AS and A level Further mathematics contents lists Contents Core Pure Mathematics Book 1/AS... 2 Core Pure Mathematics Book 2... 4 Further Pure Mathematics 1... 6 Further Pure Mathematics 2... 8 Further Statistics 1... 10 Further Statistics 2... 12 Further Mechanics 1... 14 Further Mechanics 2... 16 Decision Mathematics 1... 18 Decision Mathematics 2... 20 Version 4.0 14/3/2018
Core Pure Mathematics Book 1/AS 1 Complex numbers 1.1 Imaginary and complex numbers 1.2 Multiplying complex numbers 1.3 Complex conjugation 1.4 Roots of quadratic equations 1.5 Solving cubic and quartic equations Mixed exercise 1 2 Argand diagrams 2.1 Argand diagrams 2.2 Modulus and argument 2.3 Modulus argument form of complex numbers 2.4 Loci in the Argand diagram 2.5 Regions in the Argand diagram Mixed exercise 2 3 Series 3.1 Sums of natural numbers 3.2 Sums of squares and cubes Mixed exercise 3 4 Roots of polynomials 4.1 Roots of a quadratic equation 4.2 Roots of a cubic equation 4.3 Roots of a quartic equation 4.4 Expressions relating to the roots of a polynomial 4.5 Linear transformations of roots Mixed exercise 4 5 Volumes of revolution 5.1 Volumes of revolution around the x-axis 5.2 Volumes of revolution around the y-axis 5.3 Adding and subtracting volumes 5.4 Modelling with volumes of revolution Mixed exercise 5 Review exercise 1 2
6 Matrices 6.1 Introduction to matrices 6.2 Matrix multiplication 6.3 Determinants 6.4 Inverting a 2 2 matrix 6.5 Inverting a 3 3 matrix 6.6 Solving systems of equations using matrices Mixed exercise 6 7 Linear transformations 7.1 Linear transformations in two dimensions 7.2 Reflections and rotations 7.3 Enlargements and stretches 7.4 Successive transformations 7.5 Linear transformations in three dimensions 7.6 The inverse of a linear transformation Mixed exercise 7 8 Proof by induction 8.1 Proof by mathematical induction 8.2 Proving divisibility results 8.3 Proving statements involving matrices Mixed exercise 8 9 Vectors 9.1 Equation of a line in three dimensions 9.2 Equation of a plane in three dimensions 9.3 Scalar product 9.4 Calculating angles between lines and planes 9.5 Points of intersection 9.6 Finding perpendiculars Mixed exercise 9 Review exercise 2 Exam-style practice paper 3
Core Pure Mathematics Book 2 1 Complex numbers 1.1 Exponential form of complex numbers 1.2 Multiplying and dividing complex numbers 1.3 De Moivre s theorem 1.4 Trigonometric identities 1.5 Sums of series 1.6 nth roots of a complex number 1.7 Solving geometric problems Mixed exercise 1 2 Further series 2.1 The method of differences 2.2 Higher derivatives 2.3 Maclaurin series 2.4 Series expansions of compound functions Mixed exercise 2 3 Methods in calculus 3.1 Improper integrals 3.2 The mean value of a function 3.3 Differentiating inverse trigonometric functions 3.4 Integrating with inverse trigonometric functions 3.5 Integrating using partial fractions Mixed exercise 3 4 Volumes of revolution 4.1 Volumes of revolution around the x-axis 4.2 Volumes of revolution around the y-axis 4.3 Volumes of revolution of parametrically-defined curves 4.4 Modelling with volumes of revolution Mixed exercise 4 Review exercise 1 5 Polar coordinates 5.1 Polar coordinates and equations 5.2 Sketching curves 4
5.3 Area enclosed by a polar curve 5.4 Tangents to polar curves Mixed exercise 5 6 Hyperbolic functions 6.1 Introduction to hyperbolic functions 6.2 Inverse hyperbolic functions 6.3 Identities and equations 6.4 Differentiating hyperbolic functions 6.5 Integrating hyperbolic functions Mixed exercise 6 7 Methods in differential equations 7.1 First-order differential equations 7.2 Second-order homogeneous differential equations 7.3 Second-order non-homogeneous differential equations 7.4 Using boundary conditions Mixed exercise 7 8 Modelling with differential equations 8.1 Modelling with first-order differential equations 8.2 Simple harmonic motion 8.3 Damped and forced harmonic motion 8.4 Coupled first-order simultaneous differential equations Mixed exercise 8 Review exercise 2 Exam-style practice: Paper 1 Exam-style practice: Paper 2 5
Further Pure Mathematics 1 A level only content is indicated using italic font. 1 Vectors 1.1 Vector product 1.2 Finding areas 1.3 Scalar triple product 1.4 Straight lines 1.5 Solving geometrical problems Mixed exercise 1 2 Conic sections 1 2.1 Parametric equations 2.2 Parabolas 2.3 Rectangular hyperbolas 2.4 Tangents and normal 2.5 Loci Mixed exercise 2 3 Conic sections 2 3.1 Ellipses 3.2 Hyperbolas 3.3 Eccentricity 3.4 Tangents and normal to an ellipse 3.5 Tangents and normal to a hyperbola 3.6 Loci Mixed exercise 3 4 Inequalities 4.1 Algebraic methods 4.2 Using graphs to solve inequalities 4.3 Modulus inequalities Mixed exercise 4 Review exercise 1 6
5 The t-formulae 5.1 The t-formulae 5.2 Applying the t-formulae to trigonometric identities 5.3 Solving trigonometric equations 5.4 Modelling with trigonometry Mixed exercise 5 6 Taylor series 6.1 Taylor series 6.2 Finding limits 6.3 Series solutions of differential equations Mixed exercise 6 7 Methods in calculus 7.1 Leibnitz's theorem and nth derivatives 7.2 L'Hospital's rule 7.3 The Weierstrass substitution Mixed exercise 7 8 Numerical methods 8.1 Solving first-order differential equations 8.2 Solving second-order differential equations 8.3 Simpson's rule Mixed exercise 8 9 Reducible differential equations 9.1 First-order differential equations 9.2 Second-order differential equations 9.3 Modelling with differential equations Mixed exercise 9 Review exercise 2 Exam-style practice paper (AS level) Exam-style practice paper (A level) 7
Further Pure Mathematics 2 A level only content is indicated using italic font. 1 Number theory Further details will be available soon. 2 Group theory Further details will be available soon. 3 Complex numbers 3.1 Loci in an Argand diagram 3.2 Regions in an Argand diagram 3.3 Transformations of the complex plane Mixed exercise 3 Review exercise 1 4 Matrix algebra 4.1 Eigenvalues and eigenvectors 4.2 Reducing matrices to diagonal form 4.3 The Cayley-Hamilton Theorem Mixed exercise 4 5 Recurrence relations 5.1 Forming recurrence relations 5.2 Solving first-order recurrence relations 5.3 Solving second-order recurrence relations 5.4 Proving closed forms Mixed exercise 5 6 Integration techniques 6.1 Integration reduction formulae 6.2 Arc length 6.3 Area of a surface of revolution Mixed exercise 6 Review exercise 2 8
Exam-style practice paper (AS level) Exam-style practice paper (A level) 9
Further Statistics 1 A level only content is indicated using italic font. 1 Discrete random variables 1.1 Expected value of a discrete random variable 1.2 Variance of a discrete random variable 1.3 Expected value and variance of a function of X 1.4 Solving problems involving random variables Mixed exercise 1 2 Poisson distributions 2.1 The Poisson distribution 2.2 Modelling with the Poisson distribution 2.3 Adding Poisson distributions 2.4 Mean and variance of a Poisson distribution 2.5 Mean and variance of the binomial distribution 2.6 Using the Poisson distribution to approximate the binomial distribution Mixed exercise 2 3 Geometric and negative binomial distributions 3.1 The geometric distribution 3.2 Mean and variance of a geometric distribution 3.3 The negative binomial distribution 3.4 Mean and variance of the negative binomial distribution Mixed exercise 3 4 Hypothesis testing 4.1 Testing for the mean of a Poisson distribution 4.2 Finding critical regions for a Poisson distribution 4.3 Hypothesis testing for the parameter p of a geometric distribution 4.4 Finding critical regions for a geometric distribution Mixed exercise 4 5 Central limit theorem 5.1 The central limit theorem 5.2 Applying the central limit theorem to other distributions Mixed exercise 5 Review exercise 1 10
6 Chi-squared tests 6.1 Goodness of fit 6.2 Degrees of freedom and the chi-squared family of distributions 6.3 Testing a hypothesis 6.4 Testing the goodness of fit with discrete data 6.5 Using contingency tables 6.6 Applying goodness-of-fit tests to geometric distributions Mixed exercise 6 7 Probability generating functions 7.1 Probability generating functions 7.2 Probability generating functions of standard distributions 7.3 Mean and variance of a distribution 7.4 Sums of independent random variables Mixed exercise 7 8 Quality of tests 8.1 Type I and Type II errors 8.2 Finding Type I and Type II errors using the normal distribution 8.3 Calculate the size and power of a test 8.4 The power function Mixed exercise 8 Review exercise 2 Exam-style practice paper (AS level) Exam-style practice paper (A level) 11
Further Statistics 2 A level only content is indicated using italic font. 1 Linear regression 1.1 Least squares linear regression 1.2 Residuals Mixed exercise 1 2 Correlation 2.1 The product moment correlation coefficient 2.2 Spearman s rank correlation coefficient 2.3 Hypothesis testing for zero correlation Mixed exercise 2 3 Continuous distributions 3.1 Continuous random variables 3.2 The cumulative distribution function 3.3 Mean and variance of a continuous distribution 3.4 Mode, median, percentiles and skewness 3.5 The continuous uniform distribution 3.6 Modelling with the continuous uniform distribution Mixed exercise 3 4 Combinations of random variables 4.1 Combinations of random variables Review exercise 1 5 Estimation, confidence intervals and tests using a Normal distribution 5.1 Estimators, bias and standard error 5.2 Confidence intervals 5.3 Hypothesis testing for the difference between means 5.4 Use of large sample results for unknown population variances Mixed exercise 5 6 Further hypothesis tests 6.1 Confidence interval for the mean of a normal distribution with unknown variance 6.2 Hypothesis testing for the mean of a normal distribution with unknown variance 12
6.3 The F-test 6.4 Testing whether two independent random samples are from normal populations with equal variances Mixed exercise 6 7 Confidence intervals and tests with the t-distribution 7.1 Mean of a Normal distribution with unknown variance 7.2 The paired t-test 7.3 Difference between means of two independent normal distributions Mixed exercise 7 Review exercise 2 Exam-style practice paper (AS level) Exam-style practice paper (A level) 13
Further Mechanics 1 A level only content is indicated using italic font. 1 Momentum and impulse 1.1 Momentum in one direction 1.2 Conservation of momentum 1.3 Momentum as a vector Mixed exercise 1 2 Work, energy and power 2.1 Work done 2.2 Kinetic and potential energy 2.3 Conservation of mechanical energy and the work energy principle 2.4 Power Mixed exercise 2 3 Elastic strings and springs 3.1 Hooke s law and equilibrium problems 3.2 Hooke s law and dynamics problems 3.3 Elastic energy 3.4 Problems involving elastic energy Mixed exercise 3 Review exercise 1 4 Elastic collisions in one dimension 4.1 Direct impact and Newton s law of restitution 4.2 Direct collision with a smooth plane 4.3 Loss of kinetic energy 4.4 Successive direct impacts Mixed exercise 4 14
5 Elastic collisions in two dimensions 5.1 Oblique impact with a fixed surface 5.2 Successive oblique impacts 5.3 Oblique impact of smooth spheres Mixed exercise 5 Review exercise 2 Exam-style practice paper (AS level) Exam-style practice paper (A level) 15
Further Mechanics 2 A level only content is indicated using italic font. 1 Circular motion 1.1 Angular speed 1.2 Acceleration of an object moving on a horizontal circular path 1.3 Three-dimensional problems with objects moving in horizontal circles 1.4 Objects moving in vertical circles 1.5 Objects not constrained on a circular path Mixed exercise 1 2 Centres of mass of plane figures 2.1 Centre of mass of a set of particles on a straight line 2.2 Centre of mass of a set of particles arranged in a plane 2.3 Centres of mass of standard uniform plane laminas 2.4 Centre of mass of a composite lamina 2.5 Centre of mass of a framework 2.6 Lamina in equilibrium 2.7 Frameworks in equilibrium 2.8 Non-uniform composite lamina and frameworks Mixed exercise 2 3 Further centres of mass 3.1 Using calculus to find centres of mass 3.2 Centre of mass of a uniform body 3.3 Non-uniform bodies 3.4 Rigid bodies in equilibrium 3.5 Toppling and sliding Mixed exercise 3 Review exercise 1 4 Kinematics 4.1 Acceleration varying with time 4.2 Acceleration varying with displacement 4.3 Acceleration varying with velocity 16
Mixed exercise 4 5 Dynamics 5.1 Motion in a straight line with variable force 5.2 Newton s law of gravitation 5.3 Simple harmonic motion 5.4 Horizontal oscillation 5.5 Vertical oscillation Mixed exercise 5 Review exercise 2 Exam-style practice paper (AS level) Exam-style practice paper (A level) 17
Decision Mathematics 1 A level only content is indicated using italic font. 1 Algorithms 1.1 Using and understanding algorithms 1.2 Flow charts 1.3 Bubble sort 1.4 Quick sort 1.5 Bin-packing algorithms 1.6 Order of an algorithm Mixed exercise 1 2 Graphs and networks 2.1 Modelling with graphs 2.2 Graph theory 2.3 Special types of graph 2.4 Representing graphs and networks using matrices 2.5 The planarity algorithm Mixed exercise 2 3 Algorithms on graphs 3.1 Kruskal s algorithm 3.2 Prim s algorithm 3.3 Applying Prim s algorithm to a distance matrix 3.4 Using Dijkstra s algorithm to find the shortest path 3.5 Floyd s algorithm Mixed exercise 3 4 Route inspection 4.1 Eulerian graphs 4.2 Using the route inspection algorithm 4.3 Networks with more than four odd nodes Mixed exercise 4 5 The travelling salesman problem 5.1 The classical and practical travelling salesman problems 5.2 Using a minimum spanning tree to find an upper bound 5.3 Using a minimum spanning tree to find a lower bound 5.4 Using the nearest neighbor algorithm to find an upper bound 18
Mixed exercise 5 Review exercise 1 6 Linear programming 6.1 Linear programming problems 6.2 Graphical methods 6.3 Locating the optimal point 6.4 Solutions with integer values Mixed exercise 6 7 The simplex algorithm 7.1 Formulating linear programming problems 7.2 The simplex method 7.3 Problems requiring integer solutions 7.4 Two-stage simplex method 7.5 The Big-M method Mixed exercise 7 8 Critical path analysis 8.1 Modelling a project 8.2 Dummy activities 8.3 Early and late event times 8.4 Critical activities 8.5 The float of an activity 8.6 Gantt charts 8.7 Resource histograms 8.8 Scheduling diagrams Mixed exercise 8 Review exercise 2 Exam-style practice paper (AS level) Exam-style practice paper (A level) 19
Decision Mathematics 2 A level only content is indicated using italic font. 1 Transportation problems 1.1 The north-west corner method 1.2 Unbalanced problems and degenerate solutions 1.3 Finding an improved solution 1.4 The stepping-stone method 1.5 Linear programming Mixed exercise 1 2 Allocation (assignment) problems 2.1 The Hungarian algorithm 2.2 Using a dummy 2.3 Maximum profit allocation 2.4 Managing incomplete data 2.4 Maximum profit allocation 2.5 Linear programming Mixed exercise 2 3 Flows in networks 1 3.1 Flows in networks 3.2 Cuts and their capacities 3.3 Finding an initial flow 3.4 Flow-augmenting routes 3.5 Maximum flow minimum cut theorem Mixed exercise 3 4 Flows in networks 2 4.1 Lower capacities 4.2 Sources and sinks 4.3 Restricted capacity nodes Mixed exercise 4 Review exercise 1 5 Dynamic programming 5.1 Shortest and longest path problems 20
5.2 Minimax and maximin problems 5.3 Dynamic programming problems in table form Mixed exercise 5 6 Game theory 6.1 Play-safe strategies and stable solutions 6.2 Reducing the pay-off matrix 6.3 Optimal strategies for games with no stable solution 6.4 Converting games to linear programming problems Mixed exercise 6 7 Recurrence relations 7.1 Forming recurrence relations 7.2 Solving first-order recurrence relations 7.3 Solving second-order recurrence relations Mixed exercise 7 8 Decision analysis 8.1 Decision trees 8.2 Utility Mixed exercise 8 Review exercise 2 Exam-style practice paper (AS level) Exam-style practice paper (A level) 21