Coordinate geometry of straight lines (gradient, equations) and circles (centre-radius form, intersections), circle theorems.
Equations of circles, tangents and normals to circles, intersection of circles and lines
Solving pairs of simultaneous equations including one linear and one non-linear (quadratic/circle).
Structure of mathematical proof, disproof by counter-example, and proof by contradiction.
Proof by induction for statements involving positive integers, including divisibility and series summations.
Sine rule, cosine rule, area = ½ab sinC, triangle problems
Radian measure, s = rθ, ½r²θ
Graphs of sin, cos, tan and their transformations; exact trigonometric values
Graphs of sin/cos/tan, solving trig equations, basic identities
Trigonometric proofs including identities and standard results
Modulus function |x|, sketching graphs of modulus functions, and solving modulus equations and inequalities.
2×2 matrices, matrix operations, determinant, inverse, and solving simultaneous equations using matrices.
Linear transformations represented by 2×2 matrices, geometric effects (reflections, rotations, enlargements, shears).
Invariant lines under transformations, eigenvalues and eigenvectors for 2×2 and 3×3 matrices.
sec, cosec, cot definitions, identities (1+tan²=sec², etc.), proving identities
Compound angle sin(A±B), double angle sin2A/cos2A, product-to-sum
Rsin(θ±α), Rcos(θ±α), solving asinθ + bcosθ = c, finding max/min
Small angle approximations sinθ ≈ θ, cosθ ≈ 1 - θ²/2, tanθ ≈ θ for small θ in radians.
Chain rule, product rule, quotient rule
Differentiation using the product rule and quotient rule
Differentiating eˣ, ln x, aˣ, trig and reciprocal trig
Related rates, dA/dt = dA/dr × dr/dt
Integrating eˣ, 1/x, trig functions, reverse chain rule
Substitution method, trig substitutions
Integration by parts, repeated application
Implicit equations relating x and y, implicit differentiation to find dy/dx without rearranging.
Differential equations terminology, constructing DEs from context, solving by separation of variables, and interpreting solutions.
3×3 matrices, solving systems of 2 and 3 simultaneous equations, determinant, inverse, linear transformations in 3D, and eigenvalues.