A-level Further Mathematics for Year 12 - Course 1: Complex Numbers, Matrices, Roots of Polynomial Equations and Vectors

This course by Imperial College London is designed to help you develop the skills you need to succeed in your A-level further maths exams.

You will investigate key topic areas to gain a deeper understanding of the skills and techniques that you can apply throughout your A-level study. These skills include:

• Fluency – selecting and applying correct methods to answer with speed and efficiency
• Confidence – critically assessing mathematical methods and investigating ways to apply them
• Problem-solving – analysing the ‘unfamiliar’ and identifying which skills and techniques you require to answer questions
• Constructing mathematical argument – using mathematical tools such as diagrams, graphs, logical deduction, mathematical symbols, mathematical language, construct mathematical argument and present precisely to others
• Deep reasoning – analysing and critiquing mathematical techniques, arguments, formulae and proofs to comprehend how they can be applied

Over eight modules, you will be introduced to

• complex numbers, their modulus and argument and how they can be represented diagrammatically
• matrices, their order, determinant and inverse and their application to linear transformation
• roots of polynomial equations and their relationship to coefficients
• series, partial fractions and the method of differences
• vectors, their scalar produce and how they can be used to define straight lines and planes in 2 and 3 dimensions.

Your initial skillset will be extended to give a clear understanding of how background knowledge underpins the A-level further mathematics course. You’ll also be encouraged to consider how what you know fits into the wider mathematical world.

Module 1: Complex Numbers 1: An Introduction to Complex Numbers

• The definition of an imaginary number
• The definition of a complex number
• Solving simple quadratic equations
• Addition, subtraction and multiplication of complex numbers
• Complex conjugates and division of complex numbers
• Radian measure
• Representing complex numbers on the Argand diagram

Module 2: Matrices 1: An Introduction to Matrices

• The order of a matrix
• Addition and subtraction of conformable matrices
• Matrix multiplication
• The identity matrix
• Matrix transformations in 2 and 3 dimensions
• Invariant lines and lines of invariant points

Module 3: Further Algebra and Functions 1: Roots of Polynomial Equations

• Solving polynomial equations with real coefficients
• The relationship between roots and coefficients in a polynomial equation
• Forming a polynomial equation whose roots are a linear transformation of the roots of another polynomial equation

Module 4: Complex Numbers 2: Modulus-Argument form and Loci

• The modulus and argument of a complex number
• Writing complex numbers in modulus argument form

• The geometrical effect of multiplying by a complex number.

• Loci on the Argand diagram

Module 5: Matrices 2: Determinants and Inverse Matrices

• The determinant of a square matrix.
• The inverse of a square matrix
• Using matrices to solve simultaneous equations (5)
• The geometrical interpretation of the solution of a system of equations

Module 6: Further Algebra and Functions 2: Series, Partial Fractions and the Method of Differences

• Deriving formulae for series using standard formulae
• Separating algebraic fractions into partial fractions
• The method of differences
• Partial fractions and method of differences

Module 7: Vectors 1: The Scalar (dot) Product and Vector Equations of Lines

• The scalar product of two vectors
• The vector and Cartesian forms of an equation of a straight line in 2 and 3 dimensions
• Solving geometrical problems using vector equations of lines
• The dot product and the angle between two lines

Module 8: Vectors 2: The Vector Equations of a Plane and Geometrical Problems with Lines and Planes

• The vector and Cartesian forms of the equation of a plane
• The vector equation of a plane
• Solving geometrical problems with lines and planes using vectors
• The intersection of a line and a plane
• Perpendicular distance from a point to a plane