### Vectors Question #3

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

Find the Cartesian equation of the plane that is equidistant of the $xy$ plane and $xz$ plane.

The following should aid students to visualise.

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.

### A little reminder to students doing Calculus now

When $\frac{dy}{dx} = 0$, it implies we have a stationary point.

To determine the nature of the stationary point, we can do either the first derivative test or the second derivative.

The first derivative test:

Students should write the actual values of $\alpha^-, \alpha, \alpha^+$ and $\frac{dy}{dx}$ in the table.

We use this under these two situations:
1. $\frac{d^2y}{dx^2}$ is difficult to solve for, that is, $\frac{dy}{dx}$ is tough to be differentiated
2. $\frac{d^2y}{dx^2} = 0$

The second derivative test:

Other things students should take note is concavity and drawing of the derivative graph.

### Responsibility of Protecting the Environment

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?

### Vectors Question #2

If $c = |a| b + |b| a$, where $a$ , $b$ and $c$ are all non-zero vectors, show that $c$ bisects the angle between $a$ and $b$.

### Post-Results 2016

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.

### Things to note for 9758 H2 Mathematics

It has been awhile since the A’levels. We talked and met up with several of our students. Some students are working and some are preparing their Personal Statements for overseas University Applications.

A few of them also remarked that they wish they put more efforts into studying A’levels.  An advice to this year JC2 – Do not wait till it is too late.

Students should also have a clear understanding of their syllabus, especially their scheme of exam. I still have JC2s this year who get stunned by the applications questions I threw at them (P.S. Aside from spending time with family, I wrote many sets of applied questions.). You may read more about your syllabus here. Or see the following images.

How much practice or emphasis your school put on this, is up to them. But it is clear that application takes up 25% of your marks. The entire syllabus can be found here.

PLEASE DON’T WALK INTO THE EXAM HALL BLUR BLUR…

### Release of A’levels Results 2016

For students who took A’levels in 2016, please note that information for the release of A’levels Results 2016 can be found in the following!

Release of A’levels 2016

Grade Profile (i.e. Number of As you need to get into courses for)

SMU

NTU

NUS

P.S. Results does not define you. When one door closes, another opens.

### Checklist for Vectors

Many schools have been doing vectors recently. Thought I’ll share a little summary/ checklist I have done for my students.

Basic Concepts

• Operations on Vectors
• Scalar multiplication
• Dot Product (Scalar)
1. a • a = |a|2
2. If a ⊥ b, then a • b = 0
3. a • b = b • a
• Cross Product (Vector)
1. a × a = 0
2. a × b = − b × a
• Unit Vectors
• Parallel Vectors ( a = k)
• Collinear Vectors ( Parallel with a common point )
• Ratio Theorem ( Found in MF26)
• Midpoint Theorem
• Directional Cosines
• Angles between two Vectors
• Length of Projection
• Perpendicular Distance

Lines

• Equations
• Vector Form ( : r = a + λb, λ∈ ℜ )
• Parametric Form
• Cartesian Form
• Line & Line
• Parallel ( Directions are parallel to each other. )
• Same ( Same Equations )
• Intersecting ( There is a unique solution for λ and μ. )
• Skewed ( Not parallel AND not Intersecting. )
• Angle between two lines ( Angle between their directions )
• Point & Line
• Foot of Perpendicular
• Perpendicular (Shortest) distance
• Point on Line

Planes

• Equations
• Parametric Form ( π r = a + λb + μc, λ, μ ∈ ℜ )
• Scalar Product Form ( r • n = a • n  = d )
• Cartesian Form
• Point & Plane
• Foot of Perpendicular
• Perpendicular (Shortest) distance
• Distance from O to Plane
• Point on Plane
• Reflection of Point
• Line & Plane
• Relationships
1. Parallel
• Line intersects Plane entirely ( Infinite Solutions )
• Do not intersect ( No Solution )
2. Not Parallel
• Intersects at a point ( One Solution )
• Intersection Point
• Angle between Line & Plane
• Reflection of Line
• Plane & Plane
• Relationships
1. Parallel
• Same ( Infinite Solutions )
• Do not intersect ( No Solution )
2. Not Parallel
• Intersects at a line ( Infinite Solutions )
• Intersection Line ( Use of GC )
• Angle between two Planes ( Angle between their normals )

### Some questions that students ask recently.

I’ve been asked many times recently about what university course to take, and also what university to go too, etc. Some students ask me how does Financial Mathematics work? So I’ve come across an article here. This article illustrates how to be a quant, which is just one of the jobs available to someone who studies financial mathematics.

I should clarify that studying Finance is a far cry from studying Financial Mathematics. They are very different. For the pragmatic students, the latter earns more. Is it easy? I shared some undergraduate reviews that I’ve done previously, here. It is on some simple ideas of Financial Mathematics, its basics.

So I’ll share a bit more in near future on studying Operations Research, Bayesian Methods, Data Mining and Analytics. Hopefully it will give students a better idea of studying Mathematics in university. And please remember, that H2 Mathematics is a far cry from University Mathematics. Students are better off doing Engineering if you fancy H2 Mathematics. And I’m sorry that I can’t share too much on Pure Math, as the above mentioned are my forte.

Thanks! And now let’s all go for holidays!

### How far has modern technology made it unnecessary for individuals to possess mathematical skills?

1. How far has modern technology made it unnecessary for individuals to possess mathematical skills?

Argument 1: Modern technology has provided us with tools that automate mathematical operations for us, making possessing mathematical skills redundant.

Elaboration: All we have to do are to insert the data we want to be processed into our modern technological tools, and their software automates the required operations and calculations for us, allowing us to attain the necessary processed data effortlessly.

Example: Calculators has made skills such as the ability simple mental calculations mostly redundant in everyday life. Online free tools such as desmos and Wolframalpha. Paid tools such as MatLab, Maple and R.

Link: Modern technology has provided us with tools that make certain mathematical skills such as doing simple mental calculations seemingly redundant.

(Counter) Argument 2: We still ultimately require mathematical knowledge and skills to utilize these technological tools

Elaboration: To say that possessing mathematical skills is unnecessary altogether would be too much of a far-fetched statement

Example: We still require intermediate knowledge on math to effectively operate a Graphic Calculator for more complex math problems. Basic knowledge for syntax used in programs in Graphic Calculator, or algorithm is needed.

Link: Mathematical skills are still required to allow modern technology to operate in our favor.

(Counter) Argument 3: Recognizing patterns, reasoning and logical thinking are important life skills which are largely mathematical.

Elaboration: These life skills are largely inculcated into our children via an education in mathematics.

Example: Number or shape pattern recognition is a skill inculcated into primary school children via math education. Over the years, this analytical skill is honed and eventually applied to solve industry-relevant problems such as market trends. Famous Hedge-fund owner James Simmons, only hires Mathematicians and Physicists, who do not need any business or banking background, to work as bankers in his company, Renaissance Technologies. The top Quantitative Analyst in the world has a doctorate in Physics and to quote him, in research he deals with 23 variables, but now in life and finance, he just has 3 to handle.

Link: Mathematical skills are a prerequisite to develop higher level cognitive and analytical skills in order for us to excel in society.

(Counter) Argument 4: Being equipped with the most basic Mathematical skills provides us with much greater convenience as compared to having sole reliance on technological tools.

Elaboration: Simple day-to-day activities largely involve the use of mathematical skills and would ironically take a longer time and effort if we were to fully rely on technological tools. Moreover, it would be a hassle if we were to require such tools to physically be with us all the time as compared to solving problems on the spot with our mathematical knowledge.

Example: Mental calculations would greatly aid us in buying and selling as opposed to solely relying on a calculator even for simple operations. The absolute necessity of having a calculator around with you would often times become a hassle in that case. The basic skill of doing rough mental calculations and pattern recognition would enable us to have greater control over our finances in estimating and analyzing our spending habits and saving progress.

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