A-Level OCR Pure Core Further Maths A Videos

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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

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

De Moivre's Theorem

What is de Moivre's theorem?

Simplifying example

Expansion example

sin(nθ) and cos(nθ) in terms of sinθ and cosθ

Example expressing sin(5θ) and cos(5θ) in terms of sinθ and cosθ

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

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

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

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

Method of Differences

Factorial example

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 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

Volume of Revolution for Parametric Equations

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)

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

Polar Coordinates

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

Circles and arcs

The cardioid r = a(1 + cosθ)

The curve r = asin(2θ)

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)

Tangents

Finding tangents parallel to the initial line

Finding tangents perpendicular to the initial line

Hyperbolic Functions

Hyperbolic Functions

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)

Differential Equations

First Order Linear Differential Equations

Equations of the form dy/dx + Py = Q using an integrating factor

Examples for dy/dx + Py = Q forms

Second Order Linear Differential Equations

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)

Constant types: when f(x) = k

Linear types: when f(x) = kx

Quadratic types: when f(x) = kx²

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