### Solutions to the modified A’levels Questions

Students of mine who have been diligently doing the modified TYS and have difficulties with the questions that were added in to make the paper a full 3 hour paper, will find the following solutions helpful. Please try to do them in a single 3 hour seating, it modified to cater to the 9758 syllabus…

The rest of the solutions (that are questions from the original TYS) can be found here.

### 2017 A-level H2 Mathematics (9758) Suggested Solutions

All solutions here are SUGGESTED. KS will hold no liability for any errors. Comments are entirely personal opinions.

Paper 1

Paper 2

MF26

### 2017 A-level H2 Mathematics (9758) Paper 2 Suggested Solutions

All solutions here are SUGGESTED. KS will hold no liability for any errors. Comments are entirely personal opinions.

Numerical Answers (click the questions for workings/explanation)

Question 1:
Question 2:
Question 3:
Question 4:
Question 5:
Question 6:
Question 7:
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Question 11:

MF26

### 2017 A-level H2 Mathematics (9758) Paper 1 Suggested Solutions

All solutions here are SUGGESTED. KS will hold no liability for any errors. Comments are entirely personal opinions.

Numerical Answers (click the questions for workings/explanation)

Question 1:
Question 2:
Question 3:
Question 4:
Question 5:
Question 6:
Question 7:
Question 8:
Question 9:
Question 10:
Question 11:

MF26

### Last Hustle for A’levels 2017

As we are all busy counting down to A’levels, The Culture SG Team will like to share the preparatory course that we have for students.
The lessons will all be \$70 for each session and the max class size will be 15 students.

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

You may contact Tutor KS or Tutor Christine for further questions.

Details are as follow:

“Lets’ Hustle!”

### Trigonometry Formulae & Applications (Part 1)

Upon request by some students, I’ll discuss a few trigonometry formulae here and also some of their uses in A’levels. I’ve previously discussed the use of factor formulae here under integration.

I’ll start with the R-Formulae. It should require no introduction as it is from secondary Add Math. This formulae is not given in MF26, although students can derive it out using existing formulae in MF26.

$a \text{cos} \theta \pm b \text{sin} \theta = R \text{cos} (\theta \mp \alpha)$

$a \text{sin} \theta \pm b \text{cos} \theta = R \text{sin} (\theta \pm \alpha)$

where $R = \sqrt{a^2 + b^2}$ and $\text{tan} \alpha = \frac{b}{a}$ for $a > 0, b > 0$ and $\alpha$ is acute.

Here is a quick example,

$f(x) = 3 \text{cos}t - 2 \text{sin}t$

Write $f(x)$ as a single trigonometric function exactly.

Here, we observe, we have to use the R-Formulae where

$R = \sqrt{3^2 + 2^2} = \sqrt{13}$

$\alpha = \text{tan}^{\text{-1}} (\frac{2}{3})$

We have that

$f(x) = \sqrt{13} \text{cos} ( t + \text{tan}^{\text{-1}} (\frac{2}{3}))$.

I’ll end with a question from HCI Midyear 2017 that uses R-Formulae in one part of the question.

A curve D has parametric equations

$x = e^{t} \text{sin}t, y = e^{t} \text{cos}t, \text{~for~} 0 \le t \le \frac{\pi}{2}$

(i) Prove that $\frac{dy}{dx} = \text{tan} (\frac{\pi}{4} - t)$.

I’ll discuss about Factor Formulae soon.  And then the difference and application between this two formulae.

### 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…

### H2 Mathematics (9740) 2016 Prelim Papers

So many students have been asking for more practice. I’ll put up all the Prelim Papers for 2016 here. Do note that the syllabus is 9740 so students should practice discretion and skip questions that are out of syllabus. 🙂

Here are the Prelim Paper 2016. Have fun!

Here is the MF26.

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 $p$.