You will sit two papers, one in Pure Mathematics (P1) and one in Statistics (S1) at the end of year
12. These are followed by another Pure paper (P3) and a Mechanics paper (M1) in year 13.
The Mathematics Department is committed to ensuring that you make good progress throughout
your A level or AS course. In order that you make the best possible start to the course, we have
prepared this booklet.
It is vitally important that you spend some time working through the questions in this booklet over
the summer - you will need to have a good knowledge of these topics before you commence your
course in September. You should have met all the topics before at iGCSE. Work through the
introduction to each chapter, making sure that you understand the examples. Then tackle the
exercise – not necessarily every question, but enough to ensure you understand the topic
thoroughly. The answers are given at the back of the booklet.
A mock test is provided at the back of this booklet.
We hope that you will use this introduction to give you a good start to your AS work and that it will
help you enjoy and benefit from the course more.
Sources for further help are indicated throughout the booklet. You may also find the following
book useful
AS-Level Maths Head Start
Published by CGP Workbooks
ISBN: 978 1 84146 993 5
Cost: £4.95
CONTENTS
Chapter 1 Removing brackets page 3
Chapter 2 Linear equations 5
Chapter 3 Simultaneous equations 9
Chapter 4 Factors 11
Chapter 5 Change the subject of the formula 14
Chapter 6 Solving quadratic equations 17
Chapter 7 Indices 19
Dr A Duncombe 1
,Chapter 1: REMOVING BRACKETS
To remove a single bracket, we multiply every term in the bracket by the number or the expression
on the outside:
Examples
1) 3 (x + 2y) = 3x + 6y
2) -2(2x - 3) = (-2)(2x) + (-2)(-3)
= -4x + 6
To expand two brackets, we must multiply everything in the first bracket by everything in the
second bracket. We can do this in a variety of ways, including
* the smiley face method
* FOIL (Fronts Outers Inners Lasts)
* using a grid.
Examples:
1) (x + 1)(x + 2) = x(x + 2) + 1(x + 2)
or
(x +1)(x + 2) = x2 + 2 + 2x + x
= x2 + 3x +2
or
x 1 (x +1)(x + 2) = x2 + 2x + x + 2
x x2 x = x2 + 3x +2
2 2x 2
2) (x - 2)(2x + 3) = x(2x + 3) - 2(2x +3)
= 2x2 + 3x – 4x - 6
= 2x2 – x – 6
or
(x - 2)(2x + 3) = 2x2 – 6 + 3x – 4x = 2x2 – x – 6
or
x -2 (2x +3)(x - 2) = 2x2 + 3x - 4x - 6
2x 2x2 -4x = 2x2 - x - 6
3 3x -6
Dr A Duncombe 2
, EXERCISE A Multiply out the following brackets and simplify.
1. 7(4x + 5)
2. -3(5x - 7)
3. 5a – 4(3a - 1)
4. 4y + y(2 + 3y)
5. -3x – (x + 4)
6. 5(2x - 1) – (3x - 4)
7. (x + 2)(x + 3)
8. (t - 5)(t - 2)
9. (2x + 3y)(3x – 4y)
10. 4(x - 2)(x + 3)
11. (2y - 1)(2y + 1)
12. (3 + 5x)(4 – x)
Two Special Cases
Perfect Square: Difference of two squares:
(x + a)2 = (x + a)(x + a) = x2 + 2ax + a2 (x - a)(x + a) = x2 – a2
(2x - 3)2 = (2x – 3)(2x – 3) = 4x2 – 12x + 9 (x - 3)(x + 3) = x2 – 32
= x2 – 9
EXERCISE B Multiply out
1. (x - 1)2
2. (3x + 5)2
3. (7x - 2)2
4. (x + 2)(x - 2)
5. (3x + 1)(3x - 1)
6. (5y - 3)(5y + 3)
More help on expanding brackets is available by downloading this video tutorial:
http://stream.port.ac.uk/streams/play/play.asp?id=465&stream=MediumBand
Dr A Duncombe 3
12. These are followed by another Pure paper (P3) and a Mechanics paper (M1) in year 13.
The Mathematics Department is committed to ensuring that you make good progress throughout
your A level or AS course. In order that you make the best possible start to the course, we have
prepared this booklet.
It is vitally important that you spend some time working through the questions in this booklet over
the summer - you will need to have a good knowledge of these topics before you commence your
course in September. You should have met all the topics before at iGCSE. Work through the
introduction to each chapter, making sure that you understand the examples. Then tackle the
exercise – not necessarily every question, but enough to ensure you understand the topic
thoroughly. The answers are given at the back of the booklet.
A mock test is provided at the back of this booklet.
We hope that you will use this introduction to give you a good start to your AS work and that it will
help you enjoy and benefit from the course more.
Sources for further help are indicated throughout the booklet. You may also find the following
book useful
AS-Level Maths Head Start
Published by CGP Workbooks
ISBN: 978 1 84146 993 5
Cost: £4.95
CONTENTS
Chapter 1 Removing brackets page 3
Chapter 2 Linear equations 5
Chapter 3 Simultaneous equations 9
Chapter 4 Factors 11
Chapter 5 Change the subject of the formula 14
Chapter 6 Solving quadratic equations 17
Chapter 7 Indices 19
Dr A Duncombe 1
,Chapter 1: REMOVING BRACKETS
To remove a single bracket, we multiply every term in the bracket by the number or the expression
on the outside:
Examples
1) 3 (x + 2y) = 3x + 6y
2) -2(2x - 3) = (-2)(2x) + (-2)(-3)
= -4x + 6
To expand two brackets, we must multiply everything in the first bracket by everything in the
second bracket. We can do this in a variety of ways, including
* the smiley face method
* FOIL (Fronts Outers Inners Lasts)
* using a grid.
Examples:
1) (x + 1)(x + 2) = x(x + 2) + 1(x + 2)
or
(x +1)(x + 2) = x2 + 2 + 2x + x
= x2 + 3x +2
or
x 1 (x +1)(x + 2) = x2 + 2x + x + 2
x x2 x = x2 + 3x +2
2 2x 2
2) (x - 2)(2x + 3) = x(2x + 3) - 2(2x +3)
= 2x2 + 3x – 4x - 6
= 2x2 – x – 6
or
(x - 2)(2x + 3) = 2x2 – 6 + 3x – 4x = 2x2 – x – 6
or
x -2 (2x +3)(x - 2) = 2x2 + 3x - 4x - 6
2x 2x2 -4x = 2x2 - x - 6
3 3x -6
Dr A Duncombe 2
, EXERCISE A Multiply out the following brackets and simplify.
1. 7(4x + 5)
2. -3(5x - 7)
3. 5a – 4(3a - 1)
4. 4y + y(2 + 3y)
5. -3x – (x + 4)
6. 5(2x - 1) – (3x - 4)
7. (x + 2)(x + 3)
8. (t - 5)(t - 2)
9. (2x + 3y)(3x – 4y)
10. 4(x - 2)(x + 3)
11. (2y - 1)(2y + 1)
12. (3 + 5x)(4 – x)
Two Special Cases
Perfect Square: Difference of two squares:
(x + a)2 = (x + a)(x + a) = x2 + 2ax + a2 (x - a)(x + a) = x2 – a2
(2x - 3)2 = (2x – 3)(2x – 3) = 4x2 – 12x + 9 (x - 3)(x + 3) = x2 – 32
= x2 – 9
EXERCISE B Multiply out
1. (x - 1)2
2. (3x + 5)2
3. (7x - 2)2
4. (x + 2)(x - 2)
5. (3x + 1)(3x - 1)
6. (5y - 3)(5y + 3)
More help on expanding brackets is available by downloading this video tutorial:
http://stream.port.ac.uk/streams/play/play.asp?id=465&stream=MediumBand
Dr A Duncombe 3
