Further Mathematics

A Level Further Mathematics is designed for students with a strong aptitude and enthusiasm for mathematics who wish to deepen their understanding beyond the standard A Level course. The curriculum provides a rigorous and intellectually stimulating experience, developing advanced mathematical thinking, logical reasoning and sophisticated problem-solving skills that prepare students for the demands of university study, particularly in mathematics, engineering, physics, computer science and other STEM-related disciplines.

Building upon the content of A Level Mathematics, students explore mathematics in greater depth and encounter more abstract ideas, including complex numbers, matrices, proof, advanced calculus and further applications of mechanics and statistics. The curriculum encourages students to recognise the interconnected nature of mathematics, making links between different branches of the subject and appreciating its elegance, structure and power.

Learning is carefully sequenced to ensure that students develop secure conceptual understanding alongside technical proficiency. Regular opportunities for retrieval, review and challenging problem-solving enable students to consolidate prior learning whilst extending their mathematical knowledge. Students are encouraged to justify their reasoning, communicate complex ideas with precision and develop resilience when tackling unfamiliar and demanding questions.

A key aim of the curriculum is to foster intellectual curiosity and independent learning. Students are expected to think critically, investigate mathematical ideas and apply their knowledge creatively in a variety of contexts. Through exposure to rich and challenging mathematics, they develop the confidence to approach complex problems systematically and to evaluate different methods of solution.

By the end of the course, students will have developed the depth of understanding, analytical skills and mathematical maturity required for success in Further Mathematics examinations and for progression to highly mathematical degree courses. More importantly, they will have cultivated the habits of logical thinking, precision and perseverance that underpin lifelong success in mathematics and beyond.

Year 12

Term 1

Matrices and Transformations

Knowledge and Skills

Add, subtract and multiply conformable matrices

Multiply a matrix by a scalar.

Understand and use zero and identity matrices.

Use matrices to represent linear transformations in 2-D

Successive transformations

Single transformations in 3-D (3-D transformations confined to reflection in one of x = 0, y = 0 or rotation about one of the coordinate axes).

Find invariant points and lines for a linear transformation

Useful Links

Introduction to Complex Numbers

Knowledge and Skills

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

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

Understand and use the complex conjugate

Use and interpret Argand diagrams

Useful Links

Roots of polynomials

Knowledge and Skills

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

Solve any quadratic equation with real coefficients

Solve cubic or quartic equations with real coefficients given sufficient information to deduce at least one root for cubics or at least one complex root or quadratic factor for quartics.

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

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

Useful Links

Roots of equations 1.1

Roots of equations 1.2

Roots of equations 1.3

Roots of equations 1.4

Roots of equations 1.5

Conics

Knowledge and Skills

Sketching graphs of curves with equation y^2= 4ax , x^2/a^2 +y^2/b^2 = 1, x^2/a^2 -y^2/b^2 =1, xy=c^2

including intercepts with axes and equations of asymptotes of hyperbolas.

Single transformations of curves involving translations, stretches parallel to coordinate axes and reflections in the coordinate axes and the lines y = ± x.

Useful Links

Hyperbolic functions

Knowledge and Skills

Understand the definitions of hyperbolic functions sinhx, coshx and tanhx and be able to sketch their graphs.

Understand and be able to use the definitions of the inverse hyperbolic functions

Derive and use the lgarithmic forms of the inverse hyperbolic functions

Understand and use tanhx=sinhx/coshx

Understand and use 〖cosh〗^2 x-〖sinh〗^2 x=1

Useful Links

Hyperbolic functions 1.1

Hyperbolic functions 1.2

Hyperbolic functions 1.3

Hyperbolic functions 1.4

Sequences and series

Knowledge and Skills

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

Understand and use the method of differences for summation of series

Construct proofs using mathematical induction; contexts include sums of series, divisibility, and powers of matrices

Recognise and use the Maclaurin series for e^x, 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).

Useful Links

Sequences & series 1.1

Sequences & series 1.2

Sequences & series 1.3

Sequences & series 1.4

Sequences & series 1.5

AQA Further Pure: Sequences and series 2-1 – YouTube

AQA Further Pure: Sequences and series 2-2

AQA Further Pure: Sequences and series 2-3

AQA Further Pure: Sequences and series 2-4

AQA Further Pure: Sequences & series 3-1 – YouTube

AQA Further Pure: Sequences & series 3-2

AQA Further Pure: Sequences & series 3-3

Assessments:

End of unit written assessment for each topic.

Term 2

Applications of integration

Knowledge and Skills

Derive formulae for and calculate volumes of revolution

Understand and evaluate the mean value of a function

Useful Links

Complex numbers and geometry

Knowledge and Skills

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

Multiply and divide complex numbers in modulus-argument form

Construct and interpret simple loci in the Argand diagram such as |z-a|>r and arg⁡(z-a)=θ

Useful Links

Polar Coordinates

Knowledge and Skills

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

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

Useful Links

Algebra & graphs 3.6

Algebra & graphs 3.7

Rational functions and further algebra

Knowledge and Skills

Inequalities involving polynomial equations (cubic and quartic).

Solving inequalities such as (ax+b)/(cx+d)<ex+f algebraically.

Graphs of rational functions of form (ax+b)/(cx+d)

Asymptotes, points of intersection with coordinate axes or other straight lines

Associated inequalities.

Graphs of rational functions of form (ax^2+bx+c)/(dx^2+ex+f) , including cases when some of these coefficients are zero

Asymptotes parallel to coordinate axes.

Useful Links

Matrices and their inverses

Knowledge and Skills

Calculate determinants of 2×2 matrices

Understand and use singular and non-singular matrices; properties of inverse matrices

Calculate and use the inverse of non-singular 2×2 matrices.

Useful Links

Vectors and 3D spaces

Knowledge and Skills

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

Calculate the scalar product and use it to calculate the angle between two lines.

Check whether vectors are perpendicular by using the scalar product

Find the intersection of two lines.

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

Useful Links

AQA Further Pure: Vectors 1-1

AQA Further Pure: Vectors 1-2

AQA Further Pure: Vectors 1-3

AQA Further Pure: Vectors 2-1

AQA Further Pure: Vectors 2-2

AQA Further Pure: Vectors 2-3

AQA Further Pure: Vectors 2-4

AQA Further Pure: Vectors 2-5

AQA Further Pure: Vectors 2-6

AQA Further Pure: Vectors 2-7

AQA Further Pure: Vectors 2-8

Assessments:

End of unit written assessment for each topic.

Term 3

Mechanics – Dimensional Analysis

Knowledge and Skills

Finding dimensions of quantities

Prediction of formulae

Useful Links

AQA Mechanics: Dimensional analysis 1-1

AQA Mechanics: Dimensional analysis 1-2

AQA Mechanics: Dimensional analysis 1-3

AQA Mechanics: Dimensional analysis 1-4

Momentum and collisions

Knowledge and Skills

Conservation of momentum for linear motion and cases where velocities are given as one or two dimensional vectors (resolving will not be required at AS level, problems which require resolving will be required at A-level).

Coefficient of restitution and Newton’s Experimental Law. Use in direct collisions and impacts with a fixed smooth surface.

Impulse and its relation to momentum (in one- or two-dimensions)

Use of Ft = mv – mu

Impulse for variable forces. One dimension only. Use of I =∫▒F dt

Useful Links

AQA Mechanics: Impulse & momentum 1-1

AQA Mechanics: Impulse & momentum 1-2

AQA Mechanics: Impulse & momentum 1-3

AQA Mechanics: Impulse & momentum 1-4

AQA Mechanics: Impulse & momentum 1-5

AQA Mechanics: Impulse & momentum 2-1

AQA Mechanics: Impulse & momentum 2-2

AQA Mechanics Impulse and Momentum 2 3

AQA Mechanics: Impulse & momentum 2-4

Work, energy and power

Knowledge and Skills

Work done by a force acting in the direction of motion or directly opposing the motion.

Use of WD=Fdcosθ

Gravitational potential energy. Use in conservation of energy problems.

Kinetic energy. Use in conservation of energy problems.

Hooke’s Law including using modulus of elasticity.

Use of T = kx or T=λ/l x

Work done by a variable force.

Use of WD=∫F dx. Use in conservation of energy problems.

Elastic potential energy using modulus of elasticity. Use of EPE=(Kx^2)/2 and EPE=(λx^2)/2l

Power. Use of P=Fv

Useful Links

AQA Mechanics: Work, energy & power 1-1

AQA Mechanics: Work, energy & power 1-2

AQA Mechanics: Work, energy & power 1-3

AQA Mechanics: Work, energy & power 1-4

AQA Mechanics: Work, energy & power 1-5

AQA Mechanics: Work, energy & power 1-6

AQA Mechanics: Work, energy & power 2-1

AQA Mechanics: Work, energy & power 2-2

AQA Mechanics: Work, energy & power 2-3

AQA Mechanics: Work, energy & power 2-4

Circular Motion

Knowledge and Skills

Motion of a particle moving in a circle with constant speed (knowledge of radians assumed).

Understand the definition of angular speed. Use both radians and revolutions per unit time.

Relationships between speed, angular speed, radius and acceleration. Use of v=rω, a=rω^2 and a=v^2/r

Useful Links

AQA Mechanics: Circular motion 1-1

AQA Mechanics: Circular motion 1-2

AQA Mechanics: Circular motion 1-3

Statistics – Discrete Random Variables and Expectation

Knowledge and Skills

Understand DRVs with distributions given in the form of a table or function.

Evaluate probabilities for a DRV.

Evaluate measures of average and spread for a DRV to include mean, variance, standard deviation, mode and median.

Understand expectation and know the formulae: E(X)=∑▒〖x_i p_i 〗, E(X^2 )=∑▒〖〖x_i〗^2 p_i 〗, Var(X)=E(X^2)-E〖(X)〗^2

Understand expectation of linear functions of DRVs and know the formulae: E(aX+b)=aE(X)+b, Var(aX+b)=a^2 Var(X)

Know the discrete uniform distribution defined on the set {1, 2, …, n}. Understand when this distribution can be used as a model.

Proof of mean and variance of discrete uniform distribution.

Useful Links

AQA Statistics: Discrete random variables 1-1

AQA Statistics: Discrete random variables 1-2

AQA Statistics: Discrete random variables 1-3

AQA Statistics: Discrete random variables 1-4

AQA Statistics: Discrete random variables 1-5

Poisson Distribution

Knowledge and Skills

Understand conditions for a Poisson distribution to model a situation. Understand terminology X~Po(λ)

Know Poisson formula and calculate Poisson probabilities using the formula or equivalent calculator function

Know mean, variance and standard deviation of a Poisson distribution.

Use the result that, if X~Po(λ) then the mean and variance of X are equal

Understand the distribution of the sum of independent Poisson distributions.

Formulate hypotheses and carry out a hypothesis test of a population mean from a single observation from a Poisson distribution using direct evaluation of Poisson probabilities.

Understand Type I and Type II errors and define in context. Calculate the probability of making a Type I error from tests based on a Poisson or Binomial distributions

Useful Links

AQA Statistics: Discrete random variables 2-1

AQA Statistics: Discrete random variables 2-2

AQA Statistics: Discrete random variables 2-3

AQA Statistics: Discrete random variables 2-4

AQA Statistics: Discrete random variables 3-1

AQA Statistics: Discrete random variables 3-2

AQA Statistics: Discrete random variables 3-3

AQA Statistics: Discrete random variables 3-4

AQA Statistics: Discrete random variables 3-5

AQA Statistics: Errors in tests 1-1

AQA Statistics: Errors in tests 1-2

AQA Statistics: Errors in tests 1-3

Continuous Random Variables

Knowledge and Skills

Understand and use a probability density function, f(x), for a continuous distribution and understand the differences between discrete and continuous distributions

Find the probability of an observation lying in a specified interval

Find median and quartiles for given probability density function, f(x)

Find mean, variance and standard deviation for given CRV function, f(x).

Know the formulae:

E(X)=∫▒〖x f(x) dx〗, E(X^2 )=∫▒〖x^2 f(x) dx〗, Var(X)=E(X^2 )-E〖(X)〗^2

Understand expectation of linear functions of CRVs and know the formulae:

E(aX+b)=aE(X)+b

Var(aX+b)=a^2 Var(X)

Know the formula E(g(X))=∫▒〖g(x) f(x) 〗 dx

Find the mean, variance and standard deviation of functions of a continuous random variable such as E(5X^3), E(18X^(-3)), Var(6X^(-1))

Know that if X and Y are independent (discrete or continuous) random variables then E(X+Y)=E(X)+E(Y) and Var(X+Y)=Var(X)+Var(Y)

Useful Links

AQA Statistics: Continuous random variables 1-1

AQA Statistics: Continuous random variables 1-2

AQA Statistics: Continuous random variables 1-3

AQA Statistics: Continuous random variables 1-4

AQA Statistics: Continuous random variables 1-5

AQA Statistics: Continuous random variables 1-6

AQA Statistics: Continuous random variables 2-1

AQA Statistics: Continuous random variables 2-2

AQA Statistics: Continuous random variables 2-3

AQA Statistics: Continuous random variables 3-1

AQA Statistics: Continuous random variables 3-2

AQA Statistics: Continuous random variables 3-3

Chi-Squared Tests

Knowledge and Skills

Construction of m×n contingency tables

Use of ∑▒〖(O_i-E_i)〗^2/E_i as an approximate χ^2 statistic with appropriate degrees of freedom

Know and use the convention that all E_i should be greater than 5

Identification of sources of association in the context of a question

Knowledge of when and how to apply Yates’ correction.

Useful Links

AQA Statistics: Chi-squared tests 1-1

AQA Statistics: Chi-squared tests 1-2

AQA Statistics: Chi-squared tests 1-3

Confidence Intervals

Knowledge and Skills

Construct symmetric confidence intervals for the mean of a normal distribution with known variance.

Construct symmetric confidence intervals from large samples, of the mean of a normal distribution with unknown variance.

Make inferences from constructed or given confidence intervals

Discrete – Graphs

Knowledge and Skills

Understand and use the language of graphs including: ‘vertex’, ‘edge’, ‘trail’, ‘cycle’, ‘connected’, ‘degree’, ‘subgraph’, ‘subdivision’, ‘multiple edge’ and ‘loop’

Understand and use simple graphs, simple-connected graphs and trees.

Understand and use complete graphs and bipartite graphs, including adjacency matrices and the complement of a graph

Understand and use Euler’s formula for connected planar graphs

Identify or prove properties of a graph including that a graph is Eulerian, semi-Eulerian or Hamiltonian

Useful Links

AQA Discrete: Graphs & networks 1-1

AQA Discrete: Graphs & networks 1-2

AQA Discrete: Graphs & networks 1-3

AQA Discrete: Graphs & networks 1-4

AQA Discrete: Graphs & networks 1-5

AQA Discrete: Graphs & networks 2-1

AQA Discrete: Graphs & networks 2-2

AQA Discrete: Graphs & networks 2-3

AQA Discrete: Graphs & networks 2-4

Networks

Knowledge and Skills

Understand and use the language of networks including: ‘node’, ‘arc’ and ‘weight’.

Solve Route Inspection problems

Solve network optimisation problems using spanning trees

Find and interpret upper bounds and lower bounds for the Travelling Salesperson problem

Evaluate, modify and refine models which use networks

Interpret flow problems represented by a network of directed arcs.

Find the value of a cut and understand its meaning

Use and interpret the maximum flow-minimum cut theorem

Introduce supersources and supersinks to a network

Useful Links

AQA Discrete: Graphs & networks 3-1

AQA Discrete: Graphs & networks 3-2

AQA Discrete: Graphs & networks 3-3

AQA Discrete: Graphs & networks 3-4

Critical Path Analysis

Knowledge and Skills

Construct, represent and interpret a precedence (activity) network using activity-on-node

Determine earliest and latest start and finish times for an activity network

Identify critical activities, critical paths and the float of non-critical activities

Refine models and understand the implications of possible changes in the context of critical path analysis

Useful Links

AQA Discrete: Critical path analysis 1-1

AQA Discrete: Critical path analysis 1-2

AQA Discrete: Critical path analysis 1-3

Linear Programming

Knowledge and Skill

Formulate constrained optimisation problems

Solve constrained optimisation problems via graphical methods

Useful Links

AQA Discrete: Linear programming 1-1

AQA Discrete: Linear programming 1-2

AQA Discrete: Linear programming 1-3

AQA Discrete: Linear programming 1-4

Game theory for zero sum games

Knowledge and Skills

Understand, interpret and construct pay-off matrices

Find play-safe strategies and the value of the game

Prove the existence or non-existence of a stable solution

Identify and make use of dominated strategies

Find optimal mixed strategies for a game including use of graphical methods

Useful Links

AQA Discrete: Game theory 1-1

AQA Discrete: Game theory 1-2

AQA Discrete: Game theory 1-3

Binary operations and group theory

Knowledge and Skills

Understand and use binary operations including use of modular arithmetic and matrix multiplication

Understand, use and prove the commutativity of a binary operation

Understand, use and prove the associativity of a binary operation

Understand and prove the existence of an identity element for a given set under a given binary operation

Construct a Cayley Table for a given set and binary operation.

Useful Links

AQA Discrete: Binary operations 1-1

AQA Discrete: Binary operations 1-2

AQA Discrete: Binary operations 1-3

Assessments:

End of unit written assessment for each topic.

Year 13

Term 1

Complex Numbers

Knowledge and Skills

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

Use DMT for both finite and infinite trigonometric series.

Know and use the definition e^iθ=cosθ+isinθ and the form z=re^iθ

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.

Use complex roots of unity to solve geometric problems

Useful Links

A2 Further Maths – De Moivre’s Theorem

A2 Further Maths – Complex numbers exponential form

A2 Further Maths – Roots of unity

Series

Knowledge and Skills

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

Evaluation of limits using Maclaurin series or l’Hôpital’s rule.

Curve sketching and Hyperbolic Functions

Knowledge and Skills

Modulus of functions and associated inequalities

Graphs of y=|f(x)| and y=1/(f(x)) for given y=f(x)

Understand the concept of an oblique asymptote

Conic functions extended to composite transformations including rotations (by multiples of 90 degrees).

Use the domains and ranges of the hyperbolic functions.

Understand the definitions of hyperbolic functions sechx, cosechx and cothx, including their domains and ranges.

Use the domains and ranges of the inverse hyperbolic functions.

Differentiate and integrate hyperbolic functions.

Be able to derive logarithmic formulae for sech, cosech and coth

Understand and use the identities;

〖sech〗^2 θ=1-〖tanh〗^2 θ

〖cosech〗^2 θ=〖coth〗^2 θ-1

cosh2θ=〖cosh〗^2 θ+〖sinh〗^2 θ
sinh2θ=2sinhθcoshθ

Construct proofs involving hyperbolic functions and identities

Useful Links

A2 Further Maths – Modulus function

A2 Further Maths – Reciprocal of a function

A2 Further Maths – Graphing rational function oblique asymptotes

A2 Further Maths – Transformations of conic functions

A2 Further Maths – Inverse and reciprocal hyperbolic functions

A2 Further Maths – Hyperbolic functions calculus and identities

Integration

Knowledge and Skills

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

Differentiate inverse trigonometric and hyperbolic functions

Integrate functions of the form 〖(a^2-x^2)〗^(-1/2) , 〖(a^2+x^2)〗^(-1) , 〖(x^2+a^2)〗^(-1/2) and 〖(x^2-a^2)〗^(-1/2) and be able to choose trigonometric and hyperbolic substitutions to integrate associated functions.

Integrate using partial fractions (extend to quadratic factors ax^2+c in the denominator).

Derivation and use of reduction formulae for integration.

The limits lim┬(x→∞)⁡〖(x^k e^(-x))〗 and lim┬(x→0)⁡〖(x^k lnx)〗 where k>0, applied to improper integrals

Arc length and area of surface of revolution for curves expressed in Cartesian or parametric coordinates.

Find the area enclosed by a polar curve

Useful Links

A2 Further Maths – Improper integrals

A2 Further Maths – Differentiate inverse trigonometric functions

A2 Further Maths – Further integration

A2 Further Maths – Reduction formulae

A2 Further Maths – Maclaurin series, limits and l’Hôpital’s rule

A2 Further Maths – Arc length and area of surface of revolution

A2 Further Maths – Area of polar curves

Assessments:

End of unit written assessment for each topic.

Term 2

Differential equations

Knowledge and Skills

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.

Find both general and particular solutions to differential equations.

Solve differential equations of form y^”+ay^’+by=0 where a and b are constants by using the auxiliary equation.

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).

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

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

Solve the equation for simple harmonic motion x ̈=-ω^2 x and relate the solution to the motion.

Note and learn formulae associated with SHM.

Use of Hooke’s Law with T=kx to formulate a differential equation for simple harmonic motion, where k is a constant.

Model damped oscillations using 2nd order differential equations and interpret their solutions.

Use models for damped motion where damping force is proportional to the velocity.

Analyse and interpret models of situations with one independent variable and two dependent variables as a pair of coupled 1st order simultaneous equations and be able to solve them, for example predator-prey models

Useful Links

A2 Further Maths – Modelling with differential equations

A2 Further Maths – Simple harmonic motion and Hooke’s law

A2 Further Maths – Damped harmonic motion

A2 Further Maths – Coupled differential equations

Numerical methods

Knowledge and Skills

Mid-ordinate rule and Simpson’s rule for integration.

Euler’s step by step method for solving first order differential equations

Improvement: y_(r+1)=y_(r-1)+2hf(x_r,y_r), x_(r+1)=x_r+h

Other improvements may be used but will be clear in exam question

Useful Links

A2 Further Maths – Mid-ordinate and Simpsons rule

A2 Further Maths – Euler’s step by step method

Matrices

Knowledge and Skills

Inverse and determinant of a 3×3 Matrix

Plane geometry

Find eigenvalues and eigenvectors of 2×2 and 3×3 matrices

Find and use the characteristic equation.

Understand the geometrical significance of eigenvalues and eigenvectors

Diagonalisation of matrices

Write a matrix in the form UDU^(-1) and understand M^n=UD^n U^(-1)

Useful Links

A2 Further Maths – Inverse and determinant of a 3×3 Matrix

A2 Further Maths – Plane geometry

A2 Further Maths – Eigenvectors and Eigenvalues

A2 Further Maths – Diagonalisation

Vectors

Knowledge and Skills

Calculate and understand the properties of the vector product.

Understand and use the equation of a straight line in the form (r-a)×b=0

Use vector products to find area of a triangle

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

Express the equation of a plane in the form r.n=d

Calculate the angle between two planes and the angle between a line and a plane

Find the line of intersection of two planes or three planes

Find the intersection of a line and a plane.

Calculate the perpendicular distance from a point to a plane

Useful Links

A2 Further Maths – Vector Product

A2 Further Maths – Equations of planes

A2 Further Maths – Further Vectors Planes and Lines

Assessments:

End of unit written assessment for each topic.

Term 3

Mechanics – Dimensional analysis

Knowledge and Skills

Prediction of formulae

Finding powers in potential formulae.

Circular motion

Knowledge and Skills

Use position, velocity and acceleration as vectors in the context of circular motion.

Conical pendulum, with one or two strings.

Circular motion in a vertical plane. Includes conditions to complete vertical circles. Use conservation of energy in this context.

Centre of mass and moments

Knowledge and Skills

Centre of mass for a system of particles.

Centre of mass for a composite body.

Centre of mass of a lamina by integration.

Centres of mass of bodies formed by rotating a region about the x-axis.

Conditions for sliding and toppling. Problems including suspension and on an inclined plane.

Determine the forces acting on a rigid body in equilibrium. Use of moments and couples.

Statistics – Discrete Random Variables and Expectations

Knowledge and Skills

Know the formula E(g(X))=∑▒〖g(x_i ) p_i 〗

Find the mean, variance and standard deviation for functions of a DRV such as E(5X^3 ), E(18X^(-3) ), Var(6X^(-1) )

Useful Links

A2 Further Maths – Discrete random variables

Type I and Type II errors

Knowledge and Skills

Teach A2 Normal Distribution including Hypothesis Testing

Calculate probability of making Type I error from tests based on a normal distribution

Understand the power of a test. Calculations of P(Type II error) and power for a test for tests based on a normal, Binomial or a Poisson distribution

Useful Links

A2 Further Maths – Type I and Type II Errors

Continuous Random Variables

Knowledge and Skills

Understand and use a cumulative distribution function, F(x).

Know the relationship between f(x) and F(x).

Understand the rectangular distribution.

Know the conditions for the rectangular distribution to be used as a model.

Calculate probabilities from a rectangular distribution.

Know proofs of mean, variance and standard deviation for a rectangular distribution

Useful Links

A2 Further Maths – Cumulative distribution function

A2 Further Maths – Rectangular distribution

Exponential Distribution

Knowledge and Skills

Know conditions for an exponential distribution to be used as a model.

Know the probability density function, f(x), and the cumulative distribution function, F(x), for an exponential distribution

Calculate probabilities for an exponential distribution using F(x) or integration of f(x)

Know proofs of mean, variance and standard deviation for an exponential distribution.

Understand that the lengths of intervals between Poisson events have an exponential distribution

Useful Links

A2 Further Maths – Exponential distribution

t distribution

Knowledge and Skills

Test for the mean of a normal distribution with unknown variance using a t-statistic with appropriate degrees of freedom

Construct confidence intervals, symmetric or otherwise, from small samples, for the mean of a normal distribution with unknown variance using the t-distribution.

Discrete – Graph Theory

Knowledge and Skills

Kuratowski’s Theorem and isomorphism

Binary Operations and Group Theory

Knowledge and Skills

Understand and use the group axioms: closure, identity, inverses and associativity, including use of Cayley tables

Understand and use the language of groups including: ‘order’, ‘period’, ‘subgroup’, ‘proper’, ‘trivial’, ‘non-trivial’.

Understand and use Lagrange’s theorem
Recognise and use finite and infinite groups and their subgroups, including: groups of symmetries of regular polygons, cyclic groups and abelian groups

Identify and use the generators of groups

Recognise and find isomorphism between groups of finite order

Network Flows

Knowledge and Skills

Find the value of a cut for a network containing arcs that have both upper and lower capacities.

Use flow augmentation from an initial flow to find a maximum flow through a network with only upper capacities

State the maximum flow through a network in the context of the problem

Find a flow pattern corresponding to the maximum flow through a network.

Be able to use flow augmentation from an initial flow to find a maximum flow through a network with lower and upper capacities

Useful Links

A2 Further Maths – Network Flow Augmentation Algorithm and Student Sheet

Simplex Algorithm

Knowledge and Skills

Use the Simplex algorithm for optimising (maximising and minimising) an objective function including the use of slack variables

Interpret a Simplex tableau

Convert higher order games to linear programming problems and solve using the Simplex algorithm

Gantt Charts

Knowledge and Skills

Construct and interpret Gantt (cascade) charts and resource histograms

Carry out resource levelling (using heuristic procedures) and evaluate problems where resources are restricted

Assessments:

End of unit written assessment for each topic.

Type of Qualification:

A Level

Course entry requirements:

Mathematics GCSE Grade 7

Are you interested in:

Solving complex problems, modelling real-life situations mathematically or understanding how statistics inform decisions?

Leading to a career in:

Engineering, finance, architecture, data analysis

The subject goes well with:

• Physics
• Chemistry
• Computing
• Biology

Head of Department/course leader:

Mr S Everett

Exam Board:

AQA

Course Code:

7367

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Sociology

Politics

Art & Design – Photography

Pearson Level 3 Alternative Academic Qualiﬁcation BTEC National in Information Technology (Extended Certiﬁcate)

Further Mathematics

Term 1

Matrices and Transformations

Introduction to Complex Numbers

Roots of polynomials

Conics

Hyperbolic functions

Sequences and series

Term 2

Applications of integration

Complex numbers and geometry

Polar Coordinates

Rational functions and further algebra

Matrices and their inverses

Vectors and 3D spaces

Term 3

Mechanics – Dimensional Analysis

Momentum and collisions

Work, energy and power

Circular Motion

Statistics – Discrete Random Variables and Expectation

Poisson Distribution

Continuous Random Variables

Chi-Squared Tests

Confidence Intervals

Discrete – Graphs

Networks

Critical Path Analysis

Linear Programming

Game theory for zero sum games

Binary operations and group theory

Term 1

Complex Numbers

Series

Curve sketching and Hyperbolic Functions

Integration

Term 2

Differential equations

Numerical methods

Matrices

Vectors

Term 3

Mechanics – Dimensional analysis

Circular motion

Centre of mass and moments

Statistics – Discrete Random Variables and Expectations

Type I and Type II errors

Continuous Random Variables

Exponential Distribution

t distribution

Discrete – Graph Theory

Binary Operations and Group Theory

Network Flows

Simplex Algorithm

Gantt Charts

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Leading to a career in:

The subject goes well with:

Head of Department/course leader:

German

Level 3 Business

Geography

Design Technology: Product Design

Art & Design

French

Sociology

Politics

Art & Design – Photography

Pearson Level 3 Alternative Academic Qualiﬁcation BTEC National in Information Technology (Extended Certiﬁcate)

Chemistry

History

Psychology

English Language

Business

Health and Social Care

Music

Biology

The Wizard of Oz