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- Pure Mathematics 2 and 3: Cambridge International AS and A Level Mathematics
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Pure Mathematics 2 and 3: Cambridge International AS and A Level Mathematics
We are grateful to the following companies, institutions and individuals ou have given permission to reproduce photographs in this book. The publishers will be glad to make suitable arrangements with an copright holders whom it has not been possible to contact. Hachette UK s polic is to use papers that are natural, renewable and recclable products and made from wood grown in sustainable forests. The logging and manufacturing processes are epected to conform to the environmental regulations of the countr of origin.
Telephone: 44 Fa: 44 Lines are open , Monda to Saturda, with a 4-hour message answering service. Visit our website at Much of the material in this book was published originall as part of the MEI Structured Mathematics series.
Apart from an use permitted under UK copright law, no part of this publication ma be reproduced or transmitted in an form or b an means, electronic or mechanical, including photocoping and recording, or held within an information storage and retrieval sstem, without permission in writing from the publisher or under licence from the Copright Licensing Agenc Limited.
This smbol means that ou want to discuss a point with our teacher. If ou are working on our own there are answers in the back of the book. It is important, however, that ou have a go at answering the questions before looking up the answers if ou are to understand the mathematics full.
This smbol invites ou to join in a discussion about proof. The answers to these questions are given in the back of the book.! This is a warning sign. It is used where a common mistake, misunderstanding or trick point is being described.
This is the ICT icon. It indicates where ou could use a graphic calculator or a computer. This smbol and a dotted line down the right-hand side of the page indicates material that ou are likel to have met before. You need to be familiar with the material before ou move on to develop it further.
This smbol and a dotted line down the right-hand side of the page indicates material which is beond the sllabus for the unit but which is included for completeness. The series also contains a more advanced book for pure mathematics and one each for mechanics and statistics.
These books are based on the highl successful series for the Mathematics in Education and Industr MEI sllabus in the UK but the have been redesigned for Cambridge users; where appropriate new material has been written and the eercises contain man past Cambridge eamination questions. Throughout the series the emphasis is on understanding the mathematics as well as routine calculations.
The various eercises provide plent of scope for practising basic techniques; the also contain man tpical eamination questions. An important feature of this series is the electronic support. There is an accompaning disc containing two tpes of Personal Tutor presentation: eamination-stle questions, in which the solutions are written out, step b step, with an accompaning verbal eplanation, and test ourself questions; these are multiple-choice with eplanations of the mistakes that lead to the wrong answers as well as full solutions for the correct ones.
In addition, etensive online support is available via the MEI website, The books are written on the assumption that students have covered and understood the work in the Cambridge IGCSE sllabus. However, some of the earl material is designed to provide an overlap and this is designated Background.
There are also places where the books show how the ideas can be taken further or where fundamental underpinning work is eplored and such work is marked as Etension. The original MEI author team would like to thank Sophie Goldie who has carried out the etensive task of presenting their work in a suitable form for Cambridge International students and for her man original contributions.
The would also like to thank Cambridge International Eaminations for their detailed advice in preparing the books and for permission to use man past eamination questions. Roger Porkess Series Editor vii.
Conan Dole Background algebra Background algebra Manipulating algebraic epressions You will often wish to tid up an epression, or to rearrange it so that it is easier to read its meaning. The following eamples show ou how to do this. You should practise the techniques for ourself on the questions in Eercise A. Collecting terms Ver often ou just need to collect like terms together, in this eample those in, those in and those in z.?
What are like and unlike terms? Removing brackets Sometimes ou need to remove brackets before collecting like terms together. Simplif the epression 3 4 4 5. This usuall involves using brackets and is called factorisation. Factorisation ma make an epression easier to use and neater to write, or it ma help ou to interpret its meaning.
In the net eample the principles are the same but the epressions are not quite so simple. You might well do this line in our head. For, 0 and 4 this is 0. In this case 3, 5, and ma all be cancelled. Simplif 4 3. However, is not a factor of the top the numerator , so ma not be cancelled.
Then ou can see whether there are an common factors which can be cancelled. What is a variable? You will often need to find the value of the variable in an epression in a particular case, as in the following eample. How is this epression obtained? Tr dividing a polgon into triangles, starting from one verte. This is an equation which can be solved to find n. Linear equations Eample. An equation is formed when an epression, in this case 80 n , is set equal to a value, in this case 80 or , or to another epression.
Solving means finding the value s of the variable s in the equation. Since both sides of an equation are equal, ou ma do what ou wish to an equation provided that ou do eactl the same thing to both sides. If there is onl one variable involved like n in the above eamples , ou aim to get that on one side of the equation, and everthing else on the other.
The two eamples which follow illustrate this. In both of these eamples the working is given in full, step b step. In practice ou would epect to omit some of these lines b tiding up as ou went along.?! What happens when ou tr to solve it as an equation? This is an identit and not an equation. It is true for all values of.
To distinguish an identit from an equation, the smbol is sometimes used. Thus 5. If the are different, ou have made a mistake. Start b clearing the fractions. Since the numbers and 3 appear on the bottom line, multipl through b 6 which cancels both of them.
The third angle is The total of their ages is 3 ears. The perimeter of the field is m. One mark is deducted for an question which is not attempted. A candidate attempts q questions and gets c correct. Algebra i Write down an epression for the candidate s total mark in terms of q and c.
Write down and solve an equation for the number of questions which James gets right. Some of these are old potatoes at c per kilogram, the rest are new ones at 36c per kilogram.
What mass of new potatoes does he bu? An equation like this is often called a formula. You often need to make one of the other variables the subject of a formula. In that case, the steps involved are just the same as those in solving an equation, as the following eamples show.
It is usuall easiest if ou start b arranging the equation so that the variable ou want to be its subject is on the left-hand side. This formula gives the speed of an oscillating point. This formula gives the momentum after an impulse. Collect terms in m on the left-hand side and terms without m on the other. All the formulae in Eercise C refer to real situations. Can ou recognise them? Form an equation involving the length, m, of the field.
Since the length of the field is 40 m greater than the width, the width in m must be 40 and the area in m is This is in contrast to a linear equation. A linear equation in the variable involves onl terms in and constant terms. It is usual to write a quadratic equation with the right-hand side equal to zero.
To solve it, ou first factorise the left-hand side if possible, and this requires a particular technique. You can see this result in terms of the area of the rectangle in figure. This gives ou four terms, which correspond to the areas of the four regions in a diagram like figure. The coefficient of is 7. There are other methods of quadratic factorisation. If ou have alread learned another wa, and consistentl get our answers right, then continue to use it.
This method has one major advantage: it is self-checking. You ma have chosen the wrong numbers, or made a careless mistake, or perhaps the epression cannot be factorised.
pure mathematics 1 by sophie goldie a levels
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Bgas book pdf -
We are grateful to the following companies, institutions and individuals ou have given permission to reproduce photographs in this book. The publishers will be glad to make suitable arrangements with an copright holders whom it has not been possible to contact. Hachette UK s polic is to use papers that are natural, renewable and recclable products and made from wood grown in sustainable forests.
We are grateful to the following companies institutions and individuals you have given permission. Hachette UK s policy is to use papers that are natural renewable and recyclable products and. Telephone 44 Fax 44 Lines are open 9 00 5 00 Monday. Much of the material in this book was published originally as part of the MEI Structured. Year , All rights reserved Apart from any use permitted under UK copyright law no part of this.
Embed Size px x x x x We are grateful to the following companies, institutions and individuals you have given permission to reproduce photographs in this book. Every effort has been made to trace and acknowledge ownership of copyright. The publishers will be glad to make suitable arrangements with any copyright holders whom it has not been possible to contact. Hachette UK's policy is to use papers that are natural, renewable and recyclable products and made from wood grown in sustainable forests. The logging and manufacturing processes are expected to conform to the environmental regulations of the country of origin. Telephone: 44
Pure Mathematics 2 and 3 International ebook online. Written to match the contents of the Cambridge syllabus. Pure Mathematics 2 corresponds to units P2 and P3.
Ее снова сжали уже знакомые ей стальные руки, а ее голова была намертво прижата к груди Хейла.