Thinking [email protected] #3

JC Mathematics, Mathematics

[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

JC Mathematics, Mathematics

[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

JC Mathematics, Mathematics

[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.

Meritocracy? Junior colleges merger and its implications

Meritocracy? Junior colleges merger and its implications

JC General Paper, JC Mathematics, JC Physics

It’s pretty interesting that shortly after our post on meritocracy, we have news about the junior college (JC) mergers. For more information about the news, you could take a look at this weblink.

This drastic move by MOE has sparked a lot of concerns among the public and has brought up a few issues for us to consider. First, it would be the falling demographics of Singapore. The falling birth rates is cited as the main reason for the merger of schools, so that resources would not be wasted, and there would not be under-utilized staff in the system. With such falling birth rates, what would you think is going to happen to the future of the educational landscape (would teaching/tutoring as a profession still be lucrative? We know that MOE has cut back on the hiring of teachers from 3000 at its peak yearly to about 1000 right now).

Second question to think about would be the larger implications of these schools merger. Why are these schools selected? Some have argued that it is a strategic move by the government to level the playing field by merging these colleges so that academic standards would be streamlined? of course, we cannot merge schools like RI and HCI together as it would only further consolidate their super-elite status in society (besides strong school culture and powerful alumni).

Finally, school culture and history is being destroyed when merger takes place. If that’s the case, what does it say about how the nation values history? It is all about the future and progress right, the past no longer matters if it is holding us back. Pragmatism is the view of the Singapore’s state.

Vectors Question #4

Vectors Question #4

JC Mathematics

Another interesting vectors question.

The fixed point A has position vector a relative to a fixed point O. A variable point P has position vector r relative to O. Find the locus of P if r \bullet (ra) = 0.

Vectors Question #3

Vectors Question #3

JC Mathematics

This is a question a student sent me a few days back, and I shared with my class.

Find the Cartesian equation of the locus of all points (plane) that is equidistant of the xy plane and xz plane.

The following should aid students to visualise.

xy-, xz-, yz-planes

Sidenote: I think Vectors is a very important topic for 9758 as its applications are wide. Students should do their best to understand the topic. I will share a few more applied questions next week when I have time.

A little reminder to students doing Calculus now

A little reminder to students doing Calculus now

JC Mathematics

When \frac{dy}{dx} = 0, it implies we have a stationary point.

To determine the nature of the stationary point, we can do either the first derivative test or the second derivative.

The first derivative test:

First Derivative Test

Students should write the actual values of \alpha^-, \alpha, \alpha^+ and \frac{dy}{dx} in the table.

We use this under these two situations:
1. \frac{d^2y}{dx^2} is difficult to solve for, that is, \frac{dy}{dx} is tough to be differentiated
2. \frac{d^2y}{dx^2} = 0

The second derivative test:

Second Derivative Test

Other things students should take note is concavity and drawing of the derivative graph.