 × 0 95
+
(b)
+
÷
(d) 8 9 × 8 98
2
Unit 2: Number
33
5 Fractions
Mixed exercise
1
(a)
(b)
×
(c) 36.4 6.32
9.987
÷
2
(a)
160
200
(a)
4
9
48
72
(b)
(c)
(d)
64.25 × 3.0982
36
54
3
(f) 2
×
1
3
3
8
(b) 84 ×
9
1
2
3
4
3
(g)  4 
 4
2
(c)
5 1
÷
9 3
(h) 9 1 1 7
5
9
4
(a)
(b)
5
6
(a)
(b)
7
×
8
(a)
(b)
×
S
9
(a)
(b)
(c)
34
Unit 2: Number
(d)
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
6
Equations and transforming
formulae
6.1 Further expansions of brackets
Remember:
+×−=−
−×+=−
Exercise 6.1
1
(a) − x + y
(d) x − y
(g)
− a
(j) − x +
+×+=+
−+−=+
(b) − a − b
(e) − x x + y
(h) − x + y
(k) x x − y
(c) − − x + y)
(f) − x −
(i) x − x −
(l) − x x − y)
2
(a)
(b) − x y −
x−y + x −
(d) − x ( −
) − 2y(
(g) − x (
)+2−(
1
2
1
4
−
+
2
)
(e)
− 3)
xy −
− x + xy
+y −
−x
(h) − x − y + x x − y
(c) − x + y − x y −
(f)
x
(i)
1
−
2
− y −y − x )
(
−
)+3−(
+ 7)
6.2 Solving linear equations
Exercise 6.2
In this exercise leave answers as
fractions rather than decimals,
where necessary.
x
1
(a)
(e)
(i)
x+ = x+
x− =x+
x+ = x−
(b)
(f)
(j)
x+ =x+
x− = −x
x− = x−
(c)
(g)
(k)
x− = x+
− x= x+
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.
x
2
(a)
x− =
(e) − x − = −
(i) x + = x −
(l)
x− − x−
(b) x + =
(f)
− x =
(j) x − = x +
= x−
(c)
(g)
(k)
x+
=
x+ =x
x+ − x−
(d)
(h)
x− =
x+ = x
=
x
3
(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
(d) 28 − x = 12
6
5 2x
(h)
= −1
4
(c) 4 x = 16
6
5 2
= −1
(g)
3
(k)
10 x + 2
=6−x
3
(l)
x x
− =3
2 5
x − xy
ab − bc
( + 4)
= x+7
2
6.3 Factorising algebraic expressions
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
(a) x
(d) a
ab
(g) xy
xyz
(j) x y z
xy z
(b)
(e)
(h)
(k)
xy
pq
ab
(c)
(f)
(i)
(l)
a
ab
abc
xy
(a)
x+
(e) ab + a
(i) xy − yz
(b)
(f)
(j)
+ y
x− y
x − xy
(c)
(g)
a−
xyz − xz
(d)
(h)
(a) x + x
(e) ab + b
(i) abc − a b c
(b)
(f)
(j)
a−a
xy − x y
x − xy
(c)
(g)
(k)
x + x
x− x
ab − b
(d)
(h)
(l)
x
yz
pq
ab
b
ab
ab
xy
2
3
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.
4
(a) x + y +
+y
(c)
a+ b − a a+ b
(e) x − y + − y
(g)
+y −x y+
(i) x x − − x −
(k) x x + + y + x
36
Unit 2: Algebra
(b) x y − + y −
(d) a a − b − a − b)
(f) x x − + x −
(h) a b − c − c − b)
(j) x x − y − x − y)
(l)
x − y − x x − y)
x− x
x y − xy
a b − ab
6 Equations and transforming formulae
6.4 Transformation of a formula
A = bh.
Exercise 6.4 A
Tip
1
m
D = km
2
c
y = mx + c
3
P = ab − c
b
4
a = bx + c
b
a
5
(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)
(d) ab + c = d
c ba
(h)
=e
d
ab
+ de = f
c
(l) c +
ab
=e
d
Exercise 6.4 B
P= l+b
1
P
l
b
(a)
(b)
Tip
b
b
C=
2
(a)
(b)
(c)
Tip
r
r
r
=
=
A=
3
a
h (a b)
2
b
h
b
a=
h=
T
4
P
B
T
P
B
(a)
(i)
(ii)
Unit 2: Algebra
37
6 Equations and transforming formulae
(b)
(c)
(d)
B
m
5
t
(a)
(b)
Mixed exercise
m
t
t
x
1
(a)
x− =−
(e)
x− =
− x
(b)
x+ =−
(f)
x− =
(c)
x+
x
2
(a) m = nxp − r
(b) m =
2x 4
=2
7
(d) 5 =
(g) 3x 7 = 1 4 x
4
8
1 4x
5
(h) 3(2 x − 5) = x + 1
5
2
nx + p
q
3
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
(b) − x x −
(e)
x− + x+
(h) x x + − x x −
(c) − x − y +
(f) x x − + x −
(a) x −
(d) xy − x
(g) x + x −
(b)
(e)
(h)
(c) − x −
(f)
x−y +x x−y
4
x+
=
5
(a)
(b)
x−7
x− y
x y + xy
x x + y − x x + y)
×
(c)
2x
(d)
19x
4
x+9
REWIND
6
Use geometric properties to make
the equations (see chapter 3). !
2x + 8
12x – 45
180 – 3x
38
Unit 2: Algebra
4x – 10
x
4x + 15
6x – 45
7
Perimeter, area and volume
7.1 Perimeter and area in two dimensions
C = πd
C = 2πr
bh
A=
2
A = s2
A = bh
A = bh
A = bh
1
A = (produc f diagonals)
2
A=
(
fp
A = πr2
2
ll l d )h
Exercise 7.1 A
1
(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
m
m
92
7.2 cm
69 mm
Unit 2: Shape, space and measures
39
7 Perimeter, area and volume
π=
2
Tip
(a)
(b)
(c)
7 cm
5m
π
(e)
π
(f)
(d)
21 mm
4.5 m
4m
(g)
3m
8 mm
60°
8 cm
3
4
5
Tip
6
7
(a)
(b)
Remember, give your answer in
square units.
Exercise 7.1 B
1
(a)
12.5 cm
17 cm
(b)
1.7 m
(c)
90 cm
14 cm 19 cm
(d)
21 cm
25 cm
15 cm
19 cm
20 cm
35 cm
11 cm
21 cm
12 cm
5 cm
40
(g)
112 mm
72 mm
13 cm
Unit 2: Shape, space and measures
41 mm
67 mm
(h)
(i)
8 cm
15 cm
10 cm
(f)
12 cm
7 cm
cm
10
(e)
11 cm
7 cm
17 cm
7 Perimeter, area and volume
π=
2
(a)
(b)
(c)
(d)
(e)
300°
100 mm
20° 10 cm
140 mm
8 cm
14 mm
3
Tip
(a)
(b) 2
6
6
(c)
6
3
(d)
22
13
2
12
9
18
20
6
7
8
35
20
5
21
5
12
(f)
9
1
13
6
14
14
36
(g)
(h)
14
25
3
45
30
30
90
90
40
2
95
15
Tip
(i)
20
4
(f)
π=
(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
6.5 cm 3 cm
5.5 cm
cm
6.3
5
6
(f)
15 cm
32.4 cm
40 mm
(e)
40 cm
50 mm
30 mm
8 cm
20 cm
×
×
2
Unit 2: Shape, space and measures
41
7 Perimeter, area and volume
Exercise 7.1 C
AB
1
(a)
(b)
A
(c)
A
O
21 mm
O
120°
O
10 cm
B
40°
A
B
X
2
12 mm
B
Y
XY
637
1k
60° m
Y
X
3
(a)
(b)
12 cm
60°
(c)
20 cm
18 mm
150°
4
2
42
Unit 2: Shape, space and measures
7 Perimeter, area and volume
7.2 Three-dimensional objects
Exercise 7.2
1
(a)
(b)
×2
×6
(c)
×2
×2
(d)
×4
×1
×8
2
(a)
(b)
(c)
3
(a)
(b)
(c)
(d)
Unit 2: Shape, space and measures
43
7 Perimeter, area and volume
7.3 Surface areas and volumes of solids
h
1
V = πr 2 h
πrl
3
V=l×b×h
l
V = al
a
b
l
h
l
area of base × h
V=
3
h
4
V = πr 3
3
πr2
Exercise 7.3 A
π=
Tip
1
(a)
(b)
0.5 mm
18.4 m
1.2 mm
0.4 mm
12 m
14 m
(c)
(d)
8m
4m
12 mm
1.5 cm
2
2
(a)
(b)
Remember,
1 m2 = 10 000 cm2
44
3
×
(a)
(b)
(c)
Unit 2: Shape, space and measures
×
2
7 Perimeter, area and volume
4
Tip
(a)
(b)
45 mm
4 cm
(c)
(d)
65 mm
3 cm
80 mm
50 mm
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
6 (a)
(b)
3
7
8
3
Volume (mm3)
64 000
64 000
64 000
64 000
Length (mm)
Breadth (mm)
Height (mm)
Exercise 7.3 B
1
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
40 mm
12 cm
Unit 2: Shape, space and measures
45
7 Perimeter, area and volume
2
(a)
(i)
(ii)
(b)
Mixed exercise
1
(a)
(b)
2
2
3
170 mm
40 mm
(c)
120 mm
150 mm
2 cm
(b)
50 mm
192 mm
5 cm
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
1 cm
7 cm
5 cm
2 cm
3 cm
6 cm
3 cm
5 cm
7 Perimeter, area and volume
4 MNOP
NO
2
M
12 m
N
10 m
P
O
5
20 mm
A
40 mm
120 mm
B
20 mm
15 mm
(a)
(b)
(c)
(d)
6
7
8
Unit 2: Shape, space and measures
47
8
Introduction to probability
8.1 Basic probability
1
2
Exercise 8.1
1
Red
//// //// ////
White
//// //// //// ///
Green
//// //// //// //
(a)
(b)
(c)
(d)
2
Result
Frequency
44
48
Unit 2: Data handling
8 Introduction to probability
(a)
(b)
(i)
(ii)
(iii)
8.2 Theoretical probability
P(outcome) =
number of favourable outcomes
number of possible outcomes
P(H) =
Tip
1
2
all
Exercise 8.2
1
(a)
(b)
(i)
(iii)
(v)
(vii)
(ix)
(ii)
(iv)
(vi)
(viii)
<
<
<
<
>
2
(a)
(b)
3
(a)
(b)
(c)
4
(a)
(b) W
(c)
Unit 2: Data handling
49
8 Introduction to probability
5
(a)
(d)
(b)
(e)
<
(c)
6
8.3 The probability that an event does not happen
′
′ =
A
Exercise 8.3
1
3
8
2
3
Flavour
Strawberry
Lime
Computers
Sewing
Lemon
Blackberry
Apple
P(flavour)
(a)
(b)
(c)
(d)
4
Club
P(Club)
(a)
(b)
(c)
(d)
50
Unit 2: Data handling
Woodwork
Choir
Chess
8 Introduction to probability
8.4 Possibility diagrams
Tip
Exercise 8.4
1
(a)
(b)
2
FAST FORWARD
Tree diagrams are also probability
space diagrams. These are dealt
with in detail in chapter 24. !
(a)
(b)
(c)
(d)
3
Drinks:
Snacks:
(a)
(b)
(c)
Unit 2: Data handling
51
8 Introduction to probability
8.5 Combining independent and mutually exclusive events
–
are
–
are
–
Exercise 8.5
1
(a)
(b)
(c)
(d)
2
(a)
(b)
(c)
(d)
(e)
(f)
(g)
3
(a)
(b)
(c)
(d)
(e)
52
Unit 2: Data handling
8 Introduction to probability
Mixed exercise
1
(a)
(b)
(c)
(d)
2
(a)
(b)
(c)
(d)
(e)
(f)
(g)
3
(a)
(b)
(c)
(d)
4
(a)
(b)
(c)
(d)
5
6
Unit 2: Data handling
53
9
Sequences and sets
9.1 Sequences
1
10
n
You should recognise these
sequences of numbers:
square numbers: 1, 4, 9, 16 . . .
= a+ n
a
(a)
(e)
(b)
(f)
(c)
(g)
1 1
4 2
2
(a)
(b)
(c)
(d)
(e)
1
2
3
(a)
(b)
(c)
(d)
(e)
(f)
= n+
=n
= n
n
=n
n
=n
n
n
=
n
n
n
n
4
(a)
(b)
(c)
(d)
Unit 3: Number
n
1
triangular numbers: 1, 3, 6, 10 . . .
54
n
n
Exercise 9.1
cube numbers: 1, 8, 27, 64 . . .
Fibonacci numbers: 1, 1, 2, 3,
5, 8 . . .
d
d
n
(d)
(h)
9 Sequences and sets
5
(a)
(b) −
(c)
(d)
(e)
6
n=1
n=2
(a)
(b)
(c)
(d)
n=3
n
9.2 Rational and irrational numbers
a
b
b
b
a
b
a
b
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
a
Exercise 9.2
1
(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
(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
∈
∅
ℰ
⊂
∪
A A
A
x x
x
Exercise 9.3 A
Tip
1
(a) ∈
(b) ∈
(c)
(d)
⊂
(e)
(f)
(g) A
(h) ℰ
⊂
∩
∪
B
A
B
A
A
2 A
(a)
(b)
(c)
(d)
(e)
(f)
Sometimes listing the elements
of each set will make it easier to
answer the questions.
Unit 3: Number
B
C
B∩C
C
A
A
A
3 ℰ
C
(a)
(b)
(c)
(d)
(e)
(f)
56
A
A
A∩B
B∪C
A ∩B
B ∩ C)΄
A ∩ B΄
A∪B∪C
B
9 Sequences and sets
4
(a) x x ∈
(b) x x ∈
x
x
5
(a)
(b)
Exercise 9.3 B
Tip
1
ℰ
P
C
2
(a)
(b)
(c)
(d)
(e)
(f)
REWIND
C)
P΄)
C∩P
P∪C
P ∪ C)΄
P⊂C
3
Exam questions often combine
probability with Venn diagrams.
Revise chapter 8 if you’ve
forgotten how to work this out. !
(a)
(b)
(c)
Mixed exercise
n
1
(a)
(b)
(c)
2
n
n
(a)
(b)
(c)
3
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
(a)
! 3!
! !
(b) 0.286
5
B
C
(a)
(b)
(c)
(d)
n
(i)
(ii)
(iii)
(iv)
(v)
58
Unit 3: Number
∩
B = biology
C = chemistry
10
Straight lines and quadratic
equations
10.1 Straight lines
x y
x
y
x
m=
x
y
y
y
change in y
change in x
x
y
m
c
x+c
y
y
x
y
y
m
y
c
x
x
y
y
Exercise 10.1
Tip
x
1
−
−
x
(a) y = x + 5
(b) y
(e) x = 4
(i) 0 x 2 y − 1
(f) y = −2
1
(j) x + y = − 2
a
2
(c) y
7 2x
(g) y
2x −
(d) y
1
2
x −2
(h) 4 2x
2 − 5y
e
f
j
a
3
Remember, parallel lines have the
same gradient.
2x − 1
−
4
(a) y
3x + 3
(c) y
3x
(e) 2 y
(g) y = 8
y = x+3
(b) y
1
x−4
2
3x + 7
(d) y
0 8x − 7
y
3x + 2
y = −9
3
2
y = x+2
1
x −8
2
y
y
(f) 2 y 3x = 2
y
(h) x = −3
1
2
x=
8x + 2
1 5x + 2
Unit 3: Algebra
59
10 Straight lines and quadratic equations
5
(a)
(b)
y
6
(4, 6)
–1
4
1
4
4
–1
–2
0
–1
(g)
y
1
x
–4
2
(1, –2)
0 1 2 3 4 5 6 7
(h)
y
–3
0
3
y
x
–8
x
m
y
(a)
3x − 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
x
4
3
2
1
–3
6
y = mx + c
x
2
3
0
Tip
4
6
7
6
5
4
3
2
1
2
2
6
–2
(f)
y
(–2, 1)
x
y
0
–2
3
(3, –3)
4
–2
2
–3
0
(e)
0
(d)
y
x
–2
2
–4
(c)
y
c
(c) y
1
x+5
2
(g) y = 7
(k) x 4
y
–4
–1
–2
–3
0
(d) y
4
x
(h) y
3x
(l) 2 x
5 y
7
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
(d)
(a)
(c)
3
2
1
0
–3 –2 –1
–2
(b)
–3
(a) y
x
(d) x + y
2
60
Unit 3: Algebra
3
2
1
x
0
–3 –2 –1
–2
(e)
–3
(f)
1 2 3
x
8
y
(h)
y
(b) y
(e)
x
x
1 2 3
(g)
1x
2
y
(c)
y
x
8
10 Straight lines and quadratic equations
9
(a)
(b)
(c)
(d)
(e)
(f)
10
10.2 Quadratic expressions
x
x
x
y)
y)
x
x
xy
xy
y
y
x
x
x
x
x
x
x
x
x
x
b
a
x
x
x
x
x
a
b
x
x
x
Exercise 10.2 A
Tip
1
(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 )
(j)
(x
11) ( x + 12)
1
1 
(k)  x + 1  1
x (l)
2
 2 
(x
3)(2 3x )
(i)
2
2
+ 1 2x − 3
(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.
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.
2
(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)
(2x 3 y )2
(3 − x )2
2
3
(a)
(c)
(e)
5))(( x + 5)
(x
(i)
(j)
(
(x
(d) x 2
(4 3x )(3x 4 )
(f)
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 2 y 2
(h)
(b) (2 x 5)(2 x 5)
(3 7 y )(7 y 3)
2
4
2
2
3
2
2
y2 x3
)
y2
)
4
2
3
2
)(
)(
y2 x2 + y2
2
2
2
Exercise 10.2 B
1
Tip
(a) x 2 + 4x
4x 4
(c) x 2 + 6x
6x 9
(e) 15 + 8x
8 + x2
(g) x 2 8x
8 x 15
(i) 26 − 27x
27 + x 2
(k) x 2 + 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
(a)
(c)
(e)
(g)
(i)
5x 2 15x 10
3x 3 122 2 9 x
x3
x 2 + 20 x
x 3 5 x 2 14 x
−2 x 2 + 4 x + 48
(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
(a)
(d)
(g)
(j)
(m)
(p)
x2 − 9
49 − x 2
x2 9 y2
2 x 2 18
25 − x16
25x10 − 1
(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
Tip
3
Tip
a
b
a b a b
a b a b
62
a b a b
Unit 3: Algebra
(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.
1
(a)
(c)
(e)
(g)
(i)
(k)
(m)
(o)
(q)
Quadratic equations always have
two roots (although sometimes
the two roots are the same).
Mixed exercise
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
x2 4x 5 = 0
x 2 + 8x
8 x 20
−60 − 17 x = − x 2
x 2 − 20 x + 100 = 0
D
1
(a) y
1
x
2
x
−1
y
1
x+3
2
(b) y
x
−1
y
(c) y = 2
x
−1
y
(d) y 2 x − 4 = 0
x
−1
y
y
2
(a) y
(d) y =
2x − 1
1
−
2
(b) y + 6 x
(c) x
(e) 2 x 3y
3y = 6
(f) y
y = −8
x
3
(a)
(b)
(c)
(d)
y
y
y
x+8
4
x
5
1
2
y
y-
−
2
−
3
−
−
Unit 3: Algebra
63
10 Straight lines and quadratic equations
(e)
(f)
(g)
(h)
2 y 4x
4x + 2 0
x+ y=5
x
y
−
y
− −
4
9
8
7
6
5
4
3
2
1
y
x
–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
5
a
b
y
c
y
d)
y
y
6
4
2
4
2
–6 –3–3
–6
e
3 6
7
x
–4 –2
2 4
f)
y
–5
Tip
6 x
–10 –2
10 20 x
–4
–6
3
3 5 x
(5,–5)
–4 –2
2 4
x
–3
6
x
D= t
(a)
(b)
D
D
y = mx + c
(i)
Unit 3: Algebra
t
D= t
(c)
(d)
(e)
64
3
–4
y
5
–5 –3
–6 –3 –2
x
(ii)
(iii)
t
10 Straight lines and quadratic equations
(f)
(i)
(ii) 2
1
2
(iii)
3
4
(g)
7
(a)
(b)
(i)
(ii)
(iii)
(iv)
(v)
8
(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
2y)
2
3x )(3x 2)
2
)
2
y 1
2)
( x − 7 )2
2
(d) (1 6 y )
2
(f)
(2
5)
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
(a) a 3 4a
(c) x 2 x − 2
3 y ) − 4z 2
x2
(g) x 4 −
4
(i) 4 x 2 4 x − 48
(k) 5 20 x16
(e)
(2 x
2
(b) x 4 − 1
(d) x 2 22xx 1
(f) x 2 + 16 x + 48
(h) x 2
5x 6
14xx 24
(j) 2 x 2 14
(l) 3x 2 15x 18
x
10
(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
a
c2 = a2 + b2
b
c
c2 = a2 + b2
Exercise 11.1 A
Tip
hypotenuse
1
(a)
(b)
(c)
15 cm
x
3 cm
13 mm
y
x
8 cm
5 mm
4 cm
(d)
(e)
x
(f)
y
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
2
Tip
(a)
(b)
70 mm
(c)
6 cm
x
80 mm
y
z
20.2 cm
4 cm
8 cm
120 mm
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
3
(a)
(b)
A
10
(c)
(d) J
H
13
8
7
B
D
15
8
9
9
12
L
C
F
E
12
G
I
11
K
5
4
(a) (−2, −2)
(c) (−4, 5)
(e) (−1, −3)
Tip
(2, 2)
(0, 1)
(2, −3)
(b) (−3, −1)
(0, 2)
(d) (2, 4)
(8, 16)
(f) (−2, 0)
(4, 3)
Exercise 11.1 B
1
2
3
4
Unit 3: Shape, space and measures
67
11 Pythagoras’ theorem and similar shapes
5
11.2 Understanding similar triangles
Tip
Exercise 11.2
1
(a)
(b)
4.47 cm
x
2 cm
4 cm
β
α
β
10 cm
z
10 cm
x
(f)
z
4 cm
8 cm
β
α
43°
x
8.5 cm
α
α
β
α
27 cm
y
α
21 cm
40 cm
43°
y
α
5 cm
9.5 cm
(h)
β
x
y
8 cm
6 cm
8 cm
Unit 3: Shape, space and measures
α
β
y
y
4 mm
y
α
68
8 cm
x
3 mm
8 cm
7 cm
(d)
2 mm
6 mm
8 mm
α
4 mm
5 mm
β
y
(g)
β
9 mm
6 mm
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
Δ ABC
2
Δ ADE
A
B
C
D
E
3
building
3m
30 m
Nancy
4m
11.3 Understanding similar shapes
a b
A
a2 b2
B
2
volume A  a  3
=
volume B  b 
surface area A  a  2
=
surface area B  b 
Exercise 11.3
1
(a)
15
(b)
25
45
x
30
24
y
18
x
30
Unit 3: Shape, space and measures
69
11 Pythagoras’ theorem and similar shapes
2
3
(a)
(b)
18 cm
12 cm
69 mm
area = 113.10 cm2
23 mm
area = 4761 mm2
4
(a)
(b)
x
1 cm
area = 10 cm3
area = 40 cm3
area = 16.2 m3
5
A
6
(a)
(b)
(c)
11.4 Understanding congruence
70
Unit 3: Shape, space and measures
x
3m
B
A
B
area = 405 m3
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
y
O
Z
Y
(e)
(f)
A
B
(g)
Q
P
N
O
D
A
C
Z
87°
R
M
(h)
Y
87°
R
F
P
D
C
y
x
E
Q
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
AB
3
AC
C
B
A
D
E
Mixed exercise
1
(a)
(b)
Δ ABC AB =
2
BC =
AC =
3
(a)
(c)
(e)
(b)
(d)
4
5
(a)
4.5 cm
α
1.5 cm
(b)
β
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°
63°
x
11 Pythagoras’ theorem and similar shapes
6
(a)
(b)
1.5 m
0.75 m
3.0 m
150 m
50 m
1.5 m
7
2
18 cm
4.5 cm
area = 288 cm2
8
A
B
2
B
A
50 mm
80 mm
area = 60 800 mm2
9
2
(a)
(b)
10
(a)
H
D
A
F
C
B
(b)
D
A
G
E
B
y C
x
y
F
G
x E
y
x
I
H
I
(c)
(d)
A
D
C
6
7
5
G
x
6
F
y
B
y
x E
I
5
x
Unit 3: Shape, space and measures
H
7
73
11 Pythagoras’ theorem and similar shapes
x
11
8.6 m
xm
6.5 m
12
35 mm
17 mm
140 mm
105 mm
35 mm
(a)
(b)
74
Unit 3: Shape, space and measures
12
Averages and measures
of spread
12.1 Different types of average
Exercise 12.1
1
(a)
(b)
(c)
(d)
(e)
(f)
Tip
2
(a)
(i)
(iv)
(ii)
(v)
(iii)
(vi)
(b)
3
4
Unit 3: Data handling
75
12 Averages and measures of spread
5
Tip
6
(a)
(b)
7
Previous salary
(a)
(b)
(c)
(d)
(e)
12.2 Making comparisons using averages and ranges
Tip
Exercise 12.2
1
(a)
(b)
(c)
76
Unit 3: Data handling
Salary with increase
12 Averages and measures of spread
Tip
2
(a)
(b)
3
Runner A
Runner B
(a)
(b)
12.3 Calculating averages and ranges for frequency data
fx Mean =
total of (score frequency
f
) column
total of frequency
f
column
Exercise 12.3
1
(a)
(b)
3
(c)
3
3
3
5
3
3
(d)
2
2
2
3
2
2
2
2
3
3
3
3
2
2
2
2
(a)
(b)
Score
2
(c)
3
5
Frequency
Score
Frequency
25
22
23
Score
Frequency
Unit 3: Data handling
77
12 Averages and measures of spread
12.4 Calculating averages and ranges for grouped continuous data
–
–
–
Estimated mean =
sum of (midpoint × frequency)
sum off requencies
Exercise 12.4
m
1
Tip
Marks (m)
Midpoint
Frequency (f)
≤m
m
2
≤m
5
≤m
≤m
≤m
≤m
(a)
(b)
(c)
(d)
Tip
2
Words per minute (w)
≤w
≤w
≤w
≤w
≤w
≤w
(a)
(b)
(c)
(d)
78
Unit 3: Data handling
Frequency
Frequency × midpoint
12 Averages and measures of spread
12.5 Percentiles and quartiles
–
–
–
2
3
3
Exercise 12.5
Tip
1
1
Q
4
Q2
1
2
(
1)
(
1)
3
3
4
(
(a)
55
(b)
3
(c)
1)
(d)
2
3
Mixed exercise
3
2
5
3
5
5
2
1
(a)
(b)
(c)
2
(a)
(b)
(c)
3
Supplier
Mean (weeks)
Range (weeks)
A
B
C
Unit 3: Data handling
79
12 Averages and measures of spread
4
Volume (cm3)
2
3
5
Frequency
(a)
(b)
(c)
5
Age in years (a)
≤a
≤a
≤a
≤a
≤a
≤a
≤a
(a)
(b)
(c)
(d)
6
(a)
(b)
(c)
(d)
7
(a)
(b)
80
Unit 3: Data handling
3
Frequency
13
Understanding measurement
13.1 Understanding units
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
.. 100
.. 1000
.. 10
To change to a larger unit, divide by conversion factor.
Exercise 13.1
Tip
1
(a)
=
=
=
=
=
=
=
3
=
(b)
2
(a)
(c)
(e)
=
=
=
(b)
(d)
(f)
(a)
(c)
(e)
=
=
=
(b)
(d)
(f)
=
=
=
3
=
=
=
Unit 4: Number
81
13 Understanding measurement
4
(a)
(c)
(e)
(b)
(d)
(f)
5
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.
(a)
(c)
(e)
2
(a)
(c)
(e)
3
=
2
=
2
2
=
2
(b)
(d)
(f)
=
(b)
(d)
(f)
3
2
2
=
=
2
2
6
=
3
=
3
=
3
3
3
=
3
3
=
3
3
=
3
7
8
9
2
×
10
3
13.2 Time
15
60
1
4
Tip
=
82
Unit 4: Number
12
60
1
5
= =02
×
13 Understanding measurement
Exercise 13.2
1
(a)
Name
Time in
1
4
Time out
Lunch
Hours worked
3
4
1
2
(b)
2
3
4
(a)
(b)
(c)
(d)
5
Chavez Street
Castro Avenue
Peron Place
Marquez Lane
(a)
(b)
(c)
(d)
(e)
Unit 4: Number
83
13 Understanding measurement
13.3 Upper and lower bounds
dp means decimal places
sf means significant figures
Exercise 13.3
1
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
2
(a)
(b)
3
(a)
(b)
4
(a)
(b)
5
(a)
(b)
84
Unit 4: Number
13 Understanding measurement
13.4 Conversion graphs
x
y
y
Exercise 13.4
Tip
1
Exchange rate
Rupiah (in thousands)
600
500
400
300
200
100
0
0
10 20 30 40 50 60 70 80 90 100
Australian dollars
(a)
(b)
(i)
(ii)
(iii)
(c)
(i)
(ii)
2
Conversion graph,
Celsius to Fahrenheit
250
200
Temperature
in °F
150
100
Tip
50
–20
0
20
40
60
Temperature
in °C
80
100
(a)
(b)
(i)
°
(ii)
°
(iii)
°
Unit 4: Number
85
13 Understanding measurement
(c)
°
(d)
3
Conversion graph, pounds to kilograms
60
40
Kilograms
20
0
40
80
Pounds
120
160
(a)
(b)
(c)
(i)
(ii)
(iii)
13.5 More money
Exercise 13.5
Tip
Currency exchange rates – October 2011
Currency
1 US $
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
1 Euro
1 UK £
US
$
Euro
(€)
UK
£
Indian
rupee
Aus
$
Can
$
SA
rand
NZ
$
Yen
(¥)
13 Understanding measurement
(a)
(i)
(iii)
(v)
(ii)
(iv)
(vi)
(b)
(i)
(ii)
(iii)
(i)
(ii)
(iii)
(c)
Mixed exercise
1
(a)
(d)
(g)
(j)
3
(b)
(e)
(h)
(k)
2
3
(c)
(f)
(i)
(l)
2
3
3
2
3
4
5
(a)
(b)
6
(a)
(b)
(c)
Unit 4: Number
87
13 Understanding measurement
7
Conversion graph for
imperial gallons to litres
(a)
(b)
45
40
30
Litres
(i)
(ii)
(i)
(ii)
(c)
35
(d)
(i)
(ii)
25
20
15
10
5
0
Currency exchange rates – October 2011
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.
8
9
(a)
(b)
(c)
10
88
Unit 4: Number
14
Further solving of equations and
inequalities
14.1 Simultaneous linear equations
x
y
x
y
Exercise 14.1 A
1
x
(a)
(b)
(c)
x 3 y
x y=0
− <x<
x 2 y
x 2y = 3
− <x<
2x + 2
2 y 33xx − 1 0
− <x<
(d)
y
(e)
x=4
x −7
x 33yy = 3
y
2x + 3
−
<x<
− <x<
Unit 4: Algebra
89
14 Further solving of equations and inequalities
2
(a)
(b)
A
(i) A
F
(ii) D
C
(iii) A
F
E
(c)
A
C
10
y
D
B
8
F
6
4
2
x
–10 –8
–6
–4
–2 0
–2
2
4
6
8
10
E
–4
–6
–8
–10
x
3
y
(a) y = 2
x+ y=6
(b) y x = 3
y 3x = 5
(c) x + y = 4
2 x + 3 y = 12
(a) x + y = 5
x y=7
(b) 3
2x
(c) 2 x 33yy = 12
3x 3y
3 y = 30
(d) 2 x 33yy = 6
4 x 6 y = −4
(e) 2 x 5 y = 11
3x 2y
2y = 7
x
4
Remember that you might
need to rearrange one or both
equations before solving.
y
(a)
3x
y
7
3x
y
5
5x
+ y = 10
2
x
1
+ y=3
4
4
2x 9 y
(i) x
= 6y 9
3
Unit 4: Algebra
y 7
y 8
y 1
y 4
(f) y 2 x = 1
2 y 3x = 5
5
(e)
90
(d) 2 x
3x
(b)
2x + y
=3
2
x
+ y=3
2
(c)
(f)
3x y
+ =4
4 2
3y
x+
=7
2
7 x 6 y = 17
(g) 2 x
1
+ y=6
3
3
3x 2y
2y = 4
2x
y
3
x y
+ =2
(d) 3 2
x + 4 y 11
(h)
3x + y 1
=
2
2
5x 7 y
=2
3
14 Further solving of equations and inequalities
6
7
8
Exercise 14.1 B
1
x
(a)
y
(b)
20
(c)
– 3x
6y
2
3x
y
+6
20
2x + 4
2x
9y
2y – 4x
–3y – 6
4x – 2y
(d)
(e)
(f)
2x + 7
x+y
1
2
x
5–
3y
2
3x
5x – 2y
9–y
2x + 4
9 + 3y
2
(a)
d
c
(b)
14.2 Linear inequalities
–
–
Unit 4: Algebra
91
14 Further solving of equations and inequalities
Exercise 14.2
Tip
1
(a) x
(d)
(g)
a
(b) y ≥
(e)
n
(h)
a
(c) f
(f) –m
(i)
a
s
(b)
1
(e) −
(c)
(f)
a
2
(a)
(d)
x
h
e
3
a
2
b
3
(a)
(b) x
x
x
+8
3
r 1
+
(i)
4 8
(e)
(l)
(o)
y+4
1
≥ y−
12
2
3
8
1
3
2x
(f)
x −5
4
(j)
x
(m)
2
9
(3 x
(c)
3
33) ≥
3
5
(g)
(h)
x
(k)
x
6
(d)
x
(n)
4
3
+
g
n
3
14.3 Regions in a plane
Tip
Exercise 14.3
y
y mx
line
c
y
c
mx
mx
1
(a) y
92
x
(c) y > –x
2
(a)
(d)
y
(b) y
x
mx
Unit 4: Algebra
c
(b) x
y
x
y
(e)
(c)
y
y
(f)
x
y
x
y
4
>
3
2
2
3
e
e
g
n
1
x
4
n
14 Further solving of equations and inequalities
3
5
y
5
4
4
3
3
2
2
1
x
–5 –4 –3 –2 –1 0
–1
1
2
3
4
5
1
–2
–3
–3
–4
–4
–5
–5
y
5
4
4
3
3
2
2
1
x
–5 –4 –3 –2 –1 0
–1
1
2
3
4
5
x
–5 –4 –3 –2 –1 0
–1
–2
5
y
1
2
3
4
5
1
2
3
4
5
y
1
–5 –4 –3 –2 –1 0
–1
–2
–2
–3
–3
–4
–4
–5
–5
x
4
x
y
x
y
y
5 (a)
x
y
x
y
x
(b)
x y
14.4 Linear programming
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
y
1
x
y
x
2 (a)
y
y
(b)
x
x
y
y
x
y
x
x
y
x
x
a
y
Unit 4: Algebra
93
14 Further solving of equations and inequalities
3
Tip
4
14.5 Completing the square
2
a  a

x+
−

2  2
x2 + x
2
Exercise 14.5
( x + )2 + b
1
(a)
(d)
(g)
(j)
Tip
(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
2
x
(a) x 2 5x 6 = 0
(b) x 2 x − 6 0
2
(e) x − 16 x + 3 0 (f) x 2 + 7x
7x 1 = 0
(i) x 2 2 x − 100 = 0
x
(c) x 2 4x
4x 3 = 0
(g) x 2 + 9x
9x 1 = 0
3
Tip
ax
94
x 2 + 6x
6x 4
x 2 − 12 x + 30
x 2 + 24 x + 121
x 2 2x
2 x 10
bx
Unit 4: Algebra
c
(a) 2 x 2
(d) 2
3x − 2 0
3
5=
x
(b) x ( x − 4 ) = −3
(e)
( x + 1))(( x − 7 ) = 4
(c) 3x 2
2 (3 x 2 )
(f) x + 2 x 2 = 8
(d) x 2 6 x 7 = 0
(h) x 2 + 11x + 27 = 0
14 Further solving of equations and inequalities
14.6 Quadratic formula
ax
bx
c
−b ± b2 − 4ac
x=
2a
Exercise 14.6
Tip
1
(a)
(d)
(g)
(j)
Tip
x
(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
x 2 + 2x
2x
x 2 + 3x
3x
(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
x 2 + 18 x − 93 = 0
x 2 2 x − 100 = 0
(a)
(b) 6 x 2 11x 35 0
(c) 2 x 2
(e) 4 x 2 133
(f) 8 x 2
x 2 − 14 x + 40 = 0
x 2 8x
8 x 15 0
x 2 + 4x
4 x 12 0
x 2 4 x 12 0
6=0
4=0
8=0
40 0
2
Tip
3
Consecutive numbers are one unit
apart.
2
4x 8 = 0
(d) 2 x 2
5x = 25
(g) 9
2
1 6x
(h) x ( x − 16) + 57 = 0
(j)
2
5x = 3
5x
(k) 6 x
2x
2
9=0
13x 6 = 0
(i) 9 (
(l) 4
2
1
=0
2
4 x = 18
3x +
3x
1) x 2
3=0
4
5
Unit 4: Algebra
95
14 Further solving of equations and inequalities
14.7 Factorising quadratics where the coefficient of x2 is not 1
x
Exercise 14.7
1
Tip
(a) 2
2
(e) 4
2
(i)
−x
x−
3
(b) 9
3
(f) 14 x − 51x
51x + 7
(g) 3x
(j)
(k) 15x 2 16 x − 15
2
(c) 4 x 2 122
6x + 1
6x
2
3x 2 10 x 25
3x 2
77xx − 66
2
9
11xx 20
11
(d) 6 x 2
7x − 5
(h) 6 x 2 11x 7
(l)
8x 2
25x 3
2
(a) 4 x 2 122
(d) 6 x 2
6x
(e) 4
7 xy 5 y 2
(g) 2 ( x 1) 4 x
(j)
(b) 2 x 2 −
9
2
2
4x
(h)
2
( x + 1)
2
(
(c) 50 x 2 + 40x
40 x + 8
2
4
(k) 3 x
1 x − 132
14
14x
1
2
(f) 12 x 2
3
+ 3 ( x + 1) + 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
Exercise 14.8
Tip
1
2x
14
16z
(e)
6
3x
x
34 x
(f)
17 xy
(a)
(b)
(i)
24ab2
36a 2b
(a)
x2 + x − 6
x 2 2x
x
7x
12 x 2
(g)
18 xy
(c)
12 x
30
5a
(h)
25a 2b
(d)
2
(e)
(i)
96
Unit 4: Algebra
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
Tip
3
(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)
(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)
p.a. stands for ‘per annum’, which
means ‘per year’.
66xx 8 x + 3
×
3x 9
2−x
(i)
6
4 9x 2
2
2
4
(h)
Mixed exercise
x2
x2
2
3x
−
3
(i)
x2 + 4x
2
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
x 2 y = 12
1
x
y
x
y
2
x
y
y 4x
4x = 7
2 y 33xx − 4 0
3
4
(a) −3 < + 2
(b) 5 − 2x
2 ≥7
5
−2 < ≤ 3
6
x
y
y > 1 5 y ≤ 4 x ≥ −2 y > x + 1
y ≥ x +1
x
y
x
+y
2
7
(a) x 2 2 xy
x
(d) 2 y 2 133 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
(a) 2a 2 22aa − 6 0
(d) 3x 2 5x
5x + 2 0
(b) 3 2
(e) 5x 2
4=0
3x = −33 2
(c) x 2 + 2x
2 x 15 0
(f) 3x 2 6 x = 6 2 3
5
x
9
(a) 5x 2
8x − 4 0
8x
(b)
p
(b)
16 − 4 x 2
4 8
(c)
1
1
+
2 p2 5 p
(e)
x  2x
z 
÷
×
2 y  yz 2 xy 3 
(f)
3a 5 a a
−
+
2
3
5a
2
− qx + r = 0
10
(a)
98
Unit 4: Algebra
x2
y2
(x + y)
2
(d)
2
7 x 2 15 y
×
5 y 14 x
(g)
4
x2 + 2
(h)
3x
6 x 2 3x
8
÷
x2
4
2
−
x − 2 x2 + 4x
(i)
1
2 x 2 111
5
−
2x 2
1
x 1
15
Scale drawings, bearings
and trigonometry
15.1 Scale drawings
1
50000
Exercise 15.1
REWIND
1 (a)
Revise your metric conversions
from chapter 13. !
(b)
2 (a)
(b)
×
A5 is half A4 and has dimensions
14.8 cm × 21 cm.
3 (a)
×
(b) (i)
(ii)
×
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)
×
(b)
15.2 Bearings
Exercise 15.2
1
(a)
(b)
(c)
X
2
(a) N
(b)
Y
(c)
N
N
Y
70°
45°
Y
X
130°
X
X
Y
3
°
(a)
(b)
(c)
(d)
100
Unit 4: Shape, space and measures
15 Scale drawings, bearings and trigonometry
15.3 Understanding the tangent, cosine and sine ratios
tanθ =
sinθ =
cosθ =
opp( )
adj
d( )
opp( )
hyp
y
adj
d( )
hyp
y
the opposite side
the adjacent side
opp( ) adj(
adj( ) × tanθ
adj
d( )=
the opposite side
the hypotenuse
hyp sinθ
opp
hyp
y =
the adjacent side
the hypotenuse
hypotenuse
opp( )
sinθ
opposite side
θ
hyp cosθ
hyp
adj
d
opp( )
tanθ
hyp
y =
adjacent side
adj
d( )
cosθ
Exercise 15.3
1
(a)
(b)
(c)
B
c
e
g
a
A
A
(d)
q
f
p
A
C
b
A
r
hypotenuse
opp(A)
adj(A)
Remember, when working with
right-angled triangles you may
still need to use Pythagoras.
2
(a)
° =
° =
60°
x cm
=y
30°
y cm
(b)
p cm
50°
=p
° =q
° =q
q cm
40°
Unit 4: Shape, space and measures
101
15 Scale drawings, bearings and trigonometry
3
The memory aid,
SOHCAHTOA, or the triangle
diagrams
A
O
S
H
C
(a)
(d)
O
H
T
A
may help you remember the
trigonometric relationships.
(b)
(e)
°
°
(c)
°
°
°
4
(a)
(b)
(c)
B
A
y
55°
1.2 cm
3 cm
x
d cm
1.8 cm
C
x=
y=
(d)
1 cm
C
A
4 cm
A=
x
x=
°=
B=
(e)
Z 5m X
A
1.5 cm
12 m
y
C
2.5 cm
B
Y
y=
∠X =
X=
AC =
B=
C=
(a)
(b)
5
(c)
ym
B
3 cm
C
A
X
37° c cm
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
25 cm
x cm
C
2.5 cm
B
15 Scale drawings, bearings and trigonometry
6
(a)
(b)
1
4
(e)
(c)
13
15
(f)
(d)
(g) 5
1
2
(h)
61
63
7
(a)
(b)
(c)
13.5 m
A
a
A
2.4 cm
c
15 cm
13 m
b
B
B
17 cm
C
0.7 cm
C
(d)
(e)
Z
5m
A
X
55 cm
12 m
e
d
C
70 cm
B
θ =
(iii)
θ=
Y
8
(i)
(a)
(ii)
=
(b)
(c)
B
y
θ
x
(d)
θ
c
a
z
b
a
(e)
P
C
b
c
θ
Q
θ
y
Z
X
q
r
x
p
R
A
z
θ
Y
9
(i)
(a)
(ii)
(b)
(c)
22
13
(iii)
7
(d)
B
16
4x
5x
D
(e)
13 E
63
85
C
A
12
10
Unit 4: Shape, space and measures
103
15 Scale drawings, bearings and trigonometry
10
(a)
(b)
(c)
3
2
1
2
(d)
(e)
(f)
11
H
h
7.5 m
H
h
61º
27º
Tip
12
(a)
(b)
12
R
5 cm
cm
15 m
D
E
C
Unit 4: Shape, space and measures
y°
9m
B
D
75°
P 3 cm S
E
B
O
6 cm
x°
C
C
x°
cm
3x m
xm
104
(d)
Q
8.6
6 cm
52°
y cm
B
(c)
A
A
D
cm
x°
A
45°
5.6 cm
E
15 Scale drawings, bearings and trigonometry
15.4 Solving problems using trigonometry
Tip
Exercise 15.4
1
horizontal
angle of depression
2
lin
eo
fs
igh
t
(a)
line
ight
of s
angle of
elevation
(b)
horizontal
15.5 Angles between 0° and 180°
θ=
−θ
θ
θ
θ
θ
Exercise 15.5
1
Tip
θ
θ
(a)
(e) −
(i)
θ
–
θ
(b)
(f)
(j)
(c) −
(g)
(d)
(h)
θ
2
(a)
(e)
(i)
θ
(b)
(f)
(j)
θ
θ
θ
θ
θ
θ
−
(c)
(g)
(d)
(h)
θ
θ
θ
θ
−
15.6 The sine and cosine rules
sin A sin
s B sin C
=
=
a
b
c
a2
b2 + c 2
bc cos A
a
b
c
=
=
sin A sin
s B sin C
b2
a2 + c 2
2ac cos B
c 2 = a 2 + b2
ab cosC
Unit 4: Shape, space and measures
105
15 Scale drawings, bearings and trigonometry
Exercise 15.6
Tip
x
1
Tip
(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°
(i)
sin
sin105°
=
4
16
(h)
2
(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
23.4 cm
x cm
9.7 cm
24°
x cm
60°
3
(a)
(b)
(c)
θ
5.4 cm
4.8 cm
θ
9.5
c
m
129°
42°
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
b2
4
a2 + c 2
2ac cos B
B
θ
5
(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
3.2 cm
6
(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
1
1
2
ABC area = 2 ab sin C
1
2
area = ac sin B
area = bc sin A
Exercise 15.7
Tip
1 Draw a rough sketch of each of these figures before calculating their area.
(a)
(b)
(c)
(d)
(e)
(f)
BC
a
XZ
q
XY
p
∆ABC
∆ABC
∆XYZ
PQR
XYZ
∆PQR
∠C
∠C
∠X
∠R
∠Y
∠Q
AC
b
XY
∠Q
∠X
q
2
x
3
(a)
(b)
(c)
2.5 cm
x
x
x
4.5
5
cm
7 cm
cm
4 cm
Tip
74°
65°
5 cm
6 cm
15.8 Trigonometry in three dimensions
Exercise 15.8
1
Tip
A
D
C
B
E
H
G
108
Unit 4: Shape, space and measures
3 cm
4 cm
9 cm
F
(a)
(c)
(e)
(g)
˘
DEF
!
DFE
F
HF
CH
(b)
(d)
(f)
(h)
FD
GH
!
GHF
H
!
CHF
H
15 Scale drawings, bearings and trigonometry
2
P
60°
Q
15 m
(a) QS
(c) PQ
(d) PO
3
A
8
cm
Tip
(b) QO
E
B
D
8
F
4 cm
(a)
(b)
Mixed exercise
cm
C
BD
BD
!
BDF
CDEF
x
1
(a)
(b)
sin x sin121°
=
5. 2
7. 3
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
2
CD
3
AB
A
50°
B
21°
530 m
C
4 BD
D
AC
A
70°
B
50°
C
D
50 cm
5
X
6
Y
R S
T
X
Y
X
RS
Y
1500 m
77°
58°
R
110
Unit 4: Shape, space and measures
T
S
T
16
Scatter diagrams
and correlation
16.1 Introduction to bivariate data
Exercise16.1
1
A
B
D
E
(a)
(d)
(b)
(e)
C
(c)
Unit 4: Data handling
111
16
Scatter diagrams and correlation
2
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
(a)
(b)
(c)
(d)
(e)
3
Student
Tip
Age (years)
Distance (m)
8
9
9
Jen
(a)
(b)
(c)
(d)
(e)
(f)
(g)
112
Unit 4: Data handling
e
180
16
Mixed exercise
1
(a)
(b)
(c)
Accidents at a road junction
40
(i)
(ii)
30
(d)
(e)
20
10
d
2
0
35
70 105 140
Average speed (km/h)
Comparison of car age with re-sale price
15
Price ($ 000)
Number of accidents
50
0
Scatter diagrams and correlation
12
9
6
3
0
1
2
3
4
Age (years)
5
(a)
(b)
(c)
(d)
(e)
Unit 4: Data handling
113
17
Managing money
17.1 Earning money
Exercise 17.1
Casual workers are normally paid
an hourly rate for the hours they
work.
Tip
1
2
3
(a)
4
Some workers are paid for each
piece of work they complete. This
is called piece work.
5
6
(a)
(b)
(c)
114
Unit 5: Number
(b)
(c)
(d) 42
1
2
17 Managing money
17.2 Borrowing and investing money
=
×
PRT
I=
100
=
=
=
105
100
= 1 05
Exercise 17.2
‘per annum’ (p.a.) means
‘per year’
1
(a)
(b)
(c)
(d)
(e)
Tip
I=
PRT
100
P=
100I
RT
R=
100I
PT
T=
100I
PR
2
3
4
5
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
(a)
(b)
(c)
(d)
(e)
7
8
17.3 Buying and selling
=
=
Rate of profit (or loss) =
=
Rate of discount =
discount
× 100%
originall selling
ll
price
Exercise 17.3
1
(a)
(b)
(c)
(d)
2
3
4
116
Unit 5: Number
33
1
3
profit (or loss)
×100%
cost price
17 Managing money
5
(a)
(b)
(c)
Mixed exercise
5
1
2
1
(a)
(b)
(c)
2
(a)
(b)
1
(c)
1
2
3
(a)
(b)
4
Years
1
2
3
4
5
6
7
8
Simple interest
Compound interest
(a)
(b)
(c)
5
6
Unit 5: Number
117
17 Managing money
7
(a)
(b)
(c)
8
(a)
(b)
9
10
11
12
118
Unit 5: Number
18
Curved graphs
18.1 Plotting quadratic graphs (the parabola)
y = ax + bx + c
y
y
a
a
Exercise 18.1
Remember, the constant term
(c in the general formula) is the
y-intercept.
1
−
(a)
y
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 = −x2 + 2
(b)
x
y = x2 − 3
(c)
x
y = −x2 − 2
(d)
x
y = −x − 3
2
(e)
x
y = x2 +
1
2
Unit 5: Algebra
119
18 Curved graphs
2
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
(a)
x
−2
−1
0
1
2
3
4
5
y = x2 − 3x + 2
(b)
x
−3
−2
−1
0
1
2
3
y = x2 − 2x − 1
(c)
x
y = −x2 + 4x + 1
120
Unit 5: Algebra
−2
−1
0
1
2
3
4
5
6
18 Curved graphs
4
Height in metres
10
8
6
4
2
0
0
1 2 3 4
Time in seconds
5
(a)
(b)
(c)
(d)
(e)
18.2 Plotting reciprocal graphs (the hyperbola)
y=
x
y
x
a
x
xy = a
y
x
y=x
y
y = −x
Exercise 18.2
Tip
1
y
xy = a x
a
(a) xy = 5
(d) y = −
8
x
16
x
4
y=−
x
(b) y =
(e)
(c) xy = 9
2
2
(a)
(b)
(c)
(d)
a
2
Unit 5: Algebra
121
18 Curved graphs
18.3 Using graphs to solve quadratic equations
x
y=
x
y
Exercise 18.3
y
1
8
6
x2 − x − 6
y = x2 − x − 6
4
2
Tip
–4 –3 –2 –1 0
–2
(c)
1
2
3
4
5
–4
–6
–8
(a) x 2
x −6 0
2 (a)
(b) x 2
x −6
4
2
x −x+2
y
(c) x 2
x
x = 12
−
(b)
(i) − x 2 − x + 2 = 0
(ii) − x 2 − x + 2 = 1
(iii) − x 2 − x + 2 = −2
3 (a)
x
x
x2 − x − 6
(b)
(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
Exercise 18.4
1
4
y 2x + 2
x
2
(b) y = x + 2 x − 3
y
3
(c) y
x2 + 4
y=
x
(a) y =
122
Unit 5: Algebra
x +1
18 Curved graphs
2
(a) y 2 x 2 + 3x − 2
y
(b) y = x 2 + 2 x
2x 2 + 2x + 4
(c) y
(d) y
y = x+2
x+4
y
2x − 4
1
2
0 5 x + x + 1.5
y
x
2
18.5 Other non-linear graphs
x
x
x
x
x3
x3
y
ax3
bx2
cx
y
d
xa
Exercise 18.5
x
1
Tip
(a) y
(d) y
(g) y
y
(b)
(e)
(h)
x3 − 4x2
x3 + 4x2 − 5
x 3 − 3x 2 + 6
(c)
(f)
y = x3 + 5
y = x 3 + 2 x − 10
y
3x 3 + 5x
y
2
Tip
x
–2
–1
0
1
2 x 3 + 5x 2 + 5
2x 3 + 4 x 2 − 7
y
y
x 3 − 5 x 2 + 10
2
3
4
5
6
y
(b)
(c)
y
(i) x 3 5x 2 10 0
(iii) x 3 5x 2 10 x 5
(ii) x 3
−
≤x≤
5x 2 10 10
x
3
Tip
x 3 − 5 x 2 + 10
(a) y
(d) y
1
x
3
2x +
x
x−
1
x
2
3
x −
x
4
x
1
x2 − x +
x
(b) y = x 3 +
(c) y = x 2 + 2 −
(e)
(f) y
y
4
x
(a) y = 2 x
(b)
= 10 x
y = 2− x
y 2x − 1
x
x
Unit 5: Algebra
123
18 Curved graphs
5
(a)
(b) (i)
a
12 x + 1
(ii)
(c)
(i) A
(ii) H
18.6 Finding the gradient of a curve
=
y change
vertical change
=
x change horizontal change
Exercise 18.6
y
1
x 2 − 2x − 8
(a)
(b)
y
x
2 (a)
y
x3 − 1
x
(b)
A
3
30
y
Growth rate of beans
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)
(b) (i)
(ii)
(c)
A B
A
C
C
B
2 (a)
y
x2
y
x3
(b)
(c)
(i) 2 = 4
(ii) x 3 + 8 0
(d)
x
3
y
8
7
6
5
4
3
2
1
x
0
–8 –7 –6–5–4–3 –2–1
–1 1 2 3 4 5 6 7 8
–2
–3
–4
–5
–6
–7
–8
(a)
(b)
(c)
Unit 5: Algebra
125
19
Symmetry and loci
19.1 Symmetry in two dimensions
If a shape can only fit back into
itself after a full 360° rotation, it
has no rotational symmetry.
Exercise 19.1
1
(a)
(b)
A
B
E
D
G
2 (a)
(b)
3
126
Unit 5: Shape, space and measures
C
F
H
19 Symmetry and loci
19.2 Symmetry in three dimensions
Exercise 19.2
Tip
symmetry
Tip
symmetry
1
plane of
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
2
axis of
(a)
(b)
(c)
(d)
(e)
(f)
Unit 5: Shape, space and measures
127
19 Symmetry and loci
19.3 Symmetry properties of circles
Tip
Exercise 19.3
x
1
y
(a)
(b)
A
O
x
230°
X
x
80° Z
B
y
Y
MN
PQ MN
2
M
S
PQ
S
MN
SO
P
O
R
9 cm
N
Q
(a) MN
PQ
(b)
3
MN
(a)
N
128
Unit 5: Shape, space and measures
MN
S
(b)
M
S
PQ
SO
AB
P
O
T
C
A
O
E
Q
B
D
T
R
19 Symmetry and loci
19.4 Angle relationships in circles
Tip
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.
∠NPO
O
1
∠MOP
N
M
Q
O
70°
15°
P
(a) ∠PNO
(b) ∠PON
(c) ∠MPN
(b) ∠AOD
(c) ∠BDC
(d) ∠PMN
2 O
B
C
55°
25°
O
A
(a) ∠ACD
D
Unit 5: Shape, space and measures
129
19 Symmetry and loci
3
E
D
115°
A
C
B
4 AB
ABD
B
C
125°
D
A
5 SPT
O SR
∠PSO
∠TPR
S
29°
O
P
Q
R
T
∠BAC
O
6
A
72°
O
B
130
Unit 5: Shape, space and measures
C
D
∠BOC
19 Symmetry and loci
AB DC ∠ADC
7
∠DCA
D
A
64°
22°
B
C
(a) ∠BAC
(b) ∠ABC
(c) ∠ACB
O ∠QMO
8 SNT
∠MPN
S
M
N
18°
O
Q
56°
T
P
(a) ∠MNS
(b) ∠MOQ
(c) ∠PNT
19.5 Locus
REWIND
Make sure you know to bisect an
angle as this is often required in
loci problems (see chapter 3). !
Exercise 19.5
1
(a)
A
(b) A
B
(c) A
B
C
Unit 5: Shape, space and measures
131
19 Symmetry and loci
O
2
(a)
(b)
3
1m
4
4 km
Road
HOOL
CH
SC
School
5
132
Unit 5: Shape, space and measures
A
B
19 Symmetry and loci
6
Cell tower
A
Radio mast
C
30
m
15 m
B
Flagpole
Scale 1 cm = 5 m
(a)
AB
(b)
BC
D
D
D
X
Mixed exercise
1
(a)
(b)
(c)
(d)
(e)
(i)
(ii)
2
(a)
(b)
(c)
(d)
Unit 5: Shape, space and measures
133
19 Symmetry and loci
O
3
(a)
(b)
(c)
(d)
A
A
x
A
B
28°
O
D
w
yz
A
O
O
160°
C
C
x
29°
B
y
C
O
(a)
(b)
250 mm
18 cm
y
y
x
120 mm
O
x 15 cm
O
5
Route 66
Petrol station
AB
6
A
134
Unit 5: Shape, space and measures
B
A
O x
y
x
B
x
4
D
B
60°
C
20
Histograms and frequency
distribution diagrams
20.1 Histograms
Frequency density =
frequency
class interval
Exercise 20.1
1
Marks class interval
Remember there can be no gaps
between the bars: use the upper
and lower bounds of each class
interval to prevent gaps.
English frequency
Mathematics frequency
3
7
5
6
51
6
3
1
REWIND
You met the mode in chapter 12. !
(a)
(b)
(c)
(d)
Unit 5: Data handling
135
20 Histograms and frequency distribution diagrams
Frequency
2
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
(a)
(b)
(c)
(d)
(e)
Tip
3
Age (a)
a
a
a
a
a
No. of people
4
House prices in property magazine
30
25
20
Frequency density 15
10
5
0
(a)
(b)
(c)
136
Unit 5: Data handling
20
40
60
80
100
120
140
Price ($thousands)
160
180
200
20 Histograms and frequency distribution diagrams
20.2 Cumulative frequency
3
1
Exercise 20.2
1
Percentage
Number of students
m
m
3
m
7
m
m
m
51
m
m
m
3
m
1
(a)
Tip
(b)
(c)
(d)
(e)
1
Height of professional
basketball players
(a)
3n
4
n
4
(b)
(c)
(d)
1
3
Cumulative frequency
2
Tip
n
2
3
200
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
Heights of trees
10
Frequency
8
6
4
2
0
0
2
4
6
8 10
Height in metres
12
(a)
(b)
(c)
2
Mass (kg)
Frequency
≤m<
≤m<
7
≤m<
13
≤m<
≤m<
11
(a)
(b)
(c)
(d)
3
Minutes
≤m<
m<
No. of
teenagers
m<
m<
m<
m<
15
4
Heights (h cm)
≤h
Frequency
(a)
(b)
(c)
138
Unit 5: Data handling
3
1
3
≤h<
≤h<
15
≤h<
21
Ratio, rate and proportion
21.1 Working with ratio
a b
3
5
Remember, simplest form is also
called ‘lowest terms’.
2
5
Exercise 21.1
1
(a) 2
(c)
(e)
3
4
3
2
3
(b) 1
(d)
x
2
Tip
x
1
2
(a)
x
(b)
(e)
x
(f)
(g)
(b)
(c)
(i)
x 1
=
21 3
2 x
=
7 4
5 3
(j)
=
x 8
x
(c)
x
5 16
=
x 6
(d) x
x 10
(h)
=
4 15
3
4
(a)
5
A B C
6
Unit 6: Number
139
21
Ratio, rate and proportion
21.2 Ratio and scale
n
n
Exercise 21.2 A
Tip
n
1
n
n
n
(a)
n
n
(b)
(c)
1
n
2
(a)
(b)
(c)
Exercise 21.2 B
1
2
3
(a)
(b)
4
(a)
(b)
Map distance (mm)
Actual distance (km)
5
(a)
(b)
(c)
140
Unit 6: Number
50
50
1
2
21
Ratio, rate and proportion
21.3 Rates
n
Exercise 21.3
Tip
1
2
3
(a)
(b)
(c)
Remember, speed is a very
important rate.
distan
t ce
speed =
time
4
distan
t ce = speed × time
time =
(a)
(c)
distan
t ce
speed
(b)
(d)
2
5
(a)
(c)
1
2
(b)
(d)
21.4 Kinematic graphs
Unit 6: Number
141
21
Ratio, rate and proportion
Exercise 21.4
1
600
Distance (km)
500
400
300
200
100
0
1
2
3
4
5
Time (hours)
6
(a)
(i)
(ii)
(iii)
(b)
(c)
(d)
(e)
2
Pam
Pam
bi
lo
100
Da
Distance (km)
200
0
60
120
Time (min)
180
(a)
(b)
(c)
(d)
(i)
(e)
(f)
(g)
(h)
(i)
142
Unit 6: Number
(ii)
Dabilo
240
21
Ratio, rate and proportion
3
120
100
80
Speed (km/h) 60
40
20
0
1
2
3
4
Time (minutes)
5
6
(a)
(i)
(ii)
(b)
(c)
(d)
4
Tip
30
Speed
(m/s)
20
10
ms
−2
0
20
40
60
80
Time (seconds)
100
120
(a)
(b)
(c)
(d)
Unit 6: Number
143
21
Ratio, rate and proportion
21.5 Proportion
Exercise 21.5
Tip
x
x
y
x
y
1
(a)
y
(b)
A
600
A
5
B
(c)
A
B
2
(a)
(b)
3
4
5
(a)
(b)
(c)
6
(a)
(b)
7
(a)
(b)
(i)
(ii)
144
Unit 6: Number
6
B
21
Ratio, rate and proportion
8
(a)
(b)
9
10
21.6 Direct and inverse proportion in algebraic terms
P kQ
PQ k
k
k
Exercise 21.6
Tip
a
1
b
(a)
(b)
∝
a
b
k
a
b
m
2
T
T
m
m
T
3 F
m
(a)
(b)
F
m
xy
4
x
6
y
a
5 y∝
(a)
(b)
1
x
y
(a)
(b)
(c)
m
k
b
c
x
y
x
6 y
F
m
F
x
y
x
y
x
y x
x y
Unit 6: Number
145
21
Ratio, rate and proportion
7
8
x
y
(a)
(b)
y
x
(c)
x
y
x
1
3
a
b
(a)
(b)
a
b
a
x
y
y
b
a
b
x
9
(a)
(b)
y
y=
y
x
10
x
y
10
(a)
(b)
(c)
y
x
x
x
y
y
x
21.7 Increasing and decreasing amounts by a given ratio
new amount x
x × old amount
=
new amount =
old amount y
y
Exercise 21.7
1
2
3
(a)
(b)
4
Mixed exercise
1
(a)
(b)
2
146
Unit 6: Number
x
y
x
y
21
Ratio, rate and proportion
3
(a)
(i)
(ii) 4
(i)
(ii)
1
2
(iii)
(b)
(iii)
4
Distance (km)
500
400
300
200
100
0
1
2
3
4
5
Time (hours)
6
7
(a)
(b)
(c)
(d)
(e)
5
20
Speed
(m/s) 10
0
10
20
30
40
Time (seconds)
50
60
(a)
(b)
(c)
(d)
6
7
Unit 6: Number
147
21
Ratio, rate and proportion
P
8
(a)
(b)
9
(a)
(b)
148
Unit 6: Number
V
P
P
V
V
P
V
22
More equations, formulae and
functions
22.1 Setting up equations to solve problems
x
Tip
Exercise 22.1 A
x
1
(a)
(b)
(c)
x)
P
A
x x
2
(a)
(b)
x
S
M
x
3
(a)
(b)
x
S
x
4
(a)
(b)
(c)
Tip
x
x
Exercise 22.1 B
1
2
3
4
Unit 6: Algebra
149
22
More equations, formulae and functions
5
X Y
6
Z
Y
Z
X
X
X
Y
Z
7
8
22.2 Using and transforming formulae
–
–
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
(a) U T V W
(b) V W U T
C
(c) A =
B
B
(d) A =
C
(e) P Q
(f) P
Q
(g) P Q R
(h) Q
P
(V)
(V)
(i) Q
P
PR
(P)
(j) Q
(k) PQ
(P − R )
R
(P)
(Q)
(a)
(b)
Unit 6: Algebra
(B)
(Q)
(Q)
(Q)
(P)
V
2
150
(B)
I
I
R
V
RI
22
More equations, formulae and functions
A
3
(a)
(b)
r
r
5
9
C
4
(a)
(b)
(c)
F
K
(F − 32)
C
K
C
22.3 Functions and function notation
x
x
x
x
(d)
m
x
Exercise 22.3
Tip
1
x
x
x
(a)
y
(b)
(c)
x→ x
2
(a)
(b)
x
(i)
(ii)
(iii)
(c)
(d)
(i)
(e)
a
a
(ii)
a
b)
h:x→ 5 x
3
(a)
(b)
x
(i)
x
4
(ii)
x
x
(a)
5
(iii)
b
a
x
(b)
x
x
x
x
Unit 6: Algebra
151
22
More equations, formulae and functions
6
(a)
x
x
7
(a)
(d)
Mixed exercise
(b)
x
x x
x
g (x ) =
2
(b)
x
(e)
x
x
x
x
(c)
(c)
(f)
x
(d) f ( x ) =
x
x)
x)
1
2
3
4
5
6
b
7
(a) a =
(b) a
f (x ) =
8
9
152
Unit 6: Algebra
5 (b
9
b
x
x
(a)
(d)
x
x
) (b − 3)
+
3
b)
2x − 3
5
x
g (x ) =
x −2
5
(b)
(e)
(c)
a
a
x
−2
23
Transformations and matrices
23.1 Simple plane transformations
′
 x
 y 
x
x
The enlargement factor =
Tip
x
y
y
length of a side on the image
length of the corresponding side on the object
Exercise 23.1 A
ABCD
1
(a)
(b)
°
A′B′C′D′
D
A′B′C′D′
B′D′
Unit 6: Shape, space and measures
153
23 Transformations and matrices
2
y
8
A
6
4
2
B
–8
C
–6
–4
0
–2
H
2
4
E
(a)
(b)
(c) (i)
(ii)
F
x
–4
–6
G
8
I
–2
D
6
K
J
–8
ΔABC
y=−
DEFG
HIJK °
H ′I ′J ′K ′
−
3
 10 
 −9
(a) ΔABC
(b)
154
Unit 6: Shape, space and measures
 0
 −6
PQRS
y
P′Q′R′S ′
A′B ′C ′
23 Transformations and matrices
4
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
x
5
6
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
1
You dealt with enlargement and
scale factors in chapter 21. !
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
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
ABC
1
ΔA′B′C′
y
ΔA″ B″ C″
x
y
8
A
6
4
C
2
–1
–1
B
2
4
x
8
6
(a)
ΔABC
(b)
ΔABC
ΔA″ B″ C″
2
 −8
 1 
y
8
6
4
2
–1
–1
A
2
4
6
(a)
8
x
x
(b)
ABCD
3
A
x=
(a)
(b)
B
C
y=
D
′
ABCD
′
Unit 6: Shape, space and measures
157
23 Transformations and matrices
23.2 Vectors
A
a b c
AB
B
! "
AB AB
 x
 y 
 1
 2
a
x
! "
AB
–a
 x
 y 
y
! "
AB
 kx 
 ky 
k
 x1   x2   x1 + x2 
 y  +  y  =  y + y 
1
2
1
2
! " !" ! " ! "
AB − CA = AB + AC
 
 y 
! "
AB
x2 + y2
O
! "
OA
A
! "
OA
Exercise 23.2 A
Remember, a scalar is a quantity
without direction, basically just a
number or measurement.
10
y
1
8
6
A
(a)
(c)
(e)
(g)
(i)
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
10
x
(j)
(k)
(l)
! "
! "
!AE"
DB
! "
CD
!"
BE
! "
CD
! " ! "
AB
! " + BC
! "
! " !AE" − AB
BC = ED
(b)
(d)
(f)
(h)
! "
BC
! "
!BD"
EC
!"
BE
23 Transformations and matrices
2
Tip
! "  2
(a) AB =  
 3
! "  −1
(b) CD =  
 4
! "  4
(c) EF =  
 5
! "  3
(d) GH =  
 −4
3
Remember, A is the object and A’
is the image.
10
y
8
B'
6
C
B
4
C'
–10 –8
–6
–4
E
2
–2 0
–2
E'
2
D
–4
D'
–6
A
4
6
8
10
x
A'
–8
–10
Exercise 23.2 B
 1
 −2
a=  b= 
3
 
 4
1
(a) b
(f) a + b + c
(b)
(g)
 0
c= 
 −3
a
a+ b
(c) − c
(h) b
c
(d) a + b
(i) − a
b
(e) b + c
(j) a
b+ c
2
A
B
G
a
D
(a)
(e)
(i)
F
c
O
b
C
! "
! "
= a OC = b
! "
DE
!"
OE
!" !"
4 BF 3EF
! "
OG = c
! "
(b) AD
(c)
! "
(f) CD
(g)
! " !"
(j) 1 OC + 3GB
! "
AG
!" ! "
BF + FD
a b
c
!"
(d) OB
! " !"
(h) DE + EF
2
E
Unit 6: Shape, space and measures
159
23 Transformations and matrices
! "
OA
3
(a) –a
Tip
(b)
a
1
2
(c)
a
(d)
(e) − 1 a
2
a
4
! "  −2
(b) PQ =  
 7
! "  −13
(d) ST = 
 −12
! "  4
(a) MD =  
 5
! "  9 
(c) XY = 
 −12
O
5
X
! "  −6 ! "  −2
OA =   OB =  
 −4
 2
6
(a)
x
A B
!"
CA
! " ! "
AB BC
(b)
7
! "
XYZ
y
!"
YZ
x
! "  5
OC =  
 1
C
! "
XY
y M
! " ! "
XZ ZX
Y
x
y
M
X
8 a
Z
! "
(b) OY
! "
(d) XY
! "
(a) OX
! "
(c) OZ
Tip
Y
Z
1
 2 b
 3
 0
c
 −2
 5 
(a)
x y
(i) x
(ii) y
(iii) z
z
a b c
a b c
a b c
(b)
(i)
9
Tip
x
X
! "
AY
! "
XY AX a
Y
b
B
a
D
b
Unit 6: Shape, space and measures
Y
(iii) z
AB
X
A
160
(ii) y
C
AC
D
! "
MZ
23 Transformations and matrices
(a)
! "
(i) XY
(b)
! "
(ii) AD
a
b
! "
(iii) BC
XY BC
10
23.3 Further transformations
y
x
x
y
Exercise 23.3
1
(a)
x
(b)
y
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
(c)
Tip
y
(d)
y
y
y
10
9
8
7
6
5
4
3
2
1
k
–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
2
162
(a)
(b)
(c)
(d)
Unit 6: Shape, space and measures
x
1 2 3 4 5 6 7 8 9 10
23 Transformations and matrices
23.4 Matrices and matrix transformation
A
A
I

 –1
 c d  A
ad
bc
d
zero
b
a 
1
=
A–1
A
a
d
A–1
I
1
(ad − bc )
A
(ad − bc )
A
determinant
A
c
B
A
identity
1
B
identity
ad bc
(ad − bc )  −c
B
A
B
 a b
 c d 
A
A
b
A
A
 1 0
I 
 0 1
AI A
IA A
 0 0
O 
 0 0
A
A
B
1
0
1
0
Exercise 23.4
 −1 2 3 
 4 3 −3
A
1
(a) A
2 D
(a)
(b)
B
B
1
 3
(b) A B
4 2
1 0
(c) B A
 1 4
 5 8
D
D–1
Unit 6: Shape, space and measures
163
23 Transformations and matrices
3
 1
A
3
0
2
B
(a)
(b)
AB
(i)
A
(a) A
3 0
2 −1
B
B
6
 7
(b) 3A
C
(a)
(b)
BA
(ii) AB
1
A 
4
4
5
 1 3
 −4 2
 2 x
 1 4 
3 −3
2 5 
(c)
B
C
x
C–1
23.5 Matrices and transformations
 
 y 
BA
x y
A
AB
B
BA
Exercise 23.5
Tip
1
(i)
(ii)
(a)
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)
y
B
x
1 2 3 4 5 6
10
9
8
7
6
5
4
3
2
1
y
A
B
–1 0 1 2 3 4 5 6 7 8 9 10
–1
x
23 Transformations and matrices
2
P
 1 0
 0 4
Q
R
S
(a)
(b)
 3 0
A  0 3


A
3
(a)
(b)
B
BA
B
x
4
Mixed exercise
 −1 0
 0 1
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)
(i)
(ii)
(iii)
(b)
(i)
(ii)
(iii)
DEFG
2
y
°
D′E′F′G′
8
(a)
6
(b)
4
(iv)
 −4
 5 
DEFG
DEFG
D″ E″ F″ G″
D'
G'
2
E'
F'
2
4
6
8
x
Unit 6: Shape, space and measures
165
23 Transformations and matrices
ABCD
3
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
B
(a)
(b)
ABCD
ABCD
(c)
x=−
°
 −3
 2 
ABCD
(d)
ABCD
 −2
c= 
 −4
 3
 2
a=   b= 
 6
 −4
4
(a)
(i)
a
(ii) b + c
(iii) a
b
(iv) a + b
(ii) b c b + c
(iii) a b a b (iv) a b a + b
!"
! "
AB y
O FA x
(b)
(i) a a
x
A
F
5 ABCDEF
B
(a)
C
E
D
166
y
(b)
Unit 6: Shape, space and measures
! "
(i) ED
!"
(iv) EF
x
!"
(iii) FB
! "
(ii) DE
! "
(v) FD
!"
EB
E
y
B
23 Transformations and matrices
6
(a)
(b) y
B′
B
A
(c)
7
x
C
(d)
ABCD
1
2
x
Unit 6: Shape, space and measures
167
23 Transformations and matrices
8
ABCD
y
10
9
8
7
6
5
4
3
2
1 D
–1 0
–1
A
(a)
(b)
(c)
(d)
(e)
(f)
B
1 2 C 3 4 5 6 7 8 9 10
 1 2
P 
 2 1
9
y
x
 1 3
Q 
 2 2
P Q
P Q
PQ
QP
P
P–I
 2 5 a   a  45
 3 4 6   b  =  47

   
 −2 4 −1  3  c 
10
a b
c
11
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
ABC
x
A
A'B'C'
B
A
B
24
Probability using tree
diagrams
24.1 Using tree diagrams to show outcomes
Tip
Exercise 24.1
1
2
H
A
G
1
2
B
F
4
3
C
E
D
3
(a)
(b)
24.2 Calculating probability from tree diagrams
Exercise 24.2
1
Unit 6: Data handling
169
24 Probability using tree diagrams
H
Blue
T
H
Yellow
T
H
Black
T
(a)
(b)
(c)
(d)
2
First
toss
1
2
1
2
(a)
(b)
(c)
(d)
(e)
Mixed exercise
1
(a)
(b)
(c)
(d)
Tip
2
(a)
(b)
(c)
(d)
(e)
170
Unit 6: Data handling
Second
toss
1
2
H
1
2
T
1
2
H
1
2
T
H
T
H
T
H
T
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
(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
(iii)
6
10
13
0
10
45
14
26
2 (a) 53 200
(c) 17.4
(b) 713 000
(d) 0.00728
3 (a)
(c)
(e)
(g)
(b)
(d)
(f)
(h)
36
12 000
430 000
0.0046
5.2
0.0088
120
10
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
(b) 529 cm2
2 (a) −2 °C
Exercise 1.6
3 20 students
(l) 12
1 −3 °C
1 18 m
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
(b)
(d)
(f)
(h)
5
(i) 4 + 3x
3 (a) $
6(x − 1)
18x
x2 + 8
1
x+
3
(j) 12 – 5x
(b) p − 4
(c) 4p
x
3
x
9
(b) $ , $
2x
2x
, and $
9
3
(b) 1.875 m2
(d) 8 cm2
2 –104
(b) 91
3x 2 − 2x + 3
4x 2y − 2xy
5ab − 4ac
4x 2 + 5x − y − 5
−30mn
(f) 6x 2y
6xy 3
(h) −4x 3y
(i) 4b
(j)
(k) 3b
(o)
(q)
(s)
20 y
3x
2y
2
2
x
2
15a
4
x
6y
1
4y
(l)
9m
4
(n)
3x
y
172
5x
9
2y
3
x
2
Answers
3
(k)
16
(l)
x
8
5
y
2
5
y
16
x
22
3 (a) x
(c) x
(f)
1
2
2
y
2
1
6
(r)
−14 y
5
2
x
9
y
22
2x
1
4
(j) x y
−2
2
3
or
1
1
y2
(d) x2
2
(f) x y
29
− 16
xy − 3x
−3x + 2
3x + 1
x2 + x + 2
(b) x 2 + xy
1 (a) –1296
(b) –1
1
(g) 32
(h) 4
2
(g) 12 − x
x
4
(h) x 3 − x
2 (a) −6
(b) 24
3 (a) −2
(b)
or
1
29
x 4 y 16
4
(c)
(f)
(i)
8
4x
y
(c) 5
5 (a)
(b)
(c)
(d)
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
(e)
64 x
(g)
27 x
y
9
(f) x 9y 8
15
4y
4
(h)
3
xy
2
6
1
(b) x 2
(c) x −9 y or
y
x
9
5
3
5
2y3
x
Chapter 3
625
Exercise 3.1 A
2
2
3
−14
9
5y
2
3
1
3
(c)
7
(e) −2
9a + b
x 2 + 3x − 2
−4a 4b + 6a 2b3
−7x + 4
(d) 2 x y or
(e)
3
(f)
− 13
(d) 2
(j)
(e) 4x
7 (a) 5xy 3
1
Exercise 2.5 B
(b)
(d)
(f)
(h)
3
2
(b) x − 4
1
− 14
(f) 2
x
3
(c)
2 4
x y
(b) x2
7
3
(e) x y
x3
y
or
−4
(e) 2
(h)
(f) 5x −
1
9
1
(c) –4
(d)
(e)
8
15
(h) x 3 y
4
(b) 4
(c) 5x
(d)
4 (a)
(b)
(c)
(d)
4
(f) 2 x 3 y 3
(g) x 2 y 4
3
4
2
1
(e) 8x
(c) y
4
y
1
(d) x
3
4 (a) x
x
(b) x
(k) y −2 or
2
27 x
10
(d)
7
3
3125 x y
16
y
(b)
10
8x y
3
16
8
3
25 x y
(j)
x
x y
49
(h)
4
3
1 (a) x + 12
(f) x 7y 3
50 x
27 y
1
Mixed exercise
(d) xy 10
(i) x
(p)
(t)
2x 2 − 4x
−2x − 2
−2x 2 + 6x
x 3 − 2x 2 − x
2 (a) x 2 +
2
3 (a)
(b) 65
(d) –163
(g) 2
(b) 3x 4y
3
Exercise 2.4
1 (a)
(c)
(e)
(g)
2
(e)
(e)
Exercise 2.3
(m)
6
y
2x
3y
(c)
4 17.75
5 (a) 6
x
(c)
2 (a)
3 17
1 (a)
(b)
(c)
(d)
(e)
(g)
1 (a)
2 (a) 17
(c) 15
(d)
Exercise 2.5 A
(i) x 7y
1 (a) 54 cm2
(c) 110.25 cm2
3y
2
2
(g) –5x2 – 6x
(g)
Exercise 2.2
+
(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°
1
3
(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
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
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
No. of brothers/
sisters
0
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
Eye colour
(c) student’s own sentences
2 (a)
174
4 (a)
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
20
5 student’s own diagram
4 (a) student’s own chart
(b) student’s own chart
Frequency
1
1
7
19
12
6
4
Value ($ 000)
4 student’s own diagram
Score
0−29
30−39
40−49
50−59
60−69
70−79
80−100
08
3 (a)
20
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
Chapter 5
3 (a) 83%
(d) 37.5%
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)
63
13
108
5
(b)
28
5
(e) 3
(c) 14
(f)
6
19
(g) 120
(h)
3
14
(i) 72
(j) 3
(k)
233
50
(l)
7
4
3 (a)
13
24
(b)
19
60
(c)
19
21
(d)
35
6
(e)
183
56
(f)
161
20
(g)
18
65
−5
6
(h)
41
40
(i)
(k)
−10
3
(l)
29
21
−26
9
4 (a) 24
(b)
(c)
10
27
(e)
96
7
10
9
3
5
4
5
5
12
11
170
(c)
(j)
(d)
(g) 2
5 (a)
(d)
38
9
19
4
(g) 0
6 (a) $525
(h)
(b)
(e)
(h)
(f)
7
96
9
14
(i) 4
(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
(b)
(d)
(f)
(h)
(j)
5 (a)
(c)
(e)
(g)
+20%
+53.3%
−28.3%
+2566.7%
(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
(b) 960
(b) 27 750 (c) $114 885
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
4 × 5 = 20
70 × 5 = 350
1000 × 7 = 7000
42 ÷ 6 = 7
(b)
(d)
(f)
(h)
(j)
(l)
(b) 3
(d) 243
Mixed exercise
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%
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
1 (a) 40
(c) 22
(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%
8 (a) 5.9 × 109 km
(b) 5.753 × 109 km
9 (a) 9.4637 × 10 km
(b) 1.6 × 10−5 light years
(c) 3.975 × 1013 km
11
2
=5
(g) x = 2
(i) x = 4
(k) x =
(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
(f) x = 5
2
1
2
2 (a) x = 10
(c) x =
3
5
8
−
3
2
1 m=
(b) x = −2
=
2
−2
3
(d) x =
4
3
1
4
1
=1
3
(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
x(x + 8)
x(9x + 4)
2b(3ab + 4)
3x(2 − 3x)
3abc2(3c –ab)
b2(3a − 4c)
(d) x = 4
=3
(n) x =
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
(l) x = 10
2(1 + 4y)
x(3 − y)
3(x − 5y)
3b(3a − 4c)
2x(7 − 13y)
(b)
(d)
(f)
(h)
(j)
(l)
Exercise 6.2
=
(k) x =
3
=1
13
(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
(j) x = 9
16
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 =
(i) x =
1
2
2 (a)
(c)
(e)
(g)
(i)
−2x −2y
6x − 3y
−2x2 − 6xy
12 − 6a
3
2x2 − 2xy
1
2
9
2
3
a
4xy
xy2z
1 (a)
(c)
(e)
(g)
(i)
(k)
9
2
(h) x = = 4
1
−
3
5
6
1 (a)
(d)
(g)
(j)
Exercise 6.1
(c) x = = 4
(g) x = −1
=3
Exercise 6.3
Chapter 6
1 (a) x = 3
(f) x =
(m) x = 42
12
23
6
(e) x = 17
20
13
=1
(b) x = 27
(d) x = −44
7
13
(b)
(e)
(h)
(k)
(c)
(f)
(i)
(l)
8
3y
pq
ab3
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
b
c
a=
cd − b
2
e d
bc
c( f
d
c
b
de )
b
b
Exercise 6.4 B
P
2
1 (a) b
(b) 9 cm
2π
2A
h
3 use
(b) b = 35.5 cm
l
C
2 (a) r =
(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 =
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)
mq − p
n
−12x 2 + 8x
−16y + 2y2
5x 2 + 7x − 4
10x 2 + 24x
Chapter 7
Exercise 7.2
Exercise 7.3 A
Exercise 7.1 A
1 (a)
(b)
(c)
(d)
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 cm
(c) 4 tins
(b) 153 000 cm2
4 (a)
(c)
(e)
(g)
(b)
(d)
(f)
(h)
1 (a) 120 mm
(c) 128 mm
(e) 36.2 cm
(b) 45 cm
(d) 98 mm
(f) 233 mm
2 (a)
(c)
(e)
(g)
(b) 44.0 cm
(d) 21.6 m
(f) 151 mm
15.7 m
54.0 mm
18.8 m
24.4 cm
3 90 m
2 (a) cuboid
(b) triangular prism
(c) cylinder
3 The following are examples; there are
other possible nets.
(a)
4 164 × 45.50 = $7462
cube
cuboid
square-based pyramid
octahedron
2
90 000 mm3
20 410 mm3
960 cm3
1800 cm3
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
(b)
(d)
(f)
(h)
1.53 m2
150 cm2
71.5 cm2
243 cm2
2 (a) 7850 mm2
(c) 7693 mm2
(e) 167.47 cm2
(b) 153.86 mm2
(d) 17.44 cm2
3 (a)
(c)
(e)
(g)
(i)
(b)
(d)
(f)
(h)
288 cm2
373.5 cm2
366 cm2
272.93 cm2
5639.63 cm2
4 (a) 30 cm2
(c) 33.6 cm2
(e) 720 cm2
82 cm2
581.5 cm2
39 cm2
4000 cm2
(b)
(c)
(b) 90 cm2
(d) 61.2 cm2
(f) 2562.5 cm2
64 000
50
100
64 000
50
Breadth (mm)
40
64
80
80
Height (mm)
20
20
8
16
Mixed exercise
1 (a) 346.19 cm2
(b) 65.94 cm
2 4.55 cm
3 (a)
(c)
(e)
(g)
2000 mm2
40 cm2
106 cm2
175.84 cm2
(b) 33 000 mm2
(d) 80 cm2
(f) 35 cm2
4 15 m
Exercise 7.1 C
(b) 47.1 cm
2 6668.31 km
4 61.33 cm
64 000
80
2 (a) (i) 1.08 × 1012 km3
(ii) 5.10 × 108 km2
(b) 1.48 × 108 km2
6 70 mm = 7 cm
2
64 000
1 (a) 5.28 cm3
(b) 33 493.33 m3
(c) 25.2 cm3
(d) 169.56 cm3
(e) 65 111.04 mm3
5 11.1 m
3 (a) 75.36 cm2
(c) 92.34 mm2
Volume (mm3)
Length (mm)
Exercise 7.3 B
2
1 (a) 43.96 mm
(c) 8.37 mm
(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) 732.67 cm2
(d)
5 (a)
(b)
(c)
(d)
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
Exercise 8.1
3
9
17
red = , white = , green =
10
25
50
(b) 30%
(c) 1
(d)
2 (a) 0.61, 0.22, 0.11, 0.05, 0.01
(b) (i) highly likely
(ii) unlikely
(iii) highly unlikely
2 (a)
(c)
= 0.1 (ii)
(v)
(viii)
1
4
1
2
(b)
1
(iii)
2
5
9
10
(vi)
or 0.33
(c)
1
or 0.16
6
13
40
6
7
20
3
10
(b)
1
2
3
5
(b)
(e)
(c)
2
5
1
3
cola,
cake
cola,
muffin
water,
biscuit
water,
cake
water,
muffin
2
3
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
L
C
T
T
(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
(d)
3
Exercise 8.5
1
2
1
5
5
8
178
1
3
1, 3
2, 3
3, 3
fruit
juice,
muffin
(c)
(b)
Mixed exercise
Yellow
2
1, 2
2, 2
3, 2
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
(ix) 0
Exercise 8.3
2
(d)
Snack
(b)
1
3
(d)
1
2
(c)
3 (a)
3
10
1
2
T
HT
TT
1
1, 1
2, 1
3, 1
1
2
3
Drink
2
5
4 (a)
5 (a)
2 (a)
(b) 9
(b) no sugar; probability =
3 (a)
(b)
4
Green
1 (a) 1, 2, 3, 4, 5, 6, 7, 8, 9 or 10
(b)
3
(a)
1
3
Exercise 8.2
1
(i)
10
3
(iv)
10
3
(vii)
10
H
HH
HT
H
T
Chapter 8
1 (a)
3 (a)
1
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
3 (a)
(b)
(c)
(d)
{}
{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}
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
2n + 5
3 − 8n
6n − 4
n2
1.2n + 1.1
1
a
6
31
3
(d)
q
u
(b) 21
(d) (i)
74
99
(c)
79
90
103
900
(e)
943
999
(f)
928
4995
Exercise 9.3 A
(c) false
(f) true
w
x
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}
40
59
80
x
36 – x
(c) 0.57
Mixed exercise
T120 = 596
T120 = −694
T120 = 359
2 (a) –4, –2, 0, 2, 4, 8
(b) 174
(c) T46
3 0.213231234. . ., 2 , 4 π
(c) 16
(ii)
(v)
41
1
(iii)
80
21
5
80
−1
0
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
9
7
5
3
(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
3
1
(in fact, any five values of y are correct)
(f) x
−1
0
1
2
3
y
−2
−2
−2
−2
−2
7
1 (a) 5n − 4
(b) 26 − 6n
(c) 3n − 1
18
Exercise 10.1
S
(b) 21
16
Chapter 10
C
(e)
78 – x
C
17
1 (a) x
v
M
(b)
(b) true
(e) false
(h) true
t e
m
r
k
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. . .
4
9
1 (a) false
(d) true
(g) false
o
j
2 (a)
(b)
(c)
(d)
(e)
45 , 3 90 , π, 8
(b)
2 (a)
16 ,
n
g
h
i
s
c
y
p
286
999
25
(iv)
f
3 (a)
Exercise 9.2
1 (a)
d
P
l
5
26
b
(b)
B
Exercise 9.3 B
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
5 (a)
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 =
(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
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
9 y 14
9y
180
2
Answers
1
(k) 1 − x 2
2 (a) x 2 + 8x
8 x 16
(b) x 2 6x
6x 9
(c) x 2 + 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
(h) 4 x 2 122 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
1
1
2
4
4
, c = −2
5
(g) x = 2
(i) y = −2x
(k) y = 3x − 2
(j) x 2 + x − 132
(m) −12 x 2 + 14 x − 4
(e) m = , c =
7 (a) y
x
(c) y = 2 5
1
(e) y
x −1
3
(l) −3x + 11x − 6
m = 3, c = −4
m = −1, c = −1
1
m=− ,c=5
2
m = 1, c = 0
1
3
2
2
16
(i) m = − , c = +
(i) 2
7 xy + 2 y 2
4
4
6
7
(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 +
z   y 2
z 
y
(
(
)(
)
(p) 5x 5 1 55x 5 1
(q) 1 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
0
2 3
2 1.5
0
2
2
2
3
2
−1
2
0
4
2
8
3
10
m = −2, c = −1
m = 1, c = −6
m = 1, c = 8
1
m = 0, c = −
2
3
(i)
2
(e) m = − , c = 2
(f) m = −1, c = 0
3 (a) y
(c) y
2
1
x+
3
2
4
x −3
5
(b) y
x −3
(d) y
x −2
(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
(b) y = 7
2x − 6
(c) y = x + 4
(d) x = −10
(e) y
(f) y = −3
6 (a) t
D
x
0
0
2
14
4
28
4 6
Time
7
All four plotted on the same graph.
2 (a)
(b)
(c)
(d)
2
(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
5
(c) y = 2
−1
2
20
x
−1 0
3.5 3
x
y
25
10
1
x+3
2
(b) y
a (a + 2))((a − 2)
2
+ 1 ( x 1)( x 1)
( x 2))(( x + 1)
( x 1))(( x − 1)
3 y + 2z )(2 x − 3 y 2z )
(2x 3y
( x + 12)( x + 4 )
x  2 x

(g) x 2 +
x −

2 
2
30
1
x
2
6
42
(a)
(b)
1
(0.5, 6.5)
(ii)
2
(0, 5)
(iii)
–1
(1, 3)
(iv)
−
(v)
4
3
undefined
8 (a) x 2 − 16 x + 64
(b) 2 x 2 2
(c) 9 2 12
1 xy 4 y 2
12x
(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
1 −12y
12 36 y 2
2
9
4
4 x 2 200 x 25
9 4 y 2 6x 2 y + 1
1
(h) x 2 + xy
x + y2
4
1
(i) x 2 −
4
1
−
4
(j)
x2
(–0.5, 3)
(–1.5, 0.25)
(x
)
( 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)
(b)
(d)
(f)
(h)
5 cm
12 mm
1.09 cm
8.49 cm
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)
∴ ABC is similar to 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 cm2
(b) 529 mm2
4 (a) x = 2 cm
1
2
∴ AB = AC
Mixed exercise
1 (a) sketch
(b) x = 15 cm
3 (a) 18 = 4.24
(b) 20 = 4.47
(c) 8 2.83
(d) 5
(e) 3.5
1 (a) ABC is congruent to 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) XYZ is congruent to ONM (ASA)
(d) DEF is congruent to RQP (ASA)
(e) ACB is congruent to ACD (RHS)
(f) PMN is congruent to NOP (SSS
or SAS)
(g) PRQ is congruent to ZYX (SAS)
(h) ABC is congruent to EDC (ASA)
2 PO = PO (common)
MP = NO (given)
MO = NP (diagonals of isosceles
trapezium are equal)
∆ MPO is congruent to ∆ NOP (SSS)
Answers
1
(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.
5 Need to know how many cows there
are to work out mean litres of milk
produced per cow.
5 (a) x = 3.5 cm
(b) x = 63°, y = 87°
(c) x = 12 cm
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)
(b)
ABC is congruent to HGI (RHS)
ABC is congruent to DEF
(ASA)
(c) ACB is congruent to EDF (SAS)
(d) CAB is congruent to GIH (SAS)
11 5.63 m
140 mm
68 mm
560 mm
420 mm
140 mm
(b) 156 mm
Exercise 12.1
4 15
4 P = 2250 mm
12 (a)
Chapter 12
3 255
8 23 750 mm2
(b) 25 : 1
Exercise 11.4
182
(b) 130 m
2 102 = 62 + 82 ∴ ABC is right-angled
(converse Pythagoras)
6 (a) 4 : 1
(b) 1 : 9
5 28 000 cm3
6 (a) 5 : 1
(c) 125 : 1
3 ∠CDB = ∠DBE = ∠BEA (alt ∠ equal)
∠BAE = ∠CBD (corr ∠ equal)
AE = BD (given)
∆ BAE is congruent to ∆ 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
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).
Exercise 12.3
1
Score
Frequency
Score ×
frequency
(fx)
0
6
0
1
6
6
2
10
20
3
11
33
4
5
20
5
1
5
6
1
6
Total
(a) 2.25
(d) 6
2
40
90
(c) 2
mean
A
3.5
Frequency f × midpoint
40
1340
70
2695
41 ≤ w < 46 43.5
80
3480
46 ≤ w < 51 48.5
90
4365
51 ≤ w < 55 53.5
60
3210
Chapter 13
55 ≤ w < 60 58.5
20
1170
Exercise 13.1
Total
360
16 260
(b) 230 mm
(d) 2450.809 km
(f) 0.157 m
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)
5 (a) 1200 mm2
(c) 16 420 mm2
(e) 0.009441 km2
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
40
4.5
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
1 student’s own diagrams
2 (a) 2600 m
(c) 820 cm
(e) 20 mm
45.17
46 ≤ w < 51
41 ≤ w < 46
29
Mixed exercise
40
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.
36 ≤ w < 41 38.5
C
3
19
5
Q1 = 18, Q3 = 23, IQR = 5
fairly consistent, so data not
spread out
31 ≤ w < 36 33.5
4.12
3 and 5
Total
Mid
point
B
median
Mid
point
Words per
minute (w)
46.14
mode
Marks
(m)
6 (a)
(b)
(c)
(d)
2
(a)
(b)
(c)
(d)
(b) 3
Data set
(b) 36.74 ≈ 37
(c) 30 ≤ m < 40
(d) 30 ≤ m < 40
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) 5 cm3
(c) 5 cm3
5 (a)
(b)
(c)
(d)
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
2 6 h 25 min
(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
Exercise 13.4
3 20 min
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
(b)
13325.5
13324.5
(c)
450
350
(d)
12.245
12.235
(e)
11.495
11.485
(f)
2.55
2.45
(g)
395
(h)
1.1325
41.5
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
4 (a) 195.5 cm ≤ h < 196.5 cm
93.5 kg ≤ m < 94.5 kg
Answers
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
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
1 (a)
5
Exercise 13.5
1 (a) (i)
(ii)
(iii)
(iv)
(v)
(vi)
(b) (i)
(iii)
(c) (i)
(iii)
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
(b) 99 620
y
4
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
2700 m
6000 kg
263 000 mg
0.24 l
0.006428 km2
29 000 000 m3
=y
Time
in
1 (a)
(c)
(e)
(g)
(i)
(k)
+3
Name
Mixed exercise
2x
1
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)
=y
4
4x+2
3
2
1
–5 –4 –3 –2 –1–10 1 2 3 4
3
(–1, –2) –2 – 2y =
x
–3
x
5
–4
–5
x = −1, y = −2
+
–2x
5
4
3
2
1
y
y=
(c)
2
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
2y
x–
+3
1=
0
x = 3, y = −4
(d)
(c)
2 (a) (A) y
x+4
3
(B) y = x + 6
y
5
5
4
4 3
=
x
2
y–
1
y
2
y
4x − 2
y x
y x−4
y = −2
(−2, 6) (ii)
(4, 0)
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
–x
–7
x = −5.5, y = −1.5
(e)
5
y
3x
+
4
3
+3
1
–2 x
2
y=
=
3y
3
–5 –4 –3 –2 –1–10 1 2 3 4 5
(2, –1)
–2
–3
x = 2, y = −1
x
–2
–1
(e)
(−2, −2)
–7 –6 –5 –4 –3 –2 –1 2
(h)
0.25
0
0.75
0.5
–2 –1 0
2 (a)
(b)
(c)
(d)
(e)
(f)
–1
1.25
1
1
2
3
3
4
5
(b)
1
2
2
3
4
5
6
7
(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
10
11
(n) n > –2
1
28
1 (a)
2 (a) c + d = 15
(b) 3 desks and 12 chairs
–5 –4 –3 –2 –1 0
1
Exercise 14.3
7
2
5
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
(o) x ≤ −6
x = 3, y = –4
x = 15, y = 30
x = 4, y = 2
12
x = 15 , y =
1
4
1.75
(m) n > 31
1 (a) x = –3, y = 4
1
(b) x = –1, y =
–5 –4 –3 –2 –1 0
3
0
1.5
3 (a) x ≤ –2
(c) x ≥ 4
(e) x ≥ 9
Exercise 14.1 B
1 (a)
2
(i)
8 6 quarters and 6 dimes
–5
1
–2
(i) r < 11
Exercise 14.2
1
(g)
–4 –3.5 –3
7 number of students is 7
–4
0
(f)
–5 –4 –3 –2 –1 0
(g) x > 9
7
1
(d)
6 a chocolate bar costs $1.20 and a box
of gums $0.75
(c)
(d)
(e)
(f)
0
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
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)
–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)
5
4
3
2
1
(c)
y
y ≥ –2x + 5
x
0
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
(c)
5
4
3
2
1
y
x
1 2 3 4 5
5
4
3
2
1
y
5
4
3
2
1
0
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
186
Answers
y
5 (a)
(e)
x
–2 < y ≤ 4
1 2 3 4 5
x
1 2 3 4 5
y
5
4
3
2
1
(b) (–2, 2), (–2, 1), (–1, 2), (–1, –1),
(0, 1)
0
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
(b)
1 2 3 4 5
–3
–4
–5
0
–5 –4 –3 –2 –1
–1
–2
2y ≤ –4
x
0
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
x
3x – y ≥ 3
5
4
3
2
1
2x < 3y – 6
2
4
(d)
y > –x + 1
1
(d) y ≥ –x – 3
–3
–4
–5
0
–5 –4 –3 –2 –1
–1
–2
2 (a)
(c) y >
0
–5 –4 –3 –2 –1
–1
–2
1 2 3 4 5
3 (a) y ≥ 2x − 1
(b) y < –4x + 3
y
5
4
3
2
1
x
1 2 3 4 5
1
–3
–4
–5
y
(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
5
4
3
2
1
y
x
1 2
Exercise 14.4
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
2 (a)
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
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
y
60
x ≤ 30
it 3
of
pr
50
40
x+
30
y ≤ 20
2y
x+
20
y
10
≤
40
–1 0
–1
x
10 20 30 40 50 60
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
f
it x
x ≤ 1800
+1
.9y
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
(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)
3 (a)
(c)
(e)
(g)
(i)
4 (a)
(c)
(e)
(g)
4 (a) x > –5
(b) x ≤ –1
5
a b
a b
3 1
(d)
x −5
a b
(f)
a 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
6
7
(h)
(i)
Chapter 15
3
Exercise 15.1
x2
6
35
a2
a
c
1
42
−2 − 3x
2x + 1
y2
(b)
6
5xy
(d)
2
1 (a) length = 10 cm, width = 7 cm
(b) length = 14.4 cm, width = 7 cm
(f)
(b )(b − 1)
(h)
4−x
3
the integer solutions are:
(b)
13 p + 2
8p
29
10p
(d)
7
p +1
(f)
18 − 5m
6
5
x
+ y = 1.5
2
x
greatest, x = 2, y = 4, giving + y = 5
2
7 (a) x(x − 2y)
2
2
(b) (a + b)(a − 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
3x 2 y
xy
7
( x + 2)( x
(x
(g)
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
9
3)(
1)
17
2) ( x − 1)
4)
5 3x
)( x + 1)( x 1)
(2x 3)(
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
(c)
3 $5000 at 5% and $10 000 at 8%
(b)
17 − x
(h)
x ( x − 3)(
(i)
2 x = −2, y = 5
2x
5
2
(f)
y
1
(h)
5ab
(d)
7
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°
(e) sin E =
Exercise 15.3
1
(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
(c) 5.14
2
3
1
5
12
12
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
85
13
, tan B =
3
4
3
5
5
4
(c) sin C = , cos C = , tan C =
85
, cos E = , tan E =
(b) 64°
(e) 30°
(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°
5
–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
cos B =
(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)
(b)
(d)
(f)
(h)
90°
36.9°
9.85 cm
10.3 cm
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°
12
7
sin A = 7 , cos A = tan A =
13 ,
12
13
11
84
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
19.6
13
Exercise 15.5
(b) tan x = 3 , tan y = 2
(b) sin B = 5 , cos B =
84
1 1.68 m
2 (a) 30°
(d) tan y = , ∠X = (90 − y)°, tan X =
9 (a)
16
Exercise 15.4
(c) tan 55° = d , tan B = d
5 (a)
(b)
(c)
(d)
(e)
63
65
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
16
65
(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
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
(a)
(b)
2 $300
(c)
3 $500
2
5 (a) $179.10
(b) $40.04
(c) $963.90
(e)
2
y
x −3
y
2
(d) y
4 $64.41
+2
(b) 40 h
(c)
(b) $1386
(c) $1232
0
1
2
3
1
2
1
−2
−7
1
6
−2
−3 −2
x − 2 −11 −6
−3
−2 −3
−6 −11
−4
−3 −4
−7 −12
6
2
x − 3 −12 −7
2
y=x +
1
2
9.5 4.5 1.5 0.5 1.5 4.5 9.5
12
1
25 h
2
2 (a) $1190
−1
−7 −2
1
Mixed exercise
1 (a) 12 h
−3 −2
y
10
(e)
8
3 (a) $62 808 (b) $4149.02
6
4 (a)
4
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
7
2100
2298.74
–10
8
2400
2667.70
–12
(b)
2
–4
–2
0
–2
x
2
4
–4
–6
–8
(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
−3 −2 −1
2
x − 2 x − 1 14
0
1
2 3
y
15
y = x2 − 2x + 1
10
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
6
y
4
2
–4 –2 0
–2
x
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
–4
–6
–8
–10
–12
16 –14
y = x –16
x
–4 –3 –2 –1 0
–5
y
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
2 −1 −2 −1 2
7
y
(b)
(b)
x
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
(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
2 (a)
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
(b) −1 or 2
2 (a)
3
1 (a)
y
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
2
1
–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)
(b)
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
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
–2 –1
–2
–3
–4
–5
–6
–7
–8
Answers
2 (a)
Exercise 18.4
1 (a) −2 or 3
(c) −3 or 4
y
y = x2 − x − 6
(1, 4) and (−2, −2)
(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
(c)
(−4, 5) and (1, 0)
y = −x2 + 4
6
5
4
3
2
1
–6 –5 –4 –3 –2 –10
y
y=
3
x
(ii) y = 0
x
1 2 3 4 5
(i) y = –6
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)
y
8
7
6
y = −2x2 + 2x + 4
5
4
3
2
1
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)
(−2, 0) and (1, 3)
–3 –2 –1 0
x
–2
–3
–4
–5
–6
(iii) y = 6
(c)
y
8
y = 2x2 + 3x − 2 7
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
(d)
(0, 4) and (2, 0)
y
4
3
y =1 x
2
2
y = −0.5x2 + 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)
Exercise 18.5
(e)
50
40
30
1
x
(a) y
3
x − 4x
–5
2
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
58
(f) y
2x 3 + 4 x 2 − 7
(g) y
x 3 − 3x 2 + 6
56
22
6
(h) y
3x 3 + 5x
350
172
66
(a)
50
40
30
20
10
(b)
x
2 4 6 8 10
–10 –8 –6 –4 –2 0
–10
–20
–30
y = x3 + 5 –40
–50
2 4
6
x
8 10
–10 –8 –6 –4 –2 0
–10
–20
–30
–40
y = x3 + 2x − 10
–50
(f)
–5
–7
–1
25
343
2
4
6
2
–14 –48 –106 –194
20
10
14
–2
0
2
–14 –66 –172 –350
50
40
30
20
10
y
–10 –8 –6 –4 –2 0
–10
–20
–30
–40
3
2
y = − 2x + 5x + 5
–50
2 4
50
40
30
20
10
–10 –8 –6 –4 –2 0
–10
–20
–30
–40
3
2
y = − x + 4x − 5
–50
50
40
30
x
6 8 10
2 4
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
3
2
y = 2x + 4x − 7
–50
(g)
y
20
10
–10 –8 –6 –4 –2 0
–10
–20
–30
–40
3
2
y = −x − 3x + 6
–50
(d)
50 y
40
30
20
10
20
10
50
40
30
(c)
y
–10 –8 –6 –4 –2 0
–10
–20
–30
–40
3
3
y = x − 4x
–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
3
y = −3x + 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
30
10
(b)
50 y
40
20
10
–4
1
y = x3 +
x
–2
2 4
6
–2
(c) (i) x = –1.3, 1.8 or 4.5
(ii) x = 0 or 5
(iii) x = –1.6, 2.1 or 4.5
x
0
–10
2
–50
4
(e)
–20
50
–30
40
–40
30
–50
20
y
y = x3 −
2
x
10
3
x
–3
1
x
1
3
y
=
x
+
(b)
x
(a) y
(f)
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
3
2
y = x − 5x + 10
–50
(e)
3
x
20
y
50
40
30
(d)
y = 2x +
40
–36.875 –18 –4.625 4 8.625 10 8.875 6 2.125 –2 –5.625 –8 –6 10 46
(b)
(c)
y
x−
–2 –1 –0.5
–2.67 –1.5 0
1.5
4.8
9.9
0
0.1
0.2
0.5
1 2
3
–9.9
–4.8
–1.5
0 1.5 2.67
–2
x
0
–10
2
4
–20
–30
–40
4
y = x +2−
x 12.33 8 7 10.25
3
y 2x +
–7 –5.5 –5 –7
x
2
y x3 −
–26.33 –7 1 3.875
x
1
y x2 − x +
11. 67 5.5 1 –1.25
x
(a)
25
Answers
22.04 42.01
–50
–37.99 –17.96 –5.75 –1 4 9. 67
–15.4 –30.2
30.2
9.992 19.99
15.4
7
5 5.5
7
–19.99 –9.992 –3.875 –1 7 26.33
–4.76 –9.89
9.91
(c)
y
50
4.84
1.75
1 2.5 6.33
(f)
y
y = x2 + 2 −
50
4
x
40
15
30
30
20
20
y=x−
1
x
0
–5
40
10
5
–2
10.001 5.008 2.125 2 8.5 27.33
20
10
194
–0.1
–27.33 –8.5 –2 –2.125 –5.008 –10.001
2
–4
–0.2
–4
x
2
4
–4
–2
0
–10
4
y = x2 − x +
1
x
10
x
2
y
–4
–2
0
–10
–10
–20
–20
–15
–30
–30
–20
–40
–40
–25
–50
–50
x
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
8
18
y = 2–
–2
y
4
2
1 0.5 0.25 0.125 0.0625
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
x
0
0.2 0.4 0.6 0.8
y = 10 x
1
1.58 2.51 3.98 6.31 10
1
2 x − 1 –1 –0.6 –0.2 0.2 0.6
12
10
8
1
y
y = 10 x
6
4
2
0
–2
y = 2x − 1
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)
y
60 y
3
55 y = x − 1
50
45
40
35
30
25
20
15
10
5
x
1.2
5
x
6
(ii) ≈ 42
Exercise 18.6
1
x
1 2 3 4 5 6
(b) gradient = 12
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
0
–6 –5 –4 –3 –2 –1
–5
–10
–15
–20
–25
–30
–35
–40
–45
–50
–55
–60
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)
10
C
y
Exercise 19.4
D
1 (a) 15° (isosceles )
(b) 150° (angles in a )
(c) 35° (∠MON = 80°, and MNO 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
–2
y = x3
–1 0
–2
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
–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
3 (a) y =
(c)
4
−
x
(b) y
Exercise 19.1
1 (a)
B
A
2 (a) 2, student’s diagram
(b) 2
3 student’s own diagrams but as an
example:
6 144°
7 (a) 22°
(b) 116°
(c) 42°
8 (a) 56°
(b) 68°
(c) 52°
DIAGRAMS ARE NOT TO SCALE BUT
STUDENTS’ SHOULD BE WHERE
REQUESTED
1
(a)
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
(c)
A
A
B
C
2
O
4 cm
(c) 1
(f) 8
3
2 6.5 cm
3 (a) 49.47 cm
(b)
A 3 cm
1 (a) x = 25° (b) x = 160°, y = 20°
Answers
4 35°
Exercise 19.5
Exercise 19.3
196
3 ∠DAB = 65°, ∠ADC = 115°,
∠DCB = 115°, ∠CBA = 65°
5 59.5°
x
y
8
y = −x
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
Chapter 19
(b) A = 0, B = 3, C = 4, D = 4, E = 5,
F = 2, G = 2, H = 2
(b) 177.72 cm
1m
B
4
(ii) eight
(d) (i)
Chapter 20
Exercise 20.1
4 km
3 km
(ii) none
School
5
A
B
6 NOT TO SCALE – note that on
student drawing point X should be
2 cm (10 m) from B.
D
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
∆OCB), x = 90° (angle in semicircle), y = 62° (angles in a ∆)
z = 18° (base angle isosceles
∆ODC)
(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 ∆)
(d) x = 120° (angle at centre), y = 30°
(base angle isosceles ∆)
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
Frequency
1.5 km
A
Cell
tower
1 (a)
Frequency
Road
(e) (i)
SCHOOL
SC
SCHOOL
60
50
40
30
20
10
0
60
50
40
30
20
10
0
English exam marks
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
Frequency density
Frequency
one person
12
10
8
6
4
2
0
Heights of trees
0
2
1 (a) 3 : 4
(c) 7 : 8
2 (a)
(c)
(e)
(g)
(i)
4 6 8 10 12
Height in metres
(b) 19
(c) 6−8 metres
15 20 25 30 35 40 45 50 55
Age (years)
4 (a) 300
(b) 480
(c) 100
Frequency
2 (a)
Exercise 20.2
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%
2 (a)
(b)
(c)
(d)
166 cm
Q1 = 158, Q3 = 176
18
12.5%
198
Answers
3 60 cm and 100 cm
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
Exercise 21.2 A
56
58 60 62
Mass (kg)
64
1 (a) 1 : 2.25
(b) 1 : 3.25
2 (a) 1.5 : 1
(c) 5 : 1
(b) 5 : 1
(c) 1 : 1.8
Exercise 21.2 B
1 240 km
3
2 30 m
Histogram of airtime minutes
Q2
Q1
(e) 1 : 4
x=4
x=3
x = 1.14
x = 2.67
x = 13.33
6 810 mg
(b) 60−62 kg
(c) 7.4%
(d) 10 kg
P80
Q3
(b)
(d)
(f)
(h)
(j)
x=9
x = 16
x=4
x = 1.875
x=7
5 60°, 30° and 90°
Frequency density
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
5 teenagers
20 30 40 50 60 70 80 90 100 110 120130140150
Time (minutes)
4 (a) 6.5 cm
(b)
Cumulative
frequency
Number of students
1 (a)
Masses of students
20
18
16
14
12
10
8
6
4
2
0
54
(b) 6 : 1
(d) 1 : 5
Cumulative frequency of plant heights
y
30
20
10
0
Q1
0
1
2
3
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
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) Yes,
(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
=
B 150
8
1
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
8 (a) b = 40
(b) a = 17 7
9 (a) y = 2.5
(b) x = 2
9
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
3 (a) $51 000 (b) $34 000
2 9 cm
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) No,
(c) Yes,
2 (a) $175
A
B
=
10
1
(b) $250
5 (a) 75 km (b) 375 km
(c) 3 h 20 min
6 (a) 15 litres (b) 540 km
7 (a) inversely proportional
2 1 : 50
3 (a) (i) 85 km
(ii) 382.5 km
(iii) 21.25 km
5 breadth = 13 cm, length = 39 cm
6 X cost 90c, Y cost $1.80 and Z cost 30c
7 9 years
8 97 tickets
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) = 3x2 + 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
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
−1
f (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
(c)
y
8
D
D'
3 (a)
A
x
B''
4
2
2
4
6
8
10
X
2
y
–10 –8 –6 –4 –2 0
–2
A''
D'
D
1 (a)
u r  5
1 (a) AB =  
 0
y
A'
A
7
6
5
4
3
B'
2
1
2
C'
y
B
x
A
2
4
2 (a)
u r  −5
(g) CD =  
 −6
u r  −5
(h) BE =  
 −6
 −5
(k)  
 −6
(a)
B
10
8
4
A
6
C
B
4
2
–4
u r  9
(f) EC =  
 6
2
y
X
u r  1
(e) DB =  
 6
y
X
–4 –2 0
B
–2
u r  −1
(d) BD =  
 −6
(l) Yes
(b) rotation 180° about (0, 0)
–4
u r  0
(c) AE =  
 −6
 9
(j)  
 0
–4
–5
–6
A'' –7
2
u r  4
(b) BC =  
 0
(i) they are equal
1 2 3 4 5
–2
B'' –3
C''
4
C
x
–5 –4 –3 –2 –1 0
B'
10
Exercise 23.2 A
Exercise 23.1 C
(b)
8
(b) rotation 180° about (4, 5)
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
–2
x
6
–10
Exercise 23.1 B
0
C
4
–8
x
for MNOP: rotate 90° clockwise about
(2, 1) then reflect in the line x = 5.5
–2
2
–6
for ABC: rotate 90° clockwise about B
then reflect in the line x = −6
–4
D
–4
6 possible answers are:
A'
B
A
4
x
–10
(a)
F'
6
(d)
B
y
8
C
A'
C'
–10 –8 –6 –4 –2 0
–2
C
–4
B'
–6
D''
–8
10
C'
6
C''
(b) enlargement scale factor 2, using
(8, −1) as centre
y
2
4
A
2
x
–4
–2
0
x
2
4
6
8
–2
–4
Answers
201
(g) b
(b)
(j)
D
(c)
F
G
H
 4
C 
 −3
 2
B 
 3
 9
E 
 3
Exercise 23.2 B
 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
202
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
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)
 −8
1 (a)  
 16 
(i) −7a + 7c
6 (a) A(–6, 2), B (–2, –4), C (5, 1)
u r  4
(b) AB =  
 −6
E
 −3
D 
 −3
(h) –c
+ 3c
3 (a) – (e) student’s own diagrams
C
 8
3 A 
 1
b
2
 −3
(ii) y =  
 −3
 10 
(iii) z =  
 −4
(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
–1 0
–1
(b)
x
1 2 3 4 5 6 7 8 9 10
y
10
9
8
7
6
5
4
3
2
1
–1 0
–1
(c)
x
1 2 3 4 5 6 7 8 9 10
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
2 (a) & (b)
0
3
y
7
6
5
4
3
2
1
 −12 −6 6 
(c) 
 −14 4 −10
5 (a) x = 3
(b)
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
Exercise 23.4
0 6 5 
1 (a) 
 7 2 −3
2
(c) 
 −11
3 (a) AB =
(b)
3
 9
3 (a) enlargement, scale factor 3, centre
of enlargement the origin
 −3 0
(b) 
 0 3

3
10
(ii)
(d)
6
(c)
4
(a)
2
x
–10 –8 –6 –4 –2
–2
2
(b)
4
6
8
10
–4
Mixed exercise
8 −5
1
y
8
–10
12  −4
6
 6
 −10 −4 ; so AB ≠ BA
(b) (i) –6
(ii) –84
–2
–3
–4
–5
–6
–7
–6
(b) (i)
−1 
1 2 3 4 5
0
3
4
3
 0
x
–5 –4 –3 –2 –1
2 (a) R' (2, 8)
(b) a stretch parallel to the y-axis
(x-axis invariant), scale factor 4
–8
reflect in the line x = −1
(iii) reflect in the line y = −1
9
and BA =
1
 7 0 −3
4 (a)  11 0 4 
4 −1
2 
(ii) rotate 90° clockwise about
the origin
2 −1
4 3 
2 (a) –12

5  −3
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
1 (a) (i)
 −2 −2 1 
(b) 
4 −3
 1
1
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)
c
b
11 (a) & (c)
y
7
10
8
8
6
4
6
b+c
4
A
a
b
a
b
8
2
4
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
C 8
10
9
8
7
6
5
4
3
2
1
–8
1
(b) 
0
A
0
1
 2 0
(d) 
 0 2
2
(e) 
0
0
2
x
1 2 3 4 5 6 7 8 9 10
5
(c) 
4
7
(d) 
6
Chapter 24
Exercise 24.1
1
Card
Coin
H
T
H
T
H
T
H
T
R
7
8
Y
G
5
6
(e) |P| = –3
 1 2 
−


(f) P–1 =  3 3 
2
1

− 
 3
3
10 a = 7, b = 2 and c = –9
B x
8 10
A
–6
–10
 2 5
9 (a) 
 4 3
1
0
(b) 
1
0
2a + 3b
6
x
y
–1 0
–1
A
–4
D
–2
–2
a–b
(iv)
204
2
C
–10 –8 –6 –4 –2 0
–2
B
2
(iii)
y
B
2
1
2
3
4
3 (a) & (b)
2
3
1
3
G
H
A
B
C
D
E
F
7
11
Green
4
11
Yellow
8
11
Green
3
11
Yellow
Green
Yellow
Exercise 24.2
Mixed exercise
1 (a)
1
2
Blue
1
3
Yellow
1
6
(b)
Black
1
4
1
2
H
1
2
1
2
T
H
1
2
1
2
T
H
1
2
(c)
2 (a)
1 (a) & (b)
1
2
1
2
1
2
1
2
1
2
(d)
H
H
1
2
1
2
1
2
T
1
2
(b)
(d)
1
8
1
2
(c)
1
6
T
1
12
1
2
H
T
H
T
T
H
T
5
12
1
6
2
1
6
3
1
6
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)
2 (a) & (b)
2
7
1
2
(e) 0, not possible on three coin
tosses
1
5
7
(c)
5
7
1
2
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
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
H
T
H
T
1
2
T
1
12
1
6
5c
5
6
10c
5c
1
10c
5
5
10c
(d)
1
21
(e) 1 (there are no 5c coins left)
Answers
205