DRV questions with a twist

DRV questions with a twist

JC Mathematics, Mathematics

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.

Solutions to Review 1

Solutions to Review 1

JC Mathematics, Mathematics

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

Curve C of 1(ii)

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)}

Getting ready for JC

Getting ready for JC

JC Mathematics, Mathematics

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

JC General Paper, JC Mathematics

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 Set B

Solutions to Set B

JC Mathematics, Mathematics

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

Solutions to Set A

JC Mathematics, Mathematics

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

Thoughts on the H2 Mathematics (9758) Papers 2017

Thoughts on the H2 Mathematics (9758) Papers 2017

JC Mathematics, Mathematics

Solutions can be found here.

Personal Thoughts: The paper isn’t tedious. Students can do them so long as they know their stuffs. There are several generalising of questions, like question 6 of paper 1. We also saw how conditional probability was actually tested subtly, this tests students’ abilities to reason with guidance (not sure if after this first trial year, will they still guide the students.) Application questions were not tough and well guided. Students can solve it easily if they read it well. Statistics was well crafted and neat.

To be blunt, I’ll give credit to the 9740 H2 Mathematics paper that run concurrently, since it is too tough to set two sets of papers. Its easy to acknowledge that the 9740 (2016) paper was way harder than 9740 (2017). Next year won’t be the same.

Advice: Students should be careful when you revise, make sure you learn, and not do. Understand what you’re doing. The 2017 paper was an inquisitive paper, examiners were watching closely if you pay attention to details, and know your definitions well.

I’ll do an analysis for the paper, you can click on the individual question and read. For students that took the paper, I hope it doesn’t demoralise you.

Paper 1

Paper 2