Author: gjooheng

H2 maths tutor

Challenging Permutation question 5

A tennis club has n male players and n female players. For a tournament the players are to be arranged in n pairs, with each pair consisting of one male and one female. Find the number of possible pairings.

Answer: n!

Solution

Suppose the male got to choose his partner. The first male can choose from n females. After he has chosen, the next male can choose from (n-1) females. So the number of possible pairings is n.(n-1).(n-2)…1 = n!

For instance there are 2 male players (M1, M2) and 2 female (F1, F2) players.

One way to arrange 2 pairs of players: M1 F1 and M2 F2

Second way to arrange 2 pairs of players: M1 F2 and M2 F1

Number of ways to arrange 2 pairs of players is 2!

Comments

Some students will give the wrong answer n x n, which is the number of ways to choose a pair.

How to revise during the school holidays?

Go through the exam paper and prioritise the topics according to the number of marks lost. Start by revising the topic with the most number of marks lost. Do correction without copying solution and do 1 or 2 more similar questions until can do correctly within time limit of 1.5 min/mark. If score 0 in that topic, practice 10 questions in that topic. After topical revision, do a practice paper under exam condition. Repeat the cycle of paper topical until can achieve your desired grade.

Focus on brushing up Calculus, Sequence and Series, Vectors and Complex Numbers. These four major modules account for more than 60% of marks in A level.

Can engage a good tutor to shorten the learning cycle.

Doing tutorials is not enough!

For the average JC H2 maths student, doing tutorials alone is not enough to pass mid year or promo exams. This is because many JCs set the exams at a much higher standard than tutorials. In some JCs, about half of the cohort can fail JC 1 promo.

To do well, other than doing tutorials, students need to practise intermediate or advanced questions. If they are unable to do intermediate questions, they should seek help early from a good tutor.

If students fail or barely pass their mid year or promo, don’t wait until JC 2 to brush up. Otherwise, they have to struggle to catch up and learn new challenging topics in year 2.

Challenging Vector Problem 2

vector

Answer: 7

Solution

vector a1

vector diag

Challenging Binomial expansion problem

binomial

Reverse graph transformation

This question was posted by a student.

” What is the series of transformation from f(3-x/2) to f(x)?”

Here are 3 different methods to solve it.

Method 1

Let f(3-x/2) = g(x)

Let u=3-x/2

x= -2u+6

f(u)=g(-2u+6)

f(x)=g(-2x+6)

Therefore, the sequence of transformation is

Translate 6 units in the negative X direction.

Scale by a factor of 1/2 parallel to the X-axis.

Reflect about the Y-axis

Method 2a

Apply f(-2x) to f(3-0.5x).

f(-2x) = f[3-0.5(-2x)] = f(x+3)

ie. Scale parallel to the X axis by a factor of 0.5. Then reflect about the Y axis

Apply f(x-3) to f(x+3) to get f(x)

ie. Translate 3 units in the positive X direction

Method 2b

Apply f(x+6) to f(3-0.5x) to get f(-0.5x)

ie. Translate 6 units in the negative X direction.

Apply f(-2x) to f(-0.5x) = f(x)

ie. Scale parallel to the X axis by a factor of 0.5. Then reflect about the Y axis.

Method 3

f(3-x/2) is transforming f(x) by

Ａ：Translate 3 units in the negative X direction

B: Scale by a factor of 2 parallel to the X axis

C: Reflect about Y axis.

So to get back f(x), we reverse the transformation:

C’: Reflect about Y axis

B’: Scale by a factor of 1/2 parallel to the X-axis.

A’: Translate 3 units in the positive X direction

Comments

Method 1 and 2 are the forward approach. Method 3 is the reverse approach.

Method 1 is the preferred approach as it is easier and faster.

Preparing for exams

1. For students who have enough time, start by revising and practising questions by topic. Do basic level questions for all topics. Then do intermediate questions (slightly harder than alevel standard). If you are running out of time, go straight to doing yearly paper. Doing yearly paper and revise weak topics is the fastest way to improve.

2. For students preparing for their school internal exams, practise past year exam paper from their school, starting from the most recent. It is important to practice past year paper from your school as each school has its style of questions. For the first paper, you can refer to your notes and formulas. For any question you get incorrect, practise at least 2 more similar questions in that topic.

For subsequent paper, attempt only with the formula list MF 26 under timing. Keep practising until you can score 10% above your desired grade.

3. For students taking A level , do past 5 years alevel paper and prelim papers from YIJC, JPJC, TMJC, VJC, EJC.

Do paper 1 and paper 2 in 3 hours each, with only MF 26. Then revise weak topics (questions that you got wrong in the paper). Repeat the cycle

 

Definite Integrals involving Modulus

Definite integrals involving modulus looks complicated but can be easily evaluated using a standard approach. See the following examples.

definite integral 1

definite integral 2