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## Video Tutorials

## Complex Numbers

### Complex Numbers

- Real and imaginary numbers
- Addition, subtraction and multiplication
- Complex conjugates
- Division of a complex number by a complex number
- Division of a complex number by a complex number (example)
- Argand diagrams
- Modulus and argument
- Equating real and imaginary parts to solve equations
- Square roots of a complex number
- Solving quadratic equations with complex roots
- Solving cubic equations
- Solving quartic equations
- Reflection in the real axis
- Reflection in the real axis - example
- Modulus-argument form of a complex number
- Exponential form or Euler's form
- Rules for the Mod-Arg of two complex Numbers
- Exam questions

### Loci in the Complex Plane

## Matrix Algebra

### Matrices

- Introduction and dimension of a matrix
- Addition, subtraction and scalar multiplication
- Matrix multiplication
- Identity and inverse of a 2x2 matrix
- Non-singular matrix example
- Solving a matrix equation
- Exam questions

### Simultaneous Equations using Matrices

### Matrix Linear Transformations

- Rotations
- Reflections
- Reflections in the x-axis
- Reflections in the y-axis
- Reflection in the line y = x
- Reflection in the line y = -x
- Enlargement
- Transformation test
- Combining matrix transformations
- Inverse matrices to reverse transformations
- Determinant as the area scale factor of a transformation
- Exam questions

## Roots of Polynomial Equations

### Roots of Polynomial Equations

- Relationship between roots and coefficients of a quadratic equation
- Relationship between roots and coefficients of a quadratic equation (2)
- Roots of a cubic Equation
- Finding a cubic equation based on roots of another

## Series

### Standard Summations

- Sigma Notation
- Sum of the first n natural numbers Σr and the results for Σa and Σ(ar+b)
- Harder types for Σr and Σ(ar+b) summations
- Sum of the squares of the first n natural numbers Σr
^{2} - Sum of the cubes of the first n natural numbers Σr
^{3} - Using known formulae to sum more complex series
- Exam questions

## Proof by Mathematical Induction

### Sum of Series

- Proof of the sum of the series Σr
- Proof of the sum of the series Σr
^{2} - Proof of the sum of the series Σr
^{3} - Proofs for other series 1
- Proofs for other series 2
- Exam questions

### Divisbility and Multiple Test Proofs

- Proof that an expression is divisible by a certain integer (power type)
- Proof that an expression is divisible by a certain integer (non-power type)
- Exam questions

### Matrix Proofs

## Further Calculus

### Applications of Integration - Volumes of Revolution

- Introduction
- Example question
- Area enclosed by several curves
- Volume of about the y-axis
- Volume of about the y-axis between curves
- Exam questions

## Vectors

### Scalar Product (Dot Product)

- What is a scalar/dot product?
- Finding the interior angle of a triangle
- Perpendicular vectors
- Finding a vector that is perpendicular to 2 vectors

### Vector Equations of Lines

- Equation of a line
- Introduction
- Sketching 3D vector problems
- Method when given a point on the line and direction
- Method when line passing through 2 given points
- Angle between two lines
- Conditions for lines to be parallel
- Intersecting and skew lines
- Closest point to a line and shortest distance from origin
- Shortest distance of a point to a line

### Mixed Exam Questions on Vectors

### Planes

- Parametric vector form of a plane
- Equation of a plane
- Locating a point on a vector parametric plane
- Plane passing through 3 points
- Vector plane passing through a point parallel to 2 lines
- Scalar product forms of a plane in the form r.n=D
- Scalar product forms of a plane in the form (r-a).n=0
- Cartesian form of a plane
- Determining if a line is parallel to, or lies on a plane or intersects
- Point of intersection between a line and a plane
- Shortest distance from a point to a plane
- Shortest distance from a point to a plane (2)
- The angle between two planes

## Worked Papers

The new specification has limited resources, but these papers from the previous are still somewhat applicable.

FP1

FP2

For worked papers, see the top of this section.