### An analysis of the H2 Mathematics (9758) 2018 Paper

Let us do a breakdown and analysis of the H2 Math Paper.

Paper 1 was in general quite manageable. A few questions stood out, i.e., question 9iii and 10ii. These two are in fact “knowledge” from H2 Further Mathematics. These are considered to be application questions in the eyes of SEAB, thus, it is fair for such questions to appear.

Question 9iii introduced the idea of arc length. The question is clear on what they expect the students to do, copy the given formula, substitute everything in, simplify and integrate. It is not difficult, however students must have the confidence to do such questions.

Question 10ii was a different taste of Differential Equations. Students need to be able to relate on the spot, the many equations given and also focus properly in the face of the many unknown constants. Some students might give up and also forsake their marks in the subsequent parts. This is not wise as the papers in general, often are set such that students can still continue and attempt. T

The rest of the paper is manageable. Good students might pick up that q8 – 11 takes up 50% of the marks, while the first seven questions is another 50%. The first half of the paper is simple knowledge and skills. Students that were not careless should be able to manage them with ease. Doing the specimen paper would have helped the student slightly too. Personally, I feel that a good score for the first seven question is 43 while a good score for the next four questions should be about 37.

Paper 2 was a different ball game. Firstly, it penalized students who were spotting the topics and had Differential Equations again, while not testing Graphings entirely. Section A was rather tedious and students have to be very surgical while doing the questions. The good thing was that section A did not contain much unknown constants. Good students would be able to check their answers if they had enough time.

Section B on the other hand, was slightly different. If a student had panicked or spent too much time in Section A, they might not be able to clear Section B. Most students would have expected Section B to be a breeze given their school examinations but this time wrong, it is not the case. Questions were rather subtle in their information. There are also quite a handful of “show” questions. As what I shared with students every time, a “show” question allows us to walk out of the exam hall knowing we have the marks in the bag.

In conclusion, the papers definitely showed us what SEAB expects of students, i.e., to be thinkers and respond on the spot. Students must have strong mental fortitude now, and cannot fumble as the paper is rather rigorous. Top-tier JCs should find the paper rather ok since they ought to be used to such rigor in their regular examinations.

### Aftermath of A’levels H2 Mathematics (9758) 2018

Some students proclaimed “I’m finally done with Math!”. Just fyi, be careful of the modules you take in University because chances are that Math will still be around. 🙂

The purpose of this post is to do a breakdown of the recent paper and of course what can be observed for the upcoming paper, i.e., A’levels 2019. I am not so concerned about what had passed, but what will be coming soon. I will break it up into three parts:

Expectations for 2019 and beyond, and

How to prepare for H2 Mathematics for 2019 and beyond.

### Some tips on preparing for A’levels 2018

Personally, I find SEAB-Cambridge getting more and more creative these days. Looking at the recent practical exams, they seem to prefer students to think and perform on the spot. If we look elsewhere to O’levels, this seem to be the trend too. I will prefer to say that TYS are useful to get students used to the answering style, however students should not be over reliant on TYS. It is crucial for students to have a strong understanding of the content of H2 Mathematics. Other words, know what you are doing.

Looking at 2017, one example of think and perform would be the ratio test. This question also opens up the door to testing Maclaurin’s Series with sum to infinity, which is more like a university know-how. To quote a Friend, A’levels is Pre-U after all. Aside from this, would be the conditional probability application with regards to quick test. We also saw a handful of definition question which Cambridge was also very specific in their demands and were looking for certain keys. Examples we saw in 2017 are the definition of line, plane and random sample. Aside from these, the paper in general, was manageable, with a handful of constants to test students’ ability at an entry level.

Going into 2018, students should be aware of proper terminologies taught in H2 Mathematics, for instance, polar form of a complex number. The nitty gritty details. Definitions are very crucial, as I make my students memorise them. They are free marks if you can nail all the key words well. Given how the practicals were like, the papers can test anything. After all, application questions involves teaching a concept and having students apply on the spot.

If you are really running into a brick wall for preparation, make sure your concepts are strong. Review them again. Trust Cambridge to be Creative and not just test applications with vectors and DE. Piecewise functions are important to understand, composite functions too. Parametric equations, with unknowns or arbitrary parameter are good to understand, especially find definite integrals with them. Trigonometry is important to know, and students should do quick review on the assumed knowledge from secondary school.

Lastly, make sure you learn to manage your time well. If you observe the marks, the first five to six questions doesn’t really go more than 40 marks or so. The application questions, on the other hand, and there will be at least two, will weight at least 25% together. So make sure you get there.

### How to improve our H2 Mathematics?

Many students of mine have asked me this question numerous times. Some lamented that they work really hard, finishing off the revision package timely. However, they still cannot improve from a C/B to an A or some are still fishing for a pass.

I feel that many students have a misunderstanding about H2 Mathematics. The H2 Mathematics (9758) Syllabus is different for the previous (9740) by a lot, primarily because its assessment objective is different. On top of that, the scheme of exam has been changed to include application questions which will test students’ abilities to wrap their heads around a given situation and solve them. Basically, students have to do of the “core values” of Mathematics, that is, problem solving.

Understanding the syllabus is one thing, next is how we should be studying it. The syllabus content is a lot lesser, so its less wide but much much deeper. Students cannot finish all the drills and practices, like in O’levels. Today, they are expected to think, so how one learns each topic is very important. One thing I pick up from NIE Mathematics is definitely the idea of examples. Every little concepts and ideas we learn, can we think of an example to relate and reinforce our understanding with?

Next, is practices. When we practice, time ourselves to finish. Do not look at the answers as we are just marginalising ourselves the opportunity to learn. If we are stuck, look at one line of the solutions and see if it will help us. Personally, I never give students solutions and even in class, I do not use solutions and rather think together with the students. It is important to train up our thinking abilities when doing H2 Mathematics.

Now students, you have about two months left to A’levels, I highly advise you to really look at how you learn the topics. A’levels will be unpredictable and questions do not repeat. To me, the TYS is more like practice papers then drills cos they will not appear again in the actual exam.

### Modal value & Expected value

Let us look at the difference between modal value and expected value. We shall start by saying they are different, albeit close.

Modal value refers to the mode, that is, the value that has the highest probability (chance) of occurring.

Expected value refers to the value, we expect to have, on average.

Before we start, I’ll do a fast recap on Binomial Distribution, $X \sim \text{B}(n, p)$ by flashing the formulae that we can find on MF26.

$\text{P}(X = x) = ^n C_x (p)^x (1-p)^{n-x}$

$\mathbb{E}(X) = np$

$\text{Var}(X) = np(1-p)$

The expected value is simply given by $\mathbb{E}(X)$.

Now to find the modal value, we have to go through a slightly nasty and long working. You may click and find out.

We have that $\frac{\text{P}(X = r + 1)}{\text{P}(X = r)} = \frac{(n-r)}{(r+1)} \frac{p}{1-p}$. This is what we call the recurrence formula. We consider this to give us the ratio between successive probabilities. And to illustrate how this works, nothing beats an example question.

Consider candies are packed in packets of 20. On average the proportion of candies that are blue-colored is $p$. It is know that the most common number of blue-colored candies in a packet is 6. Use this information to find exactly the range of values that $p$ can take.

First, most common number is the same as saying the modal/ highest frequency.

This means that $\text{P}(X=6)$ is the highest/ largest probability… Let us turn our attention to the recurrence formula now. If $\text{P}(X=6)$ is the largest, then it means that $\text{P}(X=6) \textgreater \text{P}(X=7)$ and also $\text{P}(X=6) \textgreater \text{P}(X=5)$.

Lets start by looking at the first one… $\text{P}(X=6) \textgreater \text{P}(X=7)$

$\text{P}(X=6) \textgreater \text{P}(X=7)$

$1 > \frac{\text{P}(X=7)}{\text{P}(X=6)}$

$\frac{\text{P}(X=7)}{\text{P}(X=6)} \textless 1$

But hold on! This looks like the recurrence formula. (ok, in exams, its either you use the recurrence formula or derive on the spot. Both works!)

Now I’ll advice you try the second one (before clicking on answer) on your own, that is, $\text{P}(X=6) > \text{P}(X=5)$.

Now, if the question simply says that the expected number of blue-colored candies in a packet of 20 is 6. Then

$\mathbb{E}(X) = 6$

$(20)p = 6$

$p = \frac{3}{10}$

We observe that this value actually falls in the range of $p$ we found.

### DRV questions with a twist

I want to share a question that is really old (older than me, actually). It is from the June 1972 paper. As most students know, Maclaurin’s series was tested in last year A’levels Paper 1 as a sum to infinity. And this DRV did the same thing. Here is the question.

A bag contains 6 black balls and 2 white balls. Balls are drawn at random one at a time from the bag without replacement, and a white ball is drawn for the first time at R th draw.

(i) Tabulate the probability distribution for R.

(ii) Show that E( R ) = 3, and find Var( R ).

If each ball is replaced before another is drawn, show that in this case E( R ) = 4.

### Solutions to Review 1

Question 1
(i)
$y = f(x) = \frac{x^2 + 14x + 50}{3(x+7)}$

$3y(x+7) = x^2 + 14x + 50$

$x^2 + (14-3y)x + 50 - 21 y = 0$

$\text{discriminant} \ge 0$

$(14-3y)^2 - 4(1)(50-21y) \ge 0$

$196 - 84y + 9y^2 - 200 + 84y \ge 0$

$9y^2 - 4 \ge 0$

$(3y - 2)(3y + 2) \ge 0$

$y \le - \frac{2}{3} \text{~or~} y \ge \frac{2}{3}$

(ii)
Using long division, we find that

$y = \frac{x^2 + 14x + 50}{3(x+7)} = \frac{x}{3} + \frac{7}{3} + \frac{1}{3(x+7)}$

So the asymptotes are $y = \frac{x}{3} + \frac{7}{3}$ and $x = -7$

Question 2
(i)
$x^2 - 9y^2 + 18y = 18$

$x^2 - 9(y^2 - 2y) = 18$

$x^2 - 9[(y-1)^2 - 1^2] = 18$

$x^2 - 9(y-1)^2 + 9 = 18$

$x^2 - 9(y-1)^2 = 9$

$\frac{x^2}{9} - (y-1)^2 = 1$

This is a hyperbola with centre $(0, 1)$, asymptotes are $y = \pm \frac{x}{3} + 1$, and vertices $(3, 1)$ and $(-3, 1)$.

$y = \frac{1}{x^2} + 1$ is a graph with asymptotes $x = 0$ and $y=1$.

Use GC to plot.

(ii)
$\frac{x^2}{9} - (y-1)^2 = 1$—(1)

$y = \frac{1}{x^2} + 1$ —(2)

Subst (2) to (1),

$\frac{x^2}{9} - (\frac{1}{x^2} + 1 - 1)^2 = 1$

$\frac{x^2}{9} - (\frac{1}{x^2})^2 = 1$

$x^2 - \frac{9}{x^4} = 9$

$x^6 - 9 = 9x^4$

$x^6 - 9x^4 - 9 = 0$

(iii)
From graph, we observe two intersections. Thus, two roots.

Question 3
(ai)
$\sum_{r=1}^n (r+1)(3r-1)$

$= \sum_{r=1}^n (3r^2 + 2r -1)$

$= \sum_{r=1}^n 3r^2 + \sum_{r=1}^n 2r - \sum_{r=1}^n 1$

$= 3 \sum_{r=1}^n r^2 + 2 \sum_{r=1}^n r - \sum_{r=1}^n 1$

$= 3 \frac{n}{6}(n+1)(2n+1) + 2 \frac{n}{2}(1 + n) - n$

$= \frac{n}{2}(n+1)(2n+1) + n(1+n) - n$

$= \frac{n}{2}(n+1)(2n+1) + n^2$

(aii)
$2 \times 4 + 3 \times 10 + 4 \times 16 + ... + 21 \times 118$

$= 2 [2 \times 2 + 3 \times 5 + 4 \times 8 + ... + 21 \times 59]$

$= 2 [(1+1) \times (3 \cdot 1 - 1) + (2+1) \times (3 \cdot 2 -1) + (3+1) \times (3 \cdot 3 -1) + ... + (20+1) \times (3 \cdot 20 -1) ]$

$= 2 \sum_{r=1}^{20} (r+1)(3r-1)$

$= 2 [\frac{n}{2}(n+1)(2n+1) + n^2 ]$

$= n(n+1)(2n+1) + n^2$

$= n(2n^2 + 3n + 1) + n^2$

$= 2n^3 + 4n^2 + n$

(bi)
$\frac{2}{(r-1)(r+1)} = \frac{A}{r-1} - \frac{B}{r+1}$

$2 = A(r+1) - B(r-1)$

Let $r = -1$

$2 = - B(-2) \Rightarrow B = 1$

Let $r = 1$

$2 = A(2) \Rightarrow A = 1$

$\therefore \frac{2}{(r-1)(r+1)} = \frac{1}{r-1} - \frac{1}{r+1}$

(bii)
$\sum_{r=2}^n \frac{1}{(r-1)(r+1)}$

$= \frac{1}{2} \sum_{r=2}^n \frac{2}{(r-1)(r+1)}$

$= \frac{1}{2} \sum_{r=2}^n (\frac{1}{r-1} - \frac{1}{r+1})$

$= \frac{1}{2} [ 1 - \frac{1}{3}$

$+ \frac{1}{2} - \frac{1}{4}$

$+ \frac{1}{3} - \frac{1}{5}$

$...$

$+ \frac{1}{n-3} - \frac{1}{n-1}$

$+ \frac{1}{n-2} - \frac{1}{n}$

$+ \frac{1}{n-1} - \frac{1}{n+1}]$

$= \frac{1}{2} [1 + \frac{1}{2} - \frac{1}{n} - \frac{1}{n+1}]$

$= \frac{1}{2} (\frac{3}{2} - \frac{n+1+n}{n(n+1)})$

$= \frac{3}{4} - \frac{2n+1}{2n(n+1)}$

(biii)
As $n \to \infty$, $\frac{1}{n} \to 0$ and $\frac{1}{n+1} \to 0$, the sum of series tends to $\frac{3}{4}$, a constant. Thus, series is convergent.

(biv)

$\sum_{r=5}^{n+3} \frac{1}{(r-3)(r-1)}$

Replace $r$ by $r + 2$. Then we have

$\sum_{r=3}^{n+1} \frac{1}{(r-1)(r+1)}$

$= \sum_{r=2}^{n+1} \frac{1}{(r-1)(r+1)} - \frac{1}{(2-1)(2+1)}$

$= \frac{3}{4} - \frac{2(n+1)+1}{2(n+1)[(n+1)+1]} - \frac{1}{3}$

$= \frac{5}{12} - \frac{2n+3}{2(n+1)(n+2)}$

Its been awhile since we last posted. And it is good to know that JC1s have all been well inducted or settled into their schools. Of course, I do hear that many schools are severely overcrowded recently. Anyway, I thought of sharing how students can get ready for JC. I contemplated sharing how to cope with JC Mathematics, but decided to be more general this time round.

Firstly, JC life can be quite rigorous. With CCA and different subject commitments piling, students must try their best to stay healthy (get enough sleep) and juggle time (skip some dramas) efficiently. For science subjects (not just H2 Mathematics), students should avoid procrastinating. The schools do not go back to teaching the subjects again, maybe just refresh using questions or tests. Thus, seek help if you need and do not just sweep it off. For J1, your A’levels is pretty much in 22 months while for J2, it is 10 months. So the clock started ticking.

Secondly, last year’s papers were intuitive and some questions were driven to see if students do understand their content and can think on their feet. And we have a name for such questions, it is application questions. For H2 Mathematics, they have made an effort to allocate about 25% of the total marks to application questions. Thus, students need to shift their focus from doing to learning. It is important for them to appreciate the concepts in each topic.

Thirdly, I understand some students enter JC and realise that there are really some (or a lot of) smarter peers around. Do not feel pressured and just stay focus. Some of them might have found help, or developed better intuition for certain things. Comparing with your neighbour will only make yourself more stressed. This is unnecessary stress.

Lastly, JC is the last “school” you have. So do enjoy yourself. Pick a CCA that you really want to try. 🙂

Since Mr Teng teaches H2 Mathematics, here are some little tips for H2 Mathematics as I told my J1s this year.

1. Some topics from High School are still very relevant, which is why I gave a proper review test. These topics are considered under assumed knowledge for H2 Mathematics, and you can find them here. A good understanding of these topics will allow you to follow classes better. You will learn that schools are constantly rushing to clear topics.

2. Learn the topics. You do not need to master them, but learn and find out what is going on. Because you can memorise the entire Ten Years Series and realise that it will not save you.

3. You will learn that time is very precious during exams. In general, 1 mark is 1.5 mins. And you should not go beyond it for questions. Rather learn the hard way to time manage well during exams, start with your normal practices at home. Thats why I encourage my students in class to do fast. Your papers will be two 3-hour papers, so during that 3-hour, you must exhibit sufficient tenacity.

P.S. I’ve spent the last few months getting a lot of application questions up. Aside from sharing them with students in my classes, I’ll also put them here. So do check in. 🙂

Happy CNY!

### JC Talk 2018

Over the weekends, Mr. Teng had the privilege to share his knowledge about JC with parents and students of O’levels 2017, at Newtonapple Learning Hub.

The lessons for J1 2018 started on the first week of January and the schedules can be found here.

The following are the grade profiles of local universities, NUS and NTU.

NTU IGP

NUS IGP

If you do have more questions, you can contact Mr. Teng at +65 9815 6827

### Call for Registration 2018!

Registration for classes in 2018 has been opened. You can find out more about the class schedules here. Do note that JC1 class will commence in first week of January as there are registrations from IP students. For O’levels students, you can treat it like a head-start! We will also be holding a workshop for Post-O’levels Release of Results. So do stay tuned!

Students/ Parents can call Mr. Teng at +65 9815 6827, or the centre at +65 6567 3606  for further enquiries!

Lessons will be held at:
Newton Apple Learning Hub
Blk 131, Jurong Gateway Road #03-263/265/267 Singapore 600131
Tel: +65 6567 3606