H2 Mathematics Textbook CHOO YAN MIN Includes TYS & Answers. Covers both 9740 & 9758 syllabuses. This version: 27th August 2016. The latest version will always be at this link. This book is optimised for viewing in PDF format (click the above link). Other existing formats are crude conversions and may be sub-optimal. Recent changes: Minor changes since July 24th. Rewrote preface. Upcoming changes: None planned. Page 2, Table of Contents www.EconsPhDTutor.com Errors? Feedback? Email me! (cid:44) (cid:44) With your help, I plan to keep improving this textbook. Page 3, Table of Contents www.EconsPhDTutor.com Please do not be intimidated by the length of this book (~1,400 pages). The actual main content takes up only about 700+ pages. The other 700 pages are for things like front matter, TYS questions, appendices, reproductions of formula lists and syllabuses, and answers to exercises. Page 4, Table of Contents www.EconsPhDTutor.com This book is licensed under the Creative Commons license CC-BY-NC-SA 4.0. You are free to: • Share — copy and redistribute the material in any medium or format • Adapt — remix, transform, and build upon the material The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms: • Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. • NonCommercial — You may not use the material for commercial purposes. • ShareAlike — If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original. • No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Notices: You do not have to comply with the license for elements of the material in the public domain or where your use is permitted by an applicable exception or limitation. No warranties are given. The license may not give you all of the permissions necessary for your intended use. For example, other rights such as publicity, privacy, or moral rights may limit how you use the material. Author: Choo, Yan Min. Title: H2 Mathematics Textbook. ISBN: 978-981-11-0383-4 (e-book). Page 5, Table of Contents www.EconsPhDTutor.com The first thing to understand is that mathematics is an art. Paul Lockhart (2009, A Mathematician’s Lament, p. 22). A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. ... Beauty is the first test: there is no permanent place in the world for ugly mathematics. - G.H. Hardy (1940 [1967], A Mathematician’s Apology, pp. 84-85). The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. - Henri Poincaré (1908 [1914], Science and Method, English trans., p. 22). Page 6, Table of Contents www.EconsPhDTutor.com About This Book This textbook is for Singaporean H2 Maths students (hence the occasional Singlish and 1 TLAs ). Of course, I hope that anyone else in the world will also find this useful! I needed a definitive reference for my own teaching needs, but could find nothing satisfying. So I decided to just write my own textbook. This textbook is based exactly on the old (9740) and revised (9758) syllabuses (also reproduced in Part VIII). Do check to make sure which exam you’re taking. The revised syllabus (9758) is the same as the old syllabus (9740), but with noticeable chunks excised 2 and is thus easier. 9740 (old) examined? 9758 (revised) examined? 2016 Yes. No. 2017 Yes, for the last time. Yes, for the first time. 2018 No. Yes. SYLLABUS ALERT Where there are any differences between the old and revised syllabuses, I’ll let you know with a yellow box like this. • FREE! This book is free. But if you paid any money for it, I certainly hope your money is going to me! This book is free because: 1. It is a shameless advertising vehicle for my awesome tutoring services. 2. The marginal cost of reproducing this book is zero. • DONATE! This book may be free, but donations are more than welcome! Donation 3 methods in footnote. It’s irrational for Homo economicus to donate. But please consider donating because: 1. You’re a nice human being [*emotional_manipulation*]. 2. Your donations will encourag(cid:44)e me and others to continue producing awesome free content for the world. 1Three Letter Abbreviations. 2Indeed, some chunks of the old syllabus (9740) have simply been moved into the syllabus of Further Maths (9649), which the authorities have kindly resurrected for the 2017 exam season. 3Singapore POSB Savings Account 174052271; OCBC Savings Account 5523016383 (Name: Choo Yan Min). Bit- coin wallet: 1GDGNAdGZhEq9pz2SaoAdLb1uu34LFwViz. Paypal [email protected] (Name: Yan Min Choo, USD preferred because this account was set up in the US). USA. Venmo link (Name: Yanmin Choo). Or buy a copy of my book on Amazon. Not exactly the best way to support me, because Amazon eats nearly half the price you pay. But hey if you have an Amazon account set up and it’s more convenient for you than any of the preceding donation methods, go ahead and buy 100 copies. (cid:44) Page 7, Table of Contents www.EconsPhDTutor.com • HELP ME IMPROVE THIS BOOK! Feel free to email me if: 1. There are any errors in this book. Please let me know even if it’s something as trivial as a spelling mistake or a grammatical error. 2. You have absolutely any suggestions for improvement. 3. Any part of this book is less than crystal clear. Here’s an anecdote about Richard Feynman, the great teacher and physicist: Feynman was once asked by a Caltech faculty member to explain why spin 1/2 particles obey Fermi-Dirac statistics. He gauged his audience perfectly and said, “I’ll prepare a freshman lecture on it.” But a few days later he returned and said, “You know, I couldn’t do it. I couldn’t reduce it to the freshman level. That means we really don’t understand it.” 4 I agree: If you can’t explain something simply, you don’t understand it well enough. Corollary: An excellent test of whether you understand something is to see if you can explain it simply to someone else. If at any point in this textbook, you have read the same passage a few times, tried to reason it through, and still find things confusing, then it is a failure on MY part. Please let me know and I will try to rewrite it so that it’s clearer. (There is also the possibility that I simply messed up! So please let me know if there’s anything confusing!) I deeply value any feedback, because I’d like to keep improving this textbook for the benefit of everyone! I am very grateful to all the kind folks who’ve already written in, allowing me to rid this book of more than a few embarrassing errors. • LyX rocks! 5 This book was written using L X. Y • Is the font size big enough? You’re probably reading this on some device. So I’ve tried to set the font sizes and stuff so that one can comfortably read this on a device as small as a seven-inch tablet. It should also be possible to read this on a phone, though somewhat less comfortably. (Please let me know if you have any feedback about this!) (I’ll probably be contacting some publishers to see if they want to do a print version of this, for anyone who prefers it in print.) 4This quote or some similar variant is often (mis)attributed to Einstein. But as Einstein himself once said, “73% of Einstein quotes are misattributed.” 5LATEX is the typesetting program used by most economists and scientists. But LATEX can be difficult to use. LYX is a user-friendly GUI version of LATEX. LYX has boosted my productivity by countless hours over the years and you should use LYX too! Page 8, Table of Contents www.EconsPhDTutor.com Tips for the Student • Read maths slowly. Reading maths is not like reading Harry Potter. Most of Harry Potter is fluff. There is little fluff in maths. So go slowly. Dwell upon and carefully consider every sentence in this textbook. Make sure you completely understand what each statement says and why it is true. Reading maths is very different from reading any other subject matter. If you don’t quite understand some material, you might be tempted to move forward anyway. Don’t. In maths, later material usually builds on earlier material. So if you simply move forward, this will usually cost you more time and frustration in the long run. Better then to stop right there. Keep working on it until you “get” it. Ask a friend or a teacher for help. Feel free to even email me! (I’m always interested to know what the common points of confusion are and how I can better clear them up.) • Examples and exercises are your best friends. So work through them. A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one. - Paul Halmos (1983, Google Books). Work through all the examples and exercises. Merely moving your eyeballs is not the same as working. Working means having pencil and paper by your side and going through each example/exercise word-by-word, line-by-line. 2 2 For example, I might say something like “x y 0. Thus, x y x y 0.” If it’s not − = ( − )( + ) = obvious to you why the first sentence implies the second, stop right there and work on it until you understand why. Don’t just let your eyeballs fly over these sentences and pretend that your brain is “getting” it. I will often not bother to explain some steps, especially if they simply involve some simple algebra. • You get a List of Formulae during the A-level exam. So there’s no need to memorise all the formulae that are already on the list you’re getting. Note that you get a different list depending on which exam you’re taking — List of Formulae (MF15) for the old 9740 exam and List of Formulae (MF26) for the revised 9758 exam. (Both lists are reproduced in Part VIII of this book.) I cannot guarantee though that your JC will give you the List during your JC common tests and exams. Page 9, Table of Contents www.EconsPhDTutor.com • Remember your O-Level Maths & ‘A’ Maths? You’ve probably forgotten some (or most?) of it, but unfortunately, you are still assumed to know EVERYTHING from O-Level Maths & ‘A’ Maths. (To take H2 Maths, most JCs 6 require that you at least passed ‘A’ Maths. ) See in particular the lists near the end of either the 9740 (old) or the 9758 (revised) syllabus. Skim through and see if anything looks totally alien to you! Some chapters (e.g. Chapters 5 and 26) in this textbook will give a quick review of some of the O-Level Maths material that you may have forgotten but which we’ll use quite often. • Online Calculators Google is probably the quickest for simple calculations. Type in anything into your browser’s Google search bar and the answer will instantly show up: Wolfram Alpha is somewhat more advanced (but also slower). Enter “sinx” for example and you’ll get graphs, the derivative, the indefinite integral, the Maclaurin series, and a bunch of other stuff you neither know nor care about. The Derivative Calculator and the Integral Calculator are probably unbeatable for the specific purposes of differentiation and integration. Both give step-by-step solutions for anything you want to differentiate or integrate. Here is a collection of spreadsheets I made. These spreadsheets are for doing tedious and repetitive calculations you’ll often encounter in H2 maths (e.g. with vectors, complex numbers, etc.). As with anything I do, I welcome any feedback you may have about these spreadsheets. Perhaps in the future I will make a more attractive version of it. (Instructions: Click “Make a copy” to open up your own independent copy of this spreadsheet. Enter your input in the yellow cells. Output is produced in the blue cells. If you mess up anything, simply click the same link and “Make a copy” again.) 6Some JCs, like HCI, even require that you got at least a B3 for both Maths & ‘A’ Maths. Page 10, Table of Contents www.EconsPhDTutor.com