Alevel Further Mathematics for Year 12  Course 1: Complex Numbers, Matrices, Roots of Polynomial Equations and Vectors
About this course
Skip About this courseThis course by Imperial College London is designed to help you develop the skills you need to succeed in your Alevel 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 Alevel 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
 Problemsolving – 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 Alevel further mathematics course. You’ll also be encouraged to consider how what you know fits into the wider mathematical world.
At a glance
What you'll learn
Skip What you'll learnHow to extend the number system to include and the definition of a complex number.
How to add, subtract, multiply and divide complex numbers.
How to represent complex numbers on an Argand diagram and the modulus and argument of a complex number.
How to write complex numbers in modulusargument form.
How to define loci in the complex plane.
How to define a matrix by its order.
How to add and subtract conformable matrices.
How to multiply two conformable matrices.
How to use matrices to define linear transformations.
How to find invariant lines and lines of invariant points.
How to find the determinant and inverse of a 2 x 2 and 3 x 3 matrix.
How to use matrices to solve systems of linear equations.
How to use standard series formulae to find the sums of other series.
How to separate algebraic fractions into partial fractions.
How to use the method of differences to find the sum of a series.
How to find the scalar (dot) product of two vectors.
How to define the equation of a line using vectors.
How to define a plane using vectors.
How to use vectors to solve problems involving lines and planes.
Syllabus
Skip SyllabusModule 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: ModulusArgument 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
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