## A-level H2 Mathematics (9740) Suggested Solutions

After teaching H2 Mathematics for so many years and sharing my suggested solutions post A-levels, I thought I collate them nicely and type them all neatly here in Latex. 🙂 Please be kind to be should I make any careless mistakes as I typed them all out and spell check does not work with math equations. Feel free to comment.

Thoughts before 2016 A-level H2 Mathematics

Challenging A’levels Questions

MF15

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

## How to study for Gp?

Now that the A levels GP paper is drawing nearer and nearer, many students have asked about their revision strategies…

Given the limited time left, I would advise students to :

1. Consolidate their niche chosen topics for the exam- in terms of the arguments, content, evaluation and examples.
2. Make a list of the past mistakes that you have committed in your GP exams- it could range from very simple problems of hijacking the question, to overgeneralization. Commit it to memory and not repeat it again. Revise and take note of your weaknesses.
3. Take note of some of the exemplar essays that your school or tutor has given you. What are the features of these exemplar essays? Remember to include them in your essay.
4. The rest would lie on you and your question choice on that day of the exams. I have always told my students that the question choice is the most important, followed by the topics that they have prepared. There is no point doing a question even if u know the content but do not know how to tweak it to the question requirement. You will never see a quality grade.

That’s all for now, and good luck everyone!

## Thoughts about 2016 A’levels H2 Mathematics Paper

I’ve covered some things in classes, with sufficient revisions and final lap papers set. So I thought we have a little breakdown. And of course, we should review what was weeded away in the 9740 H2 Mathematics Syllabus. After all, this is the very LAST time the can test them.

1. Recurrence Relations. I’ve harped on this last year too. Conjectures! Conjectures! You can read more about it here. An a question involving conjecture should start with a recurrence relation, then a conjecture, ending with a recurrence MI. Students should know how to do both $\Sigma$ and recurrence MI. Yes, they are different.

2. Loci. You guys are the lucky last batch to do Loci. So please buy a protractor and compass. Draw them, as one of my student put it, surgically. If need be, use a graph paper (why not?). Harder loci for example can require students to draw for example, $\text{arg}(z-1+2i) = \text{tan}^{-1}(\frac{4}{3})$. You should not have trouble measuring this angle, because you could not even be able to do it. Students should be able to draw such angles with ease. One little note about Loci, will definitely their geometrical descriptions. Many students can draw this, but stumble to describe them.

3. Vectors. Truth be told, I’m still waiting for a question involving vectors in 3D, to land in A’levels. An example can be the HCI Prelims Paper 1 Question 6, which can be found here.

4. Poisson Distribution. I don’t know what this topic has been axed. So students should ready for one big Poisson Distribution questions, I say give it 12-14 marks. And it should be tested with conditional probability. I’ll practice either Demand & Supply or Inflow & Outflow questions. An example can be the one found in NYJC Prelims Paper 2 Question 11, which can be found here.

5. Correlation & Regression. Ever wondered what the $r^2$ means in the GC? Well, $r^2 = bd$ is being removed from the syllabus as well. It hasn’t surfaced before, so maybe it shall finally make its one and only LAST presence felt this year. Students should familiarise themselves with the use of $y- \bar{y} = b ( x - \bar{x})$ equation, which can be found in the MF15. I know many of you probably have not seen it before.

6. Hypothesis Testing. Students should review definitions of level of significance and p-value. Also understand what you may conclude from a Z-Test, using the results of a T-test. A little small part that students can think about, is why use a small sample size? After all, we know that have a sufficiently large $n$ allows us to perform CLT and then use a Z-Test.

7. Trigonometry. After it appeared in 2011 for a trigonometry MI, the product to sum formulas is still a problem for most students. I highly doubt its coming out again with MI, but its can easily come out again with complex numbers. An example can be this.

More examples and discussion will be made in class.

## Trigonometry used in complex Numbers

Given $z = \text{cos}\alpha + i \text{sin} \alpha$ and $w = \text{cos}\beta + i \text{sin} \beta$

$z - w = \text{cos}\alpha + i \text{sin} \alpha -(\text{cos}\beta + i \text{sin} \beta)$

$z - w = \text{cos}\alpha -\text{cos}\beta + + i \text{sin} \alpha - i \text{sin} \beta)$

$z - w = - 2 \text{sin}(\frac{\alpha-\beta}{2})\text{sin}(\frac{\alpha+\beta}{2}) + i 2 \text{sin}(\frac{\alpha-\beta}{2})\text{cos}(\frac{\alpha+\beta}{2})$

$z - w = 2 i \text{sin}(\frac{\alpha-\beta}{2}) (\text{cos}(\frac{\alpha+\beta}{2}) + i \text{sin}(\frac{\alpha+\beta}{2}))$

$z - w = 2 i \text{sin}(\frac{\alpha-\beta}{2}) e^{\frac{\alpha + \beta}{2}}$

## Random Questions from 2016 Prelims #13

NYJC/2/11

On a typical weekday morning, customers arrive at the post office independently and at a rate of 3 per 10 minute period.

(i) State, in context, a condition needed for the number of customers who arrived at the post office during a randomly chosen period of 30 minutes to be well modelled by a Poisson distribution.

(ii) Find the probability that no more than 4 customers arrive between 11.00 a.m. and 11.30 a.m.

(iii) The period from 11.00 a.m. to 11.30 a.m. on a Tuesday morning is divided into 6 periods of 5 minutes each. Find the probability that no customers arrive in at most one of these periods.

The post office opens for 3.5 hours each in the morning and afternoon and it is noted that on a typical weekday afternoon, customers arrive at the post office independently and at a rate of 1 per 10 minute period. Arrivals of customers take place independently at random times.

(iv) Show that the probability that the number of customers who arrived in the afternoon is within one standard deviation from the mean is 0.675, correct to 3 decimal places.

(v) Find the probability that more than 38 customers arrived in a morning given that a total of 40 customers arrived in a day.

(vi) Using a suitable approximation, estimate the probability that more than 100 customers arrive at the post office in a day.

## Random Questions from 2016 Prelims #12

HCI/1/6

A group of boys want to set up a camping tent. They lay down a rectangular tarp OABC on the horizontal ground with OA = 3 m and AB = 1.5 m and secure the points D and E vertically above O and B respectively, such that .

Assume that the tent takes the shape as shown above with 6 triangular surfaces and a rectangular base. The point O is taken as the origin and the unit vectors i, j and k are taken to be in the direction of , and respectively.

(i) Show that the line DE can be expressed as $r = 2k+\lambda(2i+j), \lambda \in \mathbb{R}$.

(ii) Find the Cartesian equation of the plane ADE.

(iii) Determine the acute angle between the planes ADE and OABC. Hence, or otherwise, find the acute angle between the planes ADE and CDE.

Note: Question can be made harder and trickier should Origin, O be placed in the center of the base OACB.

## ‘The young lack drive.’ Do you agree?

Question Type: Simple Polarity (no conditional words present in the question)
Key Words: The young; drive
Minimum Requirements: Students will need to unpack the key words and explore the reasons why one can argue that the young lack drive. Similarly, they must be able to also highlight why some may disagree with this claim. Better scripts will look to evaluate the arguments based on the changes that have occurred in today’s world and provide a broad and varied range of examples, not just limited to Singapore.

P: The young of today are soft, pampered and dependent because comfortable lives they enjoy today do not provide any impetus to strive.
E: The young live in a world have enjoyed the peace, prosperity, affluence, tranquillity and comfort brought by the 80s and the 90s. With greater affluence and smaller families, they are unlike earlier generations who are forced by dire circumstances to work very hard to feed their families or for survival. This lack of urgency or hunger driven by need allows them to take things easy.
Eg: In many societies worldwide, across both western and eastern cultures, in both First World countries like the USA, the EU and Japan as well as the newly developed economies of Taiwan, South Korea, Hong Kong, Singapore and Malaysia, young people normally come from one-child or two-child families where the young is being molly-coddled and spoiled. Over-indulgent parents pamper these children so much that they have no initiative and cannot do simple things like washing their own cups after drinking or carrying their own schoolbag while in Primary School.L: These young people will necessarily grow up to have no initiative and sense of urgency as everything has been done for them. With their parents carrying the full financial burden of the family, they see no need to be very independent and give up easily when things go wrong.

P: With greater affluence and family support, young people can afford to pursue their passions and try new things
E: Coming from smaller and more affluent families, young people are no longer pressed by necessity to submit to mundane jobs right after graduation and have the option to pursue their passions. While some may be hindered by fear of failure, there is now a choice to pursue something driven by desire rather than necessity for others.E: This is evident in young people who give up stable jobs to pursue their dreams such as setting up their own businesses.

P: They are risk-averse and are not willing to step out of their comfort zones.
E: In today’s competitive world, failure is a very expensive proposition. As such, in many societies, there is a fear of risk-taking and failing. Unsurprisingly, this has also affected the young who grow up in an education system that is consistently ranking and assessing them against benchmarks and model answers. Therefore, they lack the drive to venture out of their comfort zones and try new things in life, preferring to ‘play it safe’. As a result, many find themselves lacking the energy to get out of mundane jobs, leading routine lives.
Eg: For example, one common trait in Singapore is the lack of young Singaporeans who are willing to become entrepreneurs. Many are content to get themselves an education and work for others.

P: They look for instant rewards and are not willing to work hard long term to achieve their hopes and aspirations.
E: They are impatient because they never had to wait for anything as everything was instantly given, especially in the fast-pace society today where businesses compete with each other to offer the most immediate and convenient solutions to their consumers. With ‘same-day deliveries’ and ‘express results’, young people are used to instant gratification and lack the determination to persevere if things do not yield instant results.E: Young people have a tendency to quit their jobs very quickly because they do not feel fulfilled in them, choosing to find an easier or better option rather than staying and making what they have work.

p.s: this essay is contributed with most of the points from an ex student. Points are taken from his school magazine. This essay serves as a reference point only.