Mathematical Supplements

MATHEMATICALSUPPLEMENTS For H2 Math Students

MATHEMATICAL SUPPLEMENTS For H2 Math Students First Edition Yang Xu

A LETTER TO THE READER Hello, and thank you for picking up my book! I am Yang Xu from the 2021 VJC cohort, and I am really excited to share with you my journey with Mathematics! Math wasn't always my strongest subject, and I really struggled with it back in secondary school. I understand the frustration when concepts do not click despite the eort, and the disappointment at the marks lost to careless mistakes. But one thing Mathematics has taught me is that with the right mindset, appropriate guidance, and some practice, you can really start to see improvement, and even start enjoying math! Having gone through this journey myself, I rmly believe that everyone is capable of doing well in Mathematics, provided the eort and appropriate understanding of the subject. Mathematics is a subject of immense beauty, but only if you allow yourself to see it. In my humble opinion, the structure of the examined syllabus and the competitive nature of exams often overshadow the true elegance of math, making it easy to miss the profound connections between topics. Through this book, I hope to help you discover new insights and develop a deeper appreciation for math, especially for the examinable topics. But even if you don't come away with that appreciation right away, I still encourage you to keep an open mind and give your best eort. Remember, the true beauty of mathematics goes far beyond what I can express here. I urge you to continue exploring on your own if you're curious; there is truly so much more to discover! In this book, I have put together a non-exhaustive summary of the key ideas and conventional solving methods from the H2 Math syllabus. I've also included some personal tips, shortcuts, and insights that I found helpful as a student. These sections are designed to help refresh and review the H2 content while oering some new perspectives on familiar concepts. I have also dedicated a part towards the end of this book to exploring some interesting topics in the syllabus, though not explicitly taught. These concepts have personally helped me a lot in my math journey, but they're only the tip of the iceberg. I encourage you to dive deeper and explore more beyond what's presented here. Please note that this book is only supposed to serve as a supplement, not a 2

3 replacement of your school notes. Certain parts also require H2 knowledge that is not covered explicitly in this book. On a side note, I would also like to mention that Math concepts beyond the syllabus can also develop your critical thinking and problem-solving skills in H2 Math. After all, that is what you are being examined for. These topics either broaden your way of thinking, or provide valuable insights into H2 Math topics. For instance, learning about `Linear Algebra' has helped me to appreciate the interconnections between vectors, matrices, and systems of linear equations. I would occasionally share some of the insights I have gained throughout the span of this book. Meanwhile, I also encourage you to read beyond the syllabus, to see the underlying interconnection between dierent topics, and appreciate the beauty of Mathematics. As a disclaimer, some of the phrasings in this book are layman paraphrases for the sake of simplicity and ease of comprehension. They may not be accepted in working presentations, hence I would urge you to clarify the presentation aspects with your tutors. The content here is also non-exhaustive, hence don't expect to nd everything here. This just serves as a brief recap, an extension, and some hopefully-useful tips. Moreover, if you face any diculties in your studying, please do not hesitate to contact your subject tutors, or me via my VJC email (yang.xu.2021@vjc.edu.sg). With that, I wish you all the best for all your Math tests and exams; I really really believe that all of you are able to do well in them! Best Wishes, Yang Xu Class of 21S51

CONTENTS A Letter to the Reader 2 Contents 4 I Content Topics 7 1 General Tips 8 1 Analysing the Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Solving the Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Checking Your Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 System of Linear Equations 21 1 Simultaneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2 Solving Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 Tips and Tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Inequalities 25 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2 Solving Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4 Tips and Tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4 Curve Sketching 30 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 Solving Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3 Visualizing Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4 Tips and Tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5 Functions 36 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2 Key Concepts and Common Mistakes . . . . . . . . . . . . . . . . . . . . . . 36 3 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4 Why does reecting a function give its Inverse? . . . . . . . . . . . . . . . . 39 5 An Interesting Perspective on the Horizontal Line Test . . . . . . . . . . . . 41 6 Graph Transformations 44 1 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4

CONTENTS 5 2 Tips, Tricks, and Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3 Visualising Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7 Graph Deduction 49 1 Modulus Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2 Reciprocal Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3 Derivative Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4 Two Practical Tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 8 Series and Sequences 53 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2 Important Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3 Tips for Exams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 9 Arithmetic and Geometric Progressions 56 1 Important Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 Tips for Exams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 10 Summations 61 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2 Solving Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3 Tips and Tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4 Method of Dierences (MOD) . . . . . . . . . . . . . . . . . . . . . . . . . . 64 11 Dierentiation 66 1 What Exactly is Dierentiation? . . . . . . . . . . . . . . . . . . . . . . . . . 66 2 Visualising the Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3 Stationary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4 Common Problem Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 12 Integration Techniques 72 1 Integration by Inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2 Integration by Parts