### 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.

### 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…

### How to maintain your self discipline to study?

Dear all, this is a general post on how to keep your study momentum up during this december holidays. As you already know, this december holidays is not only a time for you to rejuvenate yourself from the hectic JC life, it is also an important revision time for you! It is a time when you consolidate all that you have learnt in this year, so that you would have a good foundation to start JC2. (For those who didn’t do well in the promos, you should be doing catch up. For those who did well, you could possibly revise your learning and embark on a head-start program.)

So what are the 3 tips for you to have self-discipline?

One of the reason why students do not have self-discipline is because they do not believe in themselves and their abilities to attain academic success. They attach negative talk to themselves such as they are stupid, lazy and not cut out for straight As. Hence, their inner beliefs shape their actions and their revision process. One way to change it would be to change your identity. Start thinking to yourself that you deserve Straight As everyday and you will soon internalize it and manifest these behaviours.

Secondly, you should attach pain to the notion of you not achieving your goals. Each time when you think about how you would not achieve your goals, it will automatically propel you to take action! It could possibly be a lack of self esteem, respect from your parents and society. Regret for not achieving your aims etc.

Finally, you could think about how to reward yourself whenever you achieve your goals and to get a trusted person to be accountable for you. One way to do this is to tell your friends or parents what you desire to achieve for the A levels, and to get them to monitor your revision schedule. I know this sounds unappealing to you, but this could perhaps be the best way to make sure you stay discipline to your goals!!

Have a good break everyone! Of course have a fruitful revision time too!

### 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!

### H2 Mathematics Crash Course Sept

Our team at “theculture.sg” have been really busy the past few weeks, which explains the lack of posts. We have really exciting news to share with our reads. While we are busy preparing our students for the preliminary exams on all levels such as time management and exam preparedness, we received many queries regarding the possibility for a H2 Math crash course program. We had the H1 General Paper Crash Course in June Holidays and it benefited all the students that attended it. For this September Holidays, we are kicking something different and special. We are holding a 8-hr crash course for H2 Mathematics, alongside with General Paper Crash Course.

All along, we believe that fundamentals are important, conceptual understanding is paramount for optimal learning of Sciences. Our past experiences in teaching and guiding students taught us that students know all these concepts, and they are hidden somewhere in their brain. From answering all the various questions sent to us through texts, and left online, we developed our methodology to analyze trends and behaviors. Students of Mr. Teng should know he always say, “Statistics do not lie!”. So he did his homework for A’level trends.

So are you one of those students have no trouble understanding the solutions provided, but struggle to start doing a question even? Or one of those that have difficulties finishing the paper optimally? Or one of those that want to test your knowledge?

Each crash course will be 8 hrs long from 8th-9th Sept.  The timings for J2 would be 8-12pm on both days; for J1 it would be 1-5pm. This crash course is created and planned to be very intensive and prepares the students well. Students of all abilities will be well suited to come attend the class. And since you guys are the graduating batch of 9740 H2 Mathematics, Newton Apple & theculture.sg will gift students that sign up for BOTH General Paper and Mathematics a comprehensive summary for H2 Mathematics.

You may contact me at +65 9815 6827 or Newton Apple directly at 92223423.

Do not miss this opportunity of learning with us, as we might not have the time to do this course again ahead of A’levels.

### 100 days more…

So the Midyear results are all out and Prelims are known to be in 4-5 weeks’ time. Many students are frantically searching for help and attempting to salvage their results. We are sorry to say that we aren’t able to take any more private students due to time constraints, and only the group classes are available. Our classes are all held in Newton Apple Jurong East.

### June Revision Exercise 8 Q5

(i)
$l_{AB}: r = \begin{pmatrix}4 \\ 1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix}1 \\ 2 \\ -1 \end{pmatrix}, \lambda \in \mathbb{R}$

$l_{AB}: r = \begin{pmatrix}7 \\ -3 \\ 13 \end{pmatrix} + \mu \begin{pmatrix}-2 \\ 1 \\ -5 \end{pmatrix}, \mu \in \mathbb{R}$

If they intersect, then $\begin{pmatrix}7 \\ -3 \\ 13 \end{pmatrix} + \mu \begin{pmatrix}-2 \\ 1 \\ -5 \end{pmatrix} = \begin{pmatrix}4 \\ 1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix}1 \\ 2 \\ -1 \end{pmatrix}$ for some $\lambda \text{ and } \mu$

Solving with GC, we have $\lambda = -1, \mu = 2$. Thus, the lines intersect.

(ii)
$\begin{pmatrix}-2 \\ 1 \\ -5 \end{pmatrix} \times \begin{pmatrix}1 \\ 2 \\ -1 \end{pmatrix} = \begin{pmatrix}-9 \\ 7 \\ 5 \end{pmatrix}$

Equation of plane: $r \bullet \begin{pmatrix}-9 \\ 7 \\ 5 \end{pmatrix} = \begin{pmatrix}4 \\ 1 \\ 2 \end{pmatrix} \bullet \begin{pmatrix}-9 \\ 7 \\ 5 \end{pmatrix} = -19$

(iii)
Required distance

$= \frac{|\vec{AE}\bullet \begin{pmatrix}-9 \\ 7 \\ 5 \end{pmatrix}|}{|\sqrt{9^2 + 7^2 + 5^2}|}$

$= \frac{3}{\sqrt{155}}$

(iv)

$= \text{ Area Triangle } ABC + \text{ Area Triangle } ACD$

$= \frac{1}{2}|\vec{CA} \times \vec{CD}| + \frac{1}{2}|\vec{AC} \times \vec{AB}|$

$= \frac{1}{2} |{\begin{pmatrix}-3 \\ 4 \\ -11 \end{pmatrix} \times \begin{pmatrix}-2 \\ 1 \\ -5 \end{pmatrix}}| + \frac{1}{2} |{\begin{pmatrix}3 \\ -4 \\ 1 \end{pmatrix} \times \begin{pmatrix}1 \\ 2 \\ -1 \end{pmatrix}}|$

$=\frac{1}{2} |\begin{pmatrix}-9 \\ 7 \\ 5 \end{pmatrix}| + \frac{1}{2}|\begin{pmatrix}-18 \\ 14 \\ 10 \end{pmatrix}|$

$= \frac{1}{2}(\sqrt{81 + 49 + 25} + 2\sqrt{81 + 49 + 25})$

$=\frac{3}{2}\sqrt{155}$

Back to June Revision Exercise 8.