This course is design to develop student’s key knowledge, understanding, skills and application of further mathematics appropriate for entry to a degree course at any one of the NCUK Partner Universities.

**Complex Numbers**

Imaginary and complex numbers, Argand diagram, Cartesian form Sum, difference, product and quotient of complex numbers Modulus-argument form (aka polar form or (r, ) form) Conversion between Cartesian and modulusargument form Conjugate complex roots of polynomials with real coefficients Loci and regions on an Argand diagram

**Matrices**

Definition of a matrix. Size of a matrix. Row and column vectors. Addition, subtraction and products of matrices. Special matrices: null (Z), identity (I), diagonal and transpose. Determinant and inverse of a 2x2 matrix Eigenvalues, eigenvectors and the characteristic equations of 2x2 and 3x3 matrices

**Further Curve Sketching and Inequalities**

Sketching graphs of the form = ( ) ( ) where ( ), ( ) are linear or quadratic functions, finding intercepts with the co-ordinate axes, asymptotes, behaviour near = 0 and for numerically large . Bounds on values (noncalculus method) The solution of inequalities by algebraic and graphical methods This section builds on the sketching and inequalities in Mathematics Part One Module Topic C

**Series**

Summation of simple finite series The method of differences This section builds on the series work in Mathematics Part One Module Topic D

**Roots and coefficients of Polynomials**

Relations between the roots and coefficients of quadratic and cubic equations. Evaluation of symmetric functions in 2 or 3 variables

**Mechanics**

Kinematics: equations of motion for constant acceleration. Newton’s Laws. Application of kinematics and Newton’s Laws to connected particles.

**Hyperbolic Functions**

Definition of the six hyperbolic functions in terms of exponentials Graphs and properties of hyperbolic functions Inverse hyperbolic functions, their graphs, properties and logarithm equivalents This depends on exponential functions– Mathematics Part One Module topic D

**Parametric Coordinates**

Parametric equations of curves and conversion between Cartesian and parametric forms

Calculation of tangents and normals and calculation of an area This depends on differentiation and integration – Mathematics Part One Module topics G, H

**Conic Sections**

Cartesian and parametric equations for the parabola, ellipse and rectangular hyperbola The focus-directrix properties of the parabola, ellipse and hyperbola, including the eccentricity Tangents and normals to these curves Simple loci problems This topic uses the geometry revised in Mathematics Part One Module topic A

**Maclaurin and Taylor series**

Chain Rule Third and higher order derivatives Derivation and use of Maclaurin and Taylor series This depends on differentiation –

Mathematics Part One Module topic G

**Further Mechanics**

Momentum, impulse and restitution. Kinetic and Potential Energy. Work done and power. Moments of a force.

**Further Complex Number Functions**

Euler’s relation = + Relations between trigonometric functions and hyperbolic functions. De Moivre’s theorem and its application to trigonometric identities and to roots of a complex number Integration methods using trigonometric identities This depends on exponents and trigonometric identities –Mathematics Module topics

**Further Differentiation and Integration**

Differentiation and Integration of hyperbolic functions, integration of inverse trigonometric and inverse hyperbolic functions Integration using hyperbolic and trigonometric substitutions The calculation of arc length and the area of a surface of revolution This depends on differentiation and integration – Mathematics Module topics

**Vectors**

The vector product and the scalar triple product . The use of vectors in problems involving points, lines and planes The equation of a line in the form ( − ) = 0 The equation of a plane in the forms . = , = + + This depends on vectors – Mathematics Module topics

**Calculus and Vectors**

Using calculus to find displacement, speed and acceleration in one-dimension using expressions involving time. Using calculus to find displacement, velocity and acceleration using the i, j and k system of vectors. Express forces using the i, j and k system of vectors.

**Differential Equations**

First Order: Use of the integrating factor in + ( ) = ( ) The linear second order differential equation + + = ( ) where a, b and c are real constants and the particular integral can be found by inspection or trial. This depends on calculus – Mathematics Module topics

**Confidence Intervals**

Use of the Central Limit Theorem. Finding confidence intervals of small samples with unknown standard deviation.