A-Level Mathematics — Pure Core, Stats & Mechanics

Proof

• Deduction, exhaustion, disproof by counter-example, and proof by contradiction (assume the negation, reach an absurdity - e.g. root 2 is irrational). State assumptions and finish with a clear conclusion.

Surds and indices

• Index laws (a^m a^n = a^(m+n), a^(-n) = 1/a^n, a^(m/n) = nth root of a^m); simplify surds and rationalise denominators using the conjugate.

Quadratics and equations

• Completing the square (vertex, min/max), the quadratic formula, and the discriminant b^2 - 4ac (number of real roots).
• Solve simultaneous equations (linear + quadratic by substitution) and inequalities (linear and quadratic - sketch to identify the region; represent with interval notation).

Algebraic manipulation

• Polynomial division, the factor theorem (f(a) = 0 means (x - a) is a factor) and remainder theorem; partial fractions; simplifying algebraic fractions.

A concluding statement is required to earn full proof marks, and the discriminant is your tool whenever real roots or a line-meets-curve condition is involved.

Diagram