Binomial Theorem came out as a 9 marks question in 2009 GCE ‘O’ Level Additional Mathematics Paper (Subject Code: 4038) so you know as well as I do the importance of Binomial.

Read about other useful posts on Binomial Theorems:

• Binomial Expansion Teaches how to choose the RIGHT partner (:

• A-Math Binomial Expansion: Finding Term Independent of x By A Shortcut Method

• Exam Question : Usage of Binomial Formula

I’m looking at the question now. It is testing on the usage of the Binomial formula, including the ‘n choose r’ formula. Many students call this sign: ‘!’ ‘exclamation mark’ which is known correctly as factorial.

I will be using the following question to illustrate how to simplify the ‘n choose r’ formula without memorizing. (I understand some schools want students to memorize)

Let’s begin by understanding what’s ‘n choose r’ all about:

Click on image for larger view

Do you know how to simplify ‘n choose 3′?

Here’s the question which requires us to apply what we have discussed. I would suggest you attempt it on your own before clicking here for the solution.

binomial-question

binomial · factorial sign · o level 2009

Proportion is a topic taught in Secondary 1 and 2. In fact, we have learnt about direct proportion much younger.

DIRECT PROPORTION

A real simple example of Direct Proportion would be the more money I have, the more things I can buy. When amount of money increases, the number things I can buy increase too. (Notice the increase in both things)

Another example, the less I eat, the thinner I become, so as the amount of food eaten decreases, my weight decreases too.

INVERSE PROPORTION

An example of inverse proportion most of you can relate to would be: the more time I spent on Facebook (PSP, WII, Internet), the less time I have on my books!

Allow me to add in another example of Inverse Proportion, the more I spent, the less I have in my bank.

These are some examples (simple) to understand the true meaning of Direct or Inverse proportion.

In the next post, I will be sharing with you how we can translate a statement into an equation involving proportion. I’m also going to highlight the ‘tricky’ proportion question in 2008 GCE O Level Elementary Mathematics Paper 1.

direct proportion · inverse proportion

I was looking through 2008 GCE O Level Additional Mathematics Exam Papers (Subject Code: 4038) and as expected, there was a Kinematics question (worth 6 marks) in Paper 1.

Kinematics is a application topic for Differentiation and Integration. To master this topic, you do not necessarily need to bring in your physics knowledge though it could be useful at times.

Instead, how I get my students to be a master in this topic is to be familiarize with a KINEMATICS VOCABULARY LIST.

Here’s some of the vocabulary words that are useful and common:

• Momentarily at rest, instantaneously at rest, changes direction of motion, stationary
• Initial displacement, initial velocity, initial acceleration
• Greatest displacement, greatest velocity, greatest acceleration
• Distance travelled in the 4th second VS Distance travelled in the first 4 seconds
• Maximum distance from Point O
• Particle returns to Point O
• Constant Velocity

I would say for Kinematics, it is one of the few topics in A-Math which uses extensive vocabulary. This is also the reason for you to decipher the meaning behind these words.

So do you know the meaning behind these words? I would love to hear about it in the comments section.

I have also taken a few questions from my A-Math TREQ book(Topical Real Exam Questions) to illustrate some common exam questions on Kinematics, further highlighting the importance of knowing your Kinematics well. (Click on the image for bigger view)

Click on image for a larger view

I would be sharing the step by step solutions for Question 8 in the next post. Subscribe to my blog to be updated again!

differentiation · Exam Questions · integration · kinematics · Topical Real Exam Questions · vocabulary list

Last Sunday while coaching my A-Math students on a question on Modulus Functions, we did solving of Modulus equation which is of no big problem as long as you get the basic concept correct.

|x| = x when x >= (more than and equal to) 0 or |x| = -x when x < 0

When we came to the next part of the question which involves Sketching of Modulus Graph, that’s where the interesting happens.

Read about the Differences between Drawing and Sketching in this post.

When question involves sketching of graph, we usually do not need
- a table of values.
- axis which are evenly marked out.

Sketching of Graph should however includes
- critical points (i.e x – intercept(s), y-intercept, turning point (if you are sketching a quadratic graph))

Let’s take a look at the working of 2 different students:

Student A:

• Sketch the modulus graph using table of values
• Join up the points in a straight line manner

Student B:

• Sketch the modulus graph using a series of 2 other graphs

Note the difference in the shape of the 2 graphs.

I certainly hope that my student A is convinced that using a table of values is not recommended for drawing modulus graphs. Moreover, many questions involving modulus could be that of Trigonometry graphs! So be like student B, draw modulus graph using a transformation of a series of graphs

modulus function · modulus graph · straight line graphs

In O level Additional Mathematics, there is a small section on Identity inside the topic of Factor & Remainder Theorem. Today I am going to share with you the 2 different approaches to solve this kind of questions.

• Substitution Method (My preferred method)

I am going to use the question below to show you the step by step solutions of both methods.

Given that for all values of x, find the value of A, of B and of C.

Let x = 1,

Let x = -2,

Let x = 0,

Thus A = 3, B = -2 and C = 2

Concept behind the Subsitution method: The value of x choosen will cause one or more of the unknowns to be “cancel off”, leaving just 1 unknown left. For example, when I choose x = 1 in the first subsituition, A & B are eliminated, allowing me to find ‘C’.

• Comparing Coefficients Method

Given that for all values of x, find the value of A, of B and of C.

By comparing coefficient of :

LHS: 3 = A => A = 3

By comparing coefficient of :

LHS: 1 = 2A – A + B => B = -2

By comparing coefficient of :

LHS: -2 = -2A -  B + C => C = 2

Thus A = 3, B = -2 and C = 2

Concept behind the Comparing Coefficient method: Expansion is usually required on one side of the equation. It takes up time. The reason for the insignificant working shown is due to the fact that the expansion is done mentally instead of written. This method is highly recommended if there is more than 1 unknown other than x on the left hand side of the equation. For example, there’s an unknown ‘D’ on the left hand side of the equation.

Which method do you usually use? And which method does your school teach you? Leave me your answer in the comment section below.

comparing coefficients · factor remainder theorem · identity · subsitution method