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Linear equations

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Quadratic equations

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Where your mathematical journey changes

Motivation

I completed my iGCSEs between 2022 and 2024 and faced difficulty finding resources that comprehensively explained the theory behind concepts while also being free. Recognizing that many students prefer not to spend hundreds of dollars on books, I've created this archive aimed at addressing both goals at no cost.

Content

Obviously when dealing with math at a higher level than iGCSEs, you are assumed to know most of the topics listed below. In general, I'll try my best to explain topics to an iGCSE level, but most practice questions will be harder. Also, due to the IB AA HL-nature of the content, you're going to be expected to solve these problems without a calculator (unless explicitly stated). For your convenience, when you need a calculator, there's going to be one located on the left, just next to the light/dark mode toggle! However, it should not be necessary for most of this course. At the end of every chapter, I'll try to include a "practical application" section to show how you might use the content in real-life examples.

Knowledge assumptions

When going through this course, you're required to be proficient to a certain degree in the following areas (although, I'll provide baseline knowledge when building up to more complex topics). You may hover over each topic to find out specific content knowledge.

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Solving linear equations

Including:

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Algebraic fraction manipulation
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Simultaneous equations
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Perpendicular lines

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Surds

Including:

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Simplifying basic surds
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Rationalizing denominators

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Function notation

Including:

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Inverse functions
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Composite functions

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Quadratics

Including:

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Discriminant
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Roots (or "zeros")
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Quadratic formula

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Algebraic proofs

Including:

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General numerical proofs
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Algebraic proofs

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Vector proofs

Including:

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General proofs
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Colinear proofs
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2D vector spaces

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Trigonometric functions

Including:

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Solving functions
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Unit circle
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Graphs of sin/cos/tan

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3D geometry

Including:

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Area/surface area
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Angle calculations

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Calculus

Including:

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Basic differentiation
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Basic integration

It seems like a lot of presumed knowledge, but if you know the contents above, you will find the content of this course, a lot easier to work through! Also, your best bet for revision, is to either use past papers, or the ECQs (explained below) to actually

apply

your knowledge. That's why this resource is great, because questions dotted throughout rely on your application skills, rather than adjusting a few values but maintaining the same framework, as most "helpful" math books do.

End of Chapter Questions (ECQs)

Throughout the course, I've developed some end-of-chapter questions, which will help you solidify concepts you learnt in the chapter. When I define some of these as "hard", it can mean that the content itself is difficult, or it can mean that it takes a lot of time to interpret/solve to get an answer. Below is an example of the latter (taken from Q01H).

Q01H

The line

L_2

is parallel to another line,

L_1

, with an equation

f(x)=5(x+2)-4

and passes through the point

L_{2A}(3,5)

(

a

) Find the points of intersection (

L_{2B}

and

L_{2C}

) on

L_2

created by two lines perpendicular to

L_1

at the points

L_{1A}(0,6)

and

L_{1B} \left( \frac{5}{2},\frac{5}{2} \right)

(

b

) Find the perpendicular bisector of the line segment

[L_{2B} \ L_{2C}]

(

c

) Find the solutions - on the

x

-axis - of the perpendicular bisector you just created, and any previous lines you generated (that count should be 2 additional equations).