### 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…

### Trigonometry Question #1

I saw this trigonometry question a few days ago.

$\mathrm{tan}1^{\circ} \mathrm{tan}2^{\circ} \mathrm{tan}3^{\circ} \ldots \mathrm{tan}88^{\circ} \mathrm{tan}89^{\circ} = ?$

Hint: $\mathrm{tan}(90^{\circ} - \theta) = \mathrm{cot} \theta$

### Modulus Question #1

The following is a question that involves modulus, I think something similar came out in ACJC Prelim Paper to have students consider the range of x before attempting the inequality.

For this question, students need to consider the ranges of x and y and attempt this question.

$|x|+x+y=8$
$x+|x|-y=14$
$x+y=?$

Ans: 2

### Definite Integral Question #1

This is question that was tested in ACJC H2 Math Prelim P1 2015. A few of my students know how to answer it but were uncertain how to express it.

Most students were concerned with (ii) of the question.

I think the easiest way to prove this, is to first avoid writing too much and attempt to show it mathematically.

$\int_{\pi /3}^{2\pi /3} |cos\frac{x}{2} cosx| dx$
$= \int_{\pi /3}^{pi /2} cos\frac{x}{2} cosx dx - \int_{\pi /2}^{2\pi /3} cos\frac{x}{2} cosx dx$

while

$|\int_{\pi /3}^{2\pi /3} cos\frac{x}{2} cosx dx|$
$= |\int_{\pi /3}^{pi /2} cos\frac{x}{2} cosx dx - \int_{\pi /2}^{2\pi /3} cos\frac{x}{2} cosx dx|$

Thus $|\int_{\pi /3}^{2\pi /3} cos\frac{x}{2} cosx dx|$ will be smaller in magnitude.

Students can also attempt to justify using the area under graph but they must express the answers in words carefully.

### Summation Question #3

This is an interesting and simple question.

$\sum_{n=0}^{\infty} \frac{1}{n!} = ?$

Putting the answer here will take the fun out of figuring this summation out. I will advice/ hint that you look at your standard series in MF15 carefully.

This question has come out with Poisson Distribution too. So students should take note of this particular expansion.

### Student’s Question #4

This is a very theoretical question with regards to hypothesis testing. My student asked what is meant by 1% level of significance. So I know that many would have memorised the standard format of answers, which is important. We know that level of significant refers to the P(rejecting H0 | H0 is true) which means is a false positive case here. And p-value is the smallestt level of significane.

### Student’s Question #2

This question is with regards to parametric equations. We are given $x = t^{2} -3, ~ y=t(t^{2}-3)$ and my student have difficulties finding the equation of the line of symmetry.

So let us first look at how parametric equations actually work here, since most students are really confused with it. Parametric equations are a set of equations that is “linked” or related using a parameter which is a variable. In this case, both $x~ \& ~y$ are connected using a new parameter $t$ here. And as $t~ \mathrm{varies~,~} x~\mathrm{varies,~and~so~does~}y$.

Considering that we found the cartesian equation in (i), $y=\pm x \sqrt{x+3}$, observe that if $f(x) = x \sqrt{x+3}$ then $y = \pm f(x)$, so for every x-value, we can map to two y-value, the positive and negative one. Thus the line of symmetry will be the x-axis here.

### Student’s Question #1

This is question from an AJC prelim exams, I’m not really sure which year. The question prompts students if there will be a change in the r-value should, the scientists change the unit of measurement from Celsius to Fahrenheit.

My student was uncertain if he had phrased his answers well. His answer is as follows, “the change in each value will still be the same so only the magnitude of the values themselves are affected and not their linear relations. Thus r will be the same.”

I feel that he has gotten the correct idea for this question. However, his explanation can be really confusing and misleading to examiner. A better way will just to showcase to examiner that you know that r is independent of scaling and translation.

Intuitively, any form of scaling and translation will simply move my data points, PROPORTIONATELY. So the resultant correlation observed when we measure in celsius and fahrenheit will be same. Students can refer to this webpage for a technical proof.

Now, my question back will be: Given two variables, annual wage and expenditure, will the r-value change if we consider hourly wage and expenditure, instead of using annual wage.

### Arithmetic Question #3

The average of 999 numbers is 999. From these numbers, I chose 729 of them and their average is 729. Find the average of the remaining numbers?

Hint: Try to use algebra.

Ans = 1728

### Problem Solving Question #1

These series of questions target on students’ abilities to solve it. They are common questions used in job interviews and interviewers are more interested in learning how the interviewee derive the answer, instead of the answer itself. So do feel free to share your method of solving the problem. Remember, we are interested in how you solve it. 🙂

A CUBE, which has sides 10m, is built by smaller cubes which has sides 1m. In other words, the CUBE is made up of 1000 smaller cubes. If I were to remove the external layer of 1m from each face of the CUBE, how many smaller cubes would I have removed?

Ans: 488