Table Of Content

Further Pure Mathematics 2 Edexcel AS and A Level Modular Mathematics I I y A PEARSON COMPANY —— Further Pure Mathematics 2 Edexcel AS and A-level Modular Mathematics Contents About this book cT Inequalities 1.1 Solving inequalities by manipulation 1.2 Solving inequalities graphically Series 2.1 Using the method of differences to sum simple finite series Further complex numbers 3.1 The modulus-argument form 3.2 Euler's relation 3.3. Multiplying and dividing two complex numbers 34 De Moivre's theorem 438. De Moivre's theorem applied to trigonometric identities [3.6 Using de Moivre's theorem to find the nth roots of a complex number 3.7. Using complex numbers to represent a locus of points on an Argand diagram 3.8. Using complex numbers to represent regions on an Argand diagram 3.9. Applying transformations that map points on the z-plane to points on the w-plane by applying a formula relating z= x + jy tow =u + iv Review Exercise 1 ‘uss order differential equations 4.1 Solving frst order differential equations with separable variables and the formation of differential equations and sketching members of the family of solution curves 4.2 Solving exact equations where one side is the exact derivative of a product and the other side can be integrated with respect tox 4.3. Solving first order linear differential equations ofthe type $2 + Py = Q, where ‘and Qare functions of x, by multiplying through the equation by an, Integrating factor to produce an exact equation 444 Usingaa given substitution to reduce a differential equation into one ofthe above types of equation, which you can then solve 78 81 Second order differential equations Su 52 83 34 3s 536 Finding the general solution of the linear second order differential equation ay a 8 9 = 0 wheres bandexe conn and whee > a Finding the general solution of the linear second order differential equation 2Y 5 B+ cy = 0, where a, band care constants and where b? = 4ac a2 «6S + cy = 0,wherea, band tants and where D? Finding the general solution ofthe linear second order differential equation + cy = 0, where a, b and care constants and where b? < 4ac Ping the gre ot ofthe inca scan kein qaton 85 y= hea band are consans by sng 3 Sconplenenary fron + parc mega! sng bundy condos, oa pec stton oftheir cod ok aie equton a inh odontal spec souton ofthe esa oder eta fog equation a dy + 6&4 cy =f), where a, band care constants, or ar + bE + cx=f(9, where a, band care constants Using a given substitution to transform a second order differential equation Into one of the above types of equation, and solving it ‘Maclaurin and Taylor series, 61 62 63 64 6S Finding and using higher derivatives of functions Expressing functions of x as an infinite series in ascending powers of x using. Maclaurin’s expansion Finding the series expansions of composite functions using known Maclaurin’s expansions. Finding an approximation to a function of x close tox = a, where a ¥ 0, using ‘Taylors expansion of the function Finding the solution, in the form ofa series, to a differential equation using the ‘Taylor series method. Polar coordinates 7 72 73 74 78 Polar and Cartesian coordinates Polar and Cartesian equations of curves Sketching polar equations ‘Areas using polar coordinates Finding tangents parallel and perpendicular to the initial line Review Exercise 2 Examination style paper Answers Index 87 a 93 7 100 105 106 108 42 us us 127 128 130 132 136 139 us 155 187 185 About this book ‘This book is designed to provide you with the best preparation possible for your Edexcel FP2 unit examination ‘links’ to underline the importance of mathematics: to the real world, to your study of further units and. to your career ‘+ This is Edexcel’s own course for the GCE specification. ‘+ Written by a senior examining team at Edexcel: the chair of examiners, chief examiners and principal examiners. + The LiveText CD-ROM in the back of the book contains leven more resources to support you through the unit. Finding your way around the book Sas 2 Feview exercise helps you consolidate Detailed contents aa lst shows which parts of the FP2 Speciation are covered in each fection Each section begins with a statement of what is covered in the section Each section ends. with an exercise =the questions are carefully graded so they increase in difficulty and sradualy bring you upto standard Each chapter ends with At the end of the book there is. amixed exercise anda _an examination-style paper. summary of key points ‘colour scheme, to help you find the right chapter quickly LiveText software ‘The LiveText software gives you additional resources: Solutionbank and Exam café. Simply turn the pages of the electronic book to the page you need, and explore! Unique Exam café feature: ‘+ Relax and prepare - revision planner, hints and tips; common mistakes * Refresh your memory ~ revision checklist; language of the examination; glossary * Get the result! — fully worked examination-style paper with chief examiner's commentary Solutionbank * Hints and solutions to every {question in the textbook * Solutions and commentary for all review exercises and the practice ‘examination paper Published by Pearson Education Limited a company incorporated in England and Wales, having is eistred ‘fc t Edinburgh Gate, alow, Essex, CM2O JE. Registered company number: 872828 excels a repstered tademark of Edercel Limited “Text © Greg Attwood, Lae Cope, Bronwen Mora, Laurence Pateman, Keith Plier, Geof Staley, Dave Wilkins 2009 131211100 10987654321 'elush Libary Cataloguing in Publication Data savallabe om the Bish Libary an quest ISBN 978043551921 6 Copyright notice “Allright reserved: No past ofthis publication may be reproduced in any fom or by any means (induding Dhotocopying or soring tn any medium by electronic means and whether or not ansently or ncdentally to Some ote we ofthis publication) without the writen pemsion ofthe copyright owner, except in acordance ‘ith the provision of the Copyigh, Designs and Patents fet 1988 or under the ter ofa ence sted y the Copysight Licensing Agency, alton House, 6-10 Kiby tet, London ECIN BTS (wwcla cou). Applications forthe copyright owner's writen permision should be addressed tothe publisher Ete by Catherine Riches “Typeset by Teche Ltd Mlusuated by TecSet Led Cover design by Christopher Howson Picture research by Christe Martin Cover photflustration © texee! Panted in the UK by Scotpint ‘The sauthor and publsher would like to thank the felling individuals and organisations for permission to reproduce photographs: Gesry Images / ester Lefkowitz pl Alamy / Chris Howes / Wild Places Images p41; Science Photo Library Victor abbick Visions p18; Alamy / Roger Bamber p72; Alamy / Stefan Hunziker ~ www sporsphotogaphy-ch p87; ‘Alsny’/ Andrew Paterson p08; Seence Photo Library / De Job Brachenburg p127. very efor hasbeen made to contac copyright holders of material reproduced in this bok. Any omisons wil ‘erected in subsequent printings if notice is given to the publishes, Disclaimer ‘This Edel publieation ofr high-quality suppor fr the delivery of Edexcel qulieatins Edexcel endorsement doesnot mean that this mateial is etetal to achieve any Edexcel qualification, nr does it ‘pean that hiss the only stable mater avalable wo suppor any Edexcel qualiicaton. No endorsed material wil ‘be sed verbatim in setting any Edexcel examnatin/asesment abd any resource Iss produced by Bdexcl shall, Include this and other appropriate texts. Coles offical pecfications for al Edexcel ualfcations may be found onthe Edexcel bse wdc com, After completing this chapter you should be able to: ‘© manipulate inequalities determine the critical values of an inequality ‘© find solutions of algebraic inequalities. Inequalities In manufacturing or business you will want to know what level of price will ensure that Mest epplkrtion of mathematica ‘your profit is greater than your production rade the niu of neues Costs - this means solving inequalities. some an ur 1 ‘1.1 You can manipulate inequalities to solve them. In C1 you learnt how to solve simple quadratic inequalities by rearranging them. The inequality sign can be treated like an equals sign as long as you do not divide or multiply both sides of the expression by a negative number. ‘There are three steps to solving inequalities. Solve 2x? <x +3 ‘Step Tis to find the citical values. Rearrange the expression and then replace the inequality symbol with an equals sign and solve. | 2S | $0 (2x = Bx +1) =0 So the critical values are x = 3 or —1 ‘A oketch of y = 2x® — x — 5 gives ‘Step 2s to draw a sketch, or use a table ‘of values to determine which sets of values satisty the inequality. So the solution toy 3<0 OY XESS Stop 3's to write down the answer by ie when —1<x <15 {Gang the graph to Interpret te Inequaly. In FP2 you will be dealing with algebraic fractions, and care must be taken when rearranging the inequality to make sure that you are not multiplying by a quantity that could be negative. — Solve the equality #25 <a 6 12 coag ee reomeny yh ap eeenteee eee ae Rheeee Qiemcmers (e278 x 2 < e—2F x (x +1) ‘multiply both sides of the inequality by