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Geometrical Proofs

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Introduction


Intercept theorem
There are 2 ways to apply the intercept theorem

In the first scenario, you will be able to identify a pair of parallel lines from the question
Upon identifying the pair of parallel lines,
we conclude that the ratio of AD:DB is the same as the ratio of AE:EC

In the second scenario, you identify that that the ratio of AD:DB is the same as the ratio of AE:EC
Subsequently you conclude that the lines DE and BC are parallel


Midpoint Theorem
The Midpoint theorem is a specific case of the Intercept theorem when the ratio of the lengths is 1:1

There are 2 ways to apply the Midpoint Theorem

The first way is to use Midpoint Theorem to show that 2 lines are parallel
From the information given in the question, we will be able to identify that D is the midpoint of AB and E is the midpoint of AC
Given that D and E are midpoints, we will be able to conclude that DE is parallel to BC

The second way is to use Midpoint Theorem to show that E is a midpoint
From the information given in th…

Integration techniques Part III

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Integrating fractions
Revision of anti chain rule "Increase power by 1 and divide by the new power"







Recall the differentiation of ln (ax + b)


Equation of Circles

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Formula for Equation of Circle



Completing the Square

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Completing the square is a technique required to find equations of circle



Example 1


Coordinate Geometry Part II

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Example 1

A typical coordinate geometry question from the O Levels


Given that AC = 3AB, find the coordinates of point B.

Remarks:
A lot of students will try to use the distance formula to form simultaneous equations in x and y to find the coordinates of B. This is a very long and tedious method. A much simpler method is to use ratio and proportion to find the coordinates of B

To find the x coordinate of B


We can use the same method to find the y coordinate of point B


Therefore the coordinates of point B is (4,4)

Coordinate Geometry Part I

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Technique 1: Finding the gradient of a line


Parallel lines Parallel lines have the same gradient
Perpendicular lines
If a line has a gradient of 1/2 , the gradient of the line perpendicular to it is -2.
-2 is the negative reciprocal of 1/2. In other words, we "flip" the fraction 1/2 to get 2 and attach a negative sign

Technique 2: Finding equation of a straight line
The general format of a straight line is y = mx + c
where m is the gradient and,
c is the y intercept

Example 2
Find the equation of a line that contains the points (1,2) and (3,4).

From the example above we know that gradient of the line is 1.
Hence the equation of the line is: y = 1x + c

To find the y intercept, c, we sub in one of the points
I choose the point (1,2) and sub x =1, y = 2
2 = 1 + c
Hence c = 1

The equation of the line is y = x + 1

Let us combine the techniques we have learnt so far to solve a typical O Level question

Question 1

The figure above shows a trapezium where AB is parallel to CD. Both the lines AB and CD…

Roots of Quadratic Equation Part II

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Roots of a Quadratic Equation

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What do we mean by the roots of a quadratic equation?

Example 1

We say that 2 and 1 are roots of the equation. The equation has 2 real and distinct roots

In graphical form, the roots of the equation are represented as the intersection of the curve with the x axis.


Example 2

In this case there is only 1 solution. We say that the equation has only 2 real and equal roots
Graphically it can be represented as

Notice that there is only 1 intersection point between the curve and the x axis

Example 3
Lastly we are going to look at an example where the curve does not intersect the x axis
Consider the equation:

The equation cannot be factorized. It cannot be solved using the quadratic formula.
Graphically it is represented as


It turns out that we can predict the number of roots that an equation has by looking at the discriminant

Recall the quadratic formula

Remainder theorem II

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Question 1
The coefficient of the term in a cubic polynomial f(x) is -1. The roots of f(x) are 1, 2 and k. f(x) when divided by x - 3 gives a remainder of 8.
Find the value of k

First we need to understand what is meant by roots of an equation




Remainder theorem

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Introduction

How do we find the remainder when we divide by ?

One way is to use long division


From the long division, we get a remainder of 13

Note that:
is known as the divisor
is known as the quotient
is known as the dividend

In most cases we are only interested in the remainder, there is an easier way of obtaining the remainder without using long division

The easier way is to use the Remainder Theorem

Once again we want to find the remainder when is divided by



How do we know what value to sub in ?
If we are dividing by x - 2, we let x - 2 =0 and get x = 2. So we sub in 2
If we are dividing by x + 2, we let x + 2 =0 and get x = -2. So we sub in -2
If we are dividing by x - a, we let x - a =0 and get x = a. So we sub in a

Question 1
Given that leaves a remainder of 6 when divided by and has a factor of , find the value of a and b.


Next the question says that x + 2 is a factor
Factor means that the remainder is zero
Hence we can apply the remainder theorem


By solving the 2 simultaneous equat…

Binomial Theorem

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Question

If the first 3 terms of the expansion of are , find the value of x and n

Solution

Recall the binomial expansion formula



Next we need to simplify



Going back to the question,


We can then form 2 equations involving n and x

Equation 1


Equation 2

Integration of Trig Functions Part II

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Be careful of the negative signs when integrating sin and cos functions

Question 1
a)


Note the absence of the negative sign
We divide by 2 as the differentiation of 2x is 2

b)


Note the presence of the negative sign
We divide by 2 as the differentiation of 2x is 2

Question 2
a)

Note the presence of the negative sign

b)

Note the absence of the negative sign