### H2 Mathematics (9740) 2016 Prelim Papers

So many students have been asking for more practice. I’ll put up all the Prelim Papers for 2016 here. Do note that the syllabus is 9740 so students should practice discretion and skip questions that are out of syllabus. 🙂

Here are the Prelim Paper 2016. Have fun!

Here is the MF26.

### Quick Summary (Probability)

University is starting for some students who took A’levels in 2016. And, one of my ex-students told me to share/ summarise the things to know for probability at University level. Hopefully this helps. H2 Further Mathematics Students will find some of these helpful.

Random Variables

Suppose $X$ is a random variable which can takes values $x \in \chi$.

$X$ is a discrete r.v. is $\chi$ is countable.
$\Rightarrow p(x)$ is the probability of a value of $x$ and is called the probability mass function.

$X$ is a continuous r.v. is $\chi$ is uncountable.
$\Rightarrow f(x)$ is the probability density function and can be thought of as the probability of a value $x$.

Probability Mass Function

For a discrete r.v. the probability mass function (PMF) is

$p(a) = P(X=a)$, where $a \in \mathbb{R}$.

Probability Density Function

If $B = (a, b)$

$P(X \in B) = P(a \le X \le b) = \int_a^b f(x) ~dx$.

And strictly speaking,

$P(X = a) = \int_a^a f(x) ~dx = 0$.

Intuitively,

$f(a) = P(X = a)$.

Properties of Distributions

For discrete r.v.
$p(x) \ge 0 ~ \forall x \in \chi$.
$\sum_{x \in \chi} p(x) = 1$.

For continuous r.v.
$f(x) \ge 0 ~ \forall x \in \chi$.
$\int_{x \in \chi} f(x) ~dx = 1$.

Cumulative Distribution Function

For discrete r.v., the Cumulative Distribution Function (CDF) is
$F(a) = P(X \le a) = \sum_{x \le a} p(x)$.

For continuous r.v., the CDF is
$F(a) = P(X \le a ) = \int_{- \infty}^a f(x) ~dx$.

Expected Value

For a discrete r.v. X, the expected value is
$\mathbb{E} (X) = \sum_{x \in \chi} x p(x)$.

For a continuous r.v. X, the expected value is
$\mathbb{E} (X) = \int_{x \in \chi} x f(x) ~dx$.

If $Y = g(X)$, then

For a discrete r.v. X,
$\mathbb{E} (Y) = \mathbb{E} [g(X)] = \sum_{x \in \chi} g(x) p(x)$.

For a continuous r.v. X,
$\mathbb{E} (Y) = \mathbb{E} [g(X)] = \int_{x \in \chi} g(x) f(x) ~dx$.

Properties of Expectation

For random variables $X$ and $Y$ and constants $a, b, \in \mathbb{R}$, the expected value has the following properties (applicable to both discrete and continuous r.v.s)

$\mathbb{E}(aX + b) = a \mathbb{E}(X) + b$

$\mathbb{E}(X + Y) = \mathbb{E}(X) + \mathbb{E}(Y)$

Realisations of $X$, denoted by $x$, may be larger or smaller than $\mathbb{E}(X)$,

If you observed many realisations of $X$, $\mathbb{E}(X)$ is roughly an average of the values you would observe.

$\mathbb{E} (aX + b)$
$= \int_{- \infty}^{\infty} (ax+b)f(x) ~dx$
$= \int_{- \infty}^{\infty} axf(x) ~dx + \int_{- \infty}^{\infty} bf(x) ~dx$
$= a \int_{- \infty}^{\infty} xf(x) ~dx + b \int_{- \infty}^{\infty} f(x) ~dx$
$= a \mathbb{E} (X) + b$

Variance

Generally speaking, variance is defined as

$Var(X) = \mathbb{E}[(X- \mathbb{E}(X)^2] = \mathbb{E}[X^2] - \mathbb{E}[X]^2$

If $X$ is discrete:

$Var(X) = \sum_{x \in \chi} ( x - \mathbb{E}[X])^2 p(x)$

If $X$ is continuous:

$Var(X) = \int_{x \in \chi} ( x - \mathbb{E}[X])^2 f(x) ~dx$

Using the properties of expectations, we can show $Var(X) = \mathbb{E}(X^2) - \mathbb{E}(X)^2$.

$Var(X)$
$= \mathbb{E} [(X - \mathbb{E}[X])^2]$
$= \mathbb{E} [(X^2 - 2X \mathbb{E}[X]) + \mathbb{E}[X]^2]$
$= \mathbb{E}[X^2] - 2\mathbb{E}[X]\mathbb{E}[X] + \mathbb{E}[X]^2$
$= \mathbb{E}[X^2] - \mathbb{E}[X]^2$

Standard Deviation

The standard deviation is defined as

$std(X) = \sqrt{Var(X)}$

Covariance

For two random variables $X$ and $Y$, the covariance is generally defined as

$Cov(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]$

Note that $Cov(X, X) = Var(X)$

$Cov(X, Y) = \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y]$

Properties of Variance

Given random variables $X$ and $Y$, and constants $a, b, c \in \mathbb{R}$,

$Var(aX \pm bY \pm b ) = a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X, Y)$

This proof for the above can be done using definitions of expectations and variance.

Properties of Covariance

Given random variables $W, X, Y$ and $Z$ and constants $a, b, \in \mathbb{R}$

$Cov(X, a) = 0$

$Cov(aX, bY) = ab Cov(X, Y)$

$Cov(W+X, Y+Z) = Cov(W, Y) + Cov(W, Z) + Cov(X, Y) + Cov(X, Z)$

Correlation

Correlation is defined as

$Corr(X, Y) = \dfrac{Cov(X, Y)}{Std(X) Std(Y)}$

It is clear the $-1 \le Corr(X, Y) \le 1$.

The properties of correlations of sums of random variables follow from those of covariance and standard deviations above.

### June Crash Course

The team at The Culture SG has been really busy and we have a lot of things prepared to help you guys work for that A. First up! Crash course for June…

And we know it is a bit late to be announcing this on the site now, but we have really been caught up with preparing our students lately that we don’t have the time to properly update here. So here are the details for the Math Crash Course and the Chemistry Crash Course.

P.S. For SCIENCE students who wish to chiong in October, please take note that the H2 Chem/ Phy/ Bio Paper 4 (practical) is in October. So better start soon! Here are the details!

Click to view

For 3 hr lessons, they are priced at $105. For 2 hr lessons, they are priced at$70.

Lessons will be held at:
Newton Apple Learning Hub
Blk 131, Jurong Gateway Road #03-263/265/267 Singapore 600131
Tel: +65 6567 3606

For math enquiries, you may contact Mr. Teng at +65 9815 6827.

For chem enquiries, you may contact Ms. Chan at +65 93494384.

For GP enquiries, you may contact Ms. Chen at +65 91899133.

### Thinking [email protected] #3

[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by KS. More of him can be found here.

This is a question from 1976 A’levels Paper 2. I thought it is pretty interesting to discuss the question with a little extension.

(a) In how many ways can 5 copies of a book be distributed among 10 people, if no-one gets more than one copy?

(b) In how many ways can 5 different books be distributed among 10 people if each person can get any number of books?

So now, let us modify it a bit.

(c) In how many ways can 5 copies of a book be distributed among 10 people if each person can get any number of books?

Notice that the difference between (b) and (c) is that the book distributed is not identical. So for (c), we are pretty much distributing $r$ identical balls to $n$ distinct boxes. Whereas for (b) , we are pretty much distributing $r$ distinct balls to $n$ distinct boxes.

### Thinking [email protected] #2

[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by KS. More of him can be found here.

(i) Find the two possible values of $z$ such that $z^2 = 1 + \sqrt{3}i$, leaving your answer in exact form $a + bi$, where $a$ and $b$ are real numbers.

(ii) Hence or otherwise, find the exact roots of the equation

$2w^2 + 2 \sqrt{6}w + 1 - 2 \sqrt{3} i = 0$

### Thinking [email protected] #1

[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by KS. More of him can be found here.

Each card in a deck of cards bear a single number from 1 to 5 such that there are $n$ cards bearing the number $n$, where $n = 1, 2, 3, 4, 5$. One card is randomly drawn from the deck. Let $X$ be the number on the card drawn.

(i) Find the probability distribution of $X$.

(ii) Show that $\mathbb{E}(X) = \frac{11}{3}$ and find $\text{Var}(X)$.

Andrew draws one card from the deck, notes the number and replaces it. The deck is shuffled and Beth also draws on card from the deck and notes the number. Andrew’s score is $k$ times the number on teh card he draws, while Beth’s score is the square of the number on the card she draws. Find the value of $k$ so that the game is a fair one.

### 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

### 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 $y=/frac{x^2+kx+1}{x-1}$, 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

Paper 1

Paper 2

### Relevant materials

MF26

Strictly speaking, the last 9740 H2 Mathematics Paper. Personally, I feel this paper was rather challenging and it provided a bit of perspective to what perhaps SEAB is looking for. In general, we are looking for students that are able to do and think at the same time.

2016 Paper had many unknowns or constants. This is to test students comprehensive abilities and ability to relate concepts into the questions. This paper threw a curve ball to many students. Students need to pay good attention to details and digest the informations given. This is one skill that is focused in the 9758 syllabus.

### 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.