### Thinking [email protected] #7

[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 an application question for hypothesis testing from the 9758 H2 Mathematics Specimen Paper 2 Question 10.

The average time required for the manufacture of a certain type of electronic control panel is 17 hours. An alternative manufacturing process is trialled, and the time taken, $t$ hours, for the manufacture of each of 50 randomly chosen panels using the alternative process, in hours, is recorded. The results are summarized as follows

$n = 50$
$\sum t = 835.7$
$\sum t^2 = 14067.17$

The Production Manager wishes to test whether the average time taken for the manufacture of a control panel is different using the alternative process, by carrying out a hypothesis test.
(i) Explain whether the Production Manager should use a 1-tail or a 2-tail test.
(ii) Explain why the Production Manager is able to carry out a hypothesis test without knowing anything about the distribution of the times taken to manufacture the control panels.
(iii) Find unbiased estimates of the population mean and variance, and carry out the test at the 10% level of significance for the Production Manager.
(iv) Suggest a reason why the Production Manager might be prepared to use an alternative process that takes a longer average time than the original process.
The Finance Manager wishes to test whether the average time taken for the manufacture of a control panel is shorter using the alternative process. The Finance Manger finds that the average time taken for the manufacture of each of the 40 randomly chosen control panels, using the alternative process, is 16.7 hours. He carries out a hypothesis test at 10% level of significance.
(v) Explain, with justification, how the population variance of the times will affect the conclusion made by the Finance Manager.

### Thinking [email protected] #6

[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 very interesting vectors question from a recent JC BT

Shown in the diagram is a methane molecule consisting of a carbon atom, G, with four hydrogen atoms, A, B, C, and D, symmetrically placed around it in three dimensions, such that the four hydrogen atoms form the vertices of a regular tetrahedron.

Treat A, B, C, D, and G as points. The coordinates of A, B, C, and D are given by (5, -2, 5), (5, 4, -1), (-1, -2, -1) and (-1, 4, 5) respectively, By considering the line DG and the symmetrical properties of methane, find the bond angle of methane, that is, $\angle DGA$.

### Let’s talk about money!!! (arts funding)

We all know that arts is a common topic that comes out year after year for the A levels. Should you choose to specialize in this topic, you would need to understand the issues and complexity that come along with state funding of the arts. Of course, funding does come with tangible and intangible benefits and let’s have a look at some of these.

Monetary benefits of funding: the arts put people to work; it attracts tourism revenue; creates a distinctive brand identity; it helps to develop rural development and infrastructure by allowing small businesses to be created through small handicraft sales; ability to attract the much needed foreign direct investment (FDI) into the state to improve the arts sector; link up with the overseas arts production houses for a collaboration.

Educational and work benefits of funding: arts students are apparently more critical and analytical; have better social and interpersonal skills that is needed for the new workforce, the arts industry also helps to address a shortage of creative workers; the arts help to keep students in schools by giving them a platform to express themselves etc.

Civic benefits of funding: the arts foster civic participation and a strong democracy; brings public spaces to life; contribute to community vitality etc.

As we can see there are various benefits that could be reaped through state funding of the arts. Next, this would beg the question of whether one would be overly dependent on state funding that it cripples the flourishing of the whole industry? Should states fund the arts when they are not doing well economically? Well to put things into perspective, most state funding for the arts take up around 2.8% of the total revenue of the state. So that is really up to you to decide right? 🙂

If you have any queries or comments that you would like to raise, let us know regarding this topic. We would be happy to engage.

### Let’s talk about our future! (the youths specifically)

Once in a while, A level questions would like to test something hypothetical, about the future and whether prospects are going to be more optimistic or pessimistic. Of course, most of the focus would be on the youths since they are the future. It would be advisable to attempt questions that are more general, as it allows you to have greater scope and breadth in the essay.

Well the future would be more optimistic: Poverty rates in the developing nations have been dropping; medical technology has been improving to eradicate diseases through vaccinations; creation of more jobs and opportunities through technology developments, lower start up cost for businesses with technology as a leveller; the world being more open to peace, negotiation and diplomacy;

The future could be more pessimistic: greater uncertainty and disruption due to technological advancement(artificial intelligence in displacing workers); job security would be a thing of the past, rising youth unemployment in the developed countries; fiscal imprudence and debt crisis in Europe and USA; rising income inequality; a more volatile and risky geopolitical world that is open to nuclear warfare and terrorist attacks

Thus from what we see, there remains a lot of potential for the world moving forward, but these potential can always be thwarted with these threats as well. How we are moving ahead would definitely depend on the youths to decide already and the type of government that they are electing to mitigate these crises!

As we all know, essay questions on poverty are usually popular among students. It is an easy topic that usually asks about the reasons for poverty, whether this issue can be resolved, and whether people are poor due to their own personal failings.

With that, let’s take a look at some of the reasons why ppl are poor… Of course, one has to understand that world developments are not even, and that there is a need to discuss both relative and absolute poverty, and to differentiate reasons for poverty in the first and the third world.

First world context: Poverty can always happen due to the inability to keep up with the high cost of living, personal failings such as being lazy, engaging in vices such as gambling or being addicted to alcoholism, external and unfortunate circumstances such as racial discrimination, being afflicted with a terminal illness or even being born with disabilities that cut one off opportunities

Third world context: Poverty in this sense would be in absolute terms, define to be living less than USD1.25 a day. Reasons could be due to corruption of government, presence of incompetent government that could not harness the resources of the place efficiently, cultural stereotypes such as the caste system that entraps people’s minds, natural disasters and even the presence of war.

As we could see, the reasons for why an individual is poor are aplenty.  Could we possibly say that one is poor due to their own failings? Poverty is a very complex and entrenched problem that we see in our world today, it is systemic and could possibly take generations to eradicate it. At times, an individual could also be powerless to deal with the situations that they are born into. Thus, to what extent is really poverty the fault of an individual?

For societies that follow a fair and meritocratic system, should we take on a more compassionate and humane approach towards people who are poor?

Let me know what your thoughts are on this issue! I would love to hear from you 🙂

### June Crash Course

The team at The Culture SG has been really busy and we have a lot of things prepared to help you guys work for that A. First up! Crash course for June…

And we know it is a bit late to be announcing this on the site now, but we have really been caught up with preparing our students lately that we don’t have the time to properly update here. So here are the details for the Math Crash Course and the Chemistry Crash Course.

P.S. For SCIENCE students who wish to chiong in October, please take note that the H2 Chem/ Phy/ Bio Paper 4 (practical) is in October. So better start soon! Here are the details!

Click to view

For 3 hr lessons, they are priced at $105. For 2 hr lessons, they are priced at$70.

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

For math enquiries, you may contact Mr. Teng at +65 9815 6827.

For chem enquiries, you may contact Ms. Chan at +65 93494384.

For GP enquiries, you may contact Ms. Chen at +65 91899133.

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

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 [email protected] #2

[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.

(i) Find the two possible values of $z$ such that $z^2 = 1 + \sqrt{3}i$, leaving your answer in exact form $a + bi$, where $a$ and $b$ are real numbers.

(ii) Hence or otherwise, find the exact roots of the equation

$2w^2 + 2 \sqrt{6}w + 1 - 2 \sqrt{3} i = 0$

### Thinking [email protected] #1

[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.

Each card in a deck of cards bear a single number from 1 to 5 such that there are $n$ cards bearing the number $n$, where $n = 1, 2, 3, 4, 5$. One card is randomly drawn from the deck. Let $X$ be the number on the card drawn.

(i) Find the probability distribution of $X$.

(ii) Show that $\mathbb{E}(X) = \frac{11}{3}$ and find $\text{Var}(X)$.

Andrew draws one card from the deck, notes the number and replaces it. The deck is shuffled and Beth also draws on card from the deck and notes the number. Andrew’s score is $k$ times the number on teh card he draws, while Beth’s score is the square of the number on the card she draws. Find the value of $k$ so that the game is a fair one.

### Deriving integration formulae

Some inquisitive students have asked me before, how the MF15 integration formulas come about. I thought I should share it too then.

So we want to $\int\frac {1}{\sqrt{4-x^2}}dx$ and yes, we know the formula can be plugged directly… but what if we want to avoid the formula. Now actually, the trick here involves your trigonometry identities, along with substitution methods. We have a $sin$ here, so we only know one trigo identity that involves $sin$ and that is $sin^{2}x + cos^{2}x = 1$. Hmmm, so my approach here will be to let $x=2cost$. We will see shortly why I chose to put a 2 and that using $cost$ or $sint$ will make no difference. You can try them yourself!

lets first find that $dx=2sint dt$

$\int\frac {1}{\sqrt{4-x^2}}dx= \int \frac {1}{\sqrt{4-4cos^{2}t}}(2sint dt)$

If you notice, this explains why there is a need for us to introduce $2cost$ instead of just $cost$.

Having $4-4cos^{2}t$ allows us to simplify it to $4sin^{2}t$

We have $\int \frac{1}{\sqrt{4sin^{2}t}}(2sint dt)=\int\frac{1}{2sint}(2sint)dt=\int1dt$

Finally, $\int1dt=t+C = sin^{-1}(\frac{x}{2}) + C$

That was long! But i hope it give you some insights to the formulas.