Cambridge OCR Level 3 Advanced GCE in Further Mathematics B (MEI)

Be able to form a new equation whose roots are related to the roots of a given equation by a linear transformation.

Understand and use the relationships between the roots and coefficients of quadratic, cubic and quartic equations.

Be able to derive formulae for and calculate the volumes of the solids generated by rotating a plane region about the x-axis or the y-axis.

Be able to differentiate inverse trigonometric functions.

Be able to use the method of partial fractions in integration, including where the denominator has a quadratic factor of form ax² + c and one linear term.

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

Recognise integrals of functions of the form (a² − x²)^(−1/2) and (a² + x²)⁻¹ and be able to integrate related functions by using trigonometric substitutions.

Understand and evaluate the mean value of a function.

Understand the definitions of inverse trigonometric functions.

Be able to add, subtract, multiply and divide complex numbers given in the form x + yi, x and y real.

Be able to multiply and divide complex numbers in modulus-argument form.

Be able to represent a complex number in modulus-argument form. Be able to convert between the forms z = x + yi and z = r(cos θ + i sin θ) where r is the modulus and θ is the argument of the complex number.

Be able to represent and interpret complex numbers and their conjugates on an Argand diagram.

Be able to represent and interpret sets of complex numbers as loci on an Argand diagram.

Be able to represent the sum, difference, product and quotient of two complex numbers on an Argand diagram.

Be able to solve any quadratic equation with real coefficients.

Be able to use radians in the context of complex numbers.

Know that the complex roots of polynomial equations with real coefficients occur in conjugate pairs. Be able to solve cubic or quartic equations with real coefficients.

Understand that a complex number is zero if and only if both the real and imaginary parts are zero.

Understand the language of complex numbers.

Be able to apply complex numbers to geometrical problems.

Be able to apply de Moivre’s theorem to finding multiple angle formulae and to summing suitable series.

Be able to explain why the sum of all the nth roots is zero.

Be able to represent complex roots of unity on an Argand diagram.

Know that every non-zero complex number has n distinct nth roots, and that on an Argand diagram these are the vertices of a regular n-gon.

Know that the distinct nth roots of re^(iθ) are: r^(1/n)[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] for k = 0, 1 ... n − 1.

Understand and use de Moivre’s theorem.

Understand the definition e^(iθ) = cos θ + i sin θ and hence the form z = re^(iθ).

Understand the effect of multiplication by a complex number on an Argand diagram.

Analyse and interpret model situations with one independent variable and two dependent variables which lead to coupled 1st order simultaneous linear differential equations and find the solution.

Be able to find an integrating factor and understand its significance in the solution of an equation.

Be able to find particular integrals in simple cases. Understand the relationship between different cases of the solution and the nature of the roots of the auxiliary equation.

Be able to interpret the solutions of equations modelling damped oscillations in words and graphically.

Be able to model damped oscillations using 2nd order differential equations.

Be able to relate 1st and 2nd order derivatives to verbal descriptions and so formulate differential equations.

Be able to solve an equation using an integrating factor and find both general and particular solutions.

Be able to solve differential equations of the form y″ + ay′ + by = 0, using the auxiliary equation.

Be able to solve differential equations of the form y″ + ay′ + by = f(x), by solving the homogeneous case and adding a particular integral to the complimentary function.

Be able to solve the equation for simple harmonic motion, ẍ = −ω²x, and be able to relate the solution to the motion.

Know Newton’s 2nd law of motion.

Know the difference between a general solution and a particular solution. Be able to find both general and particular solutions.

Know the language of kinematics, and the relationships between the various variables.

Recognise differential equations where the integrating factor method is appropriate.

Understand and use the relationship between different cases of the solution and the nature of the roots of the auxiliary equation.

Understand how to introduce and define variables to describe a given situation in mathematical terms.

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

Be able to derive and use the logarithmic forms of the inverse hyperbolic functions.

Be able to differentiate and integrate hyperbolic functions.

Recognise integrals of functions of the form (x² + a²)^(−1/2) and (x² − a²)^(−1/2) and be able to integrate related functions by using substitutions.

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

Understand and use the identity cosh²x − sinh²x = 1.

Understand the definitions of hyperbolic functions, know their domains and ranges and be able to sketch their graphs.

Be able to add, subtract and multiply conformable matrices, and to multiply a matrix by a scalar.

Be able to calculate the determinant of a 2 × 2 matrix and a 3 × 3 matrix. Know the meaning of the terms singular and non-singular as applied to matrices.

Be able to calculate the inverse of a non-singular 2 × 2 matrix or 3 × 3 matrix.

Be able to find the matrix associated with a linear transformation and vice versa.

Be able to use the inverse of a non-singular 2 × 2 or 3 × 3 matrix. Relate the inverse matrix to the corresponding inverse transformation.

Know that det(MN) = det M × det N and the corresponding result for scale factors of transformations.

Know that matrix multiplication is associative but not commutative.

Know that the magnitude of the determinant of a 2 × 2 matrix gives the area scale factor of the associated transformation, and understand the significance of a zero determinant. Interpret the sign of a determinant in terms of orientation of the image.

Know that the magnitude of the determinant of a 3 × 3 matrix gives the volume scale factor of the associated transformation, and understand the significance of a zero determinant. Interpret the sign of a determinant in terms of orientation of the image.

Know the meaning of, and be able to find, invariant points and invariant lines for a linear transformation.

Understand and use the product rule for inverse matrices.

Understand and use the zero and identity matrices, understand what is meant by equal matrices.

Understand successive transformations in two dimensions and the connection with matrix multiplication.

Understand the language of vectors in two dimensions and three dimensions.

Understand what is meant by an inverse matrix.

Be able to find the determinant and inverse of a 3 × 3 matrix without a calculator.

Be able to find the area enclosed by a polar curve.

Be able to sketch curves with simple polar equations where r is given as a function of θ.

Understand and use polar coordinates (r, θ) and be able to convert from polar to cartesian coordinates and vice-versa.

Be able to construct and present a proof using mathematical induction for given results for a formula for the nth term of a sequence, the sum of a series or the nth power of a matrix.

Be able to prove mathematical results by deduction and exhaustion, and disprove false conjectures by counter-example.

Be able to construct and present a proof using mathematical induction.

Be able to prove mathematical results by contradiction.

Be able to use standard formulae for Σr, Σr² and Σr³ and the method of differences to sum series.

Be able to find the Maclaurin series of a function, including the general term.

Be able to recognise and use the Maclaurin series of standard functions: eˣ, ln(1 + x), sin x, cos x and (1 + x)ⁿ.

Be able to sum a simple series using partial fractions.

Know that a Maclaurin series may converge only for a restricted set of values of x.

Know the difference between a sequence and a series.

Know the meaning of the word converge when applied to either a sequence or a series.

Be able to form and use the vector and cartesian equations of a plane. Convert between vector and cartesian forms for the equation of a plane.

Be able to solve three linear simultaneous equations in three variables by use of the inverse of the corresponding matrix. Interpret the solution or failure of solution geometrically in terms of the arrangement of three planes. Be able to find the intersection of three planes when they meet in a point.

Know how to calculate the scalar product of two vectors, and be able to use the two forms of the scalar product to find the angle between two vectors.

Know that a vector which is perpendicular to a plane is perpendicular to any vector in the plane.

Know that the angle between two planes can be found by considering the angle between their normals.

Know the different ways in which three distinct planes can be arranged in 3-D space.

Be able to calculate the angle between a line and a plane.

Be able to calculate the angle between two lines.

Be able to determine whether two lines in three dimensions are parallel, skew or intersect, and to find the point of intersection if there is one.

Be able to find the distance between two parallel lines and the shortest distance between two skew lines.

Be able to find the distance from a point to a line in 2 or 3 dimensions.

Be able to find the distance from a point to a plane.

Be able to find the intersection of a line and a plane.

Be able to form and use the equation of a line in 2-D and 3-D.

Be able to use the alternative form for the vector product. Know the significance of a × b = 0.

Be able to use the vector product in component form to give a vector perpendicular to two given vectors.

Know the different ways in which two lines can intersect or not in 3-D space.

Be able to specify an isomorphism between two groups. Know that two cyclic groups of the same order are isomorphic and be able to specify an isomorphism.

Be familiar with examples of groups, and of the use of group tables for finite groups.

For finite groups, understand the meaning of the term cyclic group, and how such a group is generated by a single element.

Know and be able to use Lagrange’s theorem.

Understand and use the common notation for sets of numbers. Know and use properties of these numbers.

Understand and use the language and notation of sets.

Understand and work with subgroups. Know the conditions for a subset of a group to be a subgroup.

Understand that different situations can give rise to the same underlying structure.

Understand the group axioms and the associated language.

Understand the terms order of a finite group, order of an element.

Be able to find powers of a 2 × 2 or 3 × 3 matrix using diagonal form.

Be able to form and solve the characteristic equation of a 2 × 2 or 3 × 3 matrix.

Be able to form the matrix of eigenvectors and use this to reduce a matrix to diagonal form.

Understand and be able to use the Cayley-Hamilton theorem.

Understand the meaning of eigenvalue and eigenvector, and be able to find these for 2 × 2 and 3 × 3 matrices.

Understand the significance of eigenvalues and eigenvectors in 2-D and 3-D transformations.

Be able to find first order partial derivatives.

Be able to find grad g, and to evaluate this at a point on the surface to give a normal vector.

Be able to sketch contours and sections, and know how these are related to the surface.

Be able to use the conditions ∂z/∂x = 0 and ∂z/∂y = 0 to find the coordinates of stationary points on a surface.

Know that the relation g(x, y, z) = c defines a surface in 3-D.

Know that the relation z = f(x, y) defines a surface in 3-D.

Understand the concepts of, and be able to find and use, the equations of the normal line and tangent plane at a point on the surface.

Be able to investigate and comment on properties of solutions of recurrence relations.

Be able to investigate and comment on the behaviour of recurrence relations, including long-run behaviour.

Be able to model appropriate problems by using recurrence relations.

Be able to solve first order linear homogeneous recurrence relations with constant coefficients.

Be able to solve first order linear non-homogeneous recurrence relations with constant coefficients.

Be able to solve second order linear homogeneous recurrence relations with constant coefficients.

Be able to solve second order linear non-homogenous recurrence relations with constant coefficients.

Be able to verify a given solution of a recurrence relation.

Understand and use the language of recurrence relations.

Be able to construct, adapt or interpret a spreadsheet to solve first order differential equations numerically.

Be able to sketch a tangent field for a first order differential equation and be able to interpret it.

Be able to solve first order differential equations using Runge-Kutta methods.

Be able to use software to produce a tangent field for a differential equation.

Be able to use software to produce analytical solutions to (families of) first order differential equations, when this is possible.

Be able to verify a given solution of a differential equation.

Be able to work with particular solutions and families of particular solutions.

Know how to solve a given first order differential equation dy/dx = f(x, y) with initial conditions to any required degree of accuracy by repeated application of the Euler method or a modified Euler method.

Understand that a smaller step length usually gives a more accurate answer. Understand that a modified Euler method usually gives a more accurate solution than an Euler method for a given step length.

Understand the concepts underlying Runge-Kutta methods.

Be able to calculate arc length using cartesian, parametric and polar coordinates.

Be able to convert equations between cartesian and polar forms in all cases. Be able to convert equations from polar to parametric form, and parametric to cartesian form.

Be able to determine asymptotes.

Be able to find and work with equations of chords, tangents and normals.

Be able to find the gradient of the tangent to a curve at a point.

Be able to find, describe and generalise properties of curves.

Be able to identify cusps by examining the behaviour nearby.

Be able to plot a family of curves in graphing software, in cartesian, polar and parametric forms.

Be able to use CAS to work with equations of curves.

Be able to use the limit of f(x) as x → a or x → ∞ to investigate and deduce properties of a curve.

Know and use the vocabulary associated with curves

Understand the meaning of an envelope of a family of curves. Be able to find an envelope of a family of curves.

Be able to find Pythagorean triples and use related equations.

Be able to identify the limitations of a short program and suggest refinements to it.

Be able to solve other Diophantine equations and use solutions to solve related problems.

Be able to solve Pell’s equation and use solutions to solve related problems.

Be able to solve problems using modular arithmetic.

Be able to write, adapt and interpret programs to solve number theory problems.

Know and use Euler’s totient function, φ(n), Euler’s theorem.

Know and use Fermat’s little theorem.

Know and use the unique prime factorisation of natural numbers.

Know and use Wilson’s theorem.

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.

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).

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.

Be able to find the centre of mass of a composite body by considering each constituent part as a particle at its centre of mass.

Be able to find the centre of mass of a system of particles of given position and mass.

Be able to use the position of the centre of mass in situations involving the equilibrium of a rigid body.

Know how to locate centre of mass by appeal to symmetry.

Know the positions of the centres of mass of a uniform rod, a rectangular lamina and a triangular lamina.

Be able to calculate the volume generated by rotating a plane region about an axis.

Be able to find the centre of mass of a compound body, parts of which are solids of revolution.

Be able to use calculus methods to calculate the centre of mass of a plane lamina.

Be able to use calculus methods to calculate the centre of mass of solid bodies formed by rotating a plane region about an axis.

Be able to use the position of the centre of mass in situations involving the equilibrium of a rigid body.

Be able to calculate acceleration towards the centre of circular motion.

Be able to calculate tangential acceleration.

Be able to model situations involving circular motion with non-uniform speed.

Be able to model situations involving circular motion with uniform speed in a horizontal plane.

Be able to model situations involving motion in a vertical circle.

Identify the conditions under which a particle departs from circular motion.

Identify the force(s) acting on a body in circular motion.

Understand the language associated with circular motion.

Be able to change the units in which a quantity is given.

Be able to determine the units of a quantity by reference to its dimensions.

Be able to find the dimensions of a quantity in terms of M, L, T.

Be able to use dimensional analysis to check the consistency of a relationship.

Understand that some quantities are dimensionless.

Use a model based on dimensional analysis.

Use dimensional analysis to determine unknown indices in a proposed formula.

Be able to apply Newton’s laws to situations involving friction.

Be able to calculate the moments about a fixed axis of forces acting on a body. Be able to calculate the moment of a couple.

Be able to derive and use the result that a body on a rough slope inclined at an angle α to the horizontal is on the point of slipping if μ = tan α.

Be able to draw a force diagram for a rigid body.

Be able to find the resultant of several concurrent forces by vector addition.

Be able to formulate and solve equations for equilibrium by resolving forces in suitable directions, or by drawing and using a polygon of forces.

Be able to identify whether equilibrium will be broken by sliding or toppling.

Be able to resolve a force into components and be able to select suitable directions for resolution.

Know that a closed figure may be drawn to represent the addition of the forces on an object in equilibrium.

Know that a particle is in equilibrium under a set of concurrent forces if and only if their resultant is zero.

Know the meaning of the term couple.

Understand and be able to apply the conditions for equilibrium of a rigid body.

Understand that a system of forces can have a turning effect on a rigid body.

Understand that bodies in contact may be subject to a frictional force as well as a normal contact force (normal reaction), and be able to represent the situation in an appropriate force diagram.

Understand that the frictional force may be modelled by F ≤ μR and that friction acts in the direction to oppose sliding. Model friction using F = μR when sliding occurs.

Understand that the total contact force between surfaces may be expressed in terms of a frictional force and a normal contact force (normal reaction).

Understand the language relating to forces. Understand that the value of the normal reaction depends on the other forces acting and why it cannot be negative.

Be able to calculate energy stored in a string or spring.

Be able to calculate the equilibrium position of a system involving elastic strings or springs.

Be able to calculate the stiffness or modulus of elasticity in a given situation.

Be able to calculate the tension in an elastic string or spring.

Be able to use energy principles to model a system involving elastic strings or springs including determining extreme positions.

Understand that Hooke’s law models the extension/compression of a material as a linear function of tension/thrust.

Understand the language associated with elasticity.

Understand when Hooke’s law is not applicable.

Be able to apply Newton’s Experimental Law in modelling direct impacts.

Be able to apply the principle of conservation of linear momentum to direct impacts within a system of bodies.

Be able to calculate the impulse of a force as a vector and in component form.

Be able to model situations involving direct impact using both conservation of linear momentum and Newton’s Experimental Law.

Know the meanings of Newton’s Experimental Law and of coefficient of restitution when applied to a direct impact.

Understand and use the concept of linear momentum and appreciate that it is a vector quantity.

Understand and use the impulse-momentum equation.

Understand and use the principle that a system subject to no external force has constant total linear momentum and that this result may be applied in any direction.

Understand that when e < 1 kinetic energy is not conserved during impacts and be able to find the loss of kinetic energy.

Understand the significance of e = 0.

Understand the significance of e = 1.

Understand the term direct impact and the assumptions made when modelling direct impact collisions.

Be able to calculate the loss of kinetic energy in an oblique impact.

Be able to model situations involving oblique impact between an object and a smooth plane by considering the components of its motion parallel and perpendicular to the line of the impulse.

Be able to model situations involving oblique impact between two bodies by considering the components of their motion in directions parallel and perpendicular to the line of the impulse.

Know the meanings of Newton’s Experimental Law and of coefficient of restitution when applied to an oblique impact.

Understand the term oblique impact and the assumptions made when modelling oblique impact collisions.

Be able to eliminate a parameter from the expressions for the position vector of a particle, thereby forming a single equation.

Be able to extend the scope of techniques from motion in 1 dimension to that in 2 and 3 dimensions by using vectors.

Be able to find the acceleration, velocity and position vector of a particle subject to a constant or variable force in 1, 2 and 3 dimensions.

Be able to find the maximum range of a projectile up or down a uniform slope, and the associated angle of projection.

Be able to find the range of a projectile up or down a uniform slope.

Be able to formulate differential equations for motion under variable acceleration in 1 and 2 dimensions.

Be able to interpret the equation resulting from the elimination of a parameter from the terms of a position vector.

Be able to recognise and formulate the simple harmonic motion equation expressed in non-standard forms and to transform it into the standard form by means of substitution.

Be able to solve the equation for simple harmonic motion, ẍ = −ω²x, and be able to relate the solution to the context.

Be able to use the acceleration, velocity and position vector of a particle to model situations in 1, 2 and 3 dimensions.

Be able to use the boundary or initial conditions to produce a particular solution from a general solution.

Be able to verify a general or particular solution of a differential equation for motion under variable acceleration.

Derive the cartesian equation of the path of a particle in 2 dimensions from an expression for its position vector.

Understand the language of kinematics appropriate to motion in 2 and 3 dimensions. Know the distinction between displacement and distance, between velocity and speed, and between acceleration and magnitude of acceleration. Know the distinction between distance from and distance travelled.

Be able to calculate gravitational potential energy.

Be able to calculate kinetic energy.

Be able to calculate the work done by a force which moves along its line of action.

Be able to calculate the work done by a force which moves at an angle to its line of action.

Understand and use the concept of the power of a force as the rate at which it does work.

Understand and use the work-energy principle.

Understand the language relating to work, energy and power.

Understand when the principle of conservation of energy may be applied and be able to use it appropriately.

Be able to find the centre of mass of a composite body by considering each constituent part as a particle at its centre of mass.

Be able to find the centre of mass of a system of particles of given position and mass.

Be able to use the position of the centre of mass in situations involving the equilibrium of a rigid body.

Know how to locate centre of mass by appeal to symmetry.

Know the positions of the centres of mass of a uniform rod, a rectangular lamina and a triangular lamina.

Be able to change the units in which a quantity is given.

Be able to determine the units of a quantity by reference to its dimensions.

Be able to find the dimensions of a quantity in terms of M, L, T.

Be able to use dimensional analysis to check the consistency of a relationship.

Understand that some quantities are dimensionless.

Use a model based on dimensional analysis.

Use dimensional analysis to determine unknown indices in a proposed formula.

Be able to apply Newton’s laws to situations involving friction.

Be able to calculate the moments about a fixed axis of forces acting on a body. Be able to calculate the moment of a couple.

Be able to derive and use the result that a body on a rough slope inclined at an angle α to the horizontal is on the point of slipping if μ = tan α.

Be able to draw a force diagram for a rigid body.

Be able to find the resultant of several concurrent forces by vector addition.

Be able to formulate and solve equations for equilibrium by resolving forces in suitable directions, or by drawing and using a polygon of forces.

Be able to identify whether equilibrium will be broken by sliding or toppling.

Be able to resolve a force into components and be able to select suitable directions for resolution.

Know that a closed figure may be drawn to represent the addition of the forces on an object in equilibrium.

Know that a particle is in equilibrium under a set of concurrent forces if and only if their resultant is zero.

Know the meaning of the term couple.

Understand and be able to apply the conditions for equilibrium of a rigid body.

Understand that a system of forces can have a turning effect on a rigid body.

Understand that bodies in contact may be subject to a frictional force as well as a normal contact force (normal reaction), and be able to represent the situation in an appropriate force diagram.

Understand that the frictional force may be modelled by F ≤ μR and that friction acts in the direction to oppose sliding. Model friction using F = μR when sliding occurs.

Understand that the total contact force between surfaces may be expressed in terms of a frictional force and a normal contact force (normal reaction).

Understand the language relating to forces. Understand that the value of the normal reaction depends on the other forces acting and why it cannot be negative.

Be able to apply Newton’s Experimental Law in modelling direct impacts.

Be able to apply the principle of conservation of linear momentum to direct impacts within a system of bodies.

Be able to calculate the impulse of a force as a vector and in component form.

Be able to model situations involving direct impact using both conservation of linear momentum and Newton’s Experimental Law.

Know the meanings of Newton’s Experimental Law and of coefficient of restitution when applied to a direct impact.

Understand and use the concept of linear momentum and appreciate that it is a vector quantity.

Understand and use the impulse-momentum equation.

Understand and use the principle that a system subject to no external force has constant total linear momentum and that this result may be applied in any direction.

Understand that when e < 1 kinetic energy is not conserved during impacts and be able to find the loss of kinetic energy.

Understand the significance of e = 0.

Understand the significance of e = 1.

Understand the term direct impact and the assumptions made when modelling direct impact collisions.

Be able to calculate gravitational potential energy.

Be able to calculate kinetic energy.

Be able to calculate the work done by a force which moves along its line of action.

Be able to calculate the work done by a force which moves at an angle to its line of action.

Understand and use the concept of the power of a force as the rate at which it does work.

Understand and use the work-energy principle.

Understand the language relating to work, energy and power.

Understand when the principle of conservation of energy may be applied and be able to use it appropriately.

Be able to count the number of comparisons and/or swaps needed in particular applications of sorting algorithms, and relate this to complexity.

Be able to count the number of comparisons needed in particular applications of packing algorithms, and relate this to complexity.

Be able to interpret and apply algorithms presented in a variety of formats.

Be able to reason about a given sorting algorithm.

Be able to repair, develop and adapt given algorithms.

Know and be able to use first fit and first fit decreasing packing algorithms and full bin strategies.

Know and be able to use the quick sort algorithm. Be able to apply other sorting algorithms which are specified.

Understand and be able to use the basic ideas of algorithmic complexity and be able to analyse the complexity of given algorithms. Know that complexity can be used, among other things, to compare algorithms.

Understand and know the importance of heuristics.

Understand that algorithms can sometimes be proved correct or incorrect.

Understand that an algorithm is a finite sequence of operations for carrying out a procedure or solving a problem. Understand that an algorithm can be the basis for a computer program.

Be able to consider the effect of modifying constraints or the objective function.

Be able to formulate a range of network problems as LPs.

Be able to graph inequalities in 2-D and identify feasible regions. Be able to recognise infeasibility.

Be able to identify and define variables from a given problem. Be able to formulate a problem as a linear program.

Be able to interpret the output from a spreadsheet optimisation routine, or other software, for the simplex method or ILPs.

Be able to recognise when an LP is in standard form.

Be able to reformulate an equality constraint as a pair of inequality constraints.

Be able to solve 2-D integer LP problems graphically.

Be able to solve a 2-D LP graphically.

Be able to use a visualisation of a 3-D LP to solve it. Be able to reduce a 3-D LP to a 2-D LP when one constraint is an equality.

Be able to use slack variables to convert an LP in standard form to augmented form.

Be able to use the simplex algorithm on an LP in augmented form.

Recognise that if an LP has variables which may take negative values or requires the objective function to be minimised then some initial reformulation is required before the simplex algorithm may be applied.

Recognise that if an LP includes ≥ constraints then the two-stage simplex method may be used; understand how this method works and be able to set up the initial tableau in such cases.

Recognise when an LP requires an integer solution.

Understand and use the language associated with linear programming.

Understand that some LPs can be solved using graphical techniques or the simplex method, but for practical problems computing power needs to be applied. Know that a spreadsheet LP solver routine, or other software, can solve an LP given in standard form or, in some cases, in non-standard form.

Understand the geometric basis for the simplex method.

Be able to model problems by using graphs.

Be able to model problems by using networks.

Be able to solve minimum connector problems using Kruskal’s and Prim’s algorithms.

Be able to specify a cut and calculate its capacity.

Be able to use a network to model a transmission system.

Know and use the fact that Kruskal’s, Prim’s and Dijkstra’s algorithms have quadratic complexity.

Model precedence problems with an activity-on-arc network.

Model shortest path problems and solve using Dijkstra’s algorithm.

Understand and be able to use graphs and associated language.

Understand and use the maximum flow/minimum cut theorem.

Understand that a network is a graph with weighted arcs.

Understand that network algorithms can be explored, understood and tested in cases in which the algorithm can be run by hand, but for practical problems the algorithm needs to be formulated in a way suitable for computing power to be applied.

Use critical path analysis and be able to interpret outcomes, including implications for criticality. Be able to analyse float (total, independent and interfering), resourcing and scheduling.

Be able to construct the interpolating polynomial of degree n given a set of n + 1 data points.

Be able to use Newton’s forward difference interpolation formula to reconstruct polynomials and to approximate functions.

Be able to use error analysis to produce an improved solution.

Know how to calculate errors in sums, differences, products and quotients. Know the meaning of absolute and relative error.

Know how to calculate the error in f(x) when there is an error in x.

Understand convergence and divergence when applied to sequences. Understand the order of convergence of an iterative sequence and the order of a method. Be able to comment on these given output from a spreadsheet.

Understand rounding and chopping and their consequences, including for calculations.

Understand that computers represent numbers to limited precision.

Understand the consequences of subtracting nearly equal quantities.

Understand the effects on errors of changing the order of a sequence of operations.

Be able to estimate a derivative using the forward and central difference methods with a suitable value (or sequence of values) of h.

Have an empirical and graphical appreciation of the greater accuracy of the central difference method. Know that the forward difference method is generally a first order method and that the central difference method is generally a second order method.

Know that, generally, the midpoint and trapezium rules are second order methods and Simpson’s rule is a fourth order method. Understand the development of Simpson’s rule from the midpoint and trapezium rules.

Midpoint rule; trapezium rule; Simpson’s rule: Be able to evaluate a given definite integral to any desired degree of accuracy using these methods.

Bisection method; False Position (linear interpolation); Secant method; Fixed point iteration; Newton-Raphson method: Be able to solve equations to any required degree of accuracy using these methods.

Bisection method; False Position (linear interpolation); Secant method; Fixed point iteration; Newton-Raphson method: Understand the graphical interpretations of these methods.

Bisection method; False Position (linear interpolation); Secant method; Fixed point iteration; Newton-Raphson method: Understand the relative computational merits and possible failure of these methods.

Know that fixed point iteration generally has first order convergence, Newton-Raphson generally has second order convergence.

Understand and be able to apply relaxation to a fixed point iteration: to accelerate convergence; to convert a divergent sequence to a convergent sequence.

Be able to use a spreadsheet to implement the methods and to explore associated ideas. Be able to interpret the output from a spreadsheet.

Be able to use the iterative capability of a calculator.

Be able to calculate Spearman’s rank correlation coefficient from raw data or summary statistics.

Be able to calculate the equation of the least squares regression line using raw data or summary statistics.

Be able to calculate the equation of the two least squares regression lines, y on x and x on y, using raw data or summary statistics. Be able to use either regression line to estimate the expected value of one variable for a given value of the other and know when it is appropriate to do so.

Be able to calculate the pmcc from raw data or summary statistics.

Be able to carry out hypothesis tests using Spearman’s rank correlation coefficient and tables of critical values or the output from software.

Be able to carry out hypothesis tests using the pmcc and tables of critical values or the p-value from software.

Be able to use and interpret a scatter diagram.

Be able to use the regression line as a model to estimate values and know when it is appropriate to do so. Know the meaning of the term residual and be able to calculate and interpret residuals.

Check how well the model fits the data.

Decide whether a test based on r or rₛ may be more appropriate, or whether neither is appropriate.

Interpret a scatter diagram produced by software.

Know the relationship between the two regression lines and when to use one rather than the other. Be able to use the correct regression line to estimate the expected value of one variable for a given value of the other and know when it is appropriate to do so.

Know when it is appropriate to carry out a hypothesis test using Pearson’s product moment correlation coefficient.

Understand what bivariate data are and know the conventions for choice of axis for variables in a scatter diagram.

Use the pmcc as an effect size.

Be able to apply the χ² test (chi-squared) to a contingency table.

Be able to carry out a χ² test for goodness of fit of a uniform, binomial or Poisson model.

Be able to interpret bivariate categorical data in a contingency table.

Be able to interpret the results of a χ² test using tables of critical values or the output from software.

Be able to find the mean and variance from a given pdf.

Be able to find the mean of any linear combination of random variables and the variance of any linear combination of independent random variables.

Be able to find the mode and median from a given pdf.

Be able to obtain a pdf from a given cdf.

Be able to use a simple continuous random variable as a model.

Be able to use linear combinations of independent Normal random variables in solving problems.

Be able to use the Normal distribution as a model, and to calculate and use probabilities from a Normal distribution.

Be able to use the Normal distribution, when appropriate, in the construction of confidence intervals.

Interpret a Normal probability plot to decide whether a Normal model might be appropriate. Interpret software output, including p-value, from the Kolmogorov-Smirnov test, to decide whether a Normal model might be appropriate.

Know and use the properties of a pdf. Be able to sketch the graph of a pdf.

Know that the Normal distribution is useful as a model in its own right, and as an approximating distribution in the context of the Central Limit Theorem (CLT).

Understand the meaning of a cdf and be able to obtain one from a given pdf. Be able to sketch a cdf.

Understand the meaning of a pdf and be able to use one to find probabilities.

Use a cdf to calculate the median and other percentiles.

Be able to calculate probabilities using a discrete uniform distribution.

Be able to calculate probabilities using a Poisson distribution.

Be able to calculate the expectation (mean), E(X), and understand its meaning.

Be able to calculate the mean and variance of any given discrete uniform distribution.

Be able to calculate the probabilities within a geometric distribution, including cumulative probabilities.

Be able to calculate the variance, Var(X), and understand its meaning.

Be able to find the expectation of any linear combination of independent random variables and the variance of any linear combination of independent random variables.

Be able to use probability functions, given algebraically or in tables. Be able to calculate the numerical probabilities for a distribution. Be able to draw and interpret graphs representing probability distributions.

Be able to use the mean and variance of a geometric distribution.

Be able to use the result E(a + bX) = a + bE(X) and understand its meaning.

Be able to use the result Var(a + bX) = b²Var(X) and understand its meaning.

Know and be able to use the mean and variance of a Poisson distribution.

Know that the sum of two or more independent Poisson distributions is also a Poisson distribution.

Recognise situations in which both the Poisson distribution and the binomial distribution might be appropriate models.

Recognise situations under which the binomial distribution is likely to be an appropriate model, and be able to calculate probabilities to use the model. Know and be able to use the mean and variance of a binomial distribution, μ = np and σ² = np(1 − p). Prove these results in particular cases.

Recognise situations under which the discrete uniform distribution is likely to be an appropriate model.

Recognise situations under which the geometric distribution is likely to be an appropriate model.

Recognise situations under which the Poisson distribution is likely to be an appropriate model.

Be able to calculate and interpret the standard error of the mean.

Be able to carry out a hypothesis test for a single population mean using the Normal or t distributions and know when it is appropriate to do so.

Be able to carry out a hypothesis test for a single population median using the Wilcoxon signed rank test and know when it is appropriate to do so.

Be able to construct and interpret a confidence interval for a single population mean using the Normal or t distributions and know when it is appropriate to do so.

Be able to construct and interpret a confidence interval for the difference in mean of two paired populations using a paired sample and a Normal or t distribution; know when it is appropriate to do so.

Be able to estimate population mean from sample data.

Be able to estimate population variance using the sample variance.

Interpret confidence intervals given by software.

Know that if the underlying distribution is Normal then the sample mean is Normally distributed.

Know the meaning of the term confidence interval for a parameter and associated language.

Know when samples from two populations should be considered as paired.

Understand how and when the Central Limit Theorem may be applied to the distribution of sample means. Use this result in probability calculations, using a continuity correction where appropriate. Be able to apply the CLT to the sum of n identically distributed independent random variables.

Understand that the sample mean is a random variable with a probability distribution.

Understand the factors which affect the width of a confidence interval.

Use a confidence interval for a population parameter to make a decision about a hypothesised value of that parameter.

Be able to explain the advantage of using a random sample when inferring properties of a population.

Be able to explain the importance of sample size in experimental design.

Be able to explain why sampling may be necessary in order to obtain information about a population, and give desirable features of a sample.

Know that simulations can be used to approximate probability distributions and to estimate probabilities, including in situations where the theory may be technically difficult. Be able to interpret output from spreadsheets investigating such situations.

Know that spreadsheets can be used to simulate probability distributions, and be able to do so for discrete and continuous uniform distributions and Normal distributions.

Be able to calculate Spearman’s rank correlation coefficient from raw data or summary statistics.

Be able to calculate the equation of the least squares regression line using raw data or summary statistics.

Be able to calculate the equation of the two least squares regression lines, y on x and x on y, using raw data or summary statistics. Be able to use either regression line to estimate the expected value of one variable for a given value of the other and know when it is appropriate to do so.

Be able to calculate the pmcc from raw data or summary statistics.

Be able to carry out hypothesis tests using Spearman’s rank correlation coefficient and tables of critical values or the output from software.

Be able to carry out hypothesis tests using the pmcc and tables of critical values or the p-value from software.

Be able to use and interpret a scatter diagram.

Be able to use the regression line as a model to estimate values and know when it is appropriate to do so. Know the meaning of the term residual and be able to calculate and interpret residuals.

Check how well the model fits the data.

Decide whether a test based on r or rₛ may be more appropriate, or whether neither is appropriate.

Interpret a scatter diagram produced by software.

Know the relationship between the two regression lines and when to use one rather than the other. Be able to use the correct regression line to estimate the expected value of one variable for a given value of the other and know when it is appropriate to do so.

Know when it is appropriate to carry out a hypothesis test using Pearson’s product moment correlation coefficient.

Understand what bivariate data are and know the conventions for choice of axis for variables in a scatter diagram.

Use the pmcc as an effect size.

Be able to apply the χ² test (chi-squared) to a contingency table.

Be able to carry out a χ² test for goodness of fit of a uniform, binomial, or Poisson model.

Be able to interpret bivariate categorical data in a contingency table.

Be able to interpret the results of a χ² test using tables of critical values or the output from software.

Be able to calculate probabilities using a discrete uniform distribution.

Be able to calculate probabilities using a Poisson distribution.

Be able to calculate the expectation (mean), E(X), and understand its meaning.

Be able to calculate the mean and variance of any given discrete uniform distribution.

Be able to calculate the probabilities within a geometric distribution, including cumulative probabilities.

Be able to calculate the variance, Var(X), and understand its meaning.

Be able to find the expectation of any linear combination of independent random variables and the variance of any linear combination of independent random variables.

Be able to use probability functions, given algebraically or in tables. Be able to calculate the numerical probabilities for a simple distribution. Be able to draw and interpret graphs representing probability distributions.

Be able to use the result E(a + bX) = a + bE(X) and understand its meaning.

Be able to use the result Var(a + bX) = b²Var(X) and understand its meaning.

Know and be able to use the mean and variance of a geometric distribution.

Know and be able to use the mean and variance of a Poisson distribution.

Know that the sum of two or more independent Poisson distributions is also a Poisson distribution.

Recognise situations in which both the Poisson distribution and the binomial distribution might be appropriate models.

Recognise situations under which the binomial distribution is likely to be an appropriate model, and be able to calculate probabilities to use the model. Know and be able to use the mean and variance of a binomial distribution, μ = np and σ² = np(1 − p). Prove these results in particular cases.

Recognise situations under which the discrete uniform distribution is likely to be an appropriate model.

Recognise situations under which the geometric distribution is likely to be an appropriate model.

Recognise situations under which the Poisson distribution is likely to be an appropriate model.

Be able to explain the advantage of using a random sample when inferring properties of a population.

Be able to explain the importance of sample size in experimental design.

Be able to explain why sampling may be necessary in order to obtain information about a population, and give desirable features of a sample.