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

### Thoughts on A’levels H2 Mathematics 2016 Paper 2

I’ll keep this short since we are all busy. One thing about paper 1 we saw, there were many unknowns.

So topics which I think will come out…

Differentiation – I think a min/max problem will come out, possibly with r and h both not given and asked to express r in terms of h. But students should revise a on the properties of curves with differentiation; given a curve equation with an unknown, for instance , find the range of k such that there is stationary points.

Complex Number – Loci will definitely come out. I’m saying they will combine with trigonometry.

Integration – Modulus integration hasn’t really been tested. Else a question on Area/ Volume could be tested, and I’ll say they need students to do some

Conics too.

For statistics, my students should have gotten the h1 stats this year. And if it’s an indicator, then it should not be a struggle.

I expect PnC and probability to be combined. Conditional Probability in a poisson question should be tested too, so do revise it well. For hypothesis testing, students should be careful of their formula and read really carefully about the alternative hypothesis. Also, :9 know that the formulas for poison PDF and binompdf are both given in mf15. Lastly, know when to use CLT.

All the best!

### 2016 A-level H2 Mathematics (9740) Suggested Solutions

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

Thoughts before 2016 A-level H2 Mathematics

### Relevant materials

### KS Comments:

### Random Questions from 2016 Prelims #13

NYJC/2/11

On a typical weekday morning, customers arrive at the post office independently and at a rate of 3 per 10 minute period.

(i) State, in context, a condition needed for the number of customers who arrived at the post office during a randomly chosen period of 30 minutes to be well modelled by a Poisson distribution.

(ii) Find the probability that no more than 4 customers arrive between 11.00 a.m. and 11.30 a.m.

(iii) The period from 11.00 a.m. to 11.30 a.m. on a Tuesday morning is divided into 6 periods of 5 minutes each. Find the probability that no customers arrive in at most one of these periods.

The post office opens for 3.5 hours each in the morning and afternoon and it is noted that on a typical weekday afternoon, customers arrive at the post office independently and at a rate of 1 per 10 minute period. Arrivals of customers take place independently at random times.

(iv) Show that the probability that the number of customers who arrived in the afternoon is within one standard deviation from the mean is 0.675, correct to 3 decimal places.

(v) Find the probability that more than 38 customers arrived in a morning given that a total of 40 customers arrived in a day.

(vi) Using a suitable approximation, estimate the probability that more than 100 customers arrive at the post office in a day.

### 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)

(ii)

Question 2

(rejected since ) or

Question 3

(i)

(ii)

Using GC, required answer (3SF)

(iii)

When

(3SF)

(iv)

Question 4

(i)

Let

When

When

Coordinates

(ii)

When . So is a minimum point.

When . So is a maximum point.

(iii)

x-intercept

(iv)

Using GC,

Question 5

(i)

Area of ABEDFCA

(ii)

Perimeter

When (rejected since )

When

Question 6

(i)

The store manager has to survey male students and 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,

Unbiased estimate of population variance,

Question 7

(i)

[Venn diagram to be inserted]

(ii)

(a)

(b)

(iii)

Question 8

(i)

Required Probability

(ii)

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

Required Probability

(iii)

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.

(a)

Required Probability

(b)

Required Probability

(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.

Required Probability

(iii)

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

Since n is large,

approximately

Required Probability

by continuity correction

Question 10

(i)

Let X be the top of speed of cheetahs.

Let be the population mean top speed of cheetahs.

Under

Test Statistic,

Using GC, is rejected.

…

(ii)

For to be not rejected,

(round down to satisfy the inequality)

Question 11

(i)

[Sketch to be inserted]

(ii)

Using GC, (3SF)

(iii)

Using GC,

(3SF)

(iv)

When (3SF)

Time taken minutes

Estimate is reliable since is within the given data range and is close to 1.

(v)

Using GC, (3SF)

(vi)

The answers in (ii) is more likely to represent since 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.

(i)

(ii)

(iii)

Let and

### 2016 A-level H2 Mathematics (9740) Paper 2 Suggested Solutions

All solutions here are SUGGESTED. KS 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:

Question 7:

Question 8:

Question 9:

Question 10:

### Relevant materials

### KS Comments

### June Revision Exercise

You can find the solutions of all ten sets of the June Revision Exercise we did in class.

Have fun!

June Revision Exercise 1

June Revision Exercise 2

June Revision Exercise 3

June Revision Exercise 4

June Revision Exercise 5

June Revision Exercise 6

June Revision Exercise 7

June Revision Exercise 8

June Revision Exercise 9

June Revision Exercise 10

### Statistics related articles

Here is a compilation of all the Statistics articles KS has done. Students should read them when they are free to improve their mathematics skills. They will come in handy! 🙂

1. What does unbiased in Math mean?

2. What is r-value?

3. Why is the r-value independent of translation and scaling? #1

4. Why is the r-value independent of translation and scaling? #2

5. Why can’t we extrapolate regression lines?

6. What is a regression line?

7. Simulation for Hypothesis Testing

8. Type I vs Type II Errors

9. A little about Central Limit Theorem

10. What is a good estimate?

11. Using symmetrical properties of Normal Curve to solve questions

12. Bayesian vs Frequentist Statistics

### JC2 Statistics 2016

Most JCs by now should be embarking on their Statistics Journey. I know a school that is finishing up really fast, and I think they will easily finish the syllabus by May. So I notice many students are thrown off the track with Statistics. It is slightly out of their comfort zone, especially Permutation and Combinations. The topic itself is really sneaky. Here’s are some advices for Statistics.

1. Read every question carefully. Words like, “at least”, “not more than”, “if” and “given”.

2. Many students seem to struggle with the idea of conditional probability and that should not the case. Students should learn what it means and how it works, instead of it being simply . Understand how this came about will really save your 6-9 marks probability question.

3. Learn to break the question down to something you’re familiar with. As what we review in class, write probabilities in a format that you understand easily. Probability is often counter-intuitive, stick to the formulas.

4. Yes, the whole statistics is one big topic by itself. What you learnt in Permutation & Combination is used in probability (e.g. A’levels 2013 P2 Q11), and probability is used in all the distributions you will be learning, lastly normal distributions is used in hypothesis testing too.

5. Statistics is rather qualitative, so students should show understanding in what they learn. Topics such as sampling requires you to think rationally and consider what limitations we face. A clear understanding of your contents and definitions will go a long way.

### 2010 A-level H2 Mathematics (9740) Paper 2 Question 7 Suggested Solutions

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

(i)

(ii)

(iii)

P(B’|A)

(iv)

since A and C are independent

(v)