### Random Questions from 2017 Prelims #4

Find the acute inclination of the reflected ray $d$ to the $z$-axis when $d$ is inclined at $60^{\circ}$ to the $x$-axis and $45^{\circ}$ to the $y$-axis.

### Making Use of this September Holidays

This is a little reminder and advice to students that are cheong-ing for their Prelims or A’levels…

For students who have not taken any H2 Math Paper 1 or 2, I strongly advise you start waking at up 730am and try some papers at 8am. I gave my own students similar advices and even hand them 4 sets of 3 hours practice papers. Students need to grind themselves to be able to handle the paper at 8am. It is really different. Not to mention, this September Holidays is probably your last chance to be able to give yourself timed practices.

For students who took H2 Math Paper 1, you might be stunned with the application questions that came out. For NJC, its Economics. For YJC, its LASER. For CJC, a wild dolphin appeared. And more. These application questions are possible, due to the inclusion of the problems in real world context in your syllabus. You can see the syllabus for yourself. I’ve attached the picture below. So for Paper 2, expect these application questions to be from statistics mainly, as suggested in your scheme of work below.

For students that have took H2 Math paper 1 & 2, and this is probably ACJC. The paper was slightly stressful, given the mark distributions, but most of the things tested are still technically “within syllabus”. For one, the directional cosine question, is a good reminder to students that they should not leave any pages un-highlighted. AC students should be able to properly identify their weaknesses and strengths this time round. If its time management, then start honing that skill this holidays – by having timed practice. A quick reminder that the TYS papers are not 3 hours, since some of the questions are out of H2 Mathematics 9758 syllabus. Students can consider the ratio of 1 mark to 1.5 min to gauge how much time they have for each paper.

R-Formulae seems to be popular about the prelims exams this time round, making waves in various schools. Perhaps it was because it appeared in the specimen paper, and if you’re keen on how it can be integrated or need a refresher. I did it recently here.

Lastly, for the students that are very concerned on application questions. Check the picture below. It contains some examples that SEAB has given. Students should also be clear about the difference between a contextual question and an application question.

With that, all the best to your revision! 🙂

### Random Questions from 2017 Prelims #3

Here is another question that is from CJC H2 Mathematics 9758 Prelim Paper 1. Its a question on differentiation. I think it is simple enough and tests student on their thinking comprehension skills. This is question 6.

A straight line passes through the point with coordinates (4, 3) cuts the positive x-axis at point P and positive y-axis at point Q. It is given that $\angle PQO = \theta$, where $0 < \theta < \frac{\pi}{2}$ and O is the origin.

(i) Show that equation of line PQ is given by $y = (4-x) \text{cot} \theta +3$.

(ii) By finding an expression for $OP + OQ$, show that as $\theta$ varies, the stationary value of $OP + OQ$ is $a + b \sqrt{3}$, where $a$ and $b$ are constants to be determined.

### Integration By Parts

Today, I’ll share a little something about Integration by Parts. I want to share this because I observe that several students are over-reliant on the LIATE to perform Integration by Parts. This caused them to overlook/ appreciate its use.

Lets start by reviewing the formula for Integration by Parts

$\int v \frac{du}{dx} ~dx = uv - \int u \frac{dv}{dx} ~dx$

I like to share with students that Integration by Parts have two interesting facts.

1. It allows us to integrate expression that we cannot integrate. Eg. $\text{ln}x$ or inverse trigonometric functions. This is also closely related to how LIATE is established.

2. This is closely related to point 1. That is, instead of trying to integrate the expression, we are differentiating it instead. And that itself, is a very important aspect.

Next, I like to point out to students that LIATE is a general rule of thumb.
GENERAL – because it does not work 100% of the times. And today, I’ll use an example to illustrate how LIATE actually fails.

$\int \frac{te^t}{(t+1)^2} ~ dt$

Here we observe two terms $\frac{t}{(t+1)^2}$ and $e^t$. Going by LIATE, we should be differentiating $\frac{t}{(t+1)^2}$ and integrating $e^t$.

$\int t^2 e^{t} ~ dt$
$= \frac{t}{(t+1)^2} e^{t} - \int \frac{(1)(t+1)^2 - t(2)(t+1)}{(t+1)^4} e^t ~dt$

Now observe what happened here, after applying integration by parts, it got “worst”. We are stuck to integrating $\frac{(1)(t+1)^2 - t(2)(t+1)}{(t+1)^4}$ with an exponential… This should raise some question marks. But we did follow LIATE.

I’ll leave students to try out this on their own. And feel free to ask questions here. Have fun!

### 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…

As the prelims examinations draw really close, many students were asking me to give questions to test their concepts for several topics. In class, I had the opportunity to explore several applications questions too. We saw several physics concepts mixed. We also have some conceptual questions that need students to be able to use the entire topic to solve it.

So I’ll share one here. This involves several concepts put together. I’ll put the solution up once I find the time. Concepts that will be involved, will be

1. Vector Product
2. Equations of Plane
3. Finding foot of perpendicular of point

The question in one a reflection of a plane in another plane. I think such questions will come out in a few guided steps in exams. But should a student be able to solve it independently, it shows that he has good understanding.

The plane $p$ has equation $x + y + z = 9$ and the plane $p_1$ contains the lines passing through $(0, 2, 3)$ and are parallel to $(1, -1, 0)$ and $(0, 1, 1)$ respectively. Find, in scalar product form, the equation of the plane which is the reflection of $p_1$ in $p$.

### 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).

### Thinking [email protected] #8

[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 interesting Complex Number Question.

The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^2$

### Thinking [email protected] #7

[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 an application question for hypothesis testing from the 9758 H2 Mathematics Specimen Paper 2 Question 10.

The average time required for the manufacture of a certain type of electronic control panel is 17 hours. An alternative manufacturing process is trialled, and the time taken, $t$ hours, for the manufacture of each of 50 randomly chosen panels using the alternative process, in hours, is recorded. The results are summarized as follows

$n = 50$
$\sum t = 835.7$
$\sum t^2 = 14067.17$

The Production Manager wishes to test whether the average time taken for the manufacture of a control panel is different using the alternative process, by carrying out a hypothesis test.
(i) Explain whether the Production Manager should use a 1-tail or a 2-tail test.
(ii) Explain why the Production Manager is able to carry out a hypothesis test without knowing anything about the distribution of the times taken to manufacture the control panels.
(iii) Find unbiased estimates of the population mean and variance, and carry out the test at the 10% level of significance for the Production Manager.
(iv) Suggest a reason why the Production Manager might be prepared to use an alternative process that takes a longer average time than the original process.
The Finance Manager wishes to test whether the average time taken for the manufacture of a control panel is shorter using the alternative process. The Finance Manger finds that the average time taken for the manufacture of each of the 40 randomly chosen control panels, using the alternative process, is 16.7 hours. He carries out a hypothesis test at 10% level of significance.
(v) Explain, with justification, how the population variance of the times will affect the conclusion made by the Finance Manager.

### June Crash Course

The team at The Culture SG has been really busy and we have a lot of things prepared to help you guys work for that A. First up! Crash course for June…

And we know it is a bit late to be announcing this on the site now, but we have really been caught up with preparing our students lately that we don’t have the time to properly update here. So here are the details for the Math Crash Course and the Chemistry Crash Course.

P.S. For SCIENCE students who wish to chiong in October, please take note that the H2 Chem/ Phy/ Bio Paper 4 (practical) is in October. So better start soon! Here are the details!

Click to view

For 3 hr lessons, they are priced at $105. For 2 hr lessons, they are priced at$70.

Lessons will be held at:
Newton Apple Learning Hub
Blk 131, Jurong Gateway Road #03-263/265/267 Singapore 600131
Tel: +65 6567 3606

For math enquiries, you may contact Mr. Teng at +65 9815 6827.

For chem enquiries, you may contact Ms. Chan at +65 93494384.

For GP enquiries, you may contact Ms. Chen at +65 91899133.