Cambridge IGCSE, AS & A Levels: Mathematics Overview

A structured guide to key topics, equations, and examples for students studying Cambridge International Mathematics.

IGCSE Mathematics (0580)

IGCSE Mathematics provides a strong foundation in core mathematical concepts, including number, algebra, geometry, and statistics.

Summary Equation: Quadratic Formula

$$ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} $$

Example: Solve the equation $$x^2 + 5x + 6 = 0$$.

Here, $$a=1, b=5, c=6$$.

$$ x = \frac{-5 \pm \sqrt{5^2 – 4(1)(6)}}{2(1)} = \frac{-5 \pm \sqrt{25 – 24}}{2} = \frac{-5 \pm 1}{2} $$

The solutions are $$x = -2$$ and $$x = -3$$.

Summary Equation: Pythagorean Theorem

$$ a^2 + b^2 = c^2 $$

Example: A right-angled triangle has sides of length 3 cm and 4 cm. Find the length of the hypotenuse.

$$ c^2 = 3^2 + 4^2 = 9 + 16 = 25 $$ $$ c = \sqrt{25} = 5 \text{ cm} $$

Summary Equation: Mean

$$ \text{Mean} = \frac{\sum x}{n} $$

Example: Find the mean of the numbers 2, 4, 6, 8.

$$ \text{Mean} = \frac{2 + 4 + 6 + 8}{4} = \frac{20}{4} = 5 $$

AS & A Level Mathematics delves into more complex areas like pure mathematics, mechanics, and probability & statistics.

Summary Equation: Differentiation (Power Rule)

$$ \frac{d}{dx} x^n = nx^{n-1} $$

Example: Differentiate $$y = 3x^4$$.

$$ \frac{dy}{dx} = 3(4)x^{4-1} = 12x^3 $$

Summary Equation: Constant Acceleration (SUVAT) Equations

$$ v^2 = u^2 + 2as $$

Example: A car accelerates from rest at $$2 \text{ m/s}^2$$. How far has it traveled when its velocity is $$10 \text{ m/s}$$?

Here, $$u=0, a=2, v=10$$. We need to find $$s$$.

$$ 10^2 = 0^2 + 2(2)s $$ $$ 100 = 4s $$ $$ s = 25 \text{ m} $$

Summary Equation: Binomial Probability Formula

$$ P(X=r) = {n \choose r}p^r(1-p)^{n-r} $$

Example: A coin is tossed 5 times. What is the probability of getting exactly 3 heads?

Here, $$n=5, r=3, p=0.5$$.

$$ P(X=3) = {5 \choose 3}(0.5)^3(0.5)^{5-3} = 10 \times 0.125 \times 0.25 = 0.3125 $$