Weekly Question

A-Math: This Plane Geometry Question Almost Drives Me To Insanity!

Plane geometry is one of the top killer questions in A-Math as there are many things to prove in addition to the many lines, circles, triangles, angles ...

Read about the post I wrote earlier What you need to know to do well in Plane Geometry Part 1

I must admit this question almost drives me to insanity ;-) but luckily I saw the light when I was on the verge of sane and insanity ...

Question: Prove that a circle can be drawn passing through the points T,P,X and Q. [3 marks]

planegeometry-circle

Let me know how you handle this question in the comments section. For those who wants the step by step solution, drop me a comment as well. I highly recommend that you think about this question first before asking for solution.

There are 2 strategies I have used to help me in this Plane Geometry question. Do you want to know? so that you can learn and use them in your own question.

Hint: You must use one of the properties of circles.

Update: Video solution Here

O Level Additional Math - How To Prove Trigonometry Identity

Dear Ms Ong,

i had just taken my class test on simple trigonometric identities and well as further trignonometric identities yesterday.

sadly to say, i left all the proving of trigonometric identities blank which is worth a total of 9 marks.

Im starting to get panic about trigonometric identities for my coming common test,end of year exams as well as the big O Level.

so im actually here to ask you if you may guide me on how to do them in a faster way, because i realised ive no time to tackle those proving questions.

Best Wishes,
Priscilla Fong

I received this email among the many others in my mailbox. I think this is a common problem face in proving Trigo Identity question. I shall provide a few strategies to handle this sort of question.

Usually start with the left hand side or whichever side which is more complicated.
When given trigo functions in the form of sec, cosec or cot x , it is advisable to change it to basic trigo functions like sin, cos, tan
When question involves power (especially square power), formulas should be considered.
Have the END in mind before you begin i.e if the end result involves only sin, your working should eventually eliminate the other trigo functions and stick on to sin.
It should take less than 8 steps for a 3-4 marks proving question.
[pmath]a^2-b^2=(a+b)(a-b)[/pmath] is sometimes used in proving Trigo Identity. (Algebra rule)

I am going to use an example to illustrate how to use the above strategies

Q: Prove the identity [pmath](sec A - tan A)^2 = {1- sin A}/{1+ sin A}[/pmath]

[pmath](sec A - tan A)^2 =( {1}/{ cos A}-{sin A}/{cos A})^2[/pmath] (Change sec and tan into sin & cos)

= [pmath]( {1 - sin A}/{cos A})^2[/pmath]

= [pmath]{(1-sin A)^2}/{(cos A)^2} [/pmath]

= [pmath] {(1 - sin A)(1 - sin A)}/{(1 - sin^2 A)} [/pmath] (applying formula)

= [pmath] {(1 - sin A)(1 - sin A)}/{(1 - sin A)(1 + sin A)} [/pmath] (applying algebraic rule)

= [pmath]{1 - sin A}/{1 + sin A} [/pmath] (Proven)

Total number of steps = 6

Additional resource: You may want to read on A-Math Trigo Graphs here.

Integration Mixed With Differentiation

In Integration, unlike Differentiation, there isn't any product rule nor quotient rule. Having said this, examiners always like to present question in that form of either product or quotient. Students who aren't able to see through their plot to confuse your mind will fall straight into their trap.
So LOOK OUT!

The question most students will ask next will be what to do when the Integration is presented in the manner as if we can use product or quotient rule aka the given question is in the form of a fraction.

These are the few ways you can work around it

Simplifying either by breaking up the numerator
Simplification by applying Indices Law (very useful when you are working with exponential or bases-powers)
Apply partial fraction concept

Look at this question:
$\int (2x+1)\sqrt{8x+4} dx$

It is expressed as two product. How do we integrate?

Discuss your approach in the comment box below.

I will reveal the step by step approach by end of this week. Be sure to subscribe to www.singaporeolevelmaths.com/feed so that you will be notified of latest updates

Cheers!

Does Stubborness Earn You More Marks?

I was doing Factor, Remainder Theorem with one of my students. She's a brilliant girl but has yet to achieve the results she truly deserves. I think she knows the reasons clearly :-)

She likes to do things her way.

And I am totally fine with students having their own style. Afterall, my best method might not be their best method. However, if their style is not helping them in terms of scoring, time management..., this is where I step in to influence and get them to better understand how my strategy (usually a much shorter way as I am a lazy person :-)) might help them to understand and get the answer faster and easily.

I get them to buy my idea. And most of the time, my idea sells out :)

Today post will be on Identity covered with Factor-Remainder Theorem.

There are 2 main methods to solve the question. And for me, I choose the "lazy" method again. Simply because it is easy, fast and accurate!

Who else wants to be like me? "Lazy" strategy which is easy, fast and accurate!

Quadratic Equation - Completing The Square Method

I have talked about the 3 main representations of a quadratic equation and each of its significance previously.

Today, I am using a real examination question to illustrate on The Completed Square Form and its significance.

Linear Law,Just 1 Strategy Makes A Grade Difference

What is Linear Law?

It is a tool which will allow you to transform non-straight line equation to straight line equation so that you can plot a straight line. Most of the time, the axis will consist of x and/or y

One of the techniques involving taking lg (log base 10) on both sides of the given equation.

Common Mistake! STOP
$y=Ax^b=>lgy=blgAx=>blgA+blgx$

This is one of the common mistakes I have highlighted in Top 7 Commonly Made Mistakes in Logarithm

How To Counter This Mistake

Very simple. Just add in a bracket on the right hand side of the equation.
$y=(Ax^b)<br> =>lgy=lg(Ax^b)<br> => lgy=lgA+lgx^b<br> => lgy=lgA+blgx<br>$

Now, your y-axis will be lg y and x-axis will be lg x. Gradient = b and y-intercept = lg A