Pure Papers 1 & 2
2 hours

Proof – Proof by deduction and exhaustion, disproof by counter example.  Further proof by deduction, proof by contradiction.

Algebra and Functions – Indices, surds, quadratic equations and graphs, discriminants, simultaneous equations and inequalities, graphs and graph transformations, factor theorem.  Simplifying rational functions, modulus function, composite and inverse functions, partial fractions, modelling using functions.

Co-ordinate Geometry – Equations of lines and circles, parallel and perpendicular lines.  Parametric and Cartesian equations of curves, modelling using parametric equations.

Sequences and Series – Binomial theorem for integer n.  Binomial theorem for rational n, arithmetic and geometric sequences, sigma notation, iterative formulae.

Trigonometry – Sine and cosine rules, graphs of the 3 basic trigonometric functions, simple equations and identities.  Radian measure, small angle approximations, exact values of trigonometric functions, reciprocal and inverse trigonometric functions, compound and double angle formulae, further equations and identities, trigonometric proofs.

Exponentials and Logarithms – Graphs of exponential functions and their transformations, logarithms and the laws of logarithms, solving exponential equations, using logarithmic graphs to estimate parameters, modelling using exponential growth/decay. 

Differentiation – Differentiation from first principles, differentiation of powers of x and constant multiples, sums and differences, application of differentiation to tangents and normals, maxima and minima, increasing and decreasing functions.  Differentiation from first principles for trigonometric functions, differentiation of exponential functions, the product and quotient rules, related rates of change, parametric and implicit differentiation, forming simple differential equations.

Integration – Indefinite integration of xn (n ≠ - 1) and constant multiples, sums and differences, definite integration and link to areas under curves.  Integration of exponential and trigonometric functions, finding the area between two curves, integration by substitution and by parts, integration using partial fractions, solution of first order differential equations and interpretation.

Numerical Methods – Root location using change of sign, root location using iteration, Newton-Raphson method, numerical integration.

Vectors – 2-dimensional vectors, magnitude and direction, vector addition/subtraction and scalar multiplication, use of vectors to solve problems (including in Mechanics).  3 dimensional vectors.

Applied Paper
2 hours

Statistics:

Statistical Sampling – Different sampling techniques and their limitations.

Data Presentation and Interpretation – Histograms, skew, scatter diagrams and correlation.  Measures of central tendency and spread, outliers.

Probability – Mutually exclusive events, independent events, conditional probability, use of tree diagrams and Venn diagrams, modeling with probability.

Statistical Distributions – Discrete probability distributions, the binomial distribution, the Normal distribution,  approximation of a binomial by a Normal.

Statistical Hypothesis Testing – Binomial hypothesis testing, Normal hypothesis testing.

Mechanics:

Quantities and Units in Mechanics – Use of S.I. units and conversions.

Kinematics – Travel graphs, constant acceleration formulae, calculus in kinematics, motion in a vertical plane including projectiles.

Forces and Newton’s laws – Use of Newton’s second law in a straight line, resolving forces, motion under gravity, equilibrium and Newton’s third law, resultant forces, friction.

Moments – Equilibrium of rigid bodies.

Further Mathematics Core Pure 1
1½ hours

Proof – Proof by induction, including series sums and divisibility.

Complex Numbers – Arithmetic with complex numbers, conjugate pairs, Argand diagrams, modulus argument form, loci in Argand diagrams, solving quadratic, cubic and quartic equations.

Matrices – Addition, subtraction and multiplication by a scalar, matrices and transformations, invariant points and lines, determinants, inverse matrices,  using matrices to solve simultaneous equations.

Further algebra and functions – Relationships between roots and coefficients of polynomial equations, linear transformations of polynomial equations, summation of series using standard formulae.

Further calculus – Volumes of revolution using Cartesian or parametric equations.

Further vectors – Vector and Cartesian equations of lines and planes in 3D, scalar product, intersection of lines and planes.

Further  Mathematics Core Pure 2
1½ hours

Complex numbers – De Moivre’s theorem and applications, complex roots of unity and application to geometric problems.

Further algebra and functions – Method of differences for summation of series, Maclaurin series.

Further calculus – Mean value of a function, integration of partial fractions with irreducible quadratic denominators, differentiation of inverse trigonometric functions, integration using trigonometric substitutions.

Polar co-ordinates – Conversion between polar and Cartesian co-ordinates, sketching curves expressed in polar for, finding the area enclosed by a polar curve.

Hyperbolic functions – Graphs of hyperbolic functions, differentiation and integration of hyperbolic functions, inverse hyperbolic functions.

Differential equations – Integrating factors, general and particular solutions, modelling using differential equations, second order differential equations and auxiliary equations, simple harmonic motion, damped oscillations, coupled first order differential equations.

Optional 

Further  Mathematics Further Pure 1
1½ hours

Further calculus – Taylor series, series expansion and limits, Leibnitz’ theorem, Weierstrass substitution for integration.

Further differential equations – Taylor series and differential equations, reducible differential equations.

Co-ordinate systems – Parametric and Cartesian equations of the parabola, ellipse and hyperbola, focus and directrix, eccentricity, tangents and normals, loci.

Further vectors – Vector product, triple scalar product, applications of vectors to 3-D geometry.

Further numerical methods – Numerical solution of first and second order differential equations, Simpson’s rule.

Inequalities – Solution of inequalities involving fractions and modulus.

Optional 

Further Mechanics
Paper 1
1½ hours

Momentum and Impulse – Conservation of momentum in direct contacts, impulse.

Collisions – Direct impact of elastic particles, Newton’s law of restitution, energy loss in impact, repeated impacts.

Centres of mass – Centre of mass of a discrete mass distribution, centre of mass of plane figures, centre of mass of a framework, equilibrium of a lamina or a framework.

Work and energy – Kinetic and potential energy, work and power, the work-energy principle, conservation of mechanical energy.

Elastic strings and springs – Hooke’s law, energy stored in an elastic string or spring.

Optional 

Further  Statistics
Paper 1
1½ hours

Linear regression – Least squares regression lines, residuals.

Statistical distributions (discrete) – Mean and variance of a discrete probability distribution, Mean and variance of functions of a variable, the Poisson distribution and its additive properties, mean and variance of the binomial and Poisson distribution, use of the Poisson distribution as an approximation to the binomial.

Statistical distributions (continuous) – Random variables, probability distribution functions and continuous distribution functions, mean and variance of continuous variables and functions of continuous variables, the continuous uniform distribution.

Correlation – Calculation of correlation coefficients, coding, Spearman’s rank correlation coefficient.

Hypothesis testing – Hypothesis test for the mean of a Poisson distribution, hypothesis test for zero correlation.

Chi squared tests – Goodness of fit tests and contingency tables, degrees of freedom.

Optional

Decision Mathematics Paper 1

(1½ hours)

Algorithms and Graph Theory – Application of algorithms defined by flow chart or text, Bin packing and sorting algorithms, the planarity algorithm.

Algorithms on Graphs – Prim’s and Kruskal’s algorithms, Dijkstra’s algorithm, the Route Inspection Algorithm, the Travelling Salesman algorithm, the Nearest Neighbour algorithm.

Critical Path Analysis – Precedence tables and activity networks, latest and earliest start and finish times for activities, float, construction of Gantt charts, construction of resource histograms, scheduling.

Linear Programming – Formulation of problems as linear programs, slack, surplus and artificial variables, Graphical solution of two variable problems using vertex method or objective line method, the Simplex algorithm for maximising problems with ≤ constraints, the two stage Simplex and big M methods for maximising and minimising problems with both ≤ and ≥ constraints.