# Maths Workbook

```Edexcel GCSE
Mathematics A Linear
Higher
REVISION
WORKBOOK
Series Director: Keith Pledger
Series Editor: Graham Cumming
Authors: Julie Bolter,
Gwenllian Burns, Jean Linsky
The Edexcel Revision Series
These revision books work in combination with Edexcel’s main GCSE Mathematics 2010 series.
The Revision Guides are designed for independent or classroom study. The Revision Workbooks
use a write-in format to provide realistic exam practice.
Specification A Linear
Specification B Modular
Higher
Foundation
1
A01_EMHL_WBK_GCSE_0154_PRE.indd 1
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Contents
NUMBER
1
Factors and primes
2
Indices 1
3
Fractions
4
Decimals
5
Recurring decimals
6
Rounding and estimation
7
Upper and lower bounds
8
Fractions and percentages
9
Percentage change
10 Reverse percentages and compound interest
11 Ratio
12 Proportion
13 Indices 2
14 Standard form
15 Calculator skills
16 Surds
17 Problem-solving practice: Number A
02 A03
18 Problem-solving practice: Number
ALGEBRA
19 Algebraic expressions
20 Arithmetic sequences
21 Expanding brackets
22 Factorising
23 Linear equations 1
24 Linear equations 2
25 Straight-line graphs
26 Parallel and perpendicular
27 3-D coordinates
28 Real-life graphs
29 Formulae
30 Rearranging formulae
31 Inequalities
32 Inequalities on graphs
34 Graphs of _xk and ax
35 Trial and improvement
36 Simultaneous equations 1
38 Completing the square
41 Equation of a circle
42 Simultaneous equations 2
43 Direct proportion
44 Proportionality formulae
45 Transformations 1
46 Transformations 2
47 Algebraic fractions
48 Proof
49 Problem-solving practice: Algebra
50 Problem-solving practice: Algebra
GEOMETRY AND MEASURES
51 Angle properties
52 Solving angle problems
53 Angles in polygons
54 Plan and elevation
55 Perimeter and area
56 Prisms
57 Circles and cylinders
58 Sectors of circles
A01_EMHL_WBK_GCSE_0154_PRE.indd 2
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
Volumes of 3-D shapes
Pythagoras’ theorem
Surface area
Converting units
Units of area and volume
Speed
Density
Congruent triangles
Similar shapes 1
Similar shapes 2
Bearings
Scale drawings and maps
Constructions
Loci
Translations, reflections and rotations
Enlargements
Combining transformations
Line segments
Trigonometry 1
Trigonometry 2
Pythagoras in 3-D
Trigonometry in 3-D
Triangles and segments
The sine rule
The cosine rule
Circle facts
Circle theorems
Vectors
Solving vector problems
Problem-solving practice: Geometry A
02
Problem-solving practice: Geometry
STATISTICS AND PROBABILITY
90 Collecting data
91 Two-way tables
92 Stratified sampling
93 Mean, median and mode
94 Frequency table averages
95 Interquartile range
96 Frequency polygons
97 Histograms
98 Cumulative frequency
99 Box plots
100 Scatter graphs
101 Probability
102 Tree diagrams
103 Problem-solving practice: Statistics
104 Problem-solving practice: Statistics
A02
A03
105
106
113
119
A02
A03
A03
Formulae page
Paper 1 Practice exam paper
Paper 2 Practice exam paper
A small bit of small print
covered by that question. Sub-parts of the question may
range that the topic is assessed at. The topic may form part
of a higher grade question if tested within the context of
another topic.
Questions in this book are targeted at the grades indicated.
24/5/11 12:01:18
NUMBER
Factors and primes
C
1
(a) Express the following numbers as products of their prime factors.
(i) 60
(ii) 150
150
60
Guided
6
2
10
10
3
…… ……
2
15
…… …… ……
Remember to circle the prime
factors as you go along.
60 5 2 3 …… 3 …… 3 ……
(2 marks)
150 5 2 3 …… 3 …… 3 ……
(2 marks)
(b) Find the highest common factor (HCF) of 60 and 150
60 5 2 3 3 3 …… 3 ……
Guided
150 5 2 3 …… 3 …… 3 ……
HCF 5 2 3 …… 3 ……
5 ………
Circle all the prime numbers
which are common to both
products of prime factors.
Multiply the circled numbers
together to find the HCF.
(1 mark)
(c) Find the lowest common multiple (LCM) of 60 and 150
LCM 5 ……… 3 …… 3 ……
Guided
5 ………
C
2
To find the LCM, multiply the
HCF by the numbers in both
products that were not circled
in part (b).
(1 mark)
(a) Express 72 as a product of its prime factors.
…………………
(2 marks)
(b) Find the highest common factor (HCF) of 72 and 120
HCF 5 …………………
(1 mark)
(c) Find the lowest common multiple (LCM) of 72 and 120
LCM 5 …………………
(1 mark)
1
M01_EMHL_WBK_GCSE_0154_U01.indd 1
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NUMBER
Indices 1
C
1
Write as a power of 7
(a) 73 3 710
73 3 710 5 73 1 10 5 …………
Guided
(1 mark)
(b) 715 4 7 9
715 4 79 5 715 2 9 5 …………
Guided
(1 mark)
712
(c) ______
74 3 7
712
712
___
5 _______
4
1 5 ......
7 37 7 37
7
712
_______
Guided
4
5 …………
(2 marks)
(d) (75)4
(75)4 5 75 3 4 5 …………
Guided
C
2
Write as a power of 5
512 3 5
(b) _______
54 3 53
(a) 58 3 54
………………
C
3
(1 mark)
(1 mark)
68 3 63 5 65 3 6x
Find the value of x.
………………
(c) (52)3
(2 marks)
………………
Use the index laws to simplify each side of the equation.
x 5 …………………
B
4
5
(2 marks)
Simplify 40
…………………
A
(1 mark)
(1 mark)
Write 93 3 272 as a single power of 3
93 3 272 5 (3……)3 3 (3……)2
Guided
5 3…… 3 3……
5 ……….
A
6
(2 marks)
Write 86 4 43 3 25 as a single power of 2
…………………
(2 marks)
2
M01_EMHL_WBK_GCSE_0154_U01.indd 2
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NUMBER
Fractions
C
1
Work out 3_23 1 1_45
4
__
2
3__
3 1 15
Guided
…
…
5 …… 1 ___ 1 ___
3
5
…
…
5 …… 1 ___ 1 ___
15
15
Write as equivalent fractions with the same denominator.
…
5 …… 1 ___
15
…
5 …… 1 1 ___
15
…
5 ……___
15
C
2
Work out
Write your final answer as a mixed number in its simplest form.
(a) 7_17 2 2_23
(3 marks)
9
(b) 8__
1 2_35
10
Give each answer as a mixed number in its simplest form.
(a) ……………………
C
3
__
3
1
2__
3 3 15
Guided
EXAM
C
Work out 2_13 3 1_35
4
(3 marks)
(b) ……………………
Exam questions similar to this
have proved especially tricky
– be prepared!
…
…
5 ___ 3 ___
3
5
Write both mixed numbers as improper fractions.
…
5 ___
…
Multiply numerators and multiply denominators.
…
5 ……___
…
Write your final answer as a mixed number in its simplest form.
Work out
(3 marks)
(a) 2_14 3 3_13
(3 marks)
(b) 5_13 4 1_29
Give each answer as a mixed number in its simplest form.
C
5
Work out
(a) ……………………
(3 marks)
(a) 8_56 2 3_25
9
(b) 4_15 4 __
10
(b) ……………………
(3 marks)
(b) ……………………
(3 marks)
Give each answer as a mixed number in its simplest form.
(a) ……………………
(3 marks)
3
M01_EMHL_WBK_GCSE_0154_U01.indd 3
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NUMBER
Decimals
C
1
Using the information that 67 3 29 5 1943
write down the value of
(a) 6.7 3 2.9
6.7 3 2.9 5 1943 4 …………
Guided
5 …………
67 has been divided by 10 and 29 has been divided by
10. So the answer needs to be divided by 100.
(1 mark)
(b) 670 3 0.0029
670 3 0.0029 5 1943 4 …………
Guided
5 …………
67 has been multiplied by 10 and 29 has been divided
by 10 000. So the answer needs to be divided by 1000.
(1 mark)
(c) 19 430 4 67
19 430 4 67 5 29 3 …………
Guided
5 …………
1943 has been multiplied by 10 and 67 is unchanged.
So multiply 29 by 10.
(1 mark)
C
2
Use the information that 127 3 84 5 10 668
to find the value of
(a) 1270 3 84
…………………
C
3
(b) 0.127 3 8.4
(1 mark)
…………………
(c) 10 668 4 1.27
(1 mark)
…………………
(1 mark)
Given that 63 3 48 5 3024
write down the value of
(a) 6300 3 4.8
…………………
(b) 0.063 3 4.8
(1 mark)
…………………
(c) 30 240 4 6.3
(1 mark)
…………………
(1 mark)
4
M01_EMHL_WBK_GCSE_0154_U01.indd 4
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NUMBER
Recurring decimals
A
1
..
Express 0.15 as a fraction in its simplest form. You must use algebra.
. .
Let x 5 0.1 5
Guided
100x 5 15.151 515…
2 x 5 0.151 515 …
99x 5 ………
………
x 5 _______
99
………
x 5 _______
………
A
2
(3 marks)
.
Change the recurring decimal 0.8 to a fraction. You must use algebra.
…………………
A
3
. .
Convert the recurring decimal 2.417 to a fraction. You must use algebra.
…………………
A
4
(2 marks)
(3 marks)
.
Convert the recurring decimal 0.47 to a fraction. You must use algebra.
.
Let x 5 0.47
Guided
10x 5 4.777 7777…
2 x 5 0.477 7777…
9x 5 …………
………
x 5 _______
9
………
x 5 _______
………
A
5
Multiply the top and bottom
of the fraction by 10.
(3 marks)
..
91
Prove that 0.827 can be written as the fraction ___
110
(3 marks)
5
M01_EMHL_WBK_GCSE_0154_U01.indd 5
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NUMBER
Rounding and estimation
D
1
Work out estimates for each of the following.
(a) 145 3 78
100 3 ………… 5 …………………
Guided
(1 mark)
(b) 19.1 4 1.51
………… 4 2 5 …………………
Guided
(1 mark)
(c) 48.9 3 2.78 3 11.9
………… 3 ………… 3 10 5 …………………
Guided
D
C
C
2
3
4
EXAM
C
3981
Work out an estimate for the value of _________
2.3 3 18.7
…………………
(2 marks)
…………………
(2 marks)
612 3 39
Work out an estimate for the value of ________
0.53
40.7 3 1.6
Work out an estimate for the value of _________
0.053
40 3 ………
………
 ____________ 5 _______
0.05
………
Guided
(1 mark)
Exam questions similar to this
have proved especially tricky
– be prepared!
First round each number
to 1 significant figure.
5 …………………
5
(2 marks)
9.73 3 4.12
Work out an estimate for the value of __________
0.0214
…………………
C
6
Work out estimates for the following calculations.
995.3
(a) _________
5.3 3 11.3
101.7
(b) _________
3.7 3 4.72
C
7
(2 marks)
…………………
(2 marks)
…………………
(2 marks)
…………………
(2 marks)
Work out an estimate for the value of 2.52 3 11.72
6
M01_EMHL_WBK_GCSE_0154_U01.indd 6
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NUMBER
Upper and lower bounds
A
1
The length of a rectangle is 9.7 cm correct to 2 significant figures.
The width of the rectangle is 6.5 cm correct to 2 significant figures.
Work out the upper bound for the area of the rectangle.
Upper bound of length 5 9.75
Guided
Upper bound of width 5 ………
Upper bound of area
5 9.75 3 …………………
5 …………………
5 ………………… cm2
A
2
(3 marks)
The length of a rectangle is 24 cm correct to 2 significant figures.
The width of the rectangle is 9.6 cm correct to 2 significant figures.
Work out the lower bound for the perimeter of the rectangle.
………………… cm
A*
3
(3 marks)
___
A ball is dropped from a window.
2s
The time that it takes to reach the ground is given by the formula t 5 __
a
where a m/s2 is the acceleration due to gravity and s m is the height of the window.
s 5 117 m correct to 3 significant figures
a 5 9.8 m/s2 correct to 2 significant figures
√
(a) Calculate the lower bound and the upper bound for the value of t.
…………………
(4 marks)
(b) Use your answers to part (a) to write down the value of t to a suitable degree of accuracy.
Guided
t 5 ………………… because the upper bound and the lower bound
both agree to 2 significant figures
(1 mark)
7
M01_EMHL_WBK_GCSE_0154_U01.indd 7
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NUMBER
Fractions and percentages
D
1
………
_______
Guided
D
Uzma invests &pound;4000 in a bank account for 1 year.
Interest is paid at a rate of 2.5% per annum.
How much interest will Uzma get at the end of 1 year?
100
2
3 &pound;4000 5 &pound;…………………
(2 marks)
A farmer has 48 llamas.
30 of the llamas are female.
(a) Work out 30 out of 48 as a percentage.
30
_______
Guided
3 100 5 …………………
(2 marks)
………
60% of the female llamas are pregnant.
(b) Write the number of pregnant female llamas as a fraction of the 48 llamas.
…………………
D
D
3
4
(2 marks)
Meera works in an electrical shop.
Each week she gets paid &pound;160 plus 15% of the value of the goods she sells.
One week Meera sold &pound;3200 of goods.
Work out the total amount she was paid this week.
&pound;…………………
(3 marks)
&pound;…………………
(4 marks)
…………………
(3 marks)
Liam’s annual income is &pound;16 000
He pays _15 of the &pound;16 000 in rent.
He spends 15% of the &pound;16 000 on food.
Work out how much of the &pound;16 000 Liam has left.
C
5
At an outdoor centre, 140 students each choose one activity.
_1 of the students choose rock climbing.
7
_3 of the students choose rafting.
7
All the rest of these students choose abseiling.
How many students choose abseiling?
8
M01_EMHL_WBK_GCSE_0154_U01.indd 8
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NUMBER
Percentage change
D
1
A washing machine costs &pound;420 plus 20% VAT.
Calculate the total cost of the washing machine.
20
VAT 5 ____ 3 ………
100
5 …………………
Guided
Total cost 5 420 1 ………
5 &pound;…………………
D
2
(3 marks)
Helen buys a jacket in a sale.
The normal price is &pound;84
The normal price of the jacket is reduced by 35%.
Work out the sale price of the jacket.
&pound;…………………
C
3
(3 marks)
Eliza went to New York.
She changed pounds (&pound;) into American dollars (\$).
The exchange rate was &pound;1 5 \$1.60
The value of the pound has decreased from \$1.60 to \$1.56
Calculate the percentage decrease in the value of the pound.
decrease
Percentage decrease 5 ____________ 3 100
original value
Guided
………
5 _______ 3 100
………
5 …………………%
C
C
4
5
(3 marks)
Ali buys 120 cans of drink for a total of &pound;30
He wants to make a profit of 40%.
Work out the price for which he should sell each can of drink.
…………………
(4 marks)
&pound;…………………
(4 marks)
Jean books a holiday.
The total cost of the holiday is &pound;1430
She pays a deposit of 35% of the total cost.
She pays the rest in 10 monthly instalments.
Work out how much she pays each month.
9
M01_EMHL_WBK_GCSE_0154_U01.indd 9
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NUMBER
Reverse percentages and
compound interest
B
1
Linda bought a new car for &pound;18 000
Each year, the car depreciated in value by 15%.
Work out the value of the car after 4 years.
15
Multiplier 5 1 2 ____ 5 ………
100
Guided
Work out the multiplier as a decimal.
Value after 4 years 5 18 000 3 (………)4
5 …………………
5 &pound;…………………
B
2
must be given to 2 decimal places.
Jalin invested &pound;3200 in a savings account for 3 years.
He was paid compound interest at a rate of 3.5% per annum.
Work out how much was in the account after 3 years.
&pound;…………………
B
3
EXAM
(3 marks)
In a sale, normal prices are reduced by 35%.
The sale price of a DVD player is &pound;403
Work out the normal price of the DVD player.
(3 marks)
Exam questions similar to this
have proved especially tricky
– be prepared!
35
Multiplier 5 1 2 ____ 5 ………
100
Guided
Normal price 5 403 4 ………
5 &pound;…………………
B
4
(3 marks)
Jill’s weekly pay this year is &pound;460
This is 15% more than her weekly pay last year.
Dave says, ‘This means Jill’s weekly pay last year was &pound;391.’
Dave is wrong. Explain why.
(2 marks)
A
5
Pete invested &pound;5100 for n years in a savings account.
He was paid 4.5% per annum compound interest.
At the end of the n years he had &pound;6641.53 in the savings account.
Work out the value of n.
Choose some values for n and work out the
amount in the savings account after n years.
n 5 …………………
(2 marks)
10
M01_EMHL_WBK_GCSE_0154_U01.indd 10
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NUMBER
Ratio
D
1
There are 60 toy cars in a box.
18 of the toy cars are blue.
The rest of the toy cars are red.
Write down the ratio of the number of red toy cars to the number of blue toy cars.
Give your ratio in its simplest form.
Number of red cars 5 60 2 ………
Guided
Make sure you put the numbers
in the ratio in the correct order.
5 ………
Ratio of red cars to blue cars 5 ……… : ………
5 ……… : ………
D
2
(2 marks)
There are 32 students in a class.
20 of the students are girls.
Rosie says, ‘The ratio of the number of girls to the number of boys in this class is 3 : 5.’
Is Rosie right?
…………………
D
3
(2 marks)
Ahmed and James share &pound;120 in the ratio 1 : 3
How much does James get?
Number of shares 5 1 1 3
Guided
5 ………
One share is worth &pound;120 4 ……… 5 &pound;…………………
James gets ………3 ……… 5 &pound;…………………
(2 marks)
C
4
Annie and Jamil share &pound;160 in the ratio 3 : 5
How much more money than Annie does Jamil get?
&pound;…………………
C
5
(3 marks)
Linda, Mel and Tomos share the driving on a journey in the ratio 2 : 3 : 4
Mel drove a distance of 240 km.
Work out the length of the journey.
………………… km
(2 marks)
11
M01_EMHL_WBK_GCSE_0154_U01.indd 11
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NUMBER
Proportion
D
1
Mike buys 6 pencils for a total cost of &pound;5.34
Work out the cost of 11 of these pencils.
Cost of 1 pencil 5 5.34 4 ………
Guided
5 &pound;…………………
Cost of 11 pencils 5 11 3 ………
5 &pound;…………………
D
2
(2 marks)
Punita buys 3 identical notebooks for a total cost of &pound;10.44
Work out the cost of 5 of these notebooks.
&pound;…………………
D
3
The total cost of 4 kg of apples is &pound;4.20
The total cost of 3 kg of apples and 2 kg of bananas is &pound;5.05
Work out the cost of 1 kg of bananas.
First work out the cost
of 1 kg of apples.
…………………
C
4
A builder lays 180 bricks in 1 hour.
He always works at the same speed.
How long will it take the builder to lay 585 bricks?
5
(2 marks)
4 workers can lay a stretch of road in 9 days.
How long would it take 6 workers to lay the same stretch of road?
1 worker would take 4 3 ……… 5 ……… days to lay the stretch of road.
Guided
So 6 workers would take ……… 4 6 5 ……… days to lay the stretch of road.
C
(3 marks)
Remember to include
…………………
C
(2 marks)
6
(2 marks)
It takes one machine at a factory 24 hours to pack 12 000 boxes of cakes.
The owner of the factory buys two more machines.
Work out the number
All the machines work at the same rate.
of boxes that 1 machine
How long would it take the 3 machines to pack a total of
can pack in 1 hour.
30 000 boxes of cakes?
…………………
(3 marks)
12
M01_EMHL_WBK_GCSE_0154_U01.indd 12
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NUMBER
Indices 2
B
1
Work out the value of
_1
(a) 422
(b) 492
1
(a) 422 5 ___2
4
……
5 _____
……
Guided
(1 mark)
___
__1
(b) 492 5 √ 49
5 …………
B
A*
2
Work out the value of
_1
3
(1 mark)
(a) 273
(b) 921
(c) 423
…………………
(1 mark)
…………………
(1 mark)
…………………
(1 mark)
_2
Work out the value of
(a) 83
(d) 80
…………………
(1 mark)
_3
(b) (__
9 22
16 )
2
(a) 83 5 (8
3)
__
2
Guided
__1
5 (……)2
5 ………
( )
9
(b) ___
16
3
2 __
2
5
(1 mark)
((__169 ) )3
__1
2
(
……
5 _____
……
……
5 _____
……
A*
4
)
3
(2 marks)
Work out the value of
2 _34
(b) 643
81
(c) ___
16
………………… (1 mark)
………………… (1 mark)
…………………
_2
2 _12
A* 5
( )
(a) 49
(2 marks)
__
___
√3
Work out the value of ___ 3 √27
9
Write each number as a power of 3 and then use the index laws.
…………………
(2 marks)
13
M01_EMHL_WBK_GCSE_0154_U01.indd 13
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NUMBER
Standard form
B
1
(a) Write 67 000 in standard form.
Exam questions similar to this
have proved especially tricky
– be prepared!
67 000 5 ………… 3 10……
Guided
EXAM
(1 mark)
(b) Write 2 3 1025 as an ordinary number.
2 3 1025 5 …………………
Guided
(1 mark)
(c) Write 760 3 104 in standard form.
760 3 104 5 …………………
Guided
First write the number as an ordinary number.
5 ………… 3 10……
B
B
2
3
(a) Write 0.54 in standard form.
…………………
(1 mark)
(b) Write 7 3 106 as an ordinary number.
…………………
(1 mark)
Write these numbers in order of size.
32 3 106
0.031 3 1010
3 3 108
…………………
A
4
(1 mark)
…………………
Write all the numbers in standard form first.
3400 3 105
…………………
…………………
(2 marks)
Work out the value of 5 3 107 3 9 3 103
5 3 107 3 9 3 103 5 (5 3 ………) 3 (107 3 10……)
Guided
5 ……… 3 10……
5 ……… 3 10……
A
A
A
5
6
7
(2 marks)
Work out the value of 1.04 3 103 4 2 3 1025
…………………
(2 marks)
…………………
(2 marks)
…………………
(2 marks)
Work out the value of 7 3 105 3 3000
The number of atoms in one kilogram of helium is 1.51 3 1026
Calculate the number of atoms in 30 kilograms of helium.
14
M01_EMHL_WBK_GCSE_0154_U01.indd 14
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NUMBER
Calculator skills
D
1
23.5 3 9.4
Use your calculator to work out the value of _________
14.6 2 5.9
Write down all the figures on your calculator display.
23.5 3 9.4
___________
Guided
14.6 2 5.9
………
5 _______
………
5 …………………
C
B
2
3
Make sure that you give your
answer as a decimal. If necessary,
use the S D button.
(2 marks)
__________
2
√ 13.5 1 3.4
Use your calculator to work out the value of ___________
2.3 3 1.5
Write down all the figures on your calculator display.
…………………
(3 marks)
…………………
(3 marks)
45.8
3 sin 34&deg;
__________
Use your calculator to work out the value of _____________
√ 8.72 1 5.22
Write down all the figures on your calculator display.
B
4
Work out (8.2 3 1024) 4 (3.1 3 1027)
…………………
A
5
a1b
y2 5 _____
ab
a 5 5 3 106
(2 marks)
b 5 4 3 103
Find y.
Guided
……… 1 ………
y2 5 __________________
5 3 106 3 4 3 103
…………………
y2 5 _______________
…………………
________________
l
y 5 √…………………
y 5 …………………
y 5 ………… 3 10……
(3 marks)
15
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NUMBER
Surds
A
1
___
Simplify
____
(a) √48
(b) √300
________
l
(a) √48 5 √ 16 3 √………
___
Guided
___
________
l
5 …… √………
________
l
(1 mark)
________
l
(b) √ 300 5 √ ……… 3 √ ………
_____
________
l
5 …… √………
A
2
Guided
(1 mark)
10
__
Rationalise the denominator of ___
√
2
__
√2
10
10
___
__ 5 ___
__ 3 ___
__
√2
√2
√2
__
5
……√ 2
_______
……
__
5 ……√ 2
A*
3
(2 marks)
__
__
Expand and simplify (2 2 √3 )(5 1 √ 3 )
__
__
__
__
(2 2 √ 3 )(5 1 √ 3 ) 5 10 1 2√3 2 ………√3 2 ………
Guided
Use FOIL (First terms, Outer terms,
Inner terms, Last terms) to expand
the brackets.
__
5 ……… 2 ………√3
A*
A*
4
5
A* 6
__
(2 marks)
__
Expand and simplify (7 2 √5 )(2 1 √5 )
…………………
(2 marks)
…………………
(2 marks)
__
Expand and simplify (3 2 √2 )2
__
12 2__5√3
Rationalise the denominator of _________
√3
__
Give your answer in the form a 1 b√3 where a and b are integers.
…………………
(3 marks)
16
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NUMBER
Problem-solving practice
D
*1 Tickets R-US and Cheap Tickets both advertise tickets for the same concert.
The question has a * next
to it, so make sure that you
in a sentence.
Tickets
Tickets
R-US
R-US
Cheap
CheapTickets
Tickets
&pound;36
&pound;36
plus
plus
5%
5%
booking
booking
feefee
&pound;35
&pound;35
plus
plus
7.5%
7.5%
booking
booking
feefee
Helen wants to pay the least money possible for a ticket.
Which shop should she buy her ticket from, Tickets R-US or Cheap Tickets?
Work out the price
plus the booking
fee for each ticket.
(4 marks)
D
Guided
2
Last year, Kevin spent
_1 of his salary on entertainment
8
_2 of his salary on rent
5
15% of his salary on living expenses.
He saved the rest of his salary.
Last year Kevin’s salary was &pound;32 000
How much money did Kevin save?
Amount spent on entertainment = 32 000 4 …………
5 &pound;…………
You should show
Amount spent on rent = 32 000 4 ………… 3 …………
5 &pound;…………
10% of 32 000 5 …………
5% of 32 000 5 …………
15% of 32 000 5 …………
Amount spent on living expenses 5 &pound;…………
Total amount spent 5 ………… 1 ………… 1 …………
5 &pound;…………
Amount of money saved 5 32 000 2 …………..
5 &pound;…………
(4 marks)
17
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NUMBER
Problem-solving practice
C
*3 Mr Li’s garden is in the shape of a rectangle.
Part of the garden is a vegetable plot in the
shape of a triangle.
The rest of the garden is grass.
Mr Li wants to spread fertiliser all over the grass.
One box of fertiliser is enough for 30 m2 of grass.
How many boxes of fertiliser will he need?
15 m
Veg.
Plot
Grass
8m
5m
First work out the area of the grass.
Divide the area by 30 to find the
number of boxes – remember that
Mr Li can only buy a whole number
of boxes.
(4 marks)
B
*4 Kevin invests &pound;6000 for 3 years at 4.5% simple interest in Simple Bank.
The interest is paid, by cheque, at the end of each year.
Kevin also invests &pound;6000 for 3 years at 4.2% compound interest in Compound Bank.
Which bank pays Kevin the greater amount of total interest, Simple Bank or Compound
Bank?
Simple interest means that the interest is the same each year.
Work out the interest for one year then multiply this amount by 3.
For compound interest, the interest will change each year. Work out
the interest for one year, add this on to the amount in the account
and then work out the interest for the next year, and so on.
(4 marks)
A
A*
5
Josip drove for 314 miles, correct to the nearest mile.
He used 34.6 litres of petrol, to the nearest tenth of a litre.
Number of miles travelled
Petrol consumption 5 ____________________________
Number of litres of petrol used
Work out the upper bound for the petrol consumption for Josip’s journey.
Use the upper bound for the number
of miles travelled and the lower bound
for the number of litres of petrol used.
…………………
(3 marks)
18
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ALGEBRA
Algebraic expressions
C
1
Guided
Simplify
(a) m3 3 m9
(b) p10 4 p2
(c) (t4)5
(a) m3 3 m9 5 m3 1 9
(b) p10 4 p2 5 p 10 2 2
(c) (t 4)5 5 t 4 3 5
5 m ………
(1 mark)
C
2
(1 mark)
………………
4
(c) ( y3)7
…………………
…………………
(1 mark)
( )
y12
(b) ______
y3 3 y
(2 marks)
Simplify
………………
z6
(c) __
z3
2
…………………
(2 marks)
28x6y5
(b) ______
7xy3
(a) 5e7f 2 3 3e4f 5
(c) (2m5p)4
5 ………e ……… f ………
28x6y5
(b) _______
7xy3
(c) (2m5p)4
(2 marks)
5 28 4 7 3 x 6 2 1 3 y 5 2 3
5 ………x ……… y ………
(2 marks)
5 24m5 3 4 p 1 3 4
5 ………m ……… p ………
5
………………
6
(2 marks)
Simplify
40a9c2
(b) ______
8a3c
(a) 6cd8 3 4c5d2
A
(2 marks)
(a) 5e7f 2 3 3e4f 5 5 5 3 3 3 e7 1 4 3 f 2 1 5
Guided
B
(1 mark)
Simplify
x5 3 x4
(a) _______
x6
B
(1 mark)
(b) k9 4 k3 3 k2
…………………
3
5 t ………
Simplify
(a) g 3 g6
C
5 p ………
(1 mark)
Simplify
(2 marks)
………………
…………………
(2 marks)
(2 marks)
1
Use the index law a2n 5 __
an
( )
1
(a) ___
2x3
(c) (5b3d2)3
(
…………………
)
25
(b) ______
64b2c8
22
(2 marks)
2 _21
…………………
(2 marks)
19
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ALGEBRA
Arithmetic sequences
C
1
Here are the first five terms of an arithmetic sequence.
1
5
9
13
17
Find an expression, in terms of n, for the nth term of the sequence.
zero term 1 … 1 … 1 … 1 …
zero term
�…
Guided
…
5
1
9
13
Work out the difference
between each term. Then
work out the zero term.
17
nth term
� ……n
� ……
nth term
5 ………
n 1 ……… 5 ………………
C
2
(2 marks)
Here are the first five terms of an arithmetic sequence.
17
12
7
2
23
Find an expression, in terms of n, for the nth term of the sequence.
…………………
C
3
(2 marks)
(a) Here are the first five terms of an arithmetic sequence.
3
7
11
15
19
Find an expression, in terms of n, for the nth term of the sequence.
…………………
(b) Paul says that 72 is a term in this sequence.
Paul is wrong. Explain why.
(2 marks)
Look at the type of numbers in the sequence.
………………………………………………………………………………………………………
(1 mark)
C
4
Guided
(a) The nth term of a sequence is 8n 1 3
Write down the first three terms of this sequence.
1st term n 5 1
8 3 1 1 3 5 ………
2nd term n 5 ………
8 3 ……… 1 3 5 ………
3rd term n 5 ………
8 3 ……… 1 3 5 ………
(b) Jenny says that 45 is a term in this sequence.
Jenny is wrong. Explain why.
Try and find a value for n
that gives a result of 45.
………………………………………………………………………………………
C
5
(2 marks)
(1 mark)
The nth term of a sequence is 3n 2 1
Work out the 50th term of this sequence.
……………………………………………………………………………………………
(1 mark)
20
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