### DRV questions with a twist

I want to share a question that is really old (older than me, actually). It is from the June 1972 paper. As most students know, Maclaurin’s series was tested in last year A’levels Paper 1 as a sum to infinity. And this DRV did the same thing. Here is the question.

A bag contains 6 black balls and 2 white balls. Balls are drawn at random one at a time from the bag without replacement, and a white ball is drawn for the first time at Rth draw.

(i) Tabulate the probability distribution for R.

(ii) Show that E( R ) = 3, and find Var( R ).

If each ball is replaced before another is drawn, show that in this case E( R ) = 4.

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

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 MF26 ### KS Comments ### 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! 🙂 ### 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. ### 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. ### 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 :) 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)

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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.