Thinking Math@TheCulture #4

Thinking [email protected] #4

JC Mathematics, Mathematics

[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by KS. More of him can be found here.

This is a question from 1993 Paper 1.

The positive integers, starting at 1, are grouped into sets containing 1, 2, 4, 8, \ldots integers, as indicated below, so that the number of integers in each set after the first is twice the number of integers in the previous set.

\{ 1 \}, \{ 2, 3 \}, \{ 4, 5, 6, 7 \}, \{ 8, 9, 10, 11, 12, 13, 14, 15 \}, \ldots

(i) Write down the expressions, in terms of r for

(a) the number of integers in the r^{th} set,

(b) the first integer in the r^{th} set,

(c) the last integer in the r^{th} set.

(ii) Given that the integer 1,000,000 occurs in the r^{th} set, find the integer value of r.

(iii) The sum of all the integers in the 20^{th} set is denoted by S, and the sum of all the integers in all of the first 20 sets is denoted by T. Show that S may be expressed as 2^{18}(3 \times 2^{19} - 1).

Hence, evaluate \frac{T}{S}, correct to 4 decimal places.


Singapore Inclusive Policies- A reflection

Singapore Inclusive Policies- A reflection

JC General Paper

“An Inclusive Society, A Stronger Singapore” Budget Plan 2012-2016: fulfilled promises, financial burden and new directions

In 2012, the Singapore government introduced a 5-year budget plan aiming to make Singapore a fair, stronger and inclusive society. The main features of this ambitious project were:

  1. Restructuring to sustain growth
  • Less dependency on foreign labour
  • More grants and support for SMEs
  1. Building a fair and inclusive society
  • Improving the social mobility of lower income families through improved financial and educational subsidy and/or grant schemes
  • Helping seniors live long and well by offering housing purchase schemes, credit incentives for hiring aged workers, increased retirement savings and more affordable healthcare
  • Supporting those with disabilities and special needs through subsidies for employing caretakers, providing educational and skills support facilities and building elder-friendly infrastructure
  • Sharing the fruits of Singapore’s economic growth by giving out GST vouchers in cash, Medisave and U-save

(taken from

It is not difficult to understand why the government initiated such a plan. In the words of PM Lee “We all have something to contribute,” (, 31 Oct 2015) hence maximising the potential of our human resource: young, old, poor, rich, abled and disabled, and keeping them healthy and happy is a strategic move to support Singapore’s future economic growth.

It is now 2017 and the government has indeed delivered most of what was promised in the budget plan. To highlight a few of these achievements, our neighbourhoods and transport systems are now more elder-friendly, we receive our GST cash vouchers as promised, we have two spanking-new hospitals in the West and are currently in the midst of integrating our healthcare groups to streamline operations and make healthcare more affordable, we built a $25million Enabling Village to help those with disabilities and special needs and we have decreased the local to foreign worker dependency ratio. However no success story is without sacrifice and just who or what were the sacrificial lambs in this budget plan? Moreover, which areas should the government focus on in the next one?

Tax payers are once again not spared from funding Singapore’s progressive aspirations and why should they? The money is redistributed and re-invested to make Singapore a fairer and stronger society. However tax rates have gone up again for YA2017 for the rich and upper middle classes, from a range of 17-20% to a range of 18-22% and with the addition of two new income tiers ( The question is, is this really fair? I heard that little “Yes!” go up in your head. Truth be told, it depends on how the rich make their money. For those who own successful local businesses, the new tax hikes are going to hurt and might even discourage further business growth or expansion. For those who make their riches through stocks and shares, guess what? These are not declared when filing income tax! Perhaps we should work towards minimising the number of those who profit from loopholes in the system if we truly wish to build a fairer – also towards the rich— and more inclusive society.

Finally, what new directions might we find in the AY2017 budget? The Committee on the Future Economy (CFE) of Singapore recently released a report highlighting key growth sectors and proposed seven strategies to sustain Singapore’s economy in the coming years.

Key growth sectors: finance, hub services, logistics, urban solutions, healthcare, the digital economy and advanced manufacturing

Seven strategies:

  • Deepen and diversify international connections
  • Acquire and utilise deep skills
  • Strengthen enterprise capabilities to innovate and scale up
  • Build strong digital capabilities
  • Develop a vibrant and connected city of opportunity
  • Develop and implement Industry Transformation Maps (ITMs)
  • Partner each other to enable innovation and growth

(taken from

One can therefore expect more money in the AY2017 budget to be channelled into re-skilling the workforce, encouraging innovation and improving digital infrastructure and technology i.e. building a Smart Nation. Support for the elderly, disabled and those with special needs might take on a minor role in the budget this year because it was the focus of AY2012-16’s budget plan.

Nonetheless, there is so only so much money can do in helping a country to survive. As pointed out by PM Lee, Singaporeans need to practice active citizenship. This means making the effort to keep oneself fit and healthy, using the incentives and support systems in place to reskill or retrain oneself to meet with changing labour demands, revising educational content to ensure our students are prepared for the future by the time they graduate and managing ones expectations and demands in times of economic difficulty. Policies, systems and budget plans will change but the brunt of it can be softened by adopting the right attitude and mentality. Thus when the AY2017 budget comes out, hold your tongue for a moment, understand the changes and safety nets in place and learn to embrace them. If not, a corner of Hong Lim Park beckons you.

2016 A Level H2 Physics (9646) Paper 1 Suggested Solutions

JC Physics

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

  1. B
  2. C
  3. B
  4. D
  5. D
  6. C
  7. D
  8. D
  9. B
  10. C
  11. A
  12. D
  13. D
  14. C
  15. C
  16. A
  17. D
  18. A
  19. B
  20. D
  21. Question 21 is a flawed question. When unpolarised light goes through a polarizer, the I is halved while the A is reduced by a factor of root 2. But based on the information Cambridge provides, the answer is C.
  22. D
  23. D
  24. C
  25. C
  26. B
  27. C
  28. A
  29. B
  30. B
  31. A
  32. C
  33. D
  34. B
  35. B
  36. B
  37. D
  38. C
  39. C
  40. C

Note to all: Casey will not respond to most of the comments as he is busy. You may contact him by SMS at  +65 9474 5005 if you have a burning question.

Feel free to explain the answers, if you are confident. Many thanks.

GP 2016 Prelim trends

GP 2016 Prelim trends

JC General Paper, Studying Tips

This post is going out to all students that are taking their A levels GP 2016 this year: the listed topics are just some of the common questions that schools have set in this prelim. You may want to take note of it and prepare for these issues just in case. Being prepared beats being caught off guard right?

  1. S&T with Artificial Intelligence/ robotics
  2. S&T with eco-cities/green cities
  3. Sports (take note it is usually cross-topical). It could be sports with arts; sports with S&T; Sports with gender; Sports with war etc.
  4. Government collaboration especially in areas like war and terrorism; migrant issues and how does government interference infringes on privacy and rights of individuals
  5. Arts and Culture (significance of museums/historical places etc).
  6. Ageism + SG society

P.S Do take note that S&T questions are usually quite narrow; while arts and culture questions tend to be broad-based. Prepare accordingly to the question types and requirements! Good luck to all who are mugging now.

2016 A-level H2 Mathematics (9740) Paper 1 Suggested Solutions

2016 A-level H2 Mathematics (9740) Paper 1 Suggested Solutions

JC Mathematics

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

As these workings and answers are rushed out asap, please pardon me for my mistakes and let me know if there is any typo. Many thanks. As these workings and answers are rushed out asap, please pardon me for my mistakes and let me know if there is any typo. Many thanks.

I’ll try my best to attend to the questions as there is H2 Math Paper 2 coming up and I have a lot of prepare. I’ll try my best to attend to the questions as there is H2 Math Paper 2 coming up and I have a lot of prepare.

Side note: I think the paper is tedious, but definitely manageable. Hard ones, could have been q3, q7a, 10, 11ib. So if you did your tutorials and past year papers well, with proper time management and no careless, 70 is manageable. To get an A, you need to fight for that 30 marks which really test you on your comprehension skills and precisions. And these questions should distinct the students who deserve an A.

Thoughts about Paper 2.

Numerical Answers (click the questions for workings/explanation)

Question 1: x \textless -2, \text{~or~} \frac{1}{3} \textless x \textless 4
Question 2: 0, -\text{ln}2;~ y=2, y = -x\text{ln}2 + \frac{\pi \text{ln}2}{2} + 1;~ (\frac{\pi}{2} - \frac{1}{\text{ln}2}, 2)
Question 3: m = b,~ l = a, ~ kl^4 + m = c
Question 4: 0.74;~ \frac{b(0.74^n)}{0.26}
Question 5: \begin{pmatrix}{2b}\\{4b-4a}\\{-2a}\end{pmatrix};~ \pm \frac{1}{6\sqrt{2}};~ 3
Question 6: 4,~ 14,~ 44;~ \frac{1}{4}n(n+1)(n^2+n+2)+2
Question 7: 2+3i;~ a=3,~ k=-30
Question 8: f(x)=1+2ax+2a^2x^2+\frac{8}{3}a^3x^3 ;~ \text{tan}2x = 2x + \frac{8}{3}x^3
Question 9: y = 5-5e^{-2t}; x = 5t + \frac{5}{2}e^{-2t} - \frac{5}{2};~ x = 5t^2 + 20\text{sin}\frac{t}{2}-10t;~ 1.47s,~ 1.05s
Question 10: f^{-1}(x) = (x-1)^2, x \in \mathbb{R}, x \ge 1;~ x = 2.62;~6,~ 8,~ 9;~\text{No}
Question 11: t = -\frac{5}{9},~ \lambda = -\frac{8}{9}, ~ \mu = \frac{19}{18};~ -2x+y +2z=35 \text{~or~} -2x+y+2z = -37;~a = 4.5


Relevant materials


KS Comments

6 weeks left to A levels!!!

6 weeks left to A levels!!!

JC General Paper, Studying Tips

Dear students,

I’m sure most of you are already done with your prelims or nearly done… In fact, some of you have already gotten back your GP papers. How did you fare? I’m sure most of your teachers would have told you that Prelim results are usually quite an accurate indication of your final A level grade (except maybe a grade jump or two if your school is pitching it at a high standard).

Whatever grade that you have received is a really a time and reality check for you. You have 6 weeks left to the final exams! How are you going to spend your time revising considering u are left with this time frame?

Talk to me if you need any help and advice. Most importantly, pls do not burn out or break down.

Special Post for A levels Students GP

Special Post for A levels Students GP

JC General Paper, Studying Tips

Dear students,

This is a special post going out to all of you. Considering that your A levels are drawing near and you are already in the midst of your prelims, what are some topics that you are still struggling with? What are some essays that you would like to see a post on?

The team will do their best to meet your requests! Just comment on the section below the post 🙂


How important is it for people in your society to retain a sense of tradition?

JC General Paper

Traditions are important because it provides people with a way to govern themselves in a world where moral standards are now often being compromised. Most traditions have moral and ethical messages embedded within them, and these serve as guidelines as to how people should govern themselves and, therefore, gain a degree of control over their lives. Before the invention of today’s modern laws, people used traditions as their guiding principles and while some traditions were abandoned, most of them are recycled and reused as the basis or foundations for some of the existing laws today. At times, traditions could have such a powerful and pervading influence, so much so that they become unspoken laws, laws of logic that the masses hold to. Examples of traditions that we hold dear would be the Chinese New Year festival, where it teaches the value of sharing and family unity and importance.

Traditions are also important to keep us rooted in our own self identity especially in this globalized world where the spread of Americanization serves to dilute our own culture and values. The unique practices that people still and continually adhere to, serve as a reminder of our origins and give us a glimpse into the lives of our ancestors. It is only through undergoing the activities that our ancestors have undergone, that we are able to understand our personal histories and be aware of the cultural identities that we each possess, particularly, in Singapore where it also serves as a binding force for a heterogeneous society, where a plethora of different cultures and traditions abound.

However, retaining a sense of tradition may not be important as it may lead to segregation and divide people especially in a multi-racial and multi-religious country such as Singapore. While having a sense of tradition unites people together, it only applies to people who belong to a particular group who share the same traditions. This inadvertently creates an “us” versus “them” kind of mind set within the different parties and it would serve to heighten the sense of suspicion and fear. Also, given that Singapore is increasingly globalized and greater influx of people are coming to this city to work and travel, it does not make sense to retain traditions. Traditions in this case no longer serve any purpose and they should be re-adapted to fit the contexts of the present day.

Finally, traditions may not even hold any role in today’s world where people’s lives are getting more and more fast-paced- simply because people do not have the time to uphold these traditions. For example, it has been observed that the annual Chinese Qing Ming festival is in the danger of being virtually extinct. The number of families that turn up on the day of Qing Ming has been decreasing most likely because of tight and busy working schedules which incidentally, has been cited as one of the common reasons by Singaporean Chinese. In a world where stress is placed on individuals to succeed and to attain a degree of sustainable income to support themselves, such traditions serve to hinder their progress and may make them lose out on the race, therefore, traditions may not always be important.



Is it ever justifiable to break the law?

JC General Paper

Is it ever justifiable to break the law?

“Ever justifiable”: set a criteria for such a question. Possible criteria that you can set: 1) law implemented is unlawful/ unreasonable e.g. when the punishments outweigh the crime 2) breaking the law suffices as the last hope/action and it actually benefits society as a whole.

Law: A set of rules/legislations that follow a social contract theory.

Assumption: Other peaceful means of handling the problems are not viable, resulting in individuals having to break the law.

Non-Justifiable Arguments

  1. It may be counter productive and may result in the country breaking down into lawlessness.
  2. It may invalidate the people efforts in breaking the law in the long run. The law exists for a reason. It exists to make people feel safe in the face of possibility of lawlessness.
  3. It may subsequently open up a myriad of “valid reasons” for breaking the law, making other laws redundant in a sense.

Justifiable Argument

  1. Provided the cause is a noble one and breaking the law is used as the last resort and this action benefits society as a whole. One should also consider if there is a high chance of success of breaking law in order to preserve the rights of individuals.

P.S These discussion points are raised during class by students themselves. If you have any other additional points or to contest any of these points, just type them in the comment section.