AS OCR Pure Core Further Maths A Videos

General

Formulae Booklet

Proof

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

Divisibility 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)

Matrix Proofs

Matrix multiplication proof 1

Matrix multiplication proof 2

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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

Multiplication rule

Loci in the Complex Plane

Perpendicular bisectors

Representing regions on an argand diagram

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

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Matrices

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

Simultaneous Equations using Matrices

Solving simultaneous equations with matrices

Matrix Linear Transformations - Rotations

Rotation 90 degrees

Rotation 180 degrees

Rotation 270 degrees

Rotation θ degrees

Matrix Linear Transformations - Reflections

Reflections in the x-axis

Reflections in the y-axis

Reflection in the line y = x

Reflection in the line y = -x

Matrix Linear Transformations - Enlargement

When scale factor k > 0

When scale factor k < 0

Transformation test

Combining matrix transformations

Inverse matrices to reverse transformations

Determinant as the area scale factor of a transformation

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Further Algebra and Functions

Standard Summations

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

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Further Calculus

Applications of Integration - Volumes of Revolution

Example question

Area enclosed by several curves

Volume of about the y-axis

Volume of about the y-axis between curves

Improper Integrals

Convergent integrals

Divergent integrals

Integrals involving Partial Fractions

Integrating Partial Fraction types

Differentiating Inverse Trigonometric Functions

Differentiating inverse sin(x/a)

Differentiating inverse cos(x/a)

Differentiating inverse tan(x/a)

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)

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Further 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

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

Planes

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

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