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.

 

Population problems eventually solve themselves-government meddling only makes things worse. Discuss

Population problems eventually solve themselves-government meddling only makes things worse. Discuss

JC General Paper

Government intervention solves population problems such as population decline, which will be left unresolved if left to the masses. With a preference for smaller families and a general unwillingness to start a family in today’s modern society, negative or zero population growth often ensues. These have detrimental impact on affected countries, such as a fall in tax revenues, a smaller workforce and a high dependence of an ageing population on the working population. As these socioeconomic perspectives are entrenched in the minds of young urban professionals, these population problems are incapable of eventually solving themselves. In this case, government intervention is beneficial. In developed countries like Italy and Spain, where fertility rates stand at a meagre 1.25, new generations are unable to replace past generations thus leading to population decline. The implementation of pro-natal policies could possibly help to increase the incentive for couples to procreate and boost total population numbers. Implemented measures include longer maternity and paternity leave in Switzerland, as well as cash incentives in Singapore. Another method of boosting population growth is through the relaxation of immigration policies, which allows for an influx of permanent residents.

Population problems such as the rampant spread of diseases are also combated more efficiently and effectively through government intervention. If left to solve by itself, this results in a higher death toll and increased spread of illnesses. The successful results of government intervention is exemplified through the World Health Organization and governments’ collaboration to wipe out smallpox, which was deadly enough to kill one in every four infected persons. With public health measures to increase hygiene standards and mandatory vaccinations, smallpox was eradicated worldwide in the 1800s.

Despite the effectiveness of government intervention in solving population problems, some policies and measures undoubtedly create new problems for countries. Firstly, policies to reduce overpopulation are often successful to the extent that they eventually lead to population decline. This is evident in Singapore, which, due to the overwhelming success of the “stop at two” policy, currently faces a replacement rate of 1.25. This has led to national concerns of unsustainable population growth and the possibility of a population decline in the near future. Furthermore,  the policy of migration to solve population problems has led to social segregation in some countries.

Thinking Math@TheCulture #3

Thinking [email protected] #3

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 1976 A’levels Paper 2. I thought it is pretty interesting to discuss the question with a little extension.

(a) In how many ways can 5 copies of a book be distributed among 10 people, if no-one gets more than one copy?

(b) In how many ways can 5 different books be distributed among 10 people if each person can get any number of books?

So now, let us modify it a bit.

(c) In how many ways can 5 copies of a book be distributed among 10 people if each person can get any number of books?

Notice that the difference between (b) and (c) is that the book distributed is not identical. So for (c), we are pretty much distributing r identical balls to n distinct boxes. Whereas for (b) , we are pretty much distributing r distinct balls to n distinct boxes.

Common Errors for Application Question(AQ)

Common Errors for Application Question(AQ)

JC General Paper

My previous posts have touched on the points that you would need to excel in your AQ. It is also important for you to understand what are some pitfalls so that you will not make these mistakes.

  • 1. Rehash of the author’s argument/ summary of the author’s argument- did not go on to elaborate or to even offer personal insight
  • 2. Lack of evaluation- paragraphs tend to be descriptive
  • 3.Poor selection of examples- they could be inaccurate or they are isolated case studies that are not representative of the whole society
  • 4.No examples given
  • 5.Evaluation lacks depth or it is done very superficially
  • 6.No link back to the requirements of the question
  • 7. No introduction or conclusion (of course this is a very obvious time management issue)

Alright, there you have it, the 7 deadly sins/pitfalls for your AQ.

Vectors Question #4

Vectors Question #4

JC Mathematics

Another interesting vectors question.

The fixed point A has position vector a relative to a fixed point O. A variable point P has position vector r relative to O. Find the locus of P if r \bullet (ra) = 0.

Evaluation for Application Question (AQ)

Evaluation for Application Question (AQ)

JC General Paper

Have you ever wonder what teachers mean by evaluation? Many students have asked me this during my classes.

Well to put it simply, evaluation for the AQ is not just simply stating whether you agree or disagree. You would need to provide a reason to support your point. So how exactly should you evaluate? Well you can consider commenting on the limitations or applicability of the author’s point or even how persuasive/convincing the author is in bringing across his point of view.

Here are some tricks that you could possibly use to help you in your evaluation to make it SUPER:

S– Span (whether the argument applies only to the past and not the present day context and vice versa)

U– Underlying assumption of the author

P– Perfect world vs Reality (does the argument exist only in theory/ in a perfect world, but not applicable in reality?)

E– Effectiveness of the argument

R– Reach (what are the different demographic groups which this argument applies to? surely it can’t apply to all?)

Using this framework, I’m sure you would be able to impress your tutors with your AQ from now on!

Vectors Question #3

Vectors Question #3

JC Mathematics

This is a question a student sent me a few days back, and I shared with my class.

Find the Cartesian equation of the locus of all points (plane) that is equidistant of the xy plane and xz plane.

The following should aid students to visualise.

xy-, xz-, yz-planes

Sidenote: I think Vectors is a very important topic for 9758 as its applications are wide. Students should do their best to understand the topic. I will share a few more applied questions next week when I have time.

Responsibility of Protecting the Environment

Responsibility of Protecting the Environment

JC General Paper

Environment is a common and easy essay question that could potentially come out for the exams… Students should always analyze their essays from a stakeholder’s perspective to give a more holistic evaluation.

Developed countries

Governments in developed countries have the necessary skills, influence and resources to aid the protection of the environment. They would have the financial means to invest in necessary science and technology which could improve the environment. At the same time, they would also possess political clout to enact policies to force companies to comply with their environmental standards. Education and campaigning to change the mindset of the people towards the environment would be within their means as people are generally more educated and aware.

Businesses could practise corporate social responsibility, and stand to gain since green consumers are more likely to purchase their products. Consumers are more concerned with the origins of their products they buy today. As such, MNCs could influence these green consumers to raise awareness about environmental issues using their products.

Individuals could make a difference by starting with themselves first. A small step could go a long way such as supporting Earth Hour or even car pooling.

NGOs no doubt would have more political clout than individuals, and do not have the same restrictions that are placed upon government agencies. They do not have to balance certain commitments such as the living standards in the country. This would result in them having greater flexibility when it comes to their operations and should be able to achieve more. \

Developing countries

Governments have the sole responsibility for the enactment and enforcement of policies to protect the environment. Unfortunately, their main priority is to ensure economic growth rather than the protection of the environment. Developing countries are more likely to have unstable government and corruption is usually rampant making it difficult to care for the environment. E.g Shell engaged in dodgy/shady dealings with the government officials in Nigeria in order to gain a foothold in the country.

Businesses are usually in the primary and secondary sector which focus on the harvesting of raw materials and manufacturing. These are highly damaging to the environment as the mining process results in large amounts of pollutants and waste. Many of their factories are owned by MNCs, and they would have little say in the production methods used.

Individuals concern would be to earn sufficient money to stay alive, they would not mind harnessing the earth for the raw materials or resorting to more efficient methods of clearing the land if they could help to save some money for them.

Considering the situation in both developed and developing countries, who do you think should be responsible for the environment?

Post-Results 2016

Post-Results 2016

Chemistry, JC Chemistry, JC General Paper, JC Mathematics, JC Physics, Mathematics, Studying Tips, University Mathematics

Let’s face it. Some of us will not get the dream results we want. Don’t give up and let fear conquer you.

For students unsure of the available courses, they can check out the following post. It contains the grade profile for local universities.

Our Team will be here if you need help/ advice. Feel free to text us.

P.S. Today, I saw an image shared by Mr Wee, which said that “You’re the architect of your own life”. So let’s not let the grades define us.

In times of economic hardship, should a country still be expected to provide financial or material aid to others?

In times of economic hardship, should a country still be expected to provide financial or material aid to others?

JC General Paper

Students are expected to address the criteria of this question throughout the essay; it being in times of economic hardship. For a quality essay, the terms “should” and “expected” should be addressed as well. For instance, a country should still give aid, but perhaps not be expected at a time when its survival is in question, and it does not have a healthy budget balance.

Financial aid- capital/loans/money

Material aid- manpower/distribution of basic goods and necessities

  • Provision of aid during economic hard times could be a political statement and commitment to the recipient country, helping to foster greater political relations in the long term. Giving of aid should be expected especially if the recipient country needs it more than the donor country. Such circumstances could be when the recipient country is facing civil war and there is urgent need of aid to cease the fighting etc. Another possible instance could be during times of natural disasters/emergencies. Also, for most of the donor country, aid takes up a small portion of their budget, hence it should not affect the current economy severely even if the country continues to give out aid during hard times. Aid accounts for 0.5% of the US budget yearly.
  • Governments should be responsible and accountable to their citizens first especially during hard times, hence aid should be allocated domestically rather than elsewhere. The dollar votes of the citizens and their voices are important, and it is only right that countries should be concerned with their own self-preservation before others. After all, an economically prosperous country will then be able to contribute more to the international community, rather than a slow and stagnate economy that is facing difficult times.
  • Perhaps a country should look at other means of support, rather than the provision of aid during bad times. It is presumptuous to assume that aid helps to alleviate the problems faced by the recipient countries. Often, aid may actually harm local industries and foster this sense of self-entitlement and dependency on the donor nations.
  • Legal obligations under international law could possibly bind countries to continue giving aid to another country. A short term economic difficulty does not suffice to repudiate this commitment, especially when contracts have been signed beforehand, and the donor country may risk ruining their legitimacy and international standing.

In dire situations, countries should still continue to give aid to one another. However, expectations to scale down in terms of aid is definitely reasonable and justifiable.