### Anderson JC GP Prelims paper 1 2017

1. Should small countries be allowed to take the lead in global affairs?
2. To what extent can the Arts effect positive social change today?
3. ‘Experiences are more valuable than material possessions.’ Do you agree?
4. ‘People in the workplace should embrace, rather than fear, technological advancements.’ Discuss.
5. ‘The news today deals with what is popular, rather than what is important.’ How far do you agree with this statement?
6. Evaluate the claim that a more connected world has resulted in greater divisions.
7. ‘Public figures today are overly concerned about what people think of them.’ What is your view?
8. Consider the view that there is no value in slowing down in today’s competitive world.
9. Discuss the appeal and value of creativity in your society.
10. Considering the increasing threat of terrorism, are governments justified in limiting people’s rights?
11. To what extent is animal testing acceptable in scientific research?
12. ‘Economic development is favoured at the expense of the welfare of people.’ How true is this of your society?

### Trigonometry Formulae & Applications (Part 2)

I meant to share more on factor Formulae today. However, a few students are not so sure how to get the R-formulae correctly during their preliminary exams recently. So I thought that I’ll share how they can derive the R-Formulae from the MF26.

The following is the R-Formulae which students should have memorised. It is under assumed knowledge, just saying…

$a \text{cos} \theta \pm b \text{sin} \theta = R \text{cos} (\theta \mp \alpha)$

$a \text{sin} \theta \pm b \text{cos} \theta = R \text{sin} (\theta \pm \alpha)$

where $R = \sqrt{a^2 + b^2}$ and $\text{tan} \alpha = \frac{b}{a}$ for $a > 0, b > 0$ and $\alpha$ is acute.

So here, I’ll write the addition formulae that’s found in MF26.

$\text{sin}(A \pm B) \equiv \text{sin}A \text{cos} B \pm \text{cos} A \text{sin} B$

$\text{cos}(A \pm B) \equiv \text{cos}A \text{cos} B \mp \text{sin} A \text{sin} B$

I’ll use an example I discussed previously.

$f(x) = 3 \text{cos}t - 2 \text{sin}t$

Write $f(x)$ as a single trigonometric function exactly.

Lets consider the formulae from MF26.

$\text{cos}(A \pm B) \equiv \text{cos}A \text{cos} B \mp \text{sin} A \text{sin} B$

$R\text{cos}(A \pm B) \equiv R \text{cos}A \text{cos} B \mp R \text{sin} A \text{sin} B$

We can let

$3 = R \text{cos} B ---(1)$

$2 = R \text{sin} B ---(2)$

$\Rightarrow \sqrt{ 3^2 + 2^2 } = \sqrt{ R^2 \text{cos}^2 B + R^2 \text{sin}^2 B}$

$\Rightarrow \sqrt{13} = R$

$\Rightarrow \frac{R \text{sin} B}{R \text{cos} B} = \frac{2}{3}$

$\Rightarrow \text{tan} B = \frac{2}{3}$

Putting things together, we have that

$f(x) = \sqrt{13} \text{cos} ( t + \text{tan}^{\text{-1}} (\frac{2}{3}))$

### Temasek JC GP Prelim Paper 1 2017

Temasek Junior College 8807 H1 General Paper Paper 1 2017

1. Can government surveillance eradicate the threat of terrorism?
2. Examine the claim that globalization creates equal opportunities for all.
3. ‘The government is not doing enough to support local sportsmen in your society.’ What is your view?
4. To what extent is a universal language desirable?
5. Should people in your society be fearful of the future?
6. ‘Graciousness is lost as society progresses.’ Is this an accurate reflection of your society?
7. How far do you agree that technology gives us greater control in life?
8. Consider the view that what is posted online is all talk and no action.
9. ‘Failure should never be acceptable.’ Discuss.
10. Do you agree that only parents should be allowed to discipline their children?
11. Is volunteerism always good?
12. ‘The world today values appearance over substance.’ Is this a fair comment?

### Tampines GP Prelim Paper 1 2017

Tampines Junior College 8807 H1 General Paper Paper 1 2017

1. How realistic is it for your society to embrace diversity?
2. Protecting the environment should be given greater priority than eradicating poverty. How far do you agree?
3. ‘Appearance can be deceiving.’ To what extent is this true of the media today?
4. Discuss the view that smart devices have not made us smarter.
5. Education is the key to solving all social problems. Discuss.
6. Should firms have the responsibility to improve the quality of life of the communities they operate in?
7. Assess the view that literature is of little use to society.
8. Wealth is no guarantee of a better life. How far do you agree?
9. History is of little significance to a modern society. Discuss.
10. ‘Failure is always an option.’ Discuss.
11. A free and unrestricted media is essential for society to progress. How far do you agree?
12. How far is the arts a reflection of your society’s level of development?

### Trigonometry Formulae & Applications (Part 1)

Upon request by some students, I’ll discuss a few trigonometry formulae here and also some of their uses in A’levels. I’ve previously discussed the use of factor formulae here under integration.

I’ll start with the R-Formulae. It should require no introduction as it is from secondary Add Math. This formulae is not given in MF26, although students can derive it out using existing formulae in MF26.

$a \text{cos} \theta \pm b \text{sin} \theta = R \text{cos} (\theta \mp \alpha)$

$a \text{sin} \theta \pm b \text{cos} \theta = R \text{sin} (\theta \pm \alpha)$

where $R = \sqrt{a^2 + b^2}$ and $\text{tan} \alpha = \frac{b}{a}$ for $a > 0, b > 0$ and $\alpha$ is acute.

Here is a quick example,

$f(x) = 3 \text{cos}t - 2 \text{sin}t$

Write $f(x)$ as a single trigonometric function exactly.

Here, we observe, we have to use the R-Formulae where

$R = \sqrt{3^2 + 2^2} = \sqrt{13}$

$\alpha = \text{tan}^{\text{-1}} (\frac{2}{3})$

We have that

$f(x) = \sqrt{13} \text{cos} ( t + \text{tan}^{\text{-1}} (\frac{2}{3}))$.

I’ll end with a question from HCI Midyear 2017 that uses R-Formulae in one part of the question.

A curve D has parametric equations

$x = e^{t} \text{sin}t, y = e^{t} \text{cos}t, \text{~for~} 0 \le t \le \frac{\pi}{2}$

(i) Prove that $\frac{dy}{dx} = \text{tan} (\frac{\pi}{4} - t)$.

I’ll discuss about Factor Formulae soon.  And then the difference and application between this two formulae.

### Random Questions from 2017 Prelims #2

Today I’ll share a question that came out of CJC Prelim 2017 Paper 1 for H2 Mathematics 9758. I think some of my student would have seen this question before and we discussed it in class before. Very technical question. This is question 11, I’ll share only the first part which is on the application of ratio theorem or mid point theorem. The second part is on application: Ray Tracing which I’ll discuss in class.

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. For the triangle show below, O, A and B are vertices, where O is the origin. $\vec{OA} = a$ and $\vec{OB} = b$. The midpoints of OB, OA and AB are M, N and T respectively.

It is given that X is the point of intersection between the medians of triangle OAB from vertices A and B.

(i) Show that $\vec{OX} = \frac{1}{3} (a +b)$

(ii) Prove that X also lies on OT, the median of triangle OAB from vertex O.

### ACJC Prelims paper 1 2017

1. ACJC 2017 H1 General Paper 8807 Prelim Paper 1
2. To what extent does education prepare the young for a world that is constantly changing?
3. ‘Science imparts knowledge but not wisdom.’ Do you agree?
4. Have we placed too much emphasis on work today?
5. Is being original always beneficial?
6. ‘It is better to be feared than to be popular.’ Discuss the view with reference to leadership.
7. Can diseases ever be eliminated?
8. ‘The media creates more problems than benefits for politicians.’ Discuss.
9. How far, in your society, are children a good investment?
10. Consider the view that the poor are more likely to commit crimes than the rich.
11. Evaluate the claim that environmental conservation is a desirable, but unrealistic, goal.
12. ‘Music breaks all barriers.’ Can music be so powerful?
13. ‘Economic development is key to a country’s stability.’ To what extent is this true in your society?

To those taking your prelims for GP really soon… here is an example of a prelim paper this year… Could you do these questions? What help would u need?

### 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…

### 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] #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).