### 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 R th 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.

### Solutions to Review 1

Question 1
(i)
$y = f(x) = \frac{x^2 + 14x + 50}{3(x+7)}$

$3y(x+7) = x^2 + 14x + 50$

$x^2 + (14-3y)x + 50 - 21 y = 0$

$\text{discriminant} \ge 0$

$(14-3y)^2 - 4(1)(50-21y) \ge 0$

$196 - 84y + 9y^2 - 200 + 84y \ge 0$

$9y^2 - 4 \ge 0$

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

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

(ii)
Using long division, we find that

$y = \frac{x^2 + 14x + 50}{3(x+7)} = \frac{x}{3} + \frac{7}{3} + \frac{1}{3(x+7)}$

So the asymptotes are $y = \frac{x}{3} + \frac{7}{3}$ and $x = -7$

Question 2
(i)
$x^2 - 9y^2 + 18y = 18$

$x^2 - 9(y^2 - 2y) = 18$

$x^2 - 9[(y-1)^2 - 1^2] = 18$

$x^2 - 9(y-1)^2 + 9 = 18$

$x^2 - 9(y-1)^2 = 9$

$\frac{x^2}{9} - (y-1)^2 = 1$

This is a hyperbola with centre $(0, 1)$, asymptotes are $y = \pm \frac{x}{3} + 1$, and vertices $(3, 1)$ and $(-3, 1)$.

$y = \frac{1}{x^2} + 1$ is a graph with asymptotes $x = 0$ and $y=1$.

Use GC to plot.

(ii)
$\frac{x^2}{9} - (y-1)^2 = 1$—(1)

$y = \frac{1}{x^2} + 1$ —(2)

Subst (2) to (1),

$\frac{x^2}{9} - (\frac{1}{x^2} + 1 - 1)^2 = 1$

$\frac{x^2}{9} - (\frac{1}{x^2})^2 = 1$

$x^2 - \frac{9}{x^4} = 9$

$x^6 - 9 = 9x^4$

$x^6 - 9x^4 - 9 = 0$

(iii)
From graph, we observe two intersections. Thus, two roots.

Question 3
(ai)
$\sum_{r=1}^n (r+1)(3r-1)$

$= \sum_{r=1}^n (3r^2 + 2r -1)$

$= \sum_{r=1}^n 3r^2 + \sum_{r=1}^n 2r - \sum_{r=1}^n 1$

$= 3 \sum_{r=1}^n r^2 + 2 \sum_{r=1}^n r - \sum_{r=1}^n 1$

$= 3 \frac{n}{6}(n+1)(2n+1) + 2 \frac{n}{2}(1 + n) - n$

$= \frac{n}{2}(n+1)(2n+1) + n(1+n) - n$

$= \frac{n}{2}(n+1)(2n+1) + n^2$

(aii)
$2 \times 4 + 3 \times 10 + 4 \times 16 + ... + 21 \times 118$

$= 2 [2 \times 2 + 3 \times 5 + 4 \times 8 + ... + 21 \times 59]$

$= 2 [(1+1) \times (3 \cdot 1 - 1) + (2+1) \times (3 \cdot 2 -1) + (3+1) \times (3 \cdot 3 -1) + ... + (20+1) \times (3 \cdot 20 -1) ]$

$= 2 \sum_{r=1}^{20} (r+1)(3r-1)$

$= 2 [\frac{n}{2}(n+1)(2n+1) + n^2 ]$

$= n(n+1)(2n+1) + n^2$

$= n(2n^2 + 3n + 1) + n^2$

$= 2n^3 + 4n^2 + n$

(bi)
$\frac{2}{(r-1)(r+1)} = \frac{A}{r-1} - \frac{B}{r+1}$

$2 = A(r+1) - B(r-1)$

Let $r = -1$

$2 = - B(-2) \Rightarrow B = 1$

Let $r = 1$

$2 = A(2) \Rightarrow A = 1$

$\therefore \frac{2}{(r-1)(r+1)} = \frac{1}{r-1} - \frac{1}{r+1}$

(bii)
$\sum_{r=2}^n \frac{1}{(r-1)(r+1)}$

$= \frac{1}{2} \sum_{r=2}^n \frac{2}{(r-1)(r+1)}$

$= \frac{1}{2} \sum_{r=2}^n (\frac{1}{r-1} - \frac{1}{r+1})$

$= \frac{1}{2} [ 1 - \frac{1}{3}$

$+ \frac{1}{2} - \frac{1}{4}$

$+ \frac{1}{3} - \frac{1}{5}$

$...$

$+ \frac{1}{n-3} - \frac{1}{n-1}$

$+ \frac{1}{n-2} - \frac{1}{n}$

$+ \frac{1}{n-1} - \frac{1}{n+1}]$

$= \frac{1}{2} [1 + \frac{1}{2} - \frac{1}{n} - \frac{1}{n+1}]$

$= \frac{1}{2} (\frac{3}{2} - \frac{n+1+n}{n(n+1)})$

$= \frac{3}{4} - \frac{2n+1}{2n(n+1)}$

(biii)
As $n \to \infty$, $\frac{1}{n} \to 0$ and $\frac{1}{n+1} \to 0$, the sum of series tends to $\frac{3}{4}$, a constant. Thus, series is convergent.

(biv)

$\sum_{r=5}^{n+3} \frac{1}{(r-3)(r-1)}$

Replace $r$ by $r + 2$. Then we have

$\sum_{r=3}^{n+1} \frac{1}{(r-1)(r+1)}$

$= \sum_{r=2}^{n+1} \frac{1}{(r-1)(r+1)} - \frac{1}{(2-1)(2+1)}$

$= \frac{3}{4} - \frac{2(n+1)+1}{2(n+1)[(n+1)+1]} - \frac{1}{3}$

$= \frac{5}{12} - \frac{2n+3}{2(n+1)(n+2)}$

Its been awhile since we last posted. And it is good to know that JC1s have all been well inducted or settled into their schools. Of course, I do hear that many schools are severely overcrowded recently. Anyway, I thought of sharing how students can get ready for JC. I contemplated sharing how to cope with JC Mathematics, but decided to be more general this time round.

Firstly, JC life can be quite rigorous. With CCA and different subject commitments piling, students must try their best to stay healthy (get enough sleep) and juggle time (skip some dramas) efficiently. For science subjects (not just H2 Mathematics), students should avoid procrastinating. The schools do not go back to teaching the subjects again, maybe just refresh using questions or tests. Thus, seek help if you need and do not just sweep it off. For J1, your A’levels is pretty much in 22 months while for J2, it is 10 months. So the clock started ticking.

Secondly, last year’s papers were intuitive and some questions were driven to see if students do understand their content and can think on their feet. And we have a name for such questions, it is application questions. For H2 Mathematics, they have made an effort to allocate about 25% of the total marks to application questions. Thus, students need to shift their focus from doing to learning. It is important for them to appreciate the concepts in each topic.

Thirdly, I understand some students enter JC and realise that there are really some (or a lot of) smarter peers around. Do not feel pressured and just stay focus. Some of them might have found help, or developed better intuition for certain things. Comparing with your neighbour will only make yourself more stressed. This is unnecessary stress.

Lastly, JC is the last “school” you have. So do enjoy yourself. Pick a CCA that you really want to try. 🙂

Since Mr Teng teaches H2 Mathematics, here are some little tips for H2 Mathematics as I told my J1s this year.

1. Some topics from High School are still very relevant, which is why I gave a proper review test. These topics are considered under assumed knowledge for H2 Mathematics, and you can find them here. A good understanding of these topics will allow you to follow classes better. You will learn that schools are constantly rushing to clear topics.

2. Learn the topics. You do not need to master them, but learn and find out what is going on. Because you can memorise the entire Ten Years Series and realise that it will not save you.

3. You will learn that time is very precious during exams. In general, 1 mark is 1.5 mins. And you should not go beyond it for questions. Rather learn the hard way to time manage well during exams, start with your normal practices at home. Thats why I encourage my students in class to do fast. Your papers will be two 3-hour papers, so during that 3-hour, you must exhibit sufficient tenacity.

P.S. I’ve spent the last few months getting a lot of application questions up. Aside from sharing them with students in my classes, I’ll also put them here. So do check in. 🙂

Happy CNY!

### JC Talk 2018

Over the weekends, we had the privilege of conducting a mini JC talk which saw Mr. Teng and Ms. Christine share their knowledge with parents and students of O’levels 2017.

The lessons for J1 2018 started on the first week of January and the schedules can be found here.

The following are the grade profiles of local universities, NUS and NTU.

NTU IGP

NUS IGP

We are very thankful for your attendance and do hope that the information was beneficial. If you do have more questions, you can contact Mr. Teng at +65 9815 6827

### Solutions to December Holiday Revision

Hopefully you find your holidays meaningful. Take the time to unwind and relax. Also, if you have completed the Set A and Set B. The solutions can be found here.

If you do have any questions, please WhatsApp me. 🙂

Solutions to Set A

Solutions to Set B

Relevant Materials: MF26

### Solutions to Set B

Hopefully, you guys have started on the Set B. You will find the following solutions useful. Click on the question. Please do attempt them during this December Holidays. 🙂

If you do have any questions, please WhatsApp me. 🙂

Relevant Materials: MF26

### Solutions to Set A

Hopefully, you guys have started on the Set A. You will find the following solutions useful. Click on the question. Please do attempt them during this December Holidays. 🙂

If you do have any questions, please WhatsApp me. 🙂

Relevant Materials: MF26

### 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: $0.773; 0.0514; 0.866; 0.362$

MF26

### 2017 A-level H1 General Paper (8807) Paper 1

Sharing the A-level H1 General Paper (8807) Paper 1…

1. ‘The past is not dead; it is not even past.’ Discuss.
2. Can the use of animals for scientific research ever be justified?
3. In your society, to what extent is it acceptable for public money to be used for the acquisition of works of art?
4. ‘Rehabilitation, not punishment, should be the purpose of the justice system.’ Discuss.
5. Is regulation of the press desirable?
6. Do events, rather than politicians, shape the future?
7. How far is science fiction becoming fact?
8. Examine the role of music in establishing a national identity in your society.
9. To what extent are people judged more by their physical appearance than by their abilities?
10. ‘Practical ability is just as important as intellectual skills.’ How far is this true in your society?
11. Assess the view that attempts to control climate change can never be truly effective.
12. The quality of written language is being destroyed by social media.’ What is your view?
13.

### 2017 A-level H2 Mathematics (9740) 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: $ax + (2a - \frac{a^2}{2})x^2 + (\frac{a^3}{3} + 2a - a^2) x^3$; $a = 4$
Question 2: $x \textgreater \frac{1}{\sqrt{b}} + a$ or $x \textless a$
Question 3: 2
Question 4: $a = 4, b =1$; translate the graph 4 units in negative y-direction and translate the graph 2 units in positive x-direction.
Question 5: $a = -1.5, b = 1.5, c = 7$; $x \approx -1.33$; $x \approx -0.145$ or $x \approx 1.15$
Question 6: $r = a + (\frac{d - a \cdot n}{b \cdot n}) b$
Question 7: $(\frac{1}{a}, \frac{1}{ae}); \frac{1}{a^2}$
Question 8: $z = -1 + 2i$ or $z = 2 - i$; $p =-6, q=-66$; $(w^2 - 2w+2)(w^2-4w+29)$
Question 9: $U_n = 2An - A +B$; $A = 3, B =-9$; $k=4$; $\frac{1}{4} (n^4 + 2n^3 + n^2)$ ; $e^x$
Question 10: $a = -4.4$; $R(1.5, 0.5, -1)$; $\frac{1}{2}\sqrt{10}$
Question 11: $\frac{dv}{dt}=c$; $v = 10t +4$; $v = \frac{1}{k}(10- 10 e^{-kt})$

MF26