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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
- sinnθ and cosnθ in terms of sin(kθ) and cos(kθ)
- New identities you will need
- cosnθ in terms of cos(kθ)
- sinnθ in terms of sin(kθ) when n is odd
- sinnθ 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 Σr2
- Sum of the cubes of the first n natural numbers Σr3
- Using known formulae to sum more complex series
- Exam questions
Method of Differences
Maclaurin's Series
- Maclaurin's series expansion
- Series expansion for ex
- 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 Σr2
- Proof of the sum of the series Σr3
- 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/(a2+x2) and 1/root(a2 - x2)
- 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(a2-x2) with limits
- How to integrate 1/root(a2-x2) 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(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
- Constant types: when f(x) = k
- Linear types: when f(x) = kx
- Quadratic types: when f(x) = kx2
- Exponential types: when f(x) = kepx
- Trig. types: when f(x) = λcos(ωx) + μsin(ωx)
- Special case for some f(x) = k types
- Special case for some f(x) = kepx 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.