#### COURSE DESCRIPTION

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.

#### LEARNING OUTCOMES

- How 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 modulus-argument 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

**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

### Course Features

- Lectures 0
- Quizzes 0
- Duration 7 weeks
- Skill level All levels
- Language English
- Students 0
- Assessments Yes