# Playlist for Further Maths / Further Pure Maths

## Other Playlists and Main Index

- Real and imaginary numbers
- Addition, Subtraction and Multiplying and simplifying powers of i
- Complex conjugates
- Division of a complex number by a complex number
- Argand diagram
- Modulus and argument of a complex number
- Mod-Arg form of a complex number
- Solving problems with complex numbers
- Square roots of a complex number
- Polynomial Equations

### Numerical solutions to equations

Solving equations of the form f(x)=0

### Coordinate systems

- Cartesian Form
- Parametric form
- Tangents and Normals
- Exam Questions

- Cartesian and Parametric Forms
- Tangents and Normals
- Exam Questions

- Introduction and dimension of a matrix
- Addition and subtraction and multiplying a matrix by a scalar
- Matrix multiplication
- Identity and Inverse of a 2x2 matrix (determinant, singular and non-singular)

Applications

### Series

- Proof of the sum of the series ∑r
- Proof of the sum of the series ∑r²
- Proof of the sum of the series ∑r³
**Further examples**- Exam Questions

**Divisibility** and Multiple Tests

- 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

### Inequalities

- Example 1
- Example 2

Modulus Inequalities (including fractional types)

- Example 1
- Example 2
- Exam Questions

- Modulus-argument form
- Exponential Form (Euler's relation)
- Multiplication rule for the mod and argument of two complex numbers
- Division rule for the mod and argument of two complex numbers
- Exam Questions

- de Moivre's theorem
**Expressing sin nθ and cos nθ in terms of sinθ and cosθ**- Expressing sin
^{n}θ and cos^{n}θ in terms of sin*k*θ and cos*k*θ - nth roots of a complex number

- The locus of a point moving in a circle : | z- z
_{1}| = r - The locus of a point moving along a perpendicular bisector : | z-z
_{1 }| = | z-z_{2 }| - The locus of a point moving along a half-line : arg(z-z
_{1}) = θ - The locus of a point moving on the arc of a circle : arg[ (z-z
_{1}) / (z-z_{2}) ] = θ - Using complex numbers to represent regions on an Argand diagram
- Exam Questions

Separating the variables (Revision)

- Finding a general solution and a particular solution
- Working with constants in log types
- Exponential and trig type (a little more challenging)

Equations of the form *a* d²y/dx² + *b* dy/dx + *c*y =
0

- Introduction
- Solving equations where
*b*² - 4*ac > 0* - Solving equations where
*b*² - 4*ac = 0* - Solving equations where
*b*² - 4*ac < 0*

Equations of the form *a* d²y/dx² + *b* dy/dx + *c*y = *f(x)*

- General solutions
- Exam Questions

- Particular solutions
- Exam Questions

Substitution types

### Maclaurin and Taylor series

Coordinates

- Defining the position of a point
- Converting cartesian coordinates to polar coordinates
- Converting polar coordinates to cartesian coordinates

Equations of Curves

- Converting the equation of a polar curve to cartesian form
- Converting the equation of a cartesian curve to polar form

- Sketching the polar equation of:

Area Bounded by a Polar Curve

Tangents

### Hyperbolic Functions

- Definitions
- Graphs of sinh(x), cosh(x) and tanh(x)
- Graphs of sech(x), cosech(x) and coth(x)
- Solving equations using inverse and exponential functions
- Hyperbolic identities
- Osborn's rule
- Inverse hyperbolic functions and their graphs
- Expressing inverse hyperbolic functions as natural logarithms
- Solving hyperbolic equations using hyperbolic identities

### Further matrix algebra

Transposed and symmetric matrices

- Still to be completed (see note at end)