Pearson Edexcel Level 3 Advanced GCE in Further Mathematics

1.1 The general ideas of algorithms and the implementation of an algorithm given by a flow chart or text.

1.2 Bin packing, bubble sort and quick sort.

1.3 Use of the order of the nodes to determine whether a graph is Eulerian, semi-Eulerian or neither.

1.4 The planarity algorithm for planar graphs.

2.1 The minimum spanning tree (minimum connector) problem. Prim’s and Kruskal’s algorithm.

2.2 Dijkstra’s and Floyd’s algorithm for finding the shortest path.

3.1 Algorithm for finding the shortest route around a network, travelling along every edge at least once and ending at the start vertex (The Route Inspection Algorithm).

3.2 Determination of upper and lower bounds using minimum spanning tree methods.

3.2 The classical problem for complete graphs satisfying the triangle inequality.

3.2 The nearest neighbour algorithm.

3.2 The practical and classical Travelling Salesman problems.

2.1 Cost matrix reduction.

2.1 Modification of the Hungarian algorithm to deal with a maximum profit allocation.

2.1 Use of the Hungarian algorithm to find a least cost allocation.

2.2 Formulation of the Hungarian algorithm as a linear programming problem.

5.1 Applications of the Central Limit Theorem to other distributions.

2.1 Moment of a force. Centre of mass of a discrete mass distribution in one and two dimensions.

2.2 Centre of mass of frameworks.

2.2 Centre of mass of uniform plane figures, and of composite plane figures.

2.2 Equilibrium of a plane lamina or framework under the action of coplanar forces.

6.1 Degrees of freedom.

6.1 Goodness of fit tests and Contingency Tables

6.1 The null and alternative hypotheses.

6.1 The use of ∑ᵢ₌₁ⁿ (Oᵢ − Eᵢ)²/Eᵢ as an approximate χ² statistic.

4.1 Distribution of linear combinations of independent Normal random variables.

2.1 Solve any quadratic equation with real coefficients.

2.1 Solve cubic or quartic equations with real coefficients.

2.10 Find the n distinct nth roots of re^(iθ) for r ≠ 0 and know that they form the vertices of a regular n-gon in the Argand diagram.

2.11 Use complex roots of unity to solve geometric problems.

2.2 Add, subtract, multiply and divide complex numbers in the form x + iy with x and y real.

2.2 Understand and use the terms ‘real part’ and ‘imaginary part’.

2.3 Know that non-real roots of polynomial equations with real coefficients occur in conjugate pairs.

2.3 Understand and use the complex conjugate.

2.4 Use and interpret Argand diagrams.

2.5 Convert between the Cartesian form and the modulus-argument form of a complex number.

2.6 Multiply and divide complex numbers in modulus argument form.

2.7 Construct and interpret simple loci in the argand diagram such as |z − a| > r and arg (z – a) = θ.

2.8 Understand de Moivre’s theorem and use it to find multiple angle formulae and sums of series.

2.9 Know and use the definition e^(iθ) = cos θ + i sin θ and the form z = re^(iθ)

7.1 Hypothesis test and confidence interval for the mean of a Normal distribution with unknown variance.

7.2 Paired t-test.

7.3 Hypothesis test and confidence interval for the difference between two means from independent Normal distributions when the variances are equal but unknown.

7.3 Use of the pooled estimate of variance.

2.1 The concept of a continuous random variable.

2.1 The probability density function and the cumulative distribution function for a continuous random variable.

2.2 Relationship between probability density and cumulative distribution functions.

2.3 Extension of expected value function to include E(g(X))

2.3 Idea of skewness

2.3 Mean and variance of continuous random variables.

2.3 Mode, median and percentiles of continuous random variables.

2.4 The continuous uniform (rectangular) distribution.

4.1 Cartesian and parametric equations for the parabola and rectangular hyperbola, ellipse and hyperbola.

4.2 The focus-directrix properties of the parabola, ellipse and hyperbola, including the eccentricity.

4.3 Tangents and normals to these curves.

4.4 Loci problems.

3.1 A knowledge of the effects of coding will be expected.

3.1 Knowledge of the conditions for the use of the product moment correlation coefficient.

3.1 Use of formulae to calculate the product moment correlation coefficient.

3.2 Spearman’s rank correlation coefficient, its use, interpretation.

3.3 Testing the hypothesis that a correlation is zero using either Spearman’s rank correlation or the product moment correlation coefficient.

4.1 Modelling of a project by an activity network, from a precedence table.

4.2 Completion of the precedence table for a given activity network.

4.3 Algorithm for finding the critical path. Earliest and latest event times. Earliest and latest start and finish times for activities. Identification of critical activities and critical path(s).

4.4 Calculation of the total float of an activity. Construction of Gantt (cascade) charts.

4.5 Construct resource histograms (including resource levelling) based on the number of workers required to complete each activity.

4.6 Scheduling the activities using the least number of workers required to complete the project.

7.1 Use, construct and interpret decision trees.

7.2 Use of expected monetary values (EMVs) and utility to compare alternative courses of action.

9.1 Find and use an integrating factor to solve differential equations of form dy/dx + P(x)y = Q(x) and recognise when it is appropriate to do so.

9.2 Find both general and particular solutions to differential equations.

9.3 Use differential equations in modelling in kinematics and in other contexts.

9.4 Solve differential equations of form y″ + ay′ + by = 0 where a and b are constants by using the auxiliary equation.

9.5 Solve differential equations of form y″ + ay′ + by = f(x) where a and b are constants by solving the homogeneous case and adding a particular integral to the complementary function (in cases where f(x) is a polynomial, exponential or trigonometric function).

9.6 Understand and use the relationship between the cases when the discriminant of the auxiliary equation is positive, zero and negative and the form of solution of the differential equation.

9.7 Solve the equation for simple harmonic motion ẍ = −ω²x and relate the solution to the motion.

9.8 Model damped oscillations using second order differential equations and interpret their solutions.

9.9 Analyse and interpret models of situations with one independent variable and two dependent variables as a pair of coupled first order simultaneous equations and be able to solve them, for example predator-prey models.

1.1 Calculation of the mean and variance of discrete probability distributions.

1.1 Extension of expected value function to include E(g(X))

4.1 Principles of dynamic programming. Bellman’s principle of optimality.

4.1 Stage variables and State variables. Use of tabulation to solve maximum, minimum, minimax or maximin problems.

4.1 Direct impact of elastic spheres. Newton’s law of restitution. Loss of kinetic energy due to impact.

4.2 Successive direct impacts of spheres and/or a sphere with a smooth plane surface.

5.1 Oblique impact of smooth elastic spheres and a smooth sphere with a fixed surface. Loss of kinetic energy due to impact.

5.2 Successive oblique impacts of a sphere with smooth plane surfaces.

3.1 Elastic strings and springs. Hooke’s law.

3.2 Energy stored in an elastic string or spring.

5.1 Concepts of standard error, estimator, bias.

5.1 Quality of estimators

5.2 Concept of a confidence interval and its interpretation.

5.3 Confidence limits for a Normal mean, with variance known.

5.4 Hypothesis test for the difference between the means of two Normal distributions with variances known.

5.5 Use of large sample results to extend to the case in which the population variances are unknown.

3.1 Cuts and their capacity.

3.2 Use of the labelling procedure to augment a flow to determine the maximum flow in a network.

3.3 Use of the max–flow min–cut theorem to prove that a flow is a maximum flow.

3.4 Multiple sources and sinks.

3.4 Vertices with restricted capacity.

3.5 Determine the optimal flow rate in a network, subject to given constraints.

4.1 Understand and use the relationship between roots and coefficients of polynomial equations up to quartic equations.

4.2 Form a polynomial equation whose roots are a linear transformation of the roots of a given polynomial equation (of at least cubic degree).

4.3 Understand and use formulae for the sums of integers, squares and cubes and use these to sum other series.

4.4 Understand and use the method of differences for summation of series including use of partial fractions.

4.5 Find the Maclaurin series of a function including the general term.

4.6 Recognise and use the Maclaurin series for eˣ, ln(1 + x), sin x, cos x and (1 + x)ⁿ, and be aware of the range of values of x for which they are valid (proof not required).

5.1 Derive formulae for and calculate volumes of revolution.

5.2 Evaluate improper integrals where either the integrand is undefined at a value in the range of integration or the range of integration extends to infinity.

5.3 Understand and evaluate the mean value of a function.

5.4 Integrate using partial fractions.

5.5 Differentiate inverse trigonometric functions.

5.6 Integrate functions of the form (a² − x²)^(−½) and (a² − x²)⁻¹ and be able to choose trigonometric substitutions to integrate associated functions.

2.1 Derivation and use of Taylor series.

2.2 Use of series expansions to find limits.

2.3 Leibnitz’s theorem.

2.4 L’Hospital’s Rule.

2.5 The Weierstrass substitution for integration.

2.1 Further Integration — Reduction formulae.

2.2 The calculation of arc length and the area of a surface of revolution.

3.1 Centre of mass of uniform and non-uniform rigid bodies and composite bodies.

3.2 Equilibrium of rigid bodies under the action of coplanar forces.

3.3 Toppling and sliding of a rigid body on a rough plane.

4.1 Further loci and regions in the Argand diagram.

4.2 Elementary transformations from the z-plane to the w-plane.

3.1 Use of Taylor series method for series solution of differential equations.

3.2 Differential equations reducible by means of a given substitution.

4.1 Newton’s laws of motion, for a particle moving in one dimension, when the applied force is variable.

4.2 Kinetic, potential and elastic energy in the context of SHM.

4.2 Oscillations of a particle attached to the end of elastic string(s) or spring(s).

4.2 Simple harmonic motion.

5.1 Kinematics of a particle moving in a straight line when the acceleration is a function of the displacement (x), or time (t) or velocity (v).

3.1 Eigenvalues and eigenvectors of 2 × 2 and 3 × 3 matrices.

3.2 Reduction of matrices to diagonal form.

3.3 The use of the Cayley-Hamilton theorem.

6.1 Numerical solution of first order and second order differential equations.

6.2 Simpson’s rule.

6.1 First and second order recurrence relations.

6.2 The solution of recurrence relations to obtain closed forms.

6.3 Proof by induction of closed forms.

1.1 The t-formulae

1.2 Applications of t-formulae to trigonometric identities

1.3 Applications of t-formulae to solve trigonometric equations

6.1 Understand and use the vector and Cartesian forms of an equation of a straight line in 3-D.

6.2 Understand and use the vector and Cartesian forms of the equation of a plane.

6.3 Calculate the scalar product and use it to express the equation of a plane, and to calculate the angle between two lines, the angle between two planes and the angle between a line and a plane.

6.4 Check whether vectors are perpendicular by using the scalar product.

6.5 Calculate the perpendicular distance between two lines, from a point to a line and from a point to a plane.

6.5 Find the intersection of a line and a plane.

5.1 The vector product a × b of two vectors.

5.2 The scalar triple product a.b × c

5.3 Applications of vectors to three dimensional geometry involving points, lines and planes.

5.1 Two person zero-sum games and the pay-off matrix.

5.2 Identification of play safe strategies and stable solutions (saddle points).

5.3 Reduction of pay-off matrices using dominance arguments.

5.4 Optimal mixed strategies for a game with no stable solution by use of graphical methods for 2 × n or n × 2 games where n = 1, 2, 3 or 4

5.5 Optimal mixed strategies for a game with no stable solution by converting games to linear programming problems that can be solved by the Simplex algorithm.

3.1 Geometric and negative binomial distributions.

3.2 Mean and variance of a geometric distribution with parameter p.

3.3 Mean and variance of negative binomial distribution with P(X = x) = ˣ⁻¹Cᵣ₋₁ pʳ(1 − p)^(x − r)

1.1 The Axioms of a group.

1.2 Cayley tables.

1.2 Cyclic groups.

1.2 Examples of groups.

1.3 Subgroups.

1.3 The order of a group and the order of an element.

1.4 Lagrange’s theorem.

1.5 Isomorphism.

8.1 Understand the definitions of hyperbolic functions sinh x, cosh x and tanh x, including their domains and ranges, and be able to sketch their graphs.

8.2 Differentiate and integrate hyperbolic functions.

8.3 Understand and be able to use the definitions of the inverse hyperbolic functions and their domains and ranges.

8.4 Derive and use the logarithmic forms of the inverse hyperbolic functions.

8.5 Integrate functions of the form (x² + a²)^(−½) and (x² − a²)^(−½) and be able to choose substitutions to integrate associated functions.

4.1 Extend ideas of hypothesis tests to test for the mean of a Poisson distribution

4.2 Extend hypothesis testing to test for the parameter p of a geometric distribution.

7.1 The manipulation and solution of algebraic inequalities and inequations, including those involving the modulus sign.

5.1 Formulation of problems as linear programs including the meaning and use of slack, surplus and artificial variables.

5.2 Graphical solution of two variable problems using objective line and vertex methods including cases where integer solutions are required.

5.3 The Simplex algorithm and tableau for maximising and minimising problems with ≤ constraints.

5.4 The two-stage Simplex and big-M methods for maximising and minimising problems which may include both ≤ and ≥ constraints.

1.1 Least squares linear regression. The concept of residuals and minimising the sum of squares of residuals.

1.2 Residuals.

1.2 The residual sum of squares (RSS)

3.1 Add, subtract and multiply conformable matrices.

3.1 Multiply a matrix by a scalar.

3.2 Understand and use zero and identity matrices.

3.3 Single transformations in 3-D.

3.3 Successive transformations.

3.3 Use matrices to represent linear transformations in 2-D.

3.4 Find invariant points and lines for a linear transformation.

3.5 Calculate determinants of 2 x 2 and 3 x 3 matrices and interpret as scale factors, including the effect on orientation.

3.6 Calculate and use the inverse of non-singular 2 x 2 matrices and 3 x 3 matrices.

3.6 Properties of inverse matrices.

3.6 Understand and use singular and non-singular matrices.

3.7 Solve three linear simultaneous equations in three variables by use of the inverse matrix.

3.8 Interpret geometrically the solution and failure of solution of three simultaneous linear equations.

1.1 Momentum and impulse. The impulse-momentum principle. The principle of conservation of momentum applied to two spheres colliding directly.

1.2 Momentum as a vector. The impulse-momentum principle in vector form.

1.1 Angular speed. v = rω.

1.1 The forms rω² and v²/r are required.

1.1 Uniform motion of a particle moving in a horizontal circle. Radial acceleration in circular motion.

1.2 Motion of a particle in a vertical circle. Radial and tangential acceleration in circular motion. Kinetic and potential energy and the conservation of energy principle applied to motion in a vertical circle.

5.1 An understanding of the division theorem and its application to the Euclidean Algorithm and congruences.

5.2 Bezout's identity.

5.3 Modular arithmetic. Understanding what is meant by two integers a and b to be congruent modulo an integer n. Properties of congruences.

5.4 Fermat's Little Theorem.

5.5 Divisibility Tests.

5.6 Solution of congruence equations.

5.7 Combinatorics: counting problems, permutations and combinations.

6.1 Hypothesis test and confidence interval for the variance of a Normal distribution.

6.2 Hypothesis test that two independent random samples are from Normal populations with equal variances.

Comprehend and critique mathematical arguments, proofs and justifications of methods and formulae, including those relating to applications of mathematics

Construct and present mathematical arguments through appropriate use of diagrams; sketching graphs; logical deduction; precise statements involving correct use of symbols and connecting language, including: constant, coefficient, expression, equation, function, identity, index, term, variable

Understand and use language and symbols associated with set theory, as set out in the glossary

Understand and use mathematical language and syntax as set out in the glossary

Understand and use the definition of a function; domain and range of functions

Construct extended arguments to solve problems presented in an unstructured form, including problems in context

Interpret and communicate solutions in the context of the original problem

Recognise the underlying mathematical structure in a situation and simplify and abstract appropriately to enable problems to be solved

Understand the concept of a mathematical problem solving cycle, including specifying the problem, collecting information, processing and representing information and interpreting results, which may identify the need to repeat the cycle

Understand, interpret and extract information from diagrams and construct mathematical diagrams to solve problems

Interpret the outputs of a mathematical model in the context of the original situation (for a given model or a model constructed or selected by the student)

Translate a situation in context into a mathematical model, making simplifying assumptions

Understand and use modelling assumptions

Understand that a mathematical model can be refined by considering its outputs and simplifying assumptions; evaluate whether the model is appropriate]

Use a mathematical model with suitable inputs to engage with and explore situations (for a given model or a model constructed or selected by the student)

2.1 The additive property of Poisson distributions

2.1 The Poisson distribution

2.2 The mean and variance of the binomial distribution and the Poisson distribution.

2.3 The use of the Poisson distribution as an approximation to the binomial distribution.

7.1 Understand and use polar coordinates and be able to convert between polar and Cartesian coordinates.

7.2 Sketch curves with r given as a function of θ, including use of trigonometric functions.

7.3 Find the area enclosed by a polar curve.

7.1 Definitions, derivations and applications.

7.1 Use of the probability generating function for the negative binomial, geometric, binomial and Poisson distributions.

7.2 Use to find the mean and variance.

7.3 Probability generating function of the sum of independent random variables.

1.1 Construct proofs using mathematical induction.

1.1 Contexts include sums of series, divisibility and powers of matrices.

8.1 Size and Power of Test.

8.1 The power function.

8.1 Type I and Type II errors.

6.1 Use of recurrence relations to model appropriate problems.

6.2 Solution of first and second order linear homogeneous and non-homogeneous recurrence relations.

1.1 The north-west corner method for finding an initial basic feasible solution.

1.2 Use of the stepping-stone method for obtaining an improved solution. Improvement indices.

1.3 Formulation of the transportation problem as a linear programming problem.

2.1 Kinetic and potential energy, work and power. The work-energy principle. The principle of conservation of mechanical energy.