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#### 2021 A-level H1 Mathematics (8865) Paper 1 Suggested Solutions

All solutions here are suggested. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. I will not be doing much formatting in this post. Q1 [Maximum marks: 4] Discriminant Q2 [Maximum marks: 5] (a) (b) Q3 Q4 Q5 Q6 [Maximum marks: 4]Let be the height of a sunflower plant, in […]

#### Vectors: Reflection of a line in a line

I was talking with a student today about the reflection of a line in a line, where both lines are in the same plane (naturally). And I thought of sharing it. One reason why I decided to discuss this in class was also because A’levels has been moving towards questions that want students to engineer […]

#### 2021 A-level H2 Mathematics (9758) Suggested Solutions

The following are the links to the 2021 A-level H2 Mathematics (9758) suggested solutions by Mr. Teng. All solutions here are suggested. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. 2021 A-level H2 Mathematics (9758) Paper 1 2021 A-level H2 Mathematics (9758) Paper 2 Relevant materials MF26

#### 2021 A-level H2 Mathematics (9758) Paper 2 Suggested Solutions

All solutions here are suggested. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Numerical Answers (workings/explanations are after the numerical answers.) Question 1: Question 2: Question 3: Question 4: ; away from the starting pointQuestion 5: Question 6: Question 7: Question 8: Question 9: Question 10: Question 11:

#### 2021 A-level H2 Mathematics (9758) Paper 1 Suggested Solutions

All solutions here are suggested. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions Numerical Answers (workings/explanations are after the numerical answers.) Question 1: Question 2: Question 3: Question 4: Question 5: Question 6: Smallest , Translate the graph of units in the negative direction Question 7: Question 8: Question […]

#### Removal of CLT topics (again) 2021

For students wondering the impact of removal of CLT. This picture should explain. 🙂 Below is a compiled list of topics removed for other subjects. H2 Mathematics: Correlation & Regression H2 Economics: International Trade & Globalisation H2 Chemistry: Transition Elements H1/H2 Physics: Nuclear Physics H2 Biology: Impact of Climate Change on Animals & Plants H1 […]

#### AY2021 June Revision J1 Revision Paper 1 Solutions

1. (a) 1. (b) is parallel to , where is a scalar. 2. Using partial fractions, we find that Using method of differences (DIY), we have that 3. differs from by less than 1 Plot the graph of and in the GC, sketch it out. * You can key in modulus by pressing (alpha)(window)(1). Observe […]

#### AY2021 June Revision P&C Solutions

1. The two empty seats have to be considered as two identical objects, hence ways. 2. 3. or 4. 5. (a) 5. (b) ; 6. (i) or 6. (ii) 6. (iii) Case 1: 4 distinct letters. Case 2: Exactly 3 distinct letters. Case 3: Exactly 2 distinct letters. Total number of ways 7. (a) He […]

#### 2020 A-level H2 Mathematics (9758) Paper 2 Suggested Solutions

All solutions here are suggested. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Numerical Answers (workings/explanations are after the numerical answers.) Question 1: Question 2: Question 3: Question 4: Question 5: Question 6: Question 7: Question 8: Question 9: Question 10:

#### 2020 A-level H2 Mathematics (9758) Paper 1 Suggested Solutions

All solutions here are suggested. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Sorry, due to copyrights issues, I cannot share the paper. Numerical Answers (workings/explanations are after the numerical answers.) Question 1: Question 2: Question 3: Question 4: Question 5: ; refer to solutions below for explanation.Question 6: […]

#### 2020 A-level H2 Mathematics (9758) Suggested Solutions

The following are the links to the 2020 A-level H2 Mathematics (9758) suggested solutions by Mr. Teng. All solutions here are suggested. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. 2020 A-level H2 Mathematics (9758) Paper 1 2020 A-level H2 Mathematics (9758) Paper 2 Relevant materials MF26

#### Impact of removal of common last topic (CLT)

It has been awhile since the last update and I hope all the students are well and studying hard for the upcoming preliminaries and A’levels. A handful of students have asked me about the impact of the removal of the common last topic (CLT). For H2 Mathematics, it will be Correlation & Regression. So I […]

#### 2019 A-level H2 Mathematics (9758) Suggested Solutions

The following are the links to the 2019 A-level H2 Mathematics (9758) suggested solutions by Mr. Teng. All solutions here are suggested. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. 2019 A-level H2 Mathematics (9758) Paper 1 2019 A-level H2 Mathematics (9758) Paper 2 Relevant materials MF26

#### 2019 A-level H2 Mathematics (9758) Paper 2 Suggested Solutions

All solutions here are suggested. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Numerical Answers (workings/explanations are after the numerical answers.) Question 1: Question 2: Question 3: Question 4: Question 5: Question 6: Question 7: Question 8: ; orange bird & yellow bird, orange bird & white rider, white […]

#### 2019 A-level H2 Mathematics (9758) Paper 1 Suggested Solutions

All solutions here are suggested. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Numerical Answers (workings/explanations are after the numerical answers.) Question 1: Question 2: Question 3: Question 4: Question 5: Question 6: Question 7: Question 8: Question 9: Question 10: Question 11: Question 12: Relevant materials MF26 KS […]

#### How to prepare for H2 Mathematics (9758) for 2019 and beyond

The last step is of course looking into how we can prepare for the papers in the upcoming years. It certainly will not be easy as Cambridge taught us not to predict and simply just be consistently ready. In class, I always stress to students about concept-learning. It is important to learn the concept well, […]

#### Expectations for H2 Mathematics (9758) 2019 and beyond

After looking at what 2018 had offered and taught us, we should now consider what we need to look forward to. The 2017 and 2018 paper both showed students that there WILL be unseen questions coming out during A’levels. Unseen questions are basically questions that expect students to be able to respond on the spot, […]

#### 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 […]

#### 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., […]

#### Complex Numbers with Graphings Problem

N95/II/12 Given that where , express the real and imaginary parts of in terms of . (i) Hence show that the point representing in an Argand diagram lies on the curve with Cartesian equation . (ii) Show that , and find a similar expression for . Deduce that .

#### What’s on my mind for H2 Mathematics Paper 2 2018?

This is just me talking to myself… With paper 1 over, seems like this time round, the paper is slightly imbalanced. Questions that were too easy at the front, and last three became rather hard. 9iii: Deviates from standard question, but a simple substitution, not that difficult if a student does not overthink it. Practice […]

#### A-level H2 Mathematics (9758) Suggested Solutions (2017 – 2021)

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 let me know should I make any careless mistakes as I typed them all out and spell check does not work with math equations. […]

#### 2018 A-level H2 Mathematics (9758) Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Paper 1 Paper 2 Relevant materials MF26 Comments on 2017 Paper

#### 2018 A-level H1 Mathematics (8865) Paper 1 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Numerical Answers (click the questions for workings/explanation) Question 1: Question 2: Question 3: Question 4: Question 5: Question 6: 0.31104; 0.54432 (exact answers) Question 7: 3360; 40; Question 8: Question 9: , 0.18, Not independent, 0.189 […]

#### What’s on my mind for H2 Mathematics Paper 1 2018?

Firstly, everything here will just be me talking to myself and I am not spotting any trends. Functions, good to know how to do piecewise composite functions. Checking for inverse algebraically. And to use discriminant to find range or domain of functions. Conics, good to be familiar and spot them. Finding graph of f(x) from […]

#### 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 […]

#### 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 […]

#### Sampling & Survey #10 – Cluster Sampling

So today we discuss about Cluster Sampling. We will begin by looking at cluster sampling with equal probabilities. Firstly, a cluster is a grouping of the members of the population. Previously, we assumed that we can directly sample the elements of the population and these sampling units are the elements of the population. However, this […]

#### 2018 A-level H2 Mathematics (9758) Paper 1 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Numerical Answers (click the questions for workings/explanation) Question 1: ; Question 2: or ; Question 3: Question 4: ; or *Question 5: , Question 6: Question 7: Question 8: ; ; Question 9: ; 8 units […]

#### 2018 A-level H2 Mathematics (9758) Paper 2 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Numerical Answers (click the questions for workings/explanation) Question 1: Question 2: , other roots , Question 3: Question 4: Question 5: Question 6: Question 7: Question 8: Question 9: Question 10: Relevant materials MF26 KS Comments

#### 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, […]

#### Differential Equations (Applications)

When Mr. Teng retired on 1 January 2018, he put a sum of 500 at the start of each month for groceries. Denote the amount of money that Mr. Teng has at the time years by . (i) The differential equation relating and can be written in the form of . State the values of […]

#### 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 […]

#### Solutions to Review 1

Question 1 (i) (ii) Using long division, we find that So the asymptotes are and Question 2 (i) This is a hyperbola with centre , asymptotes are , and vertices and . is a graph with asymptotes and . Use GC to plot. (ii) —(1) —(2) Subst (2) to (1), (iii) From graph, we observe […]

#### Scatter Diagrams

I was teaching scatter diagram to some of my students the other day. A few of them are a bit confused with correlation and causation. I gave them the typical ice cream and murder rates example, which I shared here when I discussed about the r-value. Think of correlation like a trend, it simply can […]

#### All the best for 2017 Release of A’levels Results!

Will like to take this opportunity to wish all students receiving your results tomorrow, all the best! And do not worry too much and feel free to ask for any advice! Students will find the following useful 🙂 The following are the grade profiles of local universities, NUS and NTU. NTU IGP NUS IGP Take […]

#### Release of Results for 2017 A’levels

Students should note that the release of A’levels 2017 is on 23rd Feb 2018. More details can be found here.

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 […]

#### 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 […]

#### 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 […]

#### Solutions to December Holiday Revision

Hopefully you find your holidays meaningful. Take the time to unwind and relax. Also, if you have completed the Set A and Set B. The solutions can be found here. If you do have any questions, please WhatsApp me. 🙂 Solutions to Set A Solutions to Set B Relevant Materials: MF26

#### Solutions to Set B

Hopefully, you guys have started on the Set B. You will find the following solutions useful. Click on the question. Please do attempt them during this December Holidays. 🙂 If you do have any questions, please WhatsApp me. 🙂 Relevant Materials: MF26

#### Solutions to Set A

Hopefully, you guys have started on the Set A. You will find the following solutions useful. Click on the question. Please do attempt them during this December Holidays. 🙂 If you do have any questions, please WhatsApp me. 🙂 Relevant Materials: MF26

#### Thoughts on the H2 Mathematics (9758) Papers 2017

Solutions can be found here. Personal Thoughts: The paper isn’t tedious. Students can do them so long as they know their stuffs. There are several generalising of questions, like question 6 of paper 1. We also saw how conditional probability was actually tested subtly, this tests students’ abilities to reason with guidance (not sure if […]

#### A-level H2 Mathematics (9758) Suggested Solutions (2017)

Here is the suggested solutions for H2 Mathematics (9758). They are all typed in LaTeX, so if it does not render, please leave a comment and let me know. Thank you. The suggested solutions for H2 Mathematics (9740) is here. Students of mine should obtain the modified A’levels Paper, and the solutions to the additional […]

#### 2017 A-level H1 Mathematics (8865) Paper 1 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Numerical Answers (click the questions for workings/explanation) Question 1: Question 2: Question 3: Question 4: Question 5: Question 6: Question 7: Question 8: Question 9: ; latex 0.0336; \bar{y}=0.64, s^2 = 0.0400latex \frac{48+x}{80+x}, \frac{32+x}{80+x}; x= 16; […]

#### 2017 A-level H1 General Paper (8807) Paper 1

Sharing the A-level H1 General Paper (8807) Paper 1…   ‘The past is not dead; it is not even past.’ Discuss. Can the use of animals for scientific research ever be justified? In your society, to what extent is it acceptable for public money to be used for the acquisition of works of art? ‘Rehabilitation, […]

#### Thoughts on H2 Mathematics (9758) 2017 Paper 1

This is a new syllabus and this is the first time it will be tested. Personally, I don’t think it will be easy and students should not underestimate this upcoming A’levels. And I’m referring to the A’levels, on the whole. We saw how the Science Paper 4 were… unexpected. The new H2 Mathematics (9758) syllabus […]

#### 2017 A-level H2 Mathematics (9740) Paper 2 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. This is answers for H2 Mathematics (9740). H2 Mathematics (9758), click here. Numerical Answers (click the questions for workings/explanation) Question 1: Question 2: Question 3: Question 4: Question 5: Question 6: Question 7: ; Do not […]

#### 2017 A-level H2 Mathematics (9740) Paper 1 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. This is answers for H2 Mathematics (9740). H2 Mathematics (9758), click here. Numerical Answers (click the questions for workings/explanation) Question 1: ; Question 2: or Question 3: 2 Question 4: ; translate the graph 4 units […]

#### 2017 A-level H2 Mathematics (9740) Suggested Solutions

Please take note that this is the solutions for H2 Mathematics (9740) Click here for H2 Mathematics (9758) suggested solutions Paper 1 Paper 2 Relevant materials MF26 KS Comments:

#### Interesting Trigonometry Question

We know how A’levels like to combine a few topics across. We also know how bad trigonometry can be. I was coaching my International Bachelorette (IB) class last week and came across a fairly interesting question. It tests students on their abilities to manage double angle identities. Not a tough question, but definitely good practice […]

#### Solutions to the modified A’levels Questions

Students of mine who have been diligently doing the modified TYS I sent them, and have difficulties with the questions that were added in to make the paper a full 3 hour paper, will find the following solutions helpful. Please try to do them in a single 3 hour seating, these are modified to cater […]

#### 2017 A-level H2 Mathematics (9758) Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Paper 1 Paper 2 Relevant materials MF26 Comments on 2017 Paper

#### 2017 A-level H2 Mathematics (9758) Paper 2 Suggested Solutions

All solutions here are suggested. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. This is answers for H2 Mathematics (9740). H2 Mathematics (9740), click here. Numerical Answers (click the questions for workings/explanation) Question 1: Question 2: Question 3: Question 4: Question 5: Question 6: Question 7: ; Do not […]

#### 2017 A-level H2 Mathematics (9758) Paper 1 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. This is answers for H2 Mathematics (9758). H2 Mathematics (9740), click here. Numerical Answers (click the questions for workings/explanation) Question 1: ; Question 2: or Question 3: ; Maximum point Question 4: ; translate the graph […]

#### Fun Facts #1: Poverty

This is going to be a very somber post about poverty around the world today… While considering the posts and the stats, please also consider some issues like income inequality; why is poverty concentrated only in some particular regions; Are richer nations obligated to help the poorer nations? Could this issue on poverty be resolved? […]

#### Is there any point in trying to predict future trends?

Yes there is a point in predicting future trends  Predicting future trends is not pointless especially when the lives and livelihoods of millions of people are ultimately at stake. This applies to anything from the economy of the world, to business trends for companies and even the potential of a natural disaster to occur. For […]

#### June Revision Exercise 4 Q2

(a) (i) (ii) (b) (i) (ii) Since ,

#### June Revision Exercise 4 Q1

(a) Using cover-up rule, we find that (b) Back to June Revision Exercise 4

#### June Revision Exercise

You can find the solutions of all ten sets of the June Revision Exercise we did in class. Have fun! June Revision Exercise 1 June Revision Exercise 2 June Revision Exercise 3 June Revision Exercise 4 June Revision Exercise 5 June Revision Exercise 6 June Revision Exercise 7 June Revision Exercise 8 June Revision Exercise […]

#### June Revision Exercise 3 Q11

(i) When When Using GC, When time interval between the first and second alarm. (iii) As which is impossible as the canal has only a fixed capacity of . Thus, the model is not valid for large values of (iv) Assume that the weather condition remain unchanged. Back to June Revision Exercise 3

#### June Revision Exercise 3 Q10

(i) Given Given When Back to June Revision Exercise 3

#### June Revision Exercise 3 Q9

(i) (ii) (iii) When Back to June Revision Exercise 3

#### June Revision Exercise 3 Q8

When When Back to June Revision Exercise 3

#### June Revision Exercise 3 Q7

(a) (b) (c) (i) (ii) (iii) Back to June Revision Exercise 3

#### June Revision Exercise 3 Q6

(a)(i) Consider is a circle with centre Origin and radius . Thus, required area is the area of a quadrant with radius (ii) (b) Back to June Revision Exercise 3

(i) When Hence

#### June Revision Exercise 3 Q5

(a) (b) Back to June Revision Exercise 3

#### June Revision Exercise 3 Q4

(i) (ii) Area of R As Back to June Revision Exercise 3

#### Question of the Day #1

So it was suggested in class that I start a little Integration of the Day thing, so students can have practice. We might just change the topic a bit, but for the time being, let’s stick with Integration.

#### Differential Equations Problem #1

To solve a DE with substitution, we first observe that we are given an equation with and variables, that is then we introduce . Here, our intention is to remove our variables and replace it with instead. So we should proceed to find . Since , we have that . All we need to do […]

#### All the best for your results!

All of us from theculture.sg would like to wish all the readers or ex-students the best of luck to the release of A’levels results 2015! Enjoy the open houses and do read up a bit on all the available university courses. Narrow some choices down and clear up as much doubts during the open houses […]

#### Consider the view that spoken language is more important than the written form.

Consider the view that spoken language is more important than the written form. This is one of the easier questions to attempt for this year, as it is just a simple comparison between the spoken language and the written form. So what are some of the benefits of the spoken language? We are able to […]

#### Why should we be concerned with current affairs when most of them will soon be forgotten?

Why should we be concerned with current affairs when most of them will soon be forgotten? This question requires students to think and assess about the importance of current affairs – whether there is any point in knowing and even being affected by them since these news will be outdated soon. After all, we live […]

#### A’levels Results 2015

It is FINALLY here! The day that students have been patiently expecting is fixed on 4 March 2016, a week from now. Link All the best too all the readers and my students for your results! A food for thought for all students: courses you need to apply for. Students can check up the indicative […]

#### How far, in your society, should unpopular views be open to discussion?

How far, in your society, should unpopular views be open to discussion? Students should consider whether it is possible to censor off these unpopular views given how advanced technology is nowadays. Even if it is possible for the state to monitor and censor off these views, it is going to be a very tedious process […]

#### Education should only be concerned with what is useful in life. Discuss.

Education should only be concerned with what is useful in life. Discuss. I would expect that this is a very popular examination question for this year. Education has always been a favourite topic for the cohort. Define terms Education: seen as a route to have a better life, to make more money. Usually Asian parents […]

#### How far is increased prosperity for all a realistic goal in your society?

How far is increased prosperity for all a realistic goal in your society? For the examination of this question, students will have to consider 3 main points: increased prosperity, realistic, your society (Singapore context). Students should also note that there is an absolute term “for all” which has to be carefully considered and evaluated at the […]

#### 2013 A levels H1 General Paper (8807) Paper 1 Suggested Solutions

For the benefit of our students and their revision, here’s the questions for the 2013 paper. Stay tune for our short outlines on these questions. The aim of our outlines is to create discussions online on possible arguments or perspectives on the questions. Feel free to comment on these questions when we have them up. […]

#### 2014 A levels H1 General Paper (8807) Paper 1 Suggested Solutions

For the benefit of our students and their revision, here’s the questions for the 2014 paper. Stay tune for our short outlines on these questions. The aim of our outlines is to create discussions online on possible arguments or perspectives on the questions. Feel free to comment on these questions when we have them up. […]

#### 2016 A’levels Timetable

So on the last day of CNY, SEAB announced the A’levels 2016 timetable! Below is the link. 2016 A’levels Timetable A little note: Students should take note that BOTH H2 Mathematics Paper will be a MORNING paper. So if you aren’t so much of a morning person, then you need to start doing Mathematics in […]

#### Integrating Trigonometric Functions

Trigonometric functions can be difficult to handle. So here is some intuition and idea on what to do when dealing with certain trigonometric functions. Integrating Trigonometric Function #1 Integrating Trigonometric Function #2 Integrating Trigonometric Function #3 Integrating Trigonometric Function #4 Integrating Trigonometric Function #5

#### Financial Engineering (I)

Before attempting to read what we have here, students should revise their basic probability and linear algebra first. Financial Engineering (I) #1 – Overview Financial Engineering (I) #2 – Introduction to No Arbitrage Financial Engineering (I) #3 – Interest rates and fixed income instruments Financial Engineering (I) #4 – Floating Rate Bonds and Term Structure […]

We at The Culture have been receiving calls recently since the start of the J1 orientation program about our tuition program. This post is a shoutout to those students who are interested in the services that we provide. Currently, all tutors of The Culture are working only with Newton Apple Learning Hub @ Jurong East […]

#### Arithmetic Problem #7

I came across this interesting algebra/ arithmetic problem the other day. … Do you think this will always work? Hint: We can use algebra to prove this cool sequence easily. Arithmetic Problem #7 Arithmetic Problem #6 Arithmetic Problem #5 Arithmetic Problem #4 Arithmetic Problem #3 Arithmetic Problem #2 Arithmetic Problem #1

#### Geometric Brownian Motion

This is really important for anyone interested in Finance Modelling. As what the movie Wolf on Wall Street says: They are referring to a geometric brownian motion. Firstly, we will begin with the definitions. We say that a random process, , is a geometric Brownian motion (GBM) if for all where is a Standard Brownian […]

#### Complex Number Problem #2

Show that . Hence show that For the first part, we can simply apply Euler’s Formula, that is The next part is a little more tricky, and since its hence, we will use what we solved previously to help us. Just for fun… Complex Number Problem #1

#### Introduction to Brownian Motion

Lets look at brownian motion now. And yes, its the same as what our high school teachers taught about the particles moving in random motion. Here, we attempt to give it a proper structure and definition to work with. A Brownian Motion is a random process with parameters if For , are mutually independent. This […]

#### Sampling & Survey # 9 – Regression Estimation

Today, we shall look at regression estimation. We will begin by looking at the usual & simple straight line regression model: . Let and by the ordinary least squares (OLS) regression coefficients of the slope and intercept. = = Precision is increase, that is Different estimators for population total We conclude here by observing that […]

#### Review of Basic Probability

So I understand that I lost many readers for the Sampling uploads.It is a bit difficult to the intensive use of notations and also the need for statistics knowledge. So here I’ll review a bit of basic probability. The contents here will be basic, and will involve some H2 Mathematics Statistics too. Discrete Random Variables […]

#### Sampling & Survey (Home)

Sampling & Survey #1 – Introduction Sampling & Survey #2 – Simple Probability Samples Sampling & Survey #3 – Simple Random Sampling Sampling & Survey #4 – Qualities of estimator in SRS Sampling & Survey #5 – Sampling weight, Confidence Interval and sample size in SRS Sampling & Survey #6 – Systematic Sampling Sampling & […]

#### Introduction to Martingales

This is a rather important topics for anyone interested in doing Finance. Lets look at their definition first. A Martingale is a random process with respect to the information filtration and the probability distribution , if for all for all Martingales are used widely and one example is to model fair games, thus it has […]

#### Sampling & Survey # 8 – Ratio Estimation

So last time we saw STR and here is a quick recap. Set the stratification scheme Set the stratum design Implement the sampling methods for each stratum independently Pool the strum estimates to estimate the population parameters Estimate their respective variances Construct CI, if necessary. Today, we look at ratio estimation. For starters, we will […]

#### Multivariate Distributions

Let be an n-dimensional vector of random variables. For all , the joint cumulative distribution function of X satisfies Clearly it is straightforward to generalise the previous definition to join marginal distributions. For example, the join marginal distribution of and satisfies If and is a partition of X then the conditional CDF of X_2 given […]

#### Some things that JC students should know.

In all these years of teaching, I realised that many JC students still do not know a handful of things. So I will list down a few random ones that I can recall off-hand, and of course expand a lot the way. You have MF26 formula booklet given. It contains your double angle formulas, some […]

#### Conditional Expectations and Variances

Here we look at an important concept that is an extension from Bayes Theorem, which we discussed briefly. The condition expectation identity says The condition variance identity says Here both and are both functions of Y and are therefore random variables themselves. With this, we start by considering a random sum of random variables. Let […]

#### Problem of Over-generalization in General Paper

Problem of Over-generalization in General Paper “Women are always subservient to men.” “Everyone uses Instagram nowadays.” These statements are glaring examples of over-generalization that are often too frequent in students’ essays. What is wrong with these statements? Well, to say that these statements are accurate would be akin to committing a crime for General Paper […]

#### Sampling & Survey #7 – Stratified Sampling

SRS form the basis of sampling and survey methods as it is easy to design and analyse, but it is rarely the best design. We may adopt systematic sampling or cluster sampling but we often are limited by the availability of sampling from. Thus, we need to look at stratified sampling to increase the precision. […]

#### Continuous Random Variables

We look at the definitions first. A continuous random variable, X, has a probability density function (PDF), if and for all events A The CDF and PDF are related by It is good to know that we have We X has a normal distribution, , and . And while We also have the log-normal distribution, […]

#### O’levels Results 2016!

All of us wish the students receiving the O’levels Results 2016 the best! And do not let grades define you. 🙂 If you’re keen to meet up with us for the JC Talk, you may contact Newtonapple Learning Hub at +65 9222 3423 for more details. This talk will be opened freely to the public […]

#### Financial Engineering (I) #2 – Introduction to No Arbitrage

Consider the following contract 1. Pay the price p at t=0. 2. Receive at time t=k, k = 1, …, T *Cash flow can be negative So what does the above contract mean? One pays a price p at time in order to receive this contract and get this cash flow. Thus at , and […]

#### Discrete Random Variables

As usual, definitions first. The cumulative distribution function (CDF), of a random variable, X, is defined by A discrete random variable, X, has probability mass function (PMF), , if and for all events A we have The expected value of a discrete random variable, X is given by The variance of an random variable, X, […]

#### Sampling & Survey #6 – Systematic Sampling

For today, it is going to be a short and simple session on Systematic Sampling, We can effectively consider systematic sampling as a proxy for simple random sampling. We start by taking (rounding up, if necessary), then draw a random number R from 1 to k. Then proceed to draw a systematic sample in periodic […]

#### 30 Free Awesome Resources for Students

New year, new you. Be ready to tackle on the new academic year with these awesome resources curated by Donut.sg  You’re welcome. Did we miss anything out? Let us know by leaving a comment below.

#### New Year, New Syllabus!

Happy 2016 to everyone! I’ve a really exciting news to share now with regards to H2 Mathematics. As I mentioned in a post before, I felt that the current H2 Mathematics can do some upgrading. And yes, the long wait is over, the new syllabus is here. Brace yourself JC1 of 2016 (especially those receiving […]

#### H2 Math Thur 5pm

This page contains all questions and answers asked by students from this class. The most recent questions will be at the top. MF26

#### H2 Math Tue 7pm

This page contains all questions and answers asked by students from this class. The most recent questions will be at the top. MF26 10. (i) 11. (i) Since P is on l, for some t perpendicular to l

#### H2 Math Tue 5pm

This page contains all questions and answers asked by students from this class. The most recent questions will be at the top. MF26 Differentiate with respect to t. Substitute into given differential equation, Since Using the product to sum formula as shown here, we have Note: and — (1) — (2) — (3) Using GC, […]

#### H2 Math Mon 430pm

This page contains all questions and answers asked by students from this class. The most recent questions will be at the top. MF26

#### H2 Math Sun 2pm

This page contains all questions and answers asked by students from this class. The most recent questions will be at the top. MF26 … Since the is a constant, the series convergences. The sum to infinity Let denote the term of the AP. Since they are consecutive terms of a GP, , thus its not […]

#### H2 Math Sun 1130am

This page contains all questions and answers asked by students from this class. The most recent questions will be at the top. MF26 Vectors Q7 [Homework] (i) Using ratio theorem, Since is perpendicular to (ii) To be a parallelogram, (iii) Let Since — (1) — (2) Solving, Vectors Q8 [Homework] (i) (ii) Let R be […]

#### H2 Math Sun 930am

This page contains all questions and answers asked by students from this class. The most recent questions will be at the top. MF26 Vectors Q7 [Homework] (i) Using ratio theorem, Since is perpendicular to (ii) To be a parallelogram, (iii) Let Since — (1) — (2) Solving, Vectors Q8 [Homework] (i) (ii) Let R be […]

#### H2 Math Sat 330pm

This page contains all questions and answers asked by students/ discussions from this class. The most recent questions will be at the top. MF26 Recall the If the angle is obtuse , then . Thus, Vectors Q7 [Homework] (i) Using ratio theorem, Since is perpendicular to (ii) To be a parallelogram, (iii) Let Since — […]

#### Class Q&A

The following contains questions that students from each class asked. Feel free to discuss further. It is not a tuition schedule, just a mode of communication. H2 Math Sat 130pm H2 Math Sat 330pm H2 Math Sun 930am H2 Math Sun 1130am H2 Math Sun 2pm H2 Math Mon 2pm H2 Math Mon 430pm H2 […]

#### Financial Engineering (I) #1 – Overview

As some of you observe, I’ve been posting a handful of undergraduate and postgraduate materials or courses here. Some close students also asked me to share more on financial engineering. So here, I’ll start on a simple series on Financial Engineering, and will finish up the analytics materials before I finish this definitely. I’m quite […]

#### H2 Math Sat 130pm

This page contains all questions and answers asked by students from this class. The most recent questions will be at the top. MF26 We have that or or This is a worksheet from class. There is a slight typo, we need to show that So for all proof/ show questions, it is important that we […]

#### H2 Math Mon 730pm

This page contains all questions and answers asked by students from this class. The most recent questions will be at the top. MF26 Vectors Q7 [Homework] (i) Using ratio theorem, Since is perpendicular to (ii) To be a parallelogram, (iii) Let Since — (1) — (2) Solving, Vectors Q8 [Homework] (i) (ii) Let R be […]

#### H2 Math Mon 2pm

This page contains all questions and answers asked by students from this class. The most recent questions will be at the top. MF26 Vectors Q7 [Homework] (i) Using ratio theorem, Since is perpendicular to (ii) To be a parallelogram, (iii) Let Since — (1) — (2) Solving, Vectors Q8 [Homework] (i) (ii) Let R be […]

#### Sampling & Survey #5 – Sampling weight, Confidence Interval and sample size in SRS

So let us now continue study SRS. Recall in part 3, we introduced the idea of unit inclusion probability. Subsequently, we introduced , which is the sampling weight. The sampling weight of unit i can be interpreted as the number of population units represented by unit i. It is the reciprocal of the inclusion probability […]

#### Happy Holidays!

Merry Christmas and Happy New Year to all the visitors! Thanks for following or reading. 🙂 Personally, Mr Teng has been really busy pursuing his personal interests. On a side note, I thought of sharing this link with fellow readers, be it current or ex students who are interested in Mathematics! Good news, cos Springer […]

#### Sampling & Survey #4 – Qualities of estimator in SRS

So we formally introduced the estimators used in SRS previously. Now, we are interested in how good our estimators are, and if you recall, by good, I refer to accuracy (MSE), Precision (Variance) & Bias (Expectation). Let us look at these qualities of our estimators now. Estimator of population mean – Unbiased, this implies that […]

#### Number Theory Question #1

So here is a cheeky question with regards to cryptography, its the basis of public key encryption in Cryptography. I had a student asked about cryptography once, its a really interesting and exciting field. Many students who watched the Imitation Game, should be greatly intrigued by it. 🙂 So here is the question What is […]

#### Sampling & Survey #3 – Simple Random Sampling

Today we shall look at the first sampling method we introduced: Simple Random Sampling. Recall that we mentioned that this is the most basic form of probability sampling since it provides the theoretical basis for the more complicated forms. Here we will look at how to select a simple random sample (SRS) and also estimate […]

#### Sampling & Survey #2 – Simple Probability Samples

So recall that we are interested on the statistical aspects of taking and analysing a sample, and a good sample will be representative in the sense that characteristics of interest in the population can be estimated form the sample with a known degree of accuracy. Here, we will use Probability Sampling to conduct surveys. Probability […]

#### Customer Analytics #1 – Introduction

I’ll dive straight into what is customer analytics using an example. Consider that Pixar calls me and says about a Disney film, that they want to for instance, predict how many people are going to watch Inside Out in every state of United States. This is a really interesting problem, but this isn’t customer analytics. […]

#### Vectors Question #1

Since I’m at the topic of Vectors for most of my JC1 classes, I thought I share an interesting question regarding it. Yes, some students do tell me the questions aren’t really A’levels, but I think it does make you think and this is very important in learning. Question: Can you place 5 points in […]

#### Sampling & Survey #1 – Introduction

This is a new series (this is essentially post-graduate materials, meaning you need strong statistical background at university level) on Sampling and Survey. I know many JC students will think, I thought its just a simple 2-4 marks questions in A-levels. You are correct, but here, i’ll concentrate on the statistical aspects of taking and […]

#### Application Question (Philosophy)

The Application Question (AQ) has always been a struggle for many students who were first introduced to this in J1. Many students perhaps accustomed to scoring 2-3 marks for this section (out of 10 marks) have given up on it totally from what I have heard. As such, this post is to dispel the belief […]

#### Types of Science and Technology Questions

Science and Technology has always been a popular topic among students taking the A levels. Students always think that they know a lot about this topic given their association with technology on a day to day basis but do they really? As recent trends in the A levels have shown, Science and Technology questions are […]

#### Algebra Question #1

Here is a simple and interesting algebra question for the weekend and to chase the monday blues away 🙂 Suppose How many possible pairs of m & n are there?

#### Python Codes

The following is a collection of simple codes I wrote for Python while doing a Rice University Course on Coursera. 🙂 Rock, Paper, Scissors, Spock Guess The Number Game Stopwatch: The Game Pong Memory Blackjack Spaceship Ricerocks!(Asteroids) Will upload more if I do some more interesting ones. 🙂

#### 2015 A Level H1 Chemistry (8872) Paper 1 Suggested Solutions

All solutions here are SUGGESTED. Mr. Lee will hold no liability for any errors. Comments are entirely personal opinions. Many thanks to the student who sent us the question paper. 🙂   You can post any questions regarding the answers but please do not ask me how many marks to secure an A for the […]

#### 2015 A Level H2 Chemistry (9647) Paper 1 Suggested Solutions

All solutions here are SUGGESTED. Mr. Lee will hold no liability for any errors. Comments are entirely personal opinions.   Question 1:   B The element must be from Group IV so the electronic configuration must be either 1s22s22p2 or 1s22s22p63s23p2. The four electrons of highest energy refer to the electrons at shells with higher principal […]

#### Why study Mathematics?

So the A’levels is coming to an end. Good job to all the JC2s! I did receive a few questions on studying Mathematics at undergraduate level. So I thought the following article might just answer it. https://plus.maths.org/content/how-maths-can-make-you-rich-and-famous A few (living) mathematicians you can google are, Jim Simons, Edward Thorp and James Stewart too. The article […]

#### 2015 A Level H2 Physics (9646) Paper 1 Suggested Solutions

All solutions here are SUGGESTED. Casey will hold no liability for any errors. Comments are entirely personal opinions. 1.A 2.D 3.C 4.A 5.C 6.A 7.D 8.C 9.A 10.C 11.B 12.A 13.D 14.A 15.A 16.A 17.C 18.A/ C ;I am not sure of Q18 though. Cos in reality, there is a phase change of pi on […]

#### 2015 A Level H2 Chemistry (9647) Paper 3 Suggested Solutions

All solutions here are SUGGESTED. Mr. Lee will hold no liability for any errors. Comments are entirely personal opinions. [Please do not ask me how many marks for A. The bell curve is something that is out of your control so there is no point in estimating your grade based on the number of marks […]

#### Game Theory #2

To illustrate another similar example from last post, now supposed ALL the A-level Candidates met up in the town square. Prior to meeting up, they were all told by the head examiner that if everyone gets the same grade, all of you get an A. Else, the highest will have an A while the rest […]

#### Game Theory #1

Over the past two months, I’ve been reading up on Game Theory extensively. It has always been a subject of interest for me since undergraduate days. And I’m glad that I found time to read a bit on it. Students that are keen, can also follow the open courseware provided by Yale University for free. […]

#### 2015 O Level Chemistry (5073) Paper 1 Suggested Solutions

For those of you who just completed your GCE O Level Chemistry (5073) Paper 1, here are the suggested answers for the 40 MCQ questions. The paper was relatively straightforward so I hope that everyone is careful in answering the questions and you did not make careless mistakes. A good MCQ score is 38/40 if you […]

#### Importance of Mathematics in Finance

Students over the years do ask me, why do I take Mathematics? Like whats the use other than becoming a teacher. So here is a video with Jim Simons. In this interview, he shares how he cracked wall street. Mathematics is being used widely in the world. And in this video, Jim Simons shared how […]

#### Functions Questions #2

I am currently thinking of how to solve this following question, and will encourage any undergraduates to discuss this problem together. I’ll share a bit of what I’ve thought. If is a bijective function that satisfies and , what is the minimum value of So here the definition of means that is an automorphism of […]

#### Integration Question #2

I came across the integral a few days ago. Pretty interesting, many ways to solve this integral, be it graphically or by substitution. ? Answer: Hint: Consider the graph of , it should give a semicircle which centre is

#### Post- 2015 H2 Mathematics Paper 1 Thoughts

Easier than 2014. Tricky. Very tricky. Tedious. When I did it, it felt a bit like a AJC paper slightly, so I believe AJC students might find the paper quite comfortable. Paper is not super conceptual. Highlights: Question 3 which test students on the definition of integration, not a difficult question, but some may struggle […]

#### “Human actions should be based on scientific fact, not religious faith.” How far do you agree with this statement?

“Human actions should be based on scientific fact, not religious faith.” How far do you agree with this statement? Define Scientific fact- based on logic, rational, proven Religious faith- spiritual belief, morality guides actions Human actions- developments in science and technology, business decisions, personal decisions in our lives This question requires students to discuss if […]

#### When a government’s finances for social welfare are limited, should they be directed towards the young or the old?

When a government’s finances for social welfare are limited, should they be directed towards the young or the old? This question requires students to understand that all governments have limited funds for social spending and there is a need for governments to decide if more funds should be directed towards the young or the old. […]

#### “Parents have no right to impose their own values and beliefs on their children.” Discuss.

“Parents have no right to impose their own values and beliefs on their children.” Discuss.   Define Impose- to force; dictate Values and beliefs- religion, sexual orientation, lifestyle etc This question wants students to discuss whether parents have the right and authority to impose their own lifestyles and values on their children, just because they […]

#### How effectively is public health promoted and managed in your society?

How effectively is public health promoted and managed in your society? Define Effectively- What is effective? To decide if something is effective, one has to measure the results against the intended objective. In this case, what is the state’s objective with regards to public health? Promoted- to raise awareness about public health Managed- suggests […]

#### A little on bell curve

Since many students were asking, I thought I’ll roughly explain the bell curve and its distribution properties. Before that, I thought I should mention that I’m not really sure myself if Private Candidates and School Candidates share a different bell curve. I doubt so, since all take the same subject code, there should be no […]

#### 2015 A-level H2 Mathematics (9740) Paper 2 Suggested Solutions

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

#### “No cause is ever worth dying for.” Discuss

“No cause is ever worth dying for.” Discuss Define Cause- fighting for freedom, ideologies, rights, status, wealth, power etc Dying- Can interpret in both the literal and metaphorical manner. What exactly can constitute dying in the metaphorical manner? Dying in the sense of not living the life that an individual desires and wants? This question […]

#### 2015 A-level H1 Mathematics (8864) Paper Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Students are highly encouraged to discuss freely for alternative solutions too. I typed this, so pardon me if got errors. There should be a handful of errors as I just type and solve as I go. […]

#### 2015 A-level H2 Mathematics (9740) Paper 1 Question 11 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) (ii) Let (reject since When , using trigonometry identity, we find From For since and for principal values of We have that when , its is a maximum point. (iii) Using GC, Area (iv) At […]

#### Skewes’ Massive Number

Heard of Skewes’ Number? It is the number above which must fail (assuming that the Riemann hypothesis is true), where is the prime counting function and is the logarithmic integral. Isaac Asimov featured the Skewes number in his science fact article “Skewered!”

#### 2015 A-level H2 Mathematics (9740) Paper 1 Question 10 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) (ii) Vol (iii) When When Vol Back to 2015 A-level H2 Mathematics (9740) Paper 1 Suggested Solutions KS Comments:

#### 2015 A-level H2 Mathematics (9740) Paper 1 Question 9 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (a) For to be purely imaginary, (bi) for (bii) and By Cosine Rule, Back to 2015 A-level H2 Mathematics (9740) Paper 1 Suggested Solutions KS Comments: The first part of the question is rather interesting, since […]

#### 2015 A-level H2 Mathematics (9740) Paper 1 Question 8 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) Let and denote the total time A & B take respectively, in seconds. (ii) (3 SF) (iii) For A: Time taken For B: Time taken Difference (nearest seconds) Back to 2015 A-level H2 Mathematics (9740) […]

#### 2015 A-level H2 Mathematics (9740) Paper 1 Question 7 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) (ii) (iii) At E, Solving with GC, and Back to 2015 A-level H2 Mathematics (9740) Paper 1 Suggested Solutions KS Comments: Students that faced difficulties with ratio (like me) might struggle a bit here. But […]

#### 2015 A-level H2 Mathematics (9740) Paper 1 Question 6 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) From MF15, (ii) Comparing coefficients, Coefficient of Back to 2015 A-level H2 Mathematics (9740) Paper 1 Suggested Solutions KS Comments: Hopefully you didn’t copy the formula wrongly. The next parts, just need to be really […]

#### 2015 A-level H2 Mathematics (9740) Paper 1 Question 5 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) First, translate 3 units in the positive x-direction. Then scale by a factor of units parallel to the y-axis. (ii) (iii) Explanation: Back to 2015 A-level H2 Mathematics (9740) Paper 1 Suggested Solutions KS Comments: […]

#### 2015 A-level H2 Mathematics (9740) Paper 1 Question 4 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Perimeter Area, Let Since , A is maximum when Back to 2015 A-level H2 Mathematics (9740) Paper 1 Suggested Solutions KS Comments: One important point here is that student must leave the exact answers as it […]

#### 2015 A-level H2 Mathematics (9740) Paper 1 Question 3 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) Width of each rectangle Total area of rectangles When , we have the exact area instead. (ii) where Back to 2015 A-level H2 Mathematics (9740) Paper 1 Suggested Solutions KS Comments: (i) needs some explanations […]

#### 2015 A-level H2 Mathematics (9740) Paper 1 Question 2 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) *please label graph and asymptote clearly. The app I used can’t do it. sorry (ii) Using Graphing Calculator, . (3SF) Back to 2015 A-level H2 Mathematics (9740) Paper 1 Suggested Solutions KS Comments: Students need […]

#### 2015 A-level H2 Mathematics (9740) Paper 1 Question 1 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) At At Using GC, (to 3 decimal place) (ii) Using GC, (to 3 decimal place) (iii) (to 3 sig fig) Back to 2015 A-level H2 Mathematics (9740) Paper 1 Suggested Solutions KS Comments: Students should […]

#### Should there be any controls over the production of energy when the need for it is so great?

Should there be any controls over the production of energy when the need for it is so great? Define Production of energy: refers to the production of petroleum (crude oil), natural gas, coal and other alternatives such as electricity and even nuclear energy etc. Controls- what kinds/methods of controls? Controls over the quantity of production? […]

#### How far is it possible for one country to forgive another for its past actions?

How far is it possible for one country to forgive another for its past actions? Define Past actions: war time atrocities, political feuds, economic conflicts, social and ideological conflicts Forgive: the ability to “forget” and to move on with past historical animosities and collaborate with one another on international matters or even domestic affairs.  What […]

#### In your society, how well are the demands of the economy and the environment balanced?

In your society, how well are the demands of the economy and the environment balanced? Define: demands of the economy- increasing productivity for Singaporeans, increasing economic growth of the country, being highly competitive and to have a cutting edge in the global world. This question requires students to consider if there is conflict between environment […]

#### To what extent should the arts in your society focus on local rather than foreign talent?

To what extent should the arts in your society focus on local rather than foreign talent? This question is indeed interesting considering how Singapore has stated her explicit aims of being a global arts hub on one hand, and the recent online Youtube controversy on the other hand about Stephanie Koh proclaiming that she is […]

#### “Books serve little purpose in education as technological developments become more sophisticated.” How far do you agree?

“Books serve little purpose in education as technological developments become more sophisticated.” How far do you agree? Define Sophisticated- to be more complex, challenging, makes learning easier and faster, almost instanteously Technological developments- wiring the world closer together, overload of new information, may lead to information paralysis Little purpose- losing relevance, especially in view of the […]

#### 2015 A’level Suggested Solutions

Congratulations on the completion of A’levels for the 2015 batch! As for those who still, are taking A’levels 2016, we hope you find this site helpful. Read the comments (ignore the trolls who have yet grown up) and learn from our mistakes. It has been a pleasure for all us to interact with you guys. […]

#### Consider the argument that the main purpose of television should be to educate rather than simply to entertain.

Consider the argument that the main purpose of television should be to educate rather than simply to entertain. This is a comparison question on the role of television- whether it should be of an educational role or an entertainment role.  So what is exactly an educational or even entertainment role? Define- Educate: to impart values […]

#### “There is no such thing as bad publicity.” To what extent is this true?

“There is no such thing as bad publicity.” To what extent is this true? Define: Bad publicity- negative images and news about an individual and/or company making the headlines repeatedly. It could also possibly refer to controversial news. This phrase is commonly used in by businesses and individuals especially if they want to acquire fame […]

#### 2015 A levels H1 General Paper (8807) Paper 1 Suggested Solutions

As predicted correctly, this year’s paper has a fair mix of topics that we have proudly identified for students! We have kindly highlighted the questions whose topic we have correctly predicted. “There is no such thing as bad publicity.” To what extent is this true? (Media) How far is it possible for one country to […]

#### 2015 A-level H2 Mathematics (9740) Suggested Solutions

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

#### KS’ thoughts on H2 Math 2015

After Christine shared what will likely pop up in the upcoming H1 GP paper, I thought I can share some thoughts too. Just to clarify, I am not spotting as I never believe in spotting for a Mathematics paper, the thought is pretty ambiguous. Let us first look at the effect of having the papers […]

#### Programming, why not?

As some of my closer students know, my brother is a web developer (I think that is the best I can do to introduce his job). And yes, you might have guessed that he is also the creator of this site, maintaining it haha. He shared the follow link awhile ago, and I thought I […]

#### Arithmetic Problem #6

Try this question! Hint: Simplify it first! 🙂 Arithmetic Problem #7 Arithmetic Problem #6 Arithmetic Problem #5 Arithmetic Problem #4 Arithmetic Problem #3 Arithmetic Problem #2 Arithmetic Problem #1

#### 2015 A Level General Paper Essay Predictions

Dear students, as we all know, you have currently less than 2 weeks to D-day (ok, just kidding, your GP paper). As most of you are stressing out over what to study, how to study for this remaining week, I hope this post will shed some light on what are the possible essay topics that […]

#### Learning Python #2: Object-oriented programming

Past couple of weeks, I’ve been really busy with preparation for A-levels. Over the weekends, I had enough time to spare to read up a bit on object-oriented programming while doing my Python course on Coursera. Object-oriented programming is a programming language model that revolves around objects rather than actions, and data rather than logic. […]

#### Arithmetic Problem #5

Saw this question on the internet and thought its rather interesting. A-levels students should be all buried other their workload, amidst the mock papers and ten year series. Hopefully this might help. Hint: try and group the number together. Arithmetic Problem #7 Arithmetic Problem #6 Arithmetic Problem #5 Arithmetic Problem #4 Arithmetic Problem #3 Arithmetic […]

#### Bayesian vs Frequentist Statistics

Some of you will probably hear of this two terms and wonder what’s the difference. Both of them are a statistical approach/ inference method. Personally, I am a Bayesian and use more of Bayesian Methods to do my stuffs. Firstly, the frequentist approach is a much simple model and does not consider prior knowledge. This […]

#### 2011 A-level H2 Mathematics (9740) Paper 2 Question 10 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) Let be the population mean time taken to install an electronic component. (ii) Under Test Statistic, At 5% level of significance, we reject if (iii) At 5% level of significance, we do not reject if […]

#### 2011 A-level H2 Mathematics (9740) Paper 2 Question 11 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) (ii) Since r is an integer, KS Comments: (ii) is honestly not easy. Some students didn’t know the formula which is puzzling as the formal is given in MF15. I think it is important that […]

#### H3 Mathematics

As the Promotional Examination Results are all released gradually for most schools, a few Year 5 students have came up to me to ask about H3 subjects. I can’t really comment for other subjects as I have to experience in them. Here, I’ll share more on H3 Mathematics. Firstly, there are two types of H3 […]

#### 2011 A-level H2 Mathematics (9740) Paper 2 Question 12 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) Let X denote the number of people joining the queue in 4 minutes. (ii) Let Y denotes number of people joining the queue in t seconds. Using the Graphing Calculator, t = 55 (iii) Let […]

Today, we shall look at a really brief idea of Real Business Cycle (RBC) Theory. I decided to touch a bit on this, since some students have asked me about financial mathematics and mathematical modelling topics. So I thought I can do a brief introduction of them. Business cycle theory aims to explain the fluctuations […]

#### What are stochastic and deterministic processes?

During class, I occasionally mention to students that the calculus you learnt is deterministic and they go uh huh. So like what is non deterministic?! I’ll attempt to give some simple definitions and example and also touch on stochastic process. First, a deterministic process is one that given at a particular state and an action, […]

#### Learning Python #1

I’ve been playing around with Python these days, and even participate in an online course with Coursera. I’ve previously shared that Python is a good starting point for any student that are interested in picking up mathematical programming. Here, we use codeskulptor which is an online cloud that allows us to code, run, and even […]

#### 2011 A-level H2 Mathematics (9740) Paper 2 Question 7 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) Firstly, the number of different friends is a fixed constant. Secondly, the probability of successfully contacting a friend is constant. (ii) The probability of contacting a friend will vary from friend to friend, as some […]

#### 2011 A-level H2 Mathematics (9740) Paper 2 Question 6 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) Stratify the residents of the city suburb into different age groups. Stand on a street corner and ask people who walk past until the required number of people in each age group have been interviewed. […]

#### 2011 A-level H2 Mathematics (9740) Paper 2 Question 5 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Solving, and KS Comments: A very straightforward question here, as it didn’t even try to trick unsuspecting students who often forget to change the signs before applying inverse norm function. Please use a Graphing Calculator to […]

#### 2011 A-level H2 Mathematics (9740) Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Please kindly comment at the specific questions should you have any questions.

#### 2011 A-level H2 Mathematics (9740) Paper 2 Question 3 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) Let (ii) (iii) Using Graphing calculator, KS Comments: Students should be careful to draw the curves according to the given domain. And since the question requests for equations of asymptotes, we should have label them […]

#### 2011 A-level H2 Mathematics (9740) Paper 2 Question 2 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) (ii) Let Since We reject since Required stationary value of KS Comments: Students do not need to carry out the second derivative test since question only ask for the stationary value, instead of maximum of […]

#### 2011 A-level H2 Mathematics (9740) Paper 2 Question 8 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) (ii) Using the Graphing Calculator, The product moment correlation coefficient measures the strength and direction of a linear relationship between two variables. Thus, the r-value only indicate correlation, but not causality. It is possible for […]

#### Complex Number Problem #1

I’ve come across many questions from students regarding complex numbers, and here is one thats quite fun to deal with! Now this is really interesting, but can be solved with a bit of manipulation. We all know this is akin to We know that Now our end result is a real number! Like mentioned with […]

#### What is a Markov Process?

A few students that have great interest in going into financial mathematics, be in JC students or Uni students that were taught by me before, asked me where to start. So I often recommend them to read on Markov Process first. Of course, you need a certain knowledge for statistics here to be able to […]

#### Trick to squaring numbers

Many students go wow when I evaluate workings, without a GC. I’m not showing off, but it is because I don’t really carry a calculator with me. haha. So some students do ask me how I evaluate the square of numbers so quickly. I thought, in light of A-level’s coming, I should share something interesting. […]

#### Let’s talk about Zipf’s Law

Zipf’s law /ˈzɪf/, an empirical law formulated using mathematical statistics, refers to the fact that many types of data studied in the physical and social sciences can be approximated with a Zipfian distribution, one of a family of related discrete power law probability distributions. The law is named after the American linguist George Kingsley Zipf […]

#### Pascal’s Triangle

This is a really good video that introduces students to Pascal Triangle’s. A-levels Students should be able to relate to the binomial expansion coefficients that are found in the triangle! There are much more uses to Pascal Triangle’s. I will share more when the A-levels end.

#### 2010 A-level H2 Mathematics (9740) Paper 2 Question 11 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) Let number of calls in a period of 4minutes. (ii) Let length of time Let number of calls in a period of minutes. minutes seconds (iii) Let number of calls in a working day of […]

#### 2010 A-level H2 Mathematics (9740) Paper 2 Question 10 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) (ii) (a) (b) (iii) Since the product moment correlation coefficient is close to 1 for (ib), the model is the better model. (iv) From GC, the regression line of F on is (3SF) When Since […]

#### 2010 A-level H2 Mathematics (9740) Paper 2 Question 8 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) Required Probability (ii) Required Probability (iii) Case 1: The first digit is 3 or 5 Case 2: The first digit is 4 Required Probability

#### 2010 A-level H2 Mathematics (9740) Paper 2 Question 6 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Unbiased estimate of population mean Unbiased estimate of population variance Let be the population mean time required by an employee to complete a task. against at 10% level of significance Reject if p-value Test Statistic, Applying […]

#### 2010 A-level H2 Mathematics (9740) Paper 2 Question 5 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) In order to use stratified sampling, we need to know the sampling frame. Since we do not have access to this information, it will be difficult to use a stratified sample. (ii) We line all […]

#### 2010 A-level H2 Mathematics (9740) Paper 2 Question 4 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) (ii) For to exist must be one-one, thus least value of (iii) (iv) Since (v) Range of

#### 2010 A-level H2 Mathematics (9740) Paper 2 Question 9 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) (ii) (iii) Let number of peak rate calls over the two three-month periods.

#### 2010 A-level H2 Mathematics (9740) Paper 2 Question 3 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) For the curve to have a turning point, (ii) (a) When (b) (iii)

#### 2010 A-level H2 Mathematics (9740) Paper 2 Question 2 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. (i) Let be the statement: When is true. Assume that is true for some , i.e. Want to prove