### 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.

### APGP Question #3

This is a question from a recent ACJC JC2 test.

A philanthropist started a donation matching programme to encourage more people to donate regularly to a particular charity.

(a) For a person who donates $a in the first month, and for each subsequent month donates$ $b^2$ more than the previous month, the philanthropist will donate $a in the first month and for each subsequent month, b times the amount he donated the previous month. (i) John is a regular donor of the charity. Find the values of a and b such that the philanthropist donates$20 in the third month, and ten times more than John in the seventh month.

(ii) Find the total amount of money donated by John and the philanthropist in one year, leaving your answer to the nearest dollar.

(b) In a revised donation matching programme, if a donor makes a monthly donation of $c, the philanthropist will donate according to the following plan: First month : No donation ($0\%$ donation rate) Second month : $10 \%$ of the total amount donated by the donor in the first two months Third month : $20 \%$ of the total amount donated by the donor in the first three months Fourth month : $30 \%$ of the total amount donated by the donor in the first four months and so on. (i) Show that the total amount of money the philanthropist will donate at the end of the $4^{th}$ month is$2c.

(ii) By expressing the total amount of money, the philanthropist will donate at the end of the $n^{th}$ month as a summation, and using the result $\sum_{r=1}^n r^2 = \frac{n}{6}(n+1)(2n+1)$, show that the total amount of money the philanthropist will donate at the end of the $n^{th}$ month is \$ $\frac{(n-1)n(n+1)}{30} c$.

### 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 )

### A little history of e

Some students remarked on why I actually recognise e, that is, $e=2.718281828...$. Well, e is a rather unique constants. Firstly, for all JC students, we see it our daily algebra & complex numbers. Students exposed to university statistics will see e appearing in the formula for normal distribution, that is, $f(x | \mu , \sigma^2) = \frac{1}{\sqrt{2 \sigma^2 \pi}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}}$.

Secondly, the story of how it came about is pretty cool as you will observed in the video below.

The Story of e

Hopefully it provides you with another perspective towards this constants! And now you should be more cautious when signing up savings plans that give interest per annum or per month.

### 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!

### Thoughts on A’levels H2 Mathematics 2016 Paper 2

I’ll keep this short since we are all busy. One thing about paper 1 we saw, there were many unknowns.

So topics which I think will come out…

Differentiation – I think a min/max problem will come out, possibly with r and h both not given and asked to express r in terms of h. But students should revise a on the properties of curves with differentiation; given a curve equation with an unknown, for instance $y=/frac{x^2+kx+1}{x-1}$, find the range of k such that there is stationary points.

Complex Number – Loci will definitely come out. I’m saying they will combine with trigonometry.

Integration – Modulus integration hasn’t really been tested. Else a question on Area/ Volume could be tested, and I’ll say they need students to do some
Conics too.

For statistics, my students should have gotten the h1 stats this year. And if it’s an indicator, then it should not be a struggle.

I expect PnC and probability to be combined. Conditional Probability in a poisson question should be tested too, so do revise it well. For hypothesis testing, students should be careful of their formula and read really carefully about the alternative hypothesis. Also, :9 know that the formulas for poison PDF and binompdf are both given in mf15. Lastly, know when to use CLT.

All the best!