Paper 2
Pure Mathematics 2
Proof
Proof by contradiction1 subtopic lessonsAlgebra and functions
Completing the square and the vertex · Solving quadratic equations · The discriminant and the nature of roots · Linear and quadratic inequalities · Simultaneous equations and lines meeting curves · Indices and surds · Polynomials: division, factor and remainder theorems · Partial fractions · Modulus graphs and equations · Modulus inequalities · Curve sketching and transformations · Functions, domain and range · Composite functions · One-one and inverse functions · Graphs of a function and its inverse · Transformations of graphs16 subtopic lessonsCoordinate geometry
Gradient, length, midpoint and equations of lines · Parallel and perpendicular lines · The equation of a circle · Tangents, chords and intersections4 subtopic lessonsSequences and series
Binomial expansion · Binomial expansion for any rational n · Arithmetic progressions · Geometric progressions · Sum to infinity5 subtopic lessonsTrigonometry
Sine rule, cosine rule and area · Radians · Arc length and sector area · Segments, chords and composite shapes · Exact values and the unit circle · Graphs and transformations · Trigonometric identities · Solving trigonometric equations · Inverse trigonometric functions · Reciprocal functions and their identities · Compound-angle formulae · Double-angle formulae · The R-form (harmonic form)13 subtopic lessonsExponentials and logarithms
The functions eˣ and ln x · Laws of logarithms · Exponential equations and inequalities · Logarithmic equations · Exponential models and linear forms5 subtopic lessonsDifferentiation
Gradients and the power rule · Tangents and normals · Stationary points and increasing functions · Optimisation and rates of change · Derivatives of eˣ, ln x and trigonometric functions · The chain rule · The product and quotient rules · Implicit differentiation · Parametric differentiation · Stationary points and tangents10 subtopic lessonsIntegration
Indefinite integration · Definite integrals · Areas under and between curves · Integrating (ax + b)ⁿ · Integrating eˣ, 1/x and trigonometric functions · Integration using trigonometric identities · Exact definite integrals and logarithms · Integration by substitution · Integration by parts · The trapezium rule · Forming differential equations · Separating the variables · Particular solutions with harder integrals · Modelling with differential equations14 subtopic lessonsNumerical methods
Locating roots: sketches and sign changes · Fixed-point iteration · Rearrangements, staircases and cobwebs · Newton–Raphson, bisection and interpolation4 subtopic lessonsVectors
Position vectors, magnitude and unit vectors1 subtopic lessons