2017 A-level H1 Mathematics (8865) Paper 1 Suggested Solutions

2017 A-level H1 Mathematics (8865) Paper 1 Suggested Solutions

JC Mathematics, Mathematics

All solutions here are SUGGESTED. Mr. Teng 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: \mu = 1.69, \sigma^2 = 0.0121
Question 7: 0.254; 0.194; 0.908
Question 8: 40320; 0.0142; \frac{1}{4}
Question 9: \text{r}=0.978; a=0.182, b=2.56; $293
Question 10: 0.0336; \bar{y}=0.64, s^2 = 0.0400; Sufficient evidence.
Question 11: \frac{48+x}{80+x}, \frac{32+x}{80+x}; x= 16; \frac{25}{32}; \frac{7}{16}; \frac{341}{8930}
Question 12: 0.773; 0.0514; 0.866; 0.362


Relevant materials


KS Comments

Making Use of this September Holidays

Making Use of this September Holidays

JC Mathematics, Mathematics

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.

Scheme of Examination Source: SEAB

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.

Integration & Applications Source: SEAB

With that, all the best to your revision! 🙂

Trigonometry Formulae & Applications (Part 1)

Trigonometry Formulae & Applications (Part 1)

JC Mathematics, Secondary Math

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.

Probability Question #4

Probability Question #4

JC Mathematics

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

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

JC Mathematics

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

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

\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

Graph for 3i
Graph for 3i

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

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)


\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
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)

\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.


Graph for 4iii
Graph for 4iii

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

Using GC, \int_0.5^1 y dx = 0.9375

Question 5
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


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
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.

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

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
[Venn diagram to be inserted]

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

\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
Required Probability = \frac{1}{2} \times \frac{3}{10} \times \frac{2}{9} = \frac{1}{30}

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}

\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
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)
Required Probability =\text{P}(X=8) = 0.01679616 \approx 0.0168

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

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

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
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.

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
[Sketch to be inserted]

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

Using GC, y = 0.2936681223 x - 1.88739083

y = 0.294 x - 1.89 (3SF)

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.

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

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)

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

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

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

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

2016 A’level Suggested Solutions

JC Chemistry, JC General Paper, JC Mathematics, JC Physics

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.

H1 General Paper (By Christine)

H2 Mathematics (By KS)

Read KS’ thoughts about the upcoming paper here.

H1 Mathematics (By KS)

H2 Physics (By Casey)


Welcome J1 to your new academic life :)

JC General Paper, JC Mathematics

We at The Culture have been receiving calls recently since the start of the J1 orientation program about our tuition program. This post is a shoutout to those students who are interested in the services that we provide. Currently, all tutors of The Culture are working only with Newton Apple Learning Hub @ Jurong East Gateway Road (right opp JE MRT), and we are pleased to announce some promotion of the centre for early sign ups!

Here are the details: Newton Apple is charging $280/subject, with a max class size of 8. Sign up early to avoid disappointment! Our classes are nearly 3/4 full!

Sign up by 28th Feb to enjoy 50% waiver of admin fee and material fee for sem 1 (worth up to $85)

Sign up by 31st March to enjoy 25% waiver of admin fee and material fee for sem 1 (worth up to $42.25)


10% off monthly with 2 subjects or more registered for.

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

JC Mathematics

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.

Continue reading “2015 A-level H1 Mathematics (8864) Paper Suggested Solutions”

Learning Math or Learning to do Math

JC Mathematics

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.

Credits: Universal Press Syndicate
Credits: Universal Press Syndicate

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

JC Mathematics

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!”

Credits: pinterest.com
Credits: pinterest.com

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!