### Random Questions from 2017 Prelims #1

Last year, I shared a handful of random interesting questions from the 2016 Prelims. Students feedback that they were quite helpful and gave them good exposure. I thought I share some that I’ve seen this year. I know, its a bit early for Prelims. But ACJC just had their paper 1. 🙂

This is from ACJC 2017 Prelims Paper 1 Question 7. And it is on complex numbers.

7
(a) Given that $2z + 1 = |w|$ and $2w-z = 4+8i$, solve for $w$ and $z$.

(b) Find the exact values of $x$ and $y$, where $x, y \in \mathbb{R}$ such that $2e^{-(\frac{3+x+iy}{i})} = 1 -i$

I’ll put the solutions up if I’m free.

But for students stuck, consider checking this link here for (a) and this link here for (b). These links hopefully enlightens students.

Just FYI, you cannot $\text{ln}$ complex numbers as they are not real…

### 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.

As the prelims examinations draw really close, many students were asking me to give questions to test their concepts for several topics. In class, I had the opportunity to explore several applications questions too. We saw several physics concepts mixed. We also have some conceptual questions that need students to be able to use the entire topic to solve it.

So I’ll share one here. This involves several concepts put together. I’ll put the solution up once I find the time. Concepts that will be involved, will be

1. Vector Product
2. Equations of Plane
3. Finding foot of perpendicular of point

The question in one a reflection of a plane in another plane. I think such questions will come out in a few guided steps in exams. But should a student be able to solve it independently, it shows that he has good understanding.

The plane $p$ has equation $x + y + z = 9$ and the plane $p_1$ contains the lines passing through $(0, 2, 3)$ and are parallel to $(1, -1, 0)$ and $(0, 1, 1)$ respectively. Find, in scalar product form, the equation of the plane which is the reflection of $p_1$ in $p$.

### Thinking [email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */ #9

[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 standard summation question. I’m interested in the last part only.

The answer to (ii) is written there by the student. I’ll only do the solution to (iv).

### 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 $Y = g(X), then For a discrete r.v. X,$latex \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. ### Thinking [email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */ #8 [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 interesting Complex Number Question. The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^2$ ### Thinking [email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */ #7 [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 an application question for hypothesis testing from the 9758 H2 Mathematics Specimen Paper 2 Question 10. The average time required for the manufacture of a certain type of electronic control panel is 17 hours. An alternative manufacturing process is trialled, and the time taken, $t$ hours, for the manufacture of each of 50 randomly chosen panels using the alternative process, in hours, is recorded. The results are summarized as follows $n = 50$ $\sum t = 835.7$ $\sum t^2 = 14067.17$ The Production Manager wishes to test whether the average time taken for the manufacture of a control panel is different using the alternative process, by carrying out a hypothesis test. (i) Explain whether the Production Manager should use a 1-tail or a 2-tail test. (ii) Explain why the Production Manager is able to carry out a hypothesis test without knowing anything about the distribution of the times taken to manufacture the control panels. (iii) Find unbiased estimates of the population mean and variance, and carry out the test at the 10% level of significance for the Production Manager. (iv) Suggest a reason why the Production Manager might be prepared to use an alternative process that takes a longer average time than the original process. The Finance Manager wishes to test whether the average time taken for the manufacture of a control panel is shorter using the alternative process. The Finance Manger finds that the average time taken for the manufacture of each of the 40 randomly chosen control panels, using the alternative process, is 16.7 hours. He carries out a hypothesis test at 10% level of significance. (v) Explain, with justification, how the population variance of the times will affect the conclusion made by the Finance Manager. ### Thinking [email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */ #6 [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 very interesting vectors question from a recent JC BT Shown in the diagram is a methane molecule consisting of a carbon atom, G, with four hydrogen atoms, A, B, C, and D, symmetrically placed around it in three dimensions, such that the four hydrogen atoms form the vertices of a regular tetrahedron. Treat A, B, C, D, and G as points. The coordinates of A, B, C, and D are given by (5, -2, 5), (5, 4, -1), (-1, -2, -1) and (-1, 4, 5) respectively, By considering the line DG and the symmetrical properties of methane, find the bond angle of methane, that is, $\angle DGA$. ### 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]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */ #4

[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 1993 Paper 1.

The positive integers, starting at 1, are grouped into sets containing $1, 2, 4, 8, \ldots$ integers, as indicated below, so that the number of integers in each set after the first is twice the number of integers in the previous set.

$\{ 1 \}, \{ 2, 3 \}, \{ 4, 5, 6, 7 \}, \{ 8, 9, 10, 11, 12, 13, 14, 15 \}, \ldots$

(i) Write down the expressions, in terms of $r$ for

(a) the number of integers in the $r^{th}$ set,

(b) the first integer in the $r^{th}$ set,

(c) the last integer in the $r^{th}$ set.

(ii) Given that the integer $1,000,000$ occurs in the $r^{th}$ set, find the integer value of $r$.

(iii) The sum of all the integers in the $20^{th}$ set is denoted by $S$, and the sum of all the integers in all of the first $20$ sets is denoted by $T$. Show that $S$ may be expressed as $2^{18}(3 \times 2^{19} - 1)$.

Hence, evaluate $\frac{T}{S}$, correct to 4 decimal places.