### Random Questions from 2017 Prelims #1

Last year, I shared a handful of random interesting questions from the 2016 Prelims. Students feedback that they were quite helpful and gave them good exposure. I thought I share some that I’ve seen this year. I know, its a bit early for Prelims. But ACJC just had their paper 1. 🙂

This is from ACJC 2017 Prelims Paper 1 Question 7. And it is on complex numbers.

7
(a) Given that $2z + 1 = |w|$ and $2w-z = 4+8i$, solve for $w$ and $z$.

(b) Find the exact values of $x$ and $y$, where $x, y \in \mathbb{R}$ such that $2e^{-(\frac{3+x+iy}{i})} = 1 -i$

I’ll put the solutions up if I’m free.

But for students stuck, consider checking this link here for (a) and this link here for (b). These links hopefully enlightens students.

Just FYI, you cannot $\text{ln}$ complex numbers as they are not real…

### H2 Mathematics (9740) 2016 Prelim Papers

So many students have been asking for more practice. I’ll put up all the Prelim Papers for 2016 here. Do note that the syllabus is 9740 so students should practice discretion and skip questions that are out of syllabus. 🙂

Here are the Prelim Paper 2016. Have fun!

Here is the MF26.

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

This is a past year question that has been adapted from HCI. Pretty interesting topic… Let’s see how to unpack this question together… This question assumes that population problems (demographic issues are part of the natural process on earth, which would eventually balances itself over a long period of time). Government interference would merely make it worse, since their interventions are often “artificial” and would make matters already worse than what they should be. Moreover, governments are not able to always predict future trends and outcomes, hence it is advisable for them to leave everything to nature than by chance.

As illustrated by the Demographic Transition Model (DTM), population problems can indeed solve themselves without the need for government intervention. In stage 2 of the DTM, it is asserted that developing countries experience a decline in birth rates due to the introduction of contraceptives, induced abortions and a change in socioeconomic perceptions. As countries gradually industrialize from their original agrarian societies, rationalism overrides traditionalism, thus leading to a fall in birth rates as both men and women alike desire a higher standard of living. This involves having fewer children as they are expensive to raise, according to Caldwell’s Theory of Intergenerational Wealth Flows. This hence reduces overpopulation naturally without the need for governments to step in. The model also contends that high mortality rates eventually decline as well, due to the influx of medical technologies and increase in hygiene and nutrition standards. Thus, population problems gradually solve themselves in the long run due to the advent of industrialization and inevitable changes in societal perceptions and standard of living.

Nevertheless, even though the above mentioned model claims that populations stabilize naturally in the long run, this is in part due to measures and policies implemented by governments that are in line with national interests. In this case of China, overpopulation and a stress on national resources were narrowly averted due to the government’s legislation of the “one child policy” in 1979. The reduced strain on resources thus allowed the Chinese government to focus on stimulating economic growth and developing infrastructure to attract foreign direct investment. Another country with a similar goal in mind was Singapore with its “Stop at Two” policy from 1965 to 1984, which helped to solve population problems such as overcrowding and a lack of resources.

Government intervention also solves population problems such as population decline, which will be left unsolved 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 aging 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.

Here are some reasons in tackling the demographic imbalance… What do u all think? But I would to raise some points… Many a time, the population policies done by the government are “hard to reverse” especially if they have been too successful.. An example would be Singapore’s Stop at Two policy. Even China has recently reversed its One child policy in hopes of dealing with the fast growing aging population and the male imbalance ratio.

But of course there are implications that come with these population policies… These would be for a discussion for another day.

P.s The above points have been contributed by an ex student from HCI. It has only been vetted and edited by the tutor.

### The value of humility in modern society

Came across this question in class which stumps a lot of my students in answering this… It seems so contradictory right… to be humble in our modern day?

Ok let’s see what are some characteristics of our modern society first. We are driven by our pursuit of material goals, increasingly status conscious and technologically driven. So where does the value of humility fit in?

1) Humility allows us to lower ourselves and to examine our shortcomings. This virtue is especially important when it comes to dealing with business and in reflection to improve ourselves. In this modern world where it is very competitive, it is advisable for us to always examine our business but it should not hit the point of analysis paralysis.

2) We should dare to show and flaunt our wealth, status or even ability to others in order to stand out in this competitive world, otherwise how would people even recognize and take notice of us? If we do desire social status or even respect in society, the easiest and fastest way is to of course demonstrate our signs of success and flaunt wealth. Many have turned to that on social media seen from posts from #richkidsofinstagram etc. Besides such frivolous way of gaining respect, a more important reason would be for marketing needs. Marketing needs could refer to “selling” ourselves to companies in order to get that coveted job or to sell products and services to encourage consumers to purchase. One would need to “scream” in order to stand out for others to even take notice of us. Humility definitely has no value and place.

3) Humility could be expected from leaders as they are placed in a position to serve others and to demonstrate empathy. An arrogrant leader would be a definite turn off to most people as we definitely would not want our leaders to come from a moral high ground or to even impose their power, status and influence on us. Examples of such leaders would be Mother Theresa, Ghandi just to name a few. But we should take note that humility could also backfire, especially if one presents oneself to be too meek, it could be a sign of weakness and a lack of confidence. This could bring others to exploit the situation.

To what extent is humility really relevant in our modern society? It would be dependent on the context of the societies that we are in and what is the purpose we are utilising it for 🙂

### Thinking [email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */ #9

[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 standard summation question. I’m interested in the last part only.

The answer to (ii) is written there by the student. I’ll only do the solution to (iv).

### Quick Summary (Probability)

University is starting for some students who took A’levels in 2016. And, one of my ex-students told me to share/ summarise the things to know for probability at University level. Hopefully this helps. H2 Further Mathematics Students will find some of these helpful.

Random Variables

Suppose $X$ is a random variable which can takes values $x \in \chi$.

$X$ is a discrete r.v. is $\chi$ is countable.
$\Rightarrow p(x)$ is the probability of a value of $x$ and is called the probability mass function.

$X$ is a continuous r.v. is $\chi$ is uncountable.
$\Rightarrow f(x)$ is the probability density function and can be thought of as the probability of a value $x$.

Probability Mass Function

For a discrete r.v. the probability mass function (PMF) is

$p(a) = P(X=a)$, where $a \in \mathbb{R}$.

Probability Density Function

If $B = (a, b)$

$P(X \in B) = P(a \le X \le b) = \int_a^b f(x) ~dx$.

And strictly speaking,

$P(X = a) = \int_a^a f(x) ~dx = 0$.

Intuitively,

$f(a) = P(X = a)$.

Properties of Distributions

For discrete r.v.
$p(x) \ge 0 \forall x \in \chi$.
$\sum_{x \in \chi} p(x) = 1$.

For continuous r.v.
$f(x) \ge 0 \forall x \in \chi$.
$\int_{x \in \chi} f(x) ~dx = 1$.

Cumulative Distribution Function

For discrete r.v., the Cumulative Distribution Function (CDF) is
$F(a) = P(X \le a) = \sum_{x \le a} p(x)$.

For continuous r.v., the CDF is
$F(a) = P(X \le a ) = \int_{- \infty}^a f(x) ~dx$.

Expected Value

For a discrete r.v. X, the expected value is
$\mathbb{E} (X) = \sum_{x \in \chi} x p(x)$.

For a continuous r.v. X, the expected value is
$\mathbb{E} (X) = \int_{x \in \chi} x f(x) ~dx$.

If $Y = g(X), then For a discrete r.v. X,$latex \mathbb{E} (Y) = \mathbb{E} [g(X)] = \sum_{x \in \chi} g(x) p(x)\$.

For a continuous r.v. X,
$\mathbb{E} (Y) = \mathbb{E} [g(X)] = \int_{x \in \chi} g(x) f(x) ~dx$.

Properties of Expectation

For random variables $X$ and $Y$ and constants $a, b, \in \mathbb{R}$, the expected value has the following properties (applicable to both discrete and continuous r.v.s)

$\mathbb{E}(aX + b) = a \mathbb{E}(X) + b$

$\mathbb{E}(X + Y) = \mathbb{E}(X) + \mathbb{E}(Y)$

Realisations of $X$, denoted by $x$, may be larger or smaller than $\mathbb{E}(X)$,

If you observed many realisations of $X$, $\mathbb{E}(X)$ is roughly an average of the values you would observe.

$\mathbb{E} (aX + b)$
$= \int_{- \infty}^{\infty} (ax+b)f(x) ~dx$
$= \int_{- \infty}^{\infty} axf(x) ~dx + \int_{- \infty}^{\infty} bf(x) ~dx$
$= a \int_{- \infty}^{\infty} xf(x) ~dx + b \int_{- \infty}^{\infty} f(x) ~dx$
$= a \mathbb{E} (X) + b$

Variance

Generally speaking, variance is defined as

$Var(X) = \mathbb{E}[(X- \mathbb{E}(X)^2] = \mathbb{E}[X^2] - \mathbb{E}[X]^2$

If $X$ is discrete:

$Var(X) = \sum_{x \in \chi} ( x - \mathbb{E}[X])^2 p(x)$

If $X$ is continuous:

$Var(X) = \int_{x \in \chi} ( x - \mathbb{E}[X])^2 f(x) ~dx$

Using the properties of expectations, we can show $Var(X) = \mathbb{E}(X^2) - \mathbb{E}(X)^2$.

$Var(X)$
$= \mathbb{E} [(X - \mathbb{E}[X])^2]$
$= \mathbb{E} [(X^2 - 2X \mathbb{E}[X]) + \mathbb{E}[X]^2]$
$= \mathbb{E}[X^2] - 2\mathbb{E}[X]\mathbb{E}[X] + \mathbb{E}[X]^2$
$= \mathbb{E}[X^2] - \mathbb{E}[X]^2$

Standard Deviation

The standard deviation is defined as

$std(X) = \sqrt{Var(X)}$

Covariance

For two random variables $X$ and $Y$, the covariance is generally defined as

$Cov(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]$

Note that $Cov(X, X) = Var(X)$

$Cov(X, Y) = \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[y]$

Properties of Variance

Given random variables $X$ and $Y$, and constants $a, b, c \in \mathbb{R}$,

$Var(aX \pm bY \pm b ) = a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X, Y)$

This proof for the above can be done using definitions of expectations and variance.

Properties of Covariance

Given random variables $W, X, Y$ and $Z$ and constants $a, b, \in \mathbb{R}$

$Cov(X, a) = 0$

$Cov(aX, bY) = ab Cov(X, Y)$

$Cov(W+X, Y+Z) = Cov(W, Y) + Cov(W, Z) + Cov(X, Y) + Cov(X, Z)$

Correlation

Correlation is defined as

$Corr(X, Y) = \dfrac{Cov(X, Y)}{Std(X) Std(Y)}$

It is clear the $-1 \le Corr(X, Y) \le 1$.

The properties of correlations of sums of random variables follow from those of covariance and standard deviations above.

### Thinking [email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */ #8

[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 interesting Complex Number Question.

The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^2$

### Let’s talk about the environment

A rather boringgggg yet simple topic to handle for the A levels. Most of the environment questions are asking about whether economic growth conflicts with environmental preservation… Let’s talk a look at how this issue could be approached…

No both growth and preservation could be handled together: citizens are getting more eco-conscious and could force their governments to be likewise using their votes; growth could always be attained with preservation especially if the country is interested in eco-tourism and sustainable development; some governments would only take a look at preservation of the environment after they have attained some form of growth (ie. spending money on renewable energies once they could afford them).

No growth and preservation of the environment always conflict with one another: this is especially the case for developing countries where they need to harness the environment in terms of resources to pursue growth; the burning of fossil fuels and industrialization would necessarily lead to more pollution; the culture of consumerism where individuals consume mindlessly would put factories into an overdrive to produce, to pollute and to deal with the environmental problems later on.

These are just some discussion pointers, we would need to consider the context when writing our essays! How developing countries versus developed countries would look at the issue would differ greatly!

### Let’s talk about terrorism!

Terrorism has been an ongoing threat in many parts of the world. Once in a while, we could hear of lone wolf attacks that are occurring overseas. Just yesterday Brussels was hit by an attempted terrorist attack, and recently there was a London Bridge attack as well. Gun shooting cases are also heard of once in a while in the US. Such occurrence goes to show how terrorism is more prevalent than what it seems!

With these in mind, this brings me to the question of whether it is possible to eradicate terrorism in our society… I would like to argue that it is pretty difficult to do so as these terrorists are often faceless/nameless and it would be difficult for the authorities to track them down. Second thing it would be difficult to predict the nature of these terrorist attacks given how terrorism has changed across time with the facilitation of technology. Attacks could possibly operate with greater stealth and on a faster rate than before. Finally, terrorism is often a symptom of a larger problem on hand such as discrimination that is evident in society or even poverty etc. It is merely an outlet for them to express their frustrations and gain attention to resolve the issue on hand. It is unlikely for them to resort to this mean if other means have already worked.

However, we should not be too pessimistic with regards to reducing or minimising terrorism in our society. Education could be an effective platform to dispel these issues and to disseminate information that terrorism is not the right method for one to fight for what they want. Second, terrorism in the form of radicalism and all would not bode well for those seeking peace and prosperity, hence it would be unlikely for one to support this method in the long run.

What do you guys think? Would it be possible for society to eradicate terrorism?

### Thinking [email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */ #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.