A-Level Mathematics — Pure, Statistics & Mechanics

Proof

• Deduction (algebraic argument from known facts), exhaustion (test every case), counter-example (one case disproves a statement), and proof by contradiction (assume the opposite, derive an absurdity - e.g. root 2 is irrational, there are infinitely many primes).
• State assumptions clearly and finish with a concluding statement.

Indices and surds

• Laws: a^m x a^n = a^(m+n); (a^m)^n = a^(mn); a^0 = 1; a^(-n) = 1/a^n; a^(1/n) = nth root.
• Rationalise surd denominators by multiplying by the conjugate.

Quadratics

• Completing the square gives the vertex and proves the min/max; the quadratic formula; the discriminant b^2 - 4ac tells you the number of real roots (>0 two, =0 one repeated, <0 none).
• Solve by factorising, formula or completing the square; sketch parabolas with intercepts and turning point.

Simultaneous equations and inequalities

• One linear + one quadratic: substitute and solve; the discriminant of the result tells you whether a line meets a curve.
• Inequalities: solve linear and quadratic (sketch to find the region); represent solutions with set/interval notation; beware multiplying by a negative (flip the sign).

Functions and graphs

• Domain/range, composite and inverse functions, the modulus function; transformations f(x)+a, f(x+a), af(x), f(ax) and reflections; partial fractions for algebraic division.

Show every line of a proof and always check the discriminant when a line meets a curve or a quadratic must have real roots.

Diagram