Complex numbers (1)

• 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
  • Solving quadratic equations with complex roots (complex conjugate pairs)
  • Solving cubic equations
  • Solving quartic equations

Exam Questions

Numerical solutions to equations

Solving equations of the form f(x)=0

• Bisection method
• Linear interpolation
• Newton-Raphson process
• Exam Questions

Coordinate systems

Parabola

• Cartesian Form
  • Cartesian equation of a parabola, directrix, focus and locus
• Parametric form
  • Parametric form of a parabola
• Tangents and Normals
  • Cartesian Form :
    • example 1
  • Parametric Form :
    • example 1 | example 2
• Exam Questions

Hyperbola (rectangular)

• Cartesian and Parametric Forms
  • Introduction - Cartesian and Parametric forms
• Tangents and Normals
  • Cartesian Form :
    • example
  • Parametric Form :
    • example
• Exam Questions

Matrix algebra

• 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)
  • Example on singular matrices
  • Example on solving a matrix equation
  • Example on proving a matrix identity

Applications

• Simultaneous Equations
  • Solving simultaneous equations
• Linear Transformations
  • Rotations about the origin of:
    • 90^o | 180^o | 270^o | θ^o
  • Reflections in the:
    • x-axis | y-axis | y = x | y = -x
  • Enlargement, scale factor k, centre (0,0) where:
    • | k>0 | k<0
  • Transformations Test:
    • How well do you know your transformations? : Take the Test
  • Combinations of Transformations:
    • Combinations of transformations
  • Inverse Matrices:
    • Inverse matrices to reverse linear transformations
  • Area Scale Factor:
    • Determinant as the area scale factor of a transformation
    • Proof

Exam Questions

Series

• ∑ notation (revision)
• Sum of the first n natural numbers ∑r and the results for ∑a and ∑(ar+b)
  • Harder examples using these results
• Sum of the squares of the first n natural numbers ∑r²
• Sum of the cubes of the first n natural numbers ∑r³
• Using known formulae to sum more complex series

Exam Questions

Method of Differences

• Method of differences
• Example using factorials

Proof by mathematical induction

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
  • Example 1
  • Example 2

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)

Recurrence Relations

• Proof of the general term from a recurrence relationship (example 1)
• Proof of the general term from a recurrence relationship (example 2)

Matrices

• Proof of matrix multiplication statements (example 1)
• Proof of matrix multiplication statements (example 2)

Exam Questions

Inequalities

Fractional Inequalities

• Example 1
  • Solution by graphical methods
  • Solution by analytical methods
• Example 2
  • Solution by a graphical method 1 | graphical method 2
  • Solution by analytical method

Modulus Inequalities (fractional types)

• Example 1
  • Solution by graphical methods
• Example 2
  • Solution by a graphical method

Further complex numbers

• 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
  • Mixed examples

de Moivre's Theorem

• de Moivre's theorem
  • Simplifying example
  • Expansion example
• Expressing sin nθ and cos nθ in terms of sinθ and cosθ
  • Introduction
• Example
• Expessing sinⁿθ and cosⁿθ in terms of sinkθ and coskθ
  • Introduction - new identities you will need
  • Expressing cosⁿθ in terms of coskθ
  • Expressing sinⁿθ in terms of sinkθ when n is odd
  • Expressing sinⁿθ in terms of coskθ when n is even
• nth roots of a complex number
  • Example - cube roots of 1
  • Example - 4th roots of a complex number

Loci in the complex plane

• The locus of a point moving in a circle : | z- z₁| = r
• The locus of a point moving along a perpendicular bisector : | z-z₁ | = | z-z₂ |
• The locus of a point moving along a half-line : arg(z-z₁) = θ
• The locus of a point moving on the arc of a circle : arg[ (z-z₁) / (z-z₂) ] = θ
• Using complex numbers to represent regions on an Argand diagram
• Transformations of the complex plane

First order differential equations

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)

Family of curves

• separating the variables and sketching a family of curves

Exact equations (integrating factors)

• Exact equations where one side is the exact derivative of a product
• Solving equations of the form dy/dx + Py = Q using an integrating factor
  • Examples

Substitution types

• Using substitution to reduce a differential equation to a known form

Second order differential equations

Equations of the form a d²y/dx² + b dy/dx + cy = 0

• Introduction
• Solving equations where b² - 4ac > 0
• Solving equations where b² - 4ac = 0
• Solving equations where b² - 4ac < 0

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

• General solutions
  • Introduction where f(x) = k (constant types)
  • where f(x) = kx (linear types)
  • where f(x) = kx² (quadratic types)
  • where f(x) = ke^px (exponential types)
  • where f(x) = m cosωx + n sinωx (trig types)
  • Special types of particular integrals
    • f(x) = k (constant types)
    • f(x) = ke^px (exponential types)
• Particular solutions
  • Using boundary conditions to solve differential equations

Substitution types

• Using substitution to reduce a differential equation to a known form

Maclaurin and Taylor series

Maclaurins 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

(still to be completed - target April)

• Taylor's expansion
• Solution to differential equations using Taylors series

Polar coordinates

• (still to be completed)
• Polar and Cartesian coordinates
• Polar and Cartesian equations of curves
• Sketching polar equations
• Area bounded by a polar curve
• Finding equations of tangents parallel and perpendicular to the initial line