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Imperial College London

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

Imperial College London via edX

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Overview

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.

Syllabus

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

Taught by

Philip Ramsden and Phil Chaffe

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