Tips for the final lap of preparation! #2

So I mentioned that students can consider putting aside 9H weekly to prepare for A-levels. I haven’t really discuss what if you’re a student who is totally lost and don’t know how to carry on.

For one, you should NEVER read solutions! I should say this very explicitly that there are heavy repercussions from studying solutions for H2 Mathematics. Its a written examination for a good reason. To be honest, many students can read solutions and nod their head alongside in agreement. But can you think of it during exams? And please don’t assume that because you saw it here in the solutions, you will be highly capable of reproducing in exams. There is something significant about writing things that aids our memory and understanding everything. Even to date, I prefer to write things.

Moreover, all my students know that I do not use solutions during lessons. And no, I don’t memorise the solutions and start regurgitating them on the whiteboard since I do make silly mistakes on the whiteboard too. haha. The essence of me attempting the questions alongside with them allows them to see how I solve it and also my thought process since I will be doing my workings there. Of course, I have better intuition than my students which is why I can absorb the questions faster. But from these, the students will be able to learn how I pick out tiny details in exams.

So for students that face problems, where do we go from here?

We should identify our struggles, do a timed 3H paper and observe your problems. Can you start the question promptly or you are just lost at the start? Or do you struggle with the “deduce” or “hence” parts of the questions? Both struggles here call for different solutions, and students should treat them differently.

The first struggle can be treated by clearly identifying your concepts. I always stress to students that Mathematics is a subject built on definitions and theorems. Know your definitions well and you can start your questions.

The second struggle can be treated by having more exposure. These students mostly are struggling to push their C or Ds to an A. They simply need to do beyond and be better exposed.

It should be noted that past few years, A-levels papers have included a few unusual questions that will throw you off your feet. A good understanding of concepts and healthy amount of exposure will get your by here.

For students that lack confidence, first stop using the pencil and start using pens. My top students will concur that through the June Holidays of mugging, they concur on one thing and I quote, “Math, Just do!”

This is really very important and I see that it lacks in many students. Some students who realized this in the September Holidays will share that they have more confidence for their Prelims.

All the best!

Tips for the final lap of preparation! #1

As the prelims exams draw to an end for some schools, and some even collected the grades back. Many students must be disheartened about their grades too. So there are many questions coming in from different students and I thought I can share some of my responses here.

1. How should I prepare for my A-level?

Firstly, this highly depends on your standards or how you did for prelims (and we should not focus on physical grade here but your percentile!). For students who are consistently 90th percentile and above, I told them to ensure they spend at least 9 hours weekly on H2 Mathematics. 3H without break for each paper 1 and 2. The other 3H to review their mistakes and recap the conceptual problems. They can also let me mark and review their work together. And the end of the day, presentation is really important at A-levels.

Next, for students that have problems passing their prelims. You should not be disheartened if the median mark of your school is 42 (I know of one), Work harder, look at your mistakes and find out what your struggles with the papers are. I have numerous students who consult me with their papers and ask me about their standards individually. They wanted to know where they stand and how they can improve. And I was glad to enlighten them. Times not on our side, so you need to optimise your learning here

2. How many times should I do the past year papers?

Okay, seriously, this method does not work in A-levels. It might have gotten you though O-levels, but it will not be as effective here in A-levels. What I suggest for students is that, you do them once and guess used to the rigour, know their answering style (this is very important!). At the end of the day, there will be core structure (standards) that SEAB will follow in setting the papers. So students still need to be well exposed. And questions in H2 Mathematics aren’t really recycled entirely. We are not doing some science MCQ here.

3. Are you going to spot the A-levels questions?

Well, every year I do a little bit of trend spotting, not questions spotting. It should be noted that examiner’s report had suggested that students do not attempt to spot questions. I guess the harder we try, the harder they set. haha. i do share with students what I feel will be more focused topics; and what I feel will be the questions that will serve as a benchmark to differentiate the top students with respect to the rest.

4. I failed prelims really badly. Do I still have hope? And is it too late?

First of all, how hard was your prelim? You need to have a good analysis of your paper or discussion with your tutor or teacher. It is important for you to be able to benchmark yourself, had the entire cohort took the same paper as you.

Never give up until the end! And yes, it might sound very cliche but it is never too late. I had one student 2 years ago, who got a 4/100 for prelims, and leapfrogged to an A. Of course, this student put in immense effort. I saw her daily for a month and we worked very hard together.

I wish the A-level students all the best! 🙂

Proving a function is symmetrical about y-axis

This problem should not be tough for university Math students after learning the concept of even functions and odd functions.

An even function is one that $f(-x) = f(x)$. An example will be $f(x)= x^2$

An odd function is one that $f(-x) = -f(x)$. And example will be $f(x) = x^3$

So how to we prove the symmetry with this concept, consider a simple function $f(x)=x^2$, it is obvious that its symmetrical about y-axis. from the graph. But how did we know or tell. First, we know that $f(1) = f(-1) = 1$ and $f(2) = f(-2) = 4$, etc. Thus we know that $f(x) = f(-x)$ which brings us back to the idea of even function. Students need to attempt to write out this particular relationship mathematically. Such notations help them to express their ideas much clearly and also assists the examiners to mark, instead of trying to read through a long chunk of explanation.

Now, here is a problem for students, how do we prove that a function is symmetrical about x-axis?

2013 A-level H1 Mathematics (8864) Question 12 Suggested Solutions

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

(i)

(ii)
$\mathrm{P}(A) = \frac{1}{6} \frac{1}{6} + \frac{1}{6} \frac{1}{6} + \frac{1}{6} \frac{1}{6} +\frac{1}{6} \frac{1}{6} + \frac{1}{6} + \frac{1}{6} =\frac{4}{9}$

(iii)
$\mathrm{P}(A \cap B) = \frac{1}{9}$

(iv)
$\mathrm{P}(A \cup B) = \frac{4}{9} + \frac{2}{6} - \frac{1}{9} = \frac{2}{3}$

(v)
$\mathrm{P} (B| A \prime) = \frac{\mathrm{P}(B \cap A \prime)}{\mathrm{P}(A \prime)} = \frac{\frac{1}{6} \frac{4}{6} + \frac{1}{6} \frac{4}{6}}{5/9} = \frac{2}{5}$

The tree diagram was not easy definitely.

2013 A-level H1 Mathematics (8864) Question 10 Suggested Solutions

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

(i)

$H_0: \mu = 12$

$H_1: \mu \ne 12$

Under $H_0, \bar{X} ~\sim~ \mathrm{N} (12, \frac{0.8^{2}}{20})$

For $H_0$ to be not rejected at 5% level of significance, then

$latex -1.95996 < \frac{m-12}{0.8/ \sqrt{20}} < 1.95996$ $latex \therefore, \{m \in \mathbb{R} |11.6 < m < 12.4\}$ (ii) $latex H_0: \mu = 12$ $latex H_1: \mu < 12$ Under $latex H_0, \bar{X} ~\sim~ \mathrm{N} (12, \frac{0.8^{2}}{40})$ From the graphing calculator, p-value $latex = 0.02405 < 0.05$, we reject $latex H_0$. Thus, there is sufficient evidence at 5% level of significant to conclude that the mean salt content has been reduced.

Read carefully that it is two-tailed and perform the test. Bare in mind to leave answers in set notation.

2013 A-level H1 Mathematics (8864) Question 9 Suggested Solutions

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

(i)

(ii)
From graphing calculator, $r = 0.903$

r-value is close to 1 which suggest a strong positive correlation between the variables x and y.

(iii)
From graphing calculator, required regression line: $y = 4.46x + 87.43$

(iv)
When $x = 13.2, y = 146$.

Since r is close to 1, and we used the regression line of y on x to estimate y given an x within the range of data, the estimate will be reliable.

Very standard. Some students made mistake while keying values into the GC unfortunately.

2013 A-level H1 Mathematics (8864) Question 8 Suggested Solutions

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

(i)
Let X denote the number of batteries with lifetime of at least 100 hours, out of 10.

$X ~\sim~ \mathrm{B}(10, 0.8)$
$\mathrm{P} (X=10) = 0.107$

(ii)
$\mathrm{P} (X \ge 10) = 0.678$

(iii)
Let Y denote number of packs that satisfy the customer, out of 80.

$Y ~\sim~ \mathrm{B}(80, 0.6777995)$

Since $n = 80$ is large, $np = 54.224 > 5, nq = 25.776 >5$
$\Rightarrow Y ~\sim~ \mathrm{N}(54.224, 17.471)$ approximately

$\mathrm{P} (Y \ge 60)$
$= \mathrm{P} (Y \ge 59.5)$ by continuity correction

$\approx 0.103$

Remember to do CONTINUITY CORRECTION!!!

2013 A-level H1 Mathematics (8864) Question 6 Suggested Solutions

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

(i)
Stratified random sampling.
Split the population of people into three mutually exclusive strata, $X,$Y, and $Z. Number of$X to be surveyed $= \frac{5000}{30000} \times 150 = 25$
Number of $Y to be surveyed $= \frac{10000}{30000} \times 150 = 50$ Number of$Z to be surveyed $= \frac{15000}{30000} \times 150 = 75$

Select the required number of people within each strata using simple random sampling.

(ii)
The sample she gets will be more representative of the population since she split them into mutually exclusive strata.

Students can also present answers in a neat table for (i). Do take note that you must mention the use of simple random sampling.

2013 A-level H1 Mathematics (8864) Question 5 Suggested Solutions

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

(i)
$\mathrm{ln} e^{2-2x} = \mathrm{ln}2 + \mathrm{ln}e^{-x}$
$2 - 2x = \mathrm{ln} 2 - x$
$x = 2 - ln2$

(ii)
$\frac{dy}{dx} = -2 e^{2-2x} + 2 e ^{-x} = 0$
$e^{2-2x} = e^{-x}$
$x = 2$
$y = -e^{-2}$
$\therefore, \mathrm{required~coordinates~is~} (2, -e^{-2})$

(iii)

(iv)
Area $= \int_0^1 e^{2-2x} - 2e^{-x} dx = 1.93 units^{2}$ using graphing calculator.