### Probability Question #4

A gambler bets on one of the integers from 1 to 6. Three fair dice are then rolled. If the gambler’s number appears $k$ times ($k = 1, 2, 3$), he wins $$k$. If his number fails to appear, he loses$1. Calculate the gambler’s expected winnings

### 2016 A-level H1 Mathematics (8864) Paper 1 Suggested Solutions

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

As these workings and answers are rushed out asap, please pardon me for my mistakes and let me know if there is any typo. Many thanks.
I’ll try my best to attend to the questions as there is H2 Math Paper 2 coming up.

Question 1
(i)
$\frac{d}{dx} [2 \text{ln}(3x^2 +4)]$

$= \frac{12x}{3x^2+4}$

(ii)
$\frac{d}{dx} [\frac{1}{2(1-3x)^2}]$

$= \frac{6}{2(1-3x)^3}$

$= \frac{3}{(1-3x)^3}$

Question 2

$2e^{2x} \ge 9 - 3e^x$

$2u^2 + 3u - 9 \ge 0$

$(2u-3)(u+3) \ge 0$

$\Rightarrow u \le -3 \text{~or~} u \ge \frac{3}{2}$

$e^x \le -3$ (rejected since $e^x > 0$) or $e^x \ge \frac{3}{2}$

$\therefore x \ge \text{ln} \frac{3}{2}$

Question 3
(i)

(ii)
Using GC, required answer $= -1.606531 \approx -1.61$ (3SF)

(iii)
When $x = 0.5, y = 0.35653066$

$y - 0.35653066 = \frac{-1}{-1.606531}(x-0.5)$

$y = 0.622459 x +0.04530106$

$y = 0.622 x + 0.0453$ (3SF)

(iv)

$\int_0^k e^{-x} -x^2 dx$

$= -e^{-x} - \frac{x^3}{3} \bigl|_0^k$

$= -e^{-k} - \frac{k^3}{3} + 1$

Question 4
(i)
$y = 1 + 6x - 3x^2 -4x^3$

$\frac{dy}{dx} = 6 - 6x - 12x^2$

Let $\frac{dy}{dx} = 0$

$6 - 6x - 12x^2 = 0$

$1 - x - 2x^2 = 0$

$2x^2 + x - 1 = 0$

$x = -1 or \frac{1}{2}$

When $x =-1, y = -4$

When $x = \frac{1}{2}, y = 2.75$

Coordinates $= (-1, -4) \text{~or~} (0.5, 2.75)$

(ii)
$\frac{dy}{dx} = 6 - 6x - 12x^2$

$\frac{d^2y}{dx^2} = - 6 - 24x$

When $x = -1, \frac{d^2y}{dx^2} = 18 > 0$. So $(-1, -4)$ is a minimum point.

When $x = 0.5, \frac{d^2y}{dx^2} = -18 \textless 0$. So $(0.5, 2.75)$ is a maximum point.

(iii)

x-intercept $= (-1.59, 0) \text{~and~} (1, 0) \text{~and~} (-0.157, 0)$

(iv)
Using GC, $\int_0.5^1 y dx = 0.9375$

Question 5
(i)
Area of ABEDFCA $= \frac{1}{2}(2x)(2x)\text{sin}60^{\circ} - \frac{1}{2}(y)(y)\text{sin}60^{\circ}$

$2\sqrt{3} = \sqrt{3}x^2 - \frac{\sqrt{3}}{4}y^2$

$2 = x^2 - \frac{y^2}{4}$

$4x^2 - y^2 =8$

(ii)

Perimeter $= 10$

$4x+2y + (2x-y) = 10$

$6x + y = 10$

$y = 10 - 6x$

$4x^2 - (10-6x)^2 = 8$

$4x^2 - 100 +120x -36x^2 = 8$

$32x^2 -120x+108=0$

$x=2.25 \text{~or~} 1.5$

When $x=2.25, y = -3.5$ (rejected since $y >0$)

When $x=1.5, y = 1$

Question 6
(i)
The store manager has to survey $\frac{1260}{2400} \times 80 = 42$ male students and $\frac{1140}{2400} \times 80 = 38$ female students in the college. He will do random sampling to obtain the required sample.

(ii)
Stratified sampling will give a more representative results of the students expenditure on music annually, compared to simple random sampling.

(iii)
Unbiased estimate of population mean, $\bar{x} = \frac{\sum x}{n} = \frac{312}{80} = 3.9$

Unbiased estimate of population variance, $s^2 = \frac{1}{79}[1328 - \frac{312^2}{80}] = 1.40759 \approx 1.41$

Question 7
(i)
[Venn diagram to be inserted]

(ii)
(a)
$\text{P}(A \cup B) = \text{P}(A) + \text{P}(B) - \text{P}(A \cap B) = 0.8$
(b)
$\text{P}(A \cup B) - \text{P}(A \cap B) = 0.75$

(iii)
$\text{P}(A | B')$

$= \frac{\text{P}(A \cap B')}{\text{P}(B')}$

$= \frac{\text{P}(A) - \text{P}(A \cap B)}{1 - \text{P}(B)}$

$= \frac{0.6 - 0.05}{1-0.25}$

$= \frac{11}{15}$

Question 8
(i)
Required Probability $= \frac{1}{2} \times \frac{3}{10} \times \frac{2}{9} = \frac{1}{30}$

(ii)
Find the probability that we get same color. then consider the complement.

Required Probability $= 1 - \frac{1}{30} - \frac{1}{2} \times \frac{5}{10} \times \frac{4}{9} - \frac{1}{2} \times \frac{2}{10} \times \frac{1}{9} - \frac{1}{2} \times \frac{4}{6} \times \frac{3}{5} - \frac{1}{2} \times \frac{2}{6} \times \frac{1}{5} = \frac{11}{18}$

(iii)
$\text{P}(\text{Both~balls~are~red} | \text{same~color})$

$= \frac{\text{P}(\text{Both~balls~are~red~and~same~color})}{\text{P}(\text{same~color})}$

$= \frac{\text{P}(\text{Both~balls~are~red})}{\text{P}(\text{same~color})}$

$= \frac{1/30}{7/18}$

$= \frac{3}{35}$

Question 9
(i)
Let X denote the number of batteries in a pack of 8 that has a life time of less than two years.

$X \sim B(8, 0.6)$
(a)
Required Probability $=\text{P}(X=8) = 0.01679616 \approx 0.0168$

(b)
Required Probability $=\text{P}(X \ge 4) = 1 - \text{P}(X \le 3) = 0.8263296 \approx 0.826$

(ii)
Let Y denote the number of packs of batteries, out of 4 packs, that has at least half of the batteries having a lifetime of less than two years.

$Y \sim B(4, 0.8263296)$

Required Probability $=\text{P}(Y \le 2) = 0.1417924285 \approx 0.142$

(iii)
Let W denote the number of batteries out of 80 that has a life time of less than two years.

$W \sim B(80, 0.6)$

Since n is large, $np = 48 > 5, n(1-p)=32 >5$

$W \sim N(48, 19.2)$ approximately

Required Probability

$= \text{P}(w \ge 40)$

$= \text{P}(w > 39.5)$ by continuity correction

$= 0.9738011524$
$\approx 0.974$

Question 10
(i)
Let X be the top of speed of cheetahs.
Let $\mu$ be the population mean top speed of cheetahs.

$H_0: \mu = 95$

$H_1: \mu \neq 95$

Under $H_0, \bar{X} \sim N(95, \frac{4.1^2}{40})$

Test Statistic, $Z = \frac{\bar{X}-\mu}{\frac{4.1}{\sqrt{40}}} \sim N(0,1)$

Using GC, $p=0.0449258443 \textless 0.05 \Rightarrow H_0$ is rejected.

(ii)
$H_0: \mu = 95$

$H_1: \mu > 95$

For $H_0$ to be not rejected,

$\frac{\bar{x}-95}{\frac{4.1}{\sqrt{40}}} \textless 1.644853626$

$\bar{x} \textless 96.06 \approx 96.0$ (round down to satisfy the inequality)

$\therefore \{ \bar{x} \in \mathbb{R}^+: \bar{x} \textless 96.0 \}$

Question 11
(i)
[Sketch to be inserted]

(ii)
Using GC, $r = 0.9030227 \approx 0.903$ (3SF)

(iii)
Using GC, $y = 0.2936681223 x - 1.88739083$

$y = 0.294 x - 1.89$ (3SF)

(iv)
When $x = 16.9, y = 3.0756 \approx 3.08$(3SF)

Time taken $= 3.08$ minutes

Estimate is reliable since $x = 16.9$ is within the given data range and $|r|=0.903$ is close to 1.

(v)
Using GC, $r = 0.5682278 \approx 0.568$ (3SF)

(vi)
The answers in (ii) is more likely to represent since $|r|=0.903$ is close to 1. This shows a strong positive linear correlation between x and y.

Question 12
Let X, Y denotes the mass of the individual biscuits and its empty boxes respectively.

$X \sim N(20, 1.1^2)$

$Y \sim N(5, 0.8^2)$

(i)
$\text{P}(X \textless 19) = 0.181651 \approx 0.182$

(ii)
$X_1 + ... + X_12 + Y \sim N(245, 15.16)$

$\text{P}(X_1 + ... + X_12 + Y > 248) = 0.2205021 \approx 0.221$

(iii)
$0.6X \sim N(12, 0.66^2)$

$0.2Y \sim N(1, 0.16^2)$

Let $A=0.6X$ and $B = 0.2Y$

$A_1+...+A_12+B \sim N(145, 5.2528)$

$\text{P}(142 \textless A_1+...+A_12+B \textless 149) = 0.864257 \approx 0.864$

### 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.

### H2 Mathematics (By KS)

Read KS’ thoughts about the upcoming paper here.

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### 2015 A-level H1 Mathematics (8864) Paper Suggested Solutions

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

Students are highly encouraged to discuss freely for alternative solutions too.

I typed this, so pardon me if got errors. There should be a handful of errors as I just type and solve as I go. Just refresh as you go along as I’m typing with wifi in a cafe. AND I SKIP A LOT OF STEPS, including denoting my random variables. I’ll get to them probably tomorrow morning.

### Learning Math or Learning to do Math

I’ve always struggled to accept how some students study and kept telling them off for their ways. Yes, I am no PSC scholar or top academic to comment. But seeing all the batches of students that took H2 Mathematics being so fearful of doing Mathematics in Uni is less than encouraging. Makes me wonder if I fail as an educator at times.

Every year, I have a handful of students who like math and want to pursue Mathematics at higher levels, but as time progress, that passion withers. And thats because, they become so disheartened with A-levels. Of course, it is not entirely the education system. It is also the methods that the students employ that count. I know students who can do twenty over Mathematical Induction Questions but falters at the one that comes out in exams. This shows that they don’t understand and know their concepts well. This is an example of learning to do math.

### What is learning Math, and how important is it?

Learning Math starts with understanding the definitions and concepts. We cannot jump into doing the questions head first. We need to sit on what we learnt and try to understand it. Every topic in A-levels is tested but we know topics can cross over and be tested together in one single question. As such, it is important for us to see H2 Mathematics as one single subject, or even a topic. We shouldn’t segregate too much. I can easily mix a APGP with a poisson question, and it is nothing wrong with it. It is still within boundaries of your examinations.

Here, I’m not trying to scare students off, but hoping they will start reconsidering their approach to studying mathematics. As I discussed here previously, I do think that H2 Mathematics is going for a harder route and many students will suffer.

### 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?