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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

### De Moivre's Theorem

- Introduction
- sin(nθ) and cos(nθ) in terms of sinθ and cosθ
- Example expressing sin(5θ) and cos(5θ) in terms of sinθ and cosθ
- sin
^{n}θ and cos^{n}θ in terms of sin(kθ) and cos(kθ) - New identities you will need
- cos
^{n}θ in terms of cos(kθ) - sin
^{n}θ in terms of sin(kθ) when n is odd - sin
^{n}θ in terms of sin(kθ) when n is even - Cube roots of 1
- 4th roots of a complex number
- 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

### Method of Differences

### Maclaurin's Series

- Maclaurin's series expansion
- Series expansion for e
^{x} - Series expansion for sin(x) and cos(x)
- Series expansion for ln(1+x)
- Further 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
- Volume of Revolution for Parametric Equations
- Exam questions

### Improper Integrals

### Integrals involving Partial Fractions

### Differentiating Inverse Trigonometric Functions

### Standard Integrals Involving Inverse Trigonometric Functions

- Standard Integrals 1/(a
^{2}+x^{2}) and 1/root(a^{2}- x^{2}) - Integrals which require completing the square
- Integration using completing the square - arctan type
- Integration using completing the square - arcsin type
- How to integrate 1/root(a
^{2}-x^{2}) with limits - How to integrate 1/root(a
^{2}-x^{2}) with limits (2)

## 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

## Polar Coordinates and Curves

### Polar Coordinates

- Defining the position of a point
- Converting coordinates from Cartesian to polar
- Converting coordinates from polar to Cartesian

### Equations of Curves

- Converting polar curves to Cartesian form
- Converting Cartesian curves to polar form
- Sketching polar graphs

### Area Bounded by a Polar Curve

- Area bounded by a polar curve
- Area bounded by the cardioid r = a(1 + cosθ)
- Area of a loop of the curve r = acos(3θ)
- Exam questions

### Tangents

- Finding tangents parallel to the initial line
- Finding tangents perpendicular to the initial line
- Exam questions

## Hyperbolic Functions

### Hyperbolic Functions

- Definitions
- Graphs of sinh(x), cosh(x) and tanh(x)
- Graphs of Inverse Hyperbolic Functions
- The inverse of sinh(x) expressed as a natural logarithm

### Differentiation of Hyperbolic Functions

- Differentiating hyperbolic functions sinh(x), cosh(x) and tanh(x)
- Differentiating hyperbolic functions sech(x), cosech(x) and coth(x)
- Proof of the differentials of sinh(x), cosh(x) and tanh(x)
- Proof of the differentials of cosech(x), sech(x) and coth(x)
- How to differentiate the inverse hyperbolic function arsinh (x/a) and arsinh(x)
- How to differentiate the inverse hyperbolic function arcosh (x/a) and arcosh(x)
- How to differentiate the inverse hyperbolic function artanh (x/a) and artanh(x)

## First Order Linear Differential Equations

### Exact Equations (Integrating Factors)

- Introduction
- Equations of the form dy/dx + Py = Q using an integrating factor
- Examples for dy/dx + Py = Q forms
- Exam questions

## Second Order Linear Differential Equations

### Equations of the form A(d^{2}y/dx^{2}) + B(dy/dx) + Cy = 0

- Introduction
- Solving equations where b
^{2}- 4ac > 0 - Solving equations where b
^{2}- 4ac = 0 - Solving equations where b
^{2}- 4ac < 0

### Equations of the form A(d^{2}y/dx^{2}) + B(dy/dx) + Cy = f(x)

- General Solutions
- Constant types: when f(x) = k
- Linear types: when f(x) = kx
- Quadratic types: when f(x) = kx
^{2} - Exponential types: when f(x) = ke
^{px} - Trig. types: when f(x) = λcos(ωx) + μsin(ωx)
- Special case for some f(x) = k types
- Special case for some f(x) = ke
^{px}types - Using boundary conditions to solve differential equations
- Exam questions

## 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.