## Tuesday, May 24, 2011

### SYLLABUS

So what is covered in the Further Mathematics T 1993-2012 syllabus?

In paper 1, you will be learning mostly extensions of what you learn in Maths T. Of course, this will be the harder paper of the two:

1. Logic & Proof
Still remember all those p q stuff that you learn in SPM Mathematics? Logic is a study of propositions, statements which are either true or false. One needs logic to deduce many Mathematical results. You need to know what are conjunctions (and), disjunctions (or), implications, equivalence relations, what is a converse, inverse and a contrapositive of a proposition. There are some other terminologies like tautology, contradiction or counter-example which you will eventually learn what they are. Then you will learn predicates and quantifiers, which are used to generalize (all x) or specialize (some x) the propositions that you came up with. Then in the next section, you will be studying essential methods of proofs which will help you proof some laws or formulas. Examples of proofs are direct, indirect proofs, proof by contraposition, proof by contradiction and etc. Lastly, you will learn the most important proof, which is Mathematical Induction.

2. Complex Numbers
In Maths T, you will learn the basic operations for complex numbers in the Cartesian form. In FMT, you will be focusing on its polar form, and you will learn quite a lot of geometrical significance of complex numbers. You will learn how to prove and use the applications of the famous de Moivre's theorem, which will be something new to you! After studying this chapter, you will be able to solve difficult equations involving complex numbers, or rather use complex number methods to solve some problems. The hardest part of this chapter, I think, is still learning how to illustrate complex number equations as loci in the Argand diagram.

3. Matrices
After learning matrices in Maths T, you thought that matrices is the easiest thing to score. Coming to FMT, you are wrong. Still continuing to use those 3 × 3 matrices, you are going to learn how powerful matrices can be in solving linear algebra! You will eventually find out that not all systems of equations have solutions, or rather, how to know whether they have a unique, infinitely many or no solution. You will be focusing a lot on determinants and their operations, which you touch only a little in Maths T. Then, you will learn how to use Gaussian elimination, Cramer's rule and Cayley-Hamilton theorem to solve and play around with systems of equations of 3 unknowns. In the end, you will learn how to find eigenvalues and eigenvectors of a 3 × 3 matrix, and their significance.

4, Recurrence Relations
This chapter is one of the hardest, partly because the resources on this topic is hard to find, and also a lot of understanding is required. To fully understand this chapter, prior knowledge of Sequence and Series from Maths T is required, and mastering the chapter Differential Equations later on will help too. You will get to understand what does 'recurrence' mean, learn how to find general and particular solutions for recurrence relations equations. The toughest part for you in this chapter is to learn how to solve problems that are modelled by recurrence relations.

5. Functions
This chapter really has nothing much to do with the one in Maths T. Instead, this chapter focuses on the analysis and solving of 3 kinds of functions, namely the inverse trigonometric (sin-1, cos-1, tan-1), hyperbolic (sinh, cosh, tanh) and also the inverse hyperbolic (sinh-1, cosh-1, tanh-1) functions. At last, you get to understand what is that sinh thing in your calculator! This chapter will focus a lot on memorizing of formulas (tonnes of them), proving them, sketching their graphs and solve equations which comprises of these functions.

6. Differentiation and Integration
As if differentiation and integration in Maths T wasn't enough, you will be bombarded with a whole new set of differentiation and integration formulas. You take one step further from Maths T to learn how to determine the differentiability of a function, as well as to find the second derivative of a parametric and an implicit function. You will be learning how to differentiate and integrate those nasty functions stated in the previous chapter, as well as learning how to use them in substitution. You will learn about the reduction formulae of certain definite integrals, normally learning how to prove them. In addition to the evaluation of area under the curve and the volume of revolution you learned in Maths T, you will now learn how to find the arc length and the surface area of revolution. Scary right? This is hard core calculus man….

7. Power Series
Not Power Rangers Series. This chapter will answer your doubt in Maths T paper 2, the place where you saw the proof for the Poisson Distribution. This chapter will is a continuation of the Series that you learn in Maths T, and it will introduce to you that every function can be written in the form of a series, which can be either a Taylor or a Maclaurin series (You will learn Fourier and Laplace in university). You will learn how to use the remainder theorem to approximate some functions. A lot of differentiation and integration will be used in this chapter. L' Hôpital's rule and D'Alembert's ratio test are 2 other concepts you will learn here.

8. Differential Equations
Probably the easiest chapter in the whole of Paper 1. After learning this chapter, you will say that the differential equations you learn in Maths T is really NOTHING at all. You now learn how to solve first order linear differential equations by using an integrating factor. Then you will progress to solve 2nd order linear differential equations, finding the complementary function and particular integral, and thus find the general or particular solution. The good thing about this chapter is that there is not much problems they can model for you in the exam, the questions will be more straightforward.

In paper 2, you will be learning a whole new set of chapters. You will learn 2 basic theories from discrete mathematics, 2 chapters of advanced geometry and 4 chapters of statistics. This is definitely the easier paper:

9. Number Theory
Number theory, as most people say, is the "Queen of Mathematics". It is probably the hardest chapter to score, not only because there is little low-level resources about it, also because it is hard to understand. You will be learning about divisibility of numbers, and you never knew it could be that hard! Understanding prime and composite numbers, finding greatest common divisors (gcd) and least common multiples (lcm), which you have learned in form 1 (probably you have forgotten about it totally)! Then, you will get into detail analysis to study some theorems and algorithms, and the main part will be the study of modulo congruences, stuff like a ≡ b (mod m). This is one tough chapter, and normally ignored by most people because it's hard and only comprises of a few marks.

10. Graph Theory
Bar graphs? Line graphs? No way! Graph theory has nothing to do with all those statistical graphs that you learnt in Maths T. This is one chapter that you will be doing a lot of drawing. You will learn what exactly a "graph" is, and a lot of its related terminologies. vertices, edges, walks, paths, cycles, circuits and etc. You will be exposed to Eulerian and Hamiltonian paths and learn how to identify or solve problems modeled by them. You will also learn how to use Matrix to represent graphs. Interesting right? It can be one of the easiest chapters in this paper when you fully understand it.

11. Transformation Geometry
When you learned transformation in form 4, you only learned them qualitatively. Here in FMT, you will be dealing with matrices, and how to use them to represent a transformation in 2 dimensions (only). Rotations, translations, reflections, scaling, and you will be introduced to 2 new transformations: stretching and shearing. It is good to study matrices in Maths T before you start this chapter.

12. Coordinate Geometry
In Maths T, you stop only at 2 dimensions. In FMT, you will be doing coordinate geometry in 3D, which is not only hard to visualize, but also hard to solve them. You will find out that a line can be represented by various forms, and the equations of lines are not unique at all! In 3D, you will learn about planes and how to represent them, but not solids (thank goodness). Then you will be taught how to find the intersections, angles, or shortest distances between lines, planes or both. This part is really challenging, but will be easy if you manage to memorize all the formulas.

13. Sampling & Estimation
If you would have noticed, this is the very same chapter that appeared in Maths S paper 2. Yes, this is the starting of the 4 chapters of statistics, which are the easiest parts of the paper! In real life, we need to do some surveys and samplings in order to evaluate our business, or maybe to improve the service of a newly started company. You will learn to differentiate what is a population and a random sample, and using the sampling distributions to solve problems. You will use the central limit theorem to make samples of large sizes easier to evaluate. Other than the Binomial, Poisson and Normal distributions, you will learn a new distribution called t-distribution and when to use it. This chapter will then focus more on confidence intervals, how to obtain them and interpret them based on a large sample, small sample or a population proportion. You can study this chapter immediately after doing chapter 5 in Maths T.

14. Hypothesis Testing
No, this is not Physics or Chemistry experiments. You need to test a hypothesis, which is normally a 'claim' by a company regarding their products or services, in order to know whether they are genuine, or liars. You will learn how to formulate a null and alternative hypotheses, use a systematic way to test them using different distributions you learned earlier on based on different situations. You will learn how to determine and identify critical regions based on what significance level you use, in order to decide whether to reject or accept the null hypothesis. You will also learn some concepts like type I and type II errors, and how to use or find them.

15.
χ2 Tests
It's not x2, it is "chi-squared' tests (chi, read as 'kye' is a greek alphabet).
χ2 tests are used to determine whether a sample belongs to a certain distribution. The method of doing it is very similar to hypothesis testing, but much easier. You will learn how to use χ
2 tests to determine goodness of fit or independence in a contingency table. It is quite a straightforward and short chapter, don't worry.

16. Correlation & Regression
This chapter also appears in paper 2 of Maths S. This is what I called as 'hard-core statistics'. This chapter will teach you how to make use of the full power of the statistic functions in your calculator! You will first learn about scatter diagrams. Using these scatter diagrams, you will then learn about the concept of correlation, interpret the Pearson correlation coefficient, and then the drawing of regression lines. In the end, you will learn how to use the relationships between correlation coefficient, regression coefficient and the coefficient of determination. This chapter can be very messy in calculations, so as I said earlier on, your calculator will be really useful here. This is also the only chapter whereby you might possibly need a graph paper.

If you are reading through this before you even started STPM, don't be scared by the terminologies. You must have the craving to understand everything. You can start spending time to Google about these stuff when you are free. There is still a lot to learn about Mathematics later on. As you can see, paper 2 is half statistics, half hard core mathematics. The strategy of this paper is to do perfectly in the statistics, and do your best for the rest. Not hard to get an A if you put enough effort into it. Note that I will giving you lectures on all the 16 chapters. What I will do is I will upload some tips or short notes for you to prepare for the exam. You are still required to study the textbooks and spend time understanding them.