### 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.

### 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!

### Random Questions from 2016 Prelims #11

TPJC/P2/4

(a) The complex number w is given by $3+3\sqrt{3}i$
(i) Find the modulus and argument of w, giving your answer in exact form.
(ii) Without using a calculator, find the smallest positive integer value of n for which $(\frac{w^3}{w^*})^n$ is a real number.

(b) The complex number z is such that $z^5 = - 4 \sqrt{2}$
(i) Find the value of z in the form $re^{i\theta}$, where $r > 0$ and $- \pi \textless \theta \le \pi$.
(ii) Show the roots on an argand diagram.
(iii) The roots represented by $z_1$ and $z_2$ are such that $0 \textless arg({z_1}) \textless arg({z_2}) \textless \pi$. The locus of all points z such that $|z - z_1| = |z-z_2|$ intersects the line segment joining points representing $z_1$ and $z_2$ at the point P. P represents the complex number p. Find, in exact form, the modulus and argument of p.

### Random Questions from 2016 Prelims #10

PJC P2 Q1

The complex numbers a and b are given by $2 + 3i$ and $-4-5i$ respectively.

(i) On a single Argand diagram, sketch the loci
(a) $|2z-a-b| = |a-b|$
(b) $0 \le \text{arg}(z-b) \le \text{arg}(a - b)$

(ii) Find the range of $\text{arg}(z)$ where $z$ is the complex number that satisfies the relationships in part (i)

### 2016 A’level Suggested Solutions

Congratulations on the completion of A’levels for the 2016 batch!

As for those who still, are taking A’levels 2017, we hope you find this site helpful. Read the comments (ignore the trolls who have yet grown up) and learn from our mistakes.

It has been a pleasure for all us to interact with you guys. Do check back after your release of A’levels, as we will love to hear from you! In the meantime, you can always read through some of the posts regarding undergraduate & postgraduate courses in Mathematics (KS), Economics (KS), Physics (Casey) as we share some of our past experiences, along with some of our works today.

You’ve come a long way, now party hard. We wish you all the best in your future endeavours.

### Random Questions from 2016 Prelims #4

VJC P1 Q6

A curve has equation $y^2=4x$ and a line $l$ has equation $2x-y+1=0$ as shown above.

$B(b, 2\sqrt{b})$ is a fixed point on C and A is an arbitrary point on $l$. State the geometrical relationship between the line segment AB and $l$ is the distance from B to A is the least.

Taking the coordinates of A as $(a, 2a+1)$, find an equation relating $a$ and $b$ for which AB is the least.

Deduce that when AB is the least, $(AB)^2 = m (2b - 2\sqrt{b} +1)^2$ where $m$ is a constant to be found. Hence or otherwise, find the coordinates of the point on C that is nearest on $l$.

### Random Questions from 2016 Prelims #2

NYJC P1 Q10b

The function $g: x \mapsto 2 - \frac{5x}{1+x^2}, x \in \mathbb{R}$
(i) Use an algebraic method to find the range of g.

(ii) State a sequence of transformations which transform the graph of $y=g(x)$ to the graph of $y = \frac{10x}{4+x^2}$

### Random Questions from 2016 Prelims #1

YJC P1 Q4

Given $h(x) = x \text{cos}x$ for $0 \le x \le \frac{\pi}{2}$. If $h(x)=h(-x)$ and $h(x+\pi)=-h(x)$, sketch the graph of $h(x)$ for $-2\pi \le x \le 2\pi$.