Table of contents
1. ESSENTIAL REVISION …………..……………………………. 3 - 39
a) Number & Language …..………………………….……
b) Standard Form and Approximation ..…………………...
c) Calculator
.….……….....…………………….………
d) Ordering..…………………………………………... ……
e ) Fractions ………………………………..………………..
f ) Conversion…………………….....…………………….…
2. LIMITS OF ACCURACY …………………..……………………. 40
a) Direct Problems………………..…………………………
b)(+)
& (×) Operations…..………………………………...
c) (-)
& (÷) Operations ….……….....………………………
d)Word Problems
….……...…….....………………………
3. SETS AND VENN DIAGRAM …………..…………… 61
a) Shading
……………..……...…………………….……
b)Elements shown in each region…..………….…………...
c) Problems involving sets…………... …………………..…
1
4. ALGEBRAIC REPRESENTATION &
MANIPULATION ………………………………….…………….
93
a) Expansion………………..…………….…………………
b) Factorisation…..………………………………………....
c) Substitution & Subject of the Formula ...……….………
5. SOLVING EQUATIONS………………………….……………. 116
a) Simple Linear Equations ...……..…….……………………
b) Solving Inequalities ……………………………………
c) Solving Quadratic Equations by Factorising ..………………..
d)
Solving Quadratic Equations using Quadratic Formulae .………
e) Simultaneous Equations...………….……………….……
f) Constructing Equations ..…………….……………….…
g)The Completed Square Form..……..………………..…...
6. RATIO, PROPORTION & PERCENTAGE..……………. 148
a) Ratio………………………………………………………
b)Inverse Proportion……………….………………………
c) Percentage ……………………….………………………
d)Percentage Increase & Decrease…….....…………………
e) Simple & Compound Interest ..………………...…………
f) Variation…………………………….....……………….…
2
Number
&
Language
3
a) Number & Language
1
From the list of numbers above, write down
a- A multiple of 8
Answer (a) ………….….….[1]
b- A square
Answer (b) …………..…….[1]
c- A cube
Answer (c) …………..…….[1]
d- Two prime numbers
Answer (d) …………..…….[1]
e- A factor of 156
Answer (e) ………….……..[1]
f- Square root of 784
Answer (f) ………….……...[1]
g- Two numbers whose product is 567
Answer (g) ………….…..….[1]
______________________________________________________
2
From the list of numbers above, write down
a- An integer
Answer (a) ………..….…….[1]
b- A square number
Answer (b) ………………….[1]
c- An irrational number
Answer (c) …………….…….[2]
4
3
19
225
20
8
17
15
32
35
30
From the list of numbers above, write down
a- A multiple of 7
Answer (a) ………….…….[1]
b- A square
Answer (b) ………….…….[1]
c- A cube
Answer (c) ………….…….[1]
d- Prime number
Answer (d) ………….…….[1]
e- A factor of 400
Answer (e) ………….…….[1]
f- Square root of 289
Answer (f) ………….……. [1]
_____________________________________________________________
4
From the list of numbers , find
a- A prime number
Answer (a) ………….…….[1]
b- A cube number
Answer (b) ………….…….[1]
c- A square number
Answer (c) ………….…….[1]
5
5
a) Find the value of
(i)
The square root of 64.
Answer (i) …………..…….[1]
(ii)
The cube root of 64.
Answer (ii) ………….…….[1]
b) (i) Write down a common factor of 15 and 27, which is greater than 1.
Answer (i) ………….…….[1]
(ii) Write down a common multiple of 10 and 12, which is greater than 1.
Answer (ii) ………….…….[1]
c)
Write down 40 as a product of prime numbers.
Answer (c) 40 = ………….…….…….[2]
d)
Write down 56 as a product of prime numbers.
Answer (d) 56 = …………….…….…….[2]
6
6.
From the list above, write down
Answer (a) ………….…….[1]
a- A prime number
b- A factor of 27
Answer (b) …..……….…….[1]
c- A multiple of 4
Answer (c) ……..…….…….[1]
d- An irrational number
Answer (d) …..……….…….[1]
______________________________________________________
7.
a- Find the least common multiple of 18 and 24.
Answer (a) ………..….…….[2]
b- Find the highest common factor of 18 and 24.
Answer (b) ………..….…….[2]
c- Write down the factors of 48 which are between 10 and 40.
Answer (c) ………..….…….[2]
7
8
_____________________________________________________
9
10
8
STANDARD
FORM
AND
APPROXIMATION
9
b) Standard Form and Approximation
1
2
3
10
4
5
6
7
11
8
9
10
12
11
12
13
c) Calculator
1
2
3
14
4
5
15
6
7
8
16
9
10
11
17
12
13
18
14
15
16
19
17
18
20
d) Ordering
1
2
3
21
4
5
6
22
7
For each part, choose a symbol from those above to make a correct sentence
a) 0.4 …………..……
b) 42 …………..……16
c) 0.3 ………….……
______________________________________________________________________________________________
8
_______________
______________
9
23
__________
____________
10
__________________________________________________________________________________________
11
12
24
13
25
Fractions
26
e) Fractions
1
2
3
27
4
5
28
6
7
29
8
9
30
10
11
31
12
13
32
14
15
Ali, Sami , Kahid and Fadi shared a cake.
a) Ali ate 2/9 of the cake and Sami ate 1/6 of the cake. Show that
11/18
Do not use your calculator and , show all your working.
11
/1
8
of the cake remained
.
[2]
b) Khalid ate
of the cake that remained.
Find the fraction of the cake that left was for Fadi.
33
Answer (b) ………….…….…..…………….[2]
16
17
34
18
19
35
f) Conversion:
** Distance:
1000
Km
100
m
10
cm
mm
** Area:
2
(1000)
2
Km
(100)
m2
2
(10)
** Capacity ( Volume):
100
L
10
cl
ml
cm3
1000
L
ml
cm3
** Mass:
1000
Kg
1000
gram
mg
36
2
cm2
mm2
1
2
3
4
37
5
6
7
How many glasses, each holding 200 cm3, can be filled completely from a full 4.5 Litre
bottle of water?
Answer ………....……………..………
38
[2]
A bottle of mass 480 grams contains 75 centilitres of water.
8
(i)
(ii)
(iii)
9
Write 75 centilitres in millilitres.
Answer ………....……………..………ml
[1]
Answer ………....……………..………L
[1]
Answer ………....……………..………Kg
[1]
Write 75 centilitres in Litres.
Write 480 grams in kilograms.
(a) Change 3000 mm2 to cm2.
Answer ………....……………..………
[1]
Answer ………....……………..………
[1]
Answer ………....……………..………
[1]
Answer ………....……………..………
[1]
Answer ………....……………..………
[1]
Answer ………....……………..………
[1]
(b) Change 65000 cm2 to m2.
(c) Change 2.5 m2 to cm2.
(d) Change 8500 gram to kg.
(e) Change 4.2 kg to gram.
(f) Change 4300 cm3 to Litre.
39
40
41
a) Direct Problems
1
Hana's height is given as 162 cm, correct to the nearest cm.
Find the upper and lower bounds for Hana's height?
Answer
2
cm [2]
The equilateral triangle with 6.5 cm side, all measurements are given correct to the nearest centimeter.
Find the upper and lower bound for the sides.
Answer
42
cm [2]
3
A sack of sand weighs 20 kg measured to the nearest kg.
Find the upper and Lower bound for the weigh?
Answer
4
Kg [2]
A bottle of water of 300 ml. volume measured to the nearest ml.
Find the upper and Lower bound for the volume?
Answer
5
ml [2]
The population in a country is given as 50 000, correct to the nearest 1000.
What is the lowest and highest possible number of people on the country?
Answer
6
A square of a side 3 metres, the sides are given correct to nearest metre.
43
[2]
Find the upper and lower bound for the sides.
Answer
7
m [2]
The number of people on a bus is given as 50, correct to the nearest 10.
What is the lowest and highest possible number of people on the bus?
Answer
8
44
[2]
9
10
11
45
12
13
46
b) ( + ) & ( x ) Operations
1
2
3
47
4
5
6
48
7
8
49
9
10
50
c) (−) & ( ÷ ) Operations
1
2
51
3
Hakim and Basira measure their heights.
Hakim’s height is 157 cm and Basira’s height is 163 cm, both correct to the nearest
centimeter. Find the greatest possible difference between their heights.
.
Answer
4
………….…….…..…………….[2]
A rectangular pool’s depth is 150 cm, correct to the nearest 10 centimetre.
(a)
Find the upper and lower bound for the depth of the pool.
Answer
(b)
Find the greatest difference between upper and lower bounds for it’s depth.
Answer
(c)
cm [2]
………….…….…..…………….[2]
Find the minimum value for the ratio between upper and lower bounds for it’s depth.
Answer
52
………….…….…..…………….[2]
5
Two towers has different heights. Tower A is 40 m height, and tower B is 20 m height
Both heights correct to the nearest 2 metre.
Tower B
Tower A
(a) Complete the statement about their heights, H m , of th two towers.
Answer (a)
m [2]
m [2]
(b) Find the greatest possible difference between their heights.
Answer (b)
………….…….…..…………… m [2]
(c) Find the minimum value for the ratio between their heights.
Answer (c)
53
………….…….…..……………. m [2]
d) Word Problems
1
2
3
54
4
5
6
55
7
8
56
9
10
57
11
12
7
58
13
59
60
61
SYMBOLS
• Sets listed inside a pair of curly brackets { }.
• Capital letters are usually used as names for sets
Symbol
Meaning
Empty set.
, {}
ε
Universal set
union
intersection
B
⊂A
B is a proper subset from A
B
⊄A
B is not a subset from A
B
⊆A
B is an improper subset from A
A'
Complement of a set
n (A)
Number of elements of a set
a
B
a B
** Note : B
a is an element of set B
a is not an element of set B
⊆ A : Same elements in both sets.
Or :
is improper subset from all sets.
62
{1,3,5}
⊆ {1,3,5}
⊆ {1,3,5}
a) Shading
A'
B'
A∩B
A ∩ B'
A'∩ B
A'∪ B
(A ∪ B)'
A ' ∩ B'
63
A∪B
A ∪ B'
(A ∩ B)'
A' ∪ B'
A
A ∪ B …(B⊂ A)
A∩B
A∪ B
A∩B
(A ∩ B)'
(A ∪ B)'
A ∩ B'
64
A∪B
A∪ C
A∩ B
B∩C
(A ∪ B)'
(A ∪ C)'
Aʹ ∩ Bʹ
(A ∪ B)'
B∩ C ∪ A
A∩C
Aʹ ∩ Bʹ
65
A∩B∪C
A∪B∪C
A∩B
Aʹ∪ (C∪B)
A∩ B∩C
A∩C
A∪B
Aʹ∩(C∪B)
Cʹ∩(A∪B)
Cʹ∩(A∩B)
66
Important notes for shading:
A′ = not A
B′ = not B
(A∪B)′= not all A and all B = A′ ∩ B′
(A∩B)′ = not common part of A & B = A′ ∪ B′
A ∪ B′ = all in A and rectangle
(A ∪ C) ∪ B′ = all in (A ∪ C)
A′ ∪ B = all in B and rectangle
A′ ∪ (B ∪ C) = all in (B ∪ C) and rectangle
A′ ∩ B = moon part of B
A′ ∩ (B ∪ C) = moon part of (B ∪ C)
A′ ∩ (B ∩ C) = moon part of (B ∩ C)
A ∩ B′ = moon part of A
(A ∪ C) ∩ B′ = moon part of (A ∪ C)
(A∩ C) ∩ B′ = moon part of (A ∩ C)
67
1
2
68
3
4
6
69
5
70
6
71
b) Elements shown in each region
1
72
2
73
3
74
4
75
5
76
6
77
7
78
8
79
9
80
10
11
If the n( ) = 34 , n(A B) = 29 , n (A
B' ) = 13 , n(A) = 18
Complete the Venn diagram to show this information.
ε
B
A
[2]
81
12
If the n( ) = 25 , n(A
B) = 19 , n (A
B' ) = 11 , n(A) = 15
Complete the Venn diagram to show this information.
ε
B
A
[2]
13
If the n( ) = 40 , n(A B) = 35 , n (A B' ) = 10 , n(A) = 30
Complete the Venn diagram to show this information.
ε
B
A
[2]
82
14
83
c) Problems involving sets
1 In a group of 24 students, 21 like football and 15 like swimming.
One student does not like football and does not like swimming.
Find the number of students who like both football and swimming.
ε
F
S
Answer ………....……………..……….
[2]
In a group of 45 boys, 22 like basketball and 25 like Tennis.
Three students do not like basketball and do not like Tennis.
Find the number of students who like both basketball and Tennis.
2
ε
B
T
Answer ………....……………..……….
84
[2]
3
There are 55 boys in a class, 32 study Science , 25 study Maths. And 5 neither study Science or
Maths.
ε
S
M
Complete Venn diagram to show this information.
[2]
a) Find the number of boys who study both Science and Maths.
Answer (a) ………....……………..……….
[1]
b) Find the number of boys who study only Science.
Answer (b) ………....……………..……….
[1]
c) Find the number of boys who didn’t study Maths.
Answer (c) ………....……………..……….
85
[1]
4
5
86
6
87
7
88
8
89
9
90
91
92
a) Expansion :1
a)
b)
Expand and simplify:-
= ………………………………………………………..
= ………………………………………………………..
c)
= ………………………………………………………..
d)
= ………………………………………………………..
e)
= ………………………………………………………..
f)
= ………………………………………………………..
g)
= ………………………………………………………..
h)
= ………………………………………………………..
i)
= ………………………………………………………..
………………………………………………………..
j)
= ………………………………………………………..
93
………………………………………………………..
k)
l)
m)
= ………………………………………………………..
= ………………………………………………………..
= ………………………………………………………..
n)
= ………………………………………………………..
o)
= ………………………………………………………..
………………………………………………………..
………………………………………………………..
p)
= ………………………………………………………..
………………………………………………………..
………………………………………………………..
q)
= ………………………………………………………..
………………………………………………………..
………………………………………………………..
r)
= ………………………………………………………..
………………………………………………………..
94
………………………………………………………..
2
3
95
4
5
Expand the brackets and simplify :a)
= …………………….……………………………………………..
…………………….……………………………………………..
b)
= …………………….……………………………………………..
…………………….……………………………………………..
96
6
7
97
b) Factorisation
1
Factorise completely:
= ………………………………………………………..
a)
b)
= ………………………………………………………..
c)
= ………………………………………………………..
d)
= ………………………………………………………..
= ………………………………………………………..
e)
f)
49
= ………………………………………………………..
g)
= ………………………………………………………..
h)
= ………………………………………………………..
i)
= ………………………………………………………..
j)
= ………………………………………………………..
98
k)
l)
m)
n)
o)
p)
q)
u)
= ………………………………………………………..
= ………………………………………………………..
= ………………………………………………………..
= ………………………………………………………..
= ………………………………………………………..
= ………………………………………………………..
= ………………………………………………………..
= ………………………………………………………..
99
2
3
100
3
4
101
5
6
102
7
103
8
9
104
10
11
105
12
(c)
106
c) Substitution & Subject of the Formula
1
2
107
3
4
108
5
6
109
7
8
110
9
10
11
111
12
13
Make d the subject of the formula
c=
Answer
14 Make w the subject of the formula
C
112
………....……………..……….
[3]
.
Answer
………....……………..……….
[4]
Answer
………....……………..……….
[3]
Answer
………....……………..……….
[3]
15 Make p the subject of the formula
.
16 Make c the subject of the formula
.
.
113
114
115
a) Simple Linear Equations
1
Solve the following equations
a)
Answer (a)
………....……………..……….
[2]
Answer (b)
………....……………..……….
[2]
b)
c)
Answer (c)
………....……………..……….
[2]
Answer (d)
………....……………..……….
[2]
d)
116
2
Solve the following equations
(a)
.
Answer (a)
………....……………..……….
[2]
Answer (b)
………....……………..……….
[2]
Answer (c)
………....……………..……….
[3]
Answer (d)
………....……………..……….
[3]
(b)
(c)
3
Solve the equation
117
4
5
Answer
………....……………..……….
[3]
Answer
………....……………..……….
[3]
Solve the equation
Solve the equation
Answer
118
………....……………..……….
[3]
6
7
8
119
9
10
120
11
12
\
121
b) Solving Linear Inequalities
1
Solve the inequality
.
Answer ……….......……………..……….
2
3
Solve the inequality
Solve the inequality
[2]
.
Answer ……….......……………..……….
[3]
Answer ……….......……………..……….
[2]
.
122
4
Solve the inequality
Answer ……….......……………..……….
[3]
_____________________________________________________________________________________________
5
Solve the inequality
Answer ……….......……………..……….
6
[3]
Solve the inequality
Answer ……….......……………..………. [3]
123
7. Solve the inequality:
Answer ……….......……………..………. [3]
8 Solve the inequality:
A
n
s
w
e
r
…
124
Formula
________________________________________________
Example :
125
Notes :
126
c) Solving Quadratic Equations by Factorising
1 (a)
2
127
3
4
128
5
Solve the following equations by factorising:
a)
Answer (a)
………....…… or
………..……….
[3]
Answer (b)
………....…… or
………..……….
[3]
Answer (c)
………....…… or
………..……….
[3]
b)
c)
c)
d)
129
d) Solving Quadratic Equations using Quadratic Formulae
1
Solve the equation.
Show all your working and give your answers correct to 2 decimal places.
Answer
2
………....…… or
………..……….
Solve the equation.
Show all your working and give your answers correct to 2 decimal places.
………....…… or
Answer
130
………..……….
[4]
[4]
3
Solve the equation.
Show all your working and give your answers correct to 2 decimal places.
Answer
4
………....…… or
………..……….
Solve the equation.
Show all your working and give your answers correct to 2 decimal places.
………....…… or
Answer
131
………..……….
[4]
[4]
5
Solve the equation.
Show all your working and give your answers correct to 2 decimal places.
………....…… or
Answer
6
………..……….
[4]
………..……….
[4]
Solve the equation.
Show all your working and give your answers correct to 2 decimal places.
………....…… or
Answer
132
e) Simultaneous Equations
1
2
133
3
4
134
5
Solve the simultaneous equations.
Answer
…………………..…………....
……………...…………..……
135
[3]
Equations
and
Inequalities
136
e) Constructing Equations :1
Ravindra scores x marks in a test.
Manpreet scores 4 more marks than Ravindra.
(i)
Write down Manpreet’s mark in terms of .
Answer (i)
(ii)
………....……………..……….
[1]
………....……………..……….
[1]
Tamsin scores 3 times as many marks as Ravindra.
Write down Tamsin’s mark in terms of .
Answer (ii)
(iii)
Write down and simplify the total of the three marks in terms of .
Answer (iii)
137
………....……………..……….
[3]
2
138
3
4
139
5
a)
50º
xº
2yº
Complete the equation
………....……………..……….
[2]
………....……………..……….
[2]
b)
160º
100º
xº
yº
Complete the equation
140
6
The length, in centimeters of the sides of sides of a triangle are
and
The perimeter of the triangle is 52 cm.
(a) Use this information to write down an equation in .
(b) Find the value of
Answer (a)
………....……………..……….
[3]
Answer (b)
………....……………..……….
[2]
.
141
7
x cm
2x cm
The perimeter of the rectangle in the diagram above is 36 centimeters.
a) Find the value of .
Answer (a)
………....……………..……….
[3]
………....……………..……….
[2]
b) Using this value of x, calculate the area of the rectangle.
Answer (b)
142
8
3 cm
17 cm
The diagram above shows another rectangle.
a) Write down two equations in a and b.
Answer (a)
………....……………..……….
[2]
b) Solve these two equations simultaneously to find a and b.
Answer
……………………………..…....
……………..…………..……….
143
[3]
9
In this question the diagrams are not to scale.
a) Calculate the value of .
800
500
1100
0
Answer (a)
………....……………..……….
[2]
………....……………..……….
[2]
b) Write down an equation in x and y.
800
3y0
0
950
Answer (b)
144
10
145
11
146
12
147
f) The Completed Square Form
1
Write
in the completed square form
.
Answer ……………………………………..……….
2
Write
in the completed square form
.
Answer ……………………………………..……….
148
[3]
[3]
3
Write
in the completed square form
.
Answer ……………………………………..……….
4
Write
in the completed square form
.
Answer ……………………………………..……….
149
[3]
[3]
5
Write
in the completed square form
.
Answer ……………………………………..……….
6
Write
in the completed square form
.
Answer ……………………………………..……….
150
[3]
[3]
151
152
a) Ratio
1. Direct Proportion
1
Mark and Amy share $600 in the ratio 5: 1
Calculate how much money does Mark receives?
Answer
2
Chris and Mark share $45 in the ratio 7:2
[2]
Calculate how much Chris receives.
Answer
3
………....……………..……….
………....……………..……….
[2]
A packet of sweets costs $ 2.25.
Felipe and his brother share the cost in the ratio of 5:4.
How much does Felipe pay?
Answer
153
………....……………..……….
[2]
4
Mortar is a mixture of cement, sand, and lime in the ratio
Cement : sand : lime = 1 : 5 : 2.
Calculate how much sand there is in a 12 kg bag of mortar.
Answer
5
………....……………..……….
[2]
The ratio of teachers : male students : female students in a school are 2:17:18.
The total number of students is 665.
Find the number of teachers.
Answer
6
………....……………..……….
The train fare is $ 24 for an adult. The train fare for a child is
a)
for an adult fare.
Find the fare for a child.
Answer (a) ………....……………..……….
b)
[2]
[1]
the total fare for Mr. and Mrs. Sayed and their 3 children.
Answer (b) ………....……………..……….
154
[1]
7
8
155
9
10
156
Inverse Proportion
1
Eleven taps fill a tank in three hours. How long would it take to fill the tank if only six
taps are working?
Answer
2
………....……………..………. [2]
Nine children share out equally the chocolates in a large tin and get eight each.
If there were only six children, how many would each get?
Answer
4
………....……………..………. [2]
A field of grass feeds 24 cows for six days. How long would the same field feed 18 cows?
Answer
157
………....……………..………. [2]
c) Percentage
1
2
3
4 Calculate 30% of 270 m
Answer
158
………....……………..………. m [2]
5
Calculate 60% of 200 .
Answer
6
[1]
A Write 0.36 as a fraction. Give your answer in its lowest terms.
Answer
7
………....……………..……….
………....……………..……….
[2]
A Write 0.48 as a fraction. Give your answer in its lowest terms.
Answer
8
159
………....……………..……….
[2]
9
10
10
160
11
12
12
13
When Jon opened a packet containing 40 biscuits, he found 8 biscuits were broken.
What percentage of biscuits were broken?
Answer
161
………....……………..……….
[2]
d) Percentage Increase & Decrease
1
2
162
3
4
5
163
6
7
164
8
9
10
165
11
12
166
13 The $72 is 60% of the cost of a ticket.
Calculate the cost of the ticket.
Answer
………....……………..……….
Simple
And
Compound
Interest
167
[2]
e) Simple & Compound Interest
** Simple Interest
1
Sophie invests $450 at a rate of 1.5% per year simple interest.
Calculate the interest she earns after 8 years.
Answer
2
………....……………..……….
[2]
………....……………..……….
[2]
Ally invests $720 at a rate of 3% per year simple interest.
Calculate the total amount she earns after 5 years.
Answer
_____________________________________________________________________________________
3
Maria puts $600 into a bank account for 3 years at a rate of 3.5% per year simple interest.
Calculate the total amount she earns after 3 years.
Answer
168
………....……………..……….
[2]
4
Mark invests $1200 at a rate of 4.5% per year simple interest.
Calculate the total amount he earns after 6 months.
Answer
5
169
………....……………..……….
[3]
** Compound Interest
1
Amiria invests $200 for 2 years at a rate of 1.5% per year compound interest.
Calculate the total amount Amiria has at the end of the two years.
Answer
2
………....……………..……….
[3]
John invests $600 for 3 years at 4% per year compound interest.
Calculate the total amount he has at the end of the three years.
Answer
3
………....……………..……….
[3]
Sally invests $1200 at a rate of 5.5% per year compound interest.
Calculate the interest she earns after 10 years.
Answer
170
………....……………..……….
[3
4
5
171
6
172
** Mixed Questions
1
173
2
3
174
4
175
Variation
176
f) Variation
Key works
for
** Direct Proportional
direct variation:
* Proportional Directly
1
* Proportional
* Varies
* Varies directly
2
177
3
4
178
5
6
179
7
8
180
** Inverse Proportional
1
2
181
3
_________________________________________________________________________________________________
4
182
5
6
183
.
184