A-Level Mathematics — Pure, Mechanics & Statistics

Indices and surds

• Laws: a^m x a^n = a^(m+n); a^m / a^n = a^(m-n); (a^m)^n = a^(mn); a^0 = 1; a^(-n) = 1/a^n; a^(1/n) = the nth root; a^(m/n) = the nth root of a^m.
• Surds: simplify using root(ab) = root(a) root(b); rationalise a denominator by multiplying top and bottom by its conjugate.

Quadratics

• Solve by factorising, the formula x = (-b +/- root(b^2 - 4ac)) / 2a, or completing the square (x + p)^2 + q, which gives the vertex (-p, q).
• Discriminant b^2 - 4ac: greater than 0 = two real roots; equal to 0 = one repeated root (curve touches the x-axis); less than 0 = no real roots.

Functions, graphs, transformations

• Composite fg(x) = f(g(x)); inverse f inverse reflects the graph in y = x (domain and range swap).
• Transformations: f(x) + a moves up; f(x + a) moves left; a f(x) stretches vertically by a; f(ax) stretches horizontally by 1/a; -f(x) reflects in the x-axis; f(-x) reflects in the y-axis.
• Sketch quadratics, cubics, reciprocals (1/x, 1/x^2) and the modulus |x|; mark intercepts and asymptotes.

Simultaneous equations and inequalities

• Substitute a linear equation into a quadratic; the discriminant of the result gives the number of intersections (2, 1 tangent, or 0).
• Solve quadratic inequalities by sketching and reading off where the curve is above/below the axis.

Factor and remainder theorems

• Factor theorem: (x - a) is a factor if f(a) = 0. Remainder theorem: dividing by (x - a) leaves remainder f(a).

Whenever a question asks about the number of roots or a tangent, quote and evaluate the discriminant.

Diagram