A-Level AQA Pure Maths Videos

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General

Formulae Booklet

Algebra and Functions

Indices

Practice question

Multiplication Indices

Rule 1 tutorial

Rule 1 practice questions

Rule 2 tutorial

Rule 2 practice questions

Division Indices

Practice questions

Negative Indices

Practice questions

Fractions with Negative Indices

Practice questions

Rational Indices

Practice questions

More Indices

Simplifying negative powers

Expressions terms in the form axⁿ

Surds

Addition and subtraction

Factorising

Quadratic Expressions

Difference of two square types

Trinomials by grouping

Trinomials by inspection 1

Trinomials by inspection 2

Trinomials by inspection 3

Trinomials by inspection 4

Trinomials by inspection 5

Trinomials by inspection 6

Trinomials by inspection 7

Completing the Square

Completing the square for x² terms

Completing the square for 2x², 3x², ... terms

Completing the square for negative terms

Sketching positive graphs

Sketching negative graphs

Quadratic Equations

Solve by factorising

Solve by completing the square

Solve by the quadratic formula

Solving in some function of x

Roots and discriminant

Common types of discriminant questions

Simultaneous Equations

Elimination method 1

Elimination method 2

Elimination method 3

Elimination method 4

Elimination method 5

Elimination method 6

Substitution method

Substitution method example

Inequalities

Rules for reversing the inequality sign

Solving linear inequalities

Solving double inequalities

Linear inequalities in two variables

Solving quadratic inequalities

Polynomials

Algebraic Long Division

Handling missing terms

Factor Theorem

The factor theorem

Showing that x-1 is a factor of a cubic polynomial

Factorising a cubic polynomial (Method 1)

Factorising a cubic polynomial (Method 2)

Solving a cubic equation

Finding constants in a polynomial given the factors

Rational Expressions

Simplifying algebraic fractions

Addition and subtraction of algebraic fractions 1

Addition and subtraction of algebraic fractions 2

Multiplication of algebraic fractions

Simplifying "stacked" fractions

Division of algebraic fractions

Coordinate Geometry

Algebra and Functions 2

Sketching Curves

Sketching quadratic curves

Sketching cubic curves

Sketching y = k/x curves

Modulus Functions

The Modulus function: |x|

Graphing y = |f(x)|

Graphing y = f(|x|)

Modulus Equations

Example: How to solve |x + 1| = -2x - 5

Example: How to solve |x - 2| = 3

Example: How to solve |3x + 9| = |2x + 1|

Modulus Inequalities

Example: Inequalities type 1

Example: Inequalities type 2

Example: Inequalities type 3

Working with Functions

Domain and range 1

Domain and range 2

Combination of functions

Combination of functions - Example 1

Combination of functions - Example 2

The inverse of a function

The inverse of a function - Example 1

The inverse of a function - Example 2

The inverse of a function - Example 3

Graphical relationship between f(x) and its inverse

Graphical relationship between f(x) and its inverse - Example 1

Graphical relationship between f(x) and its inverse - Example 2

Graph Transformations

Basic graphs used in transformations

Translations

y = f(x + a) and y = f(x) + a

y = f(x + a): how it works

y = f(x) + a: how it works

Reflections

y= -f(x) and y=f(-x)

Stretches

y = af(x) and y = f(ax)

How y = af(x) stretches y = f(x) by scale factor a parallel to the y-axis

How y = f(ax) stretches y = f(x) by scale factor 1/a parallel to the x-axis

Partial Fractions

Summary exercise

Denominator Contains 2 or 3 Linear Factors

Denominator contains 2 linear factors

Summary Exercise: 2 linear factors

Denominator contains 3 linear factors

Summary Exercise: 3 linear factors

Repeated Linear Factors

Repeated Linear Factors (Alternative Method)

Trigonometry

Trigonometric Ratios

Simple way to learn the trig. ratios for 30, 45 and 60 degrees

Simple way to work out the trig. ratios for multiples of 30, 45 and 60 degrees

Graphs and Transformations

Trigonometric graphs

Translations parallel to the y-axis

Translations parallel to the x-axis

Reflection in the x-axis

Reflection in the y-axis

Stretch parallel to the y-axis: y = kf(x)

Stretch parallel to the x-axis: y = f(kx)

Combining transformations

Applications

Finding the area of a triangle with two sides and an included angle

Sine Rule

Finding the length of a side of a non-right triangle

Finding an angle of a non-right triangle

The Ambiguous Case

Cosine Rule

Finding the length of a side of a non-right triangle

Finding an angle of a non-right triangle

Arcs, Sectors and Segments

What is a radian?

Arc length and sector area (Degrees)

Arc length and sector area (Radians)

Area of a segment (Degrees)

Area of a segment (Radians)

Trig. Equations and the Quadrant Rule

What is the Quadrant Rule/CAST diagram?

Using the Quadrant Rule to solve trig. equations

Solving trig. equations in various ranges

Solving trig. equations with multiple angles

Solving trig. equations that can be factorised 1

Solving trig. equations that can be factorised 2

Solving trig. equations that can be factorised 3

A common mistake

Identities

tanθ = sinθ/cosθ and sin²θ+cos²θ = 1

Proving trig. identities 1

Proving trig. identities 2

Proving trig. identities 3

cos(θ) = cos(-θ) and sin(θ) = -sin(-θ)

Solving trig. equations using identities 1

Solving trig. equations using identities 2

Solving trig. equations using identities 3

Secθ, Cosecθ and Cotθ

What are secθ, cosecθ and cotθ?

Graphs of secθ, cosecθ and cotθ

Inverse Trigonometric Functions

arcsin(x) or sin^-1(x)

arccos(x) or cos^-1(x)

arctan(x) or tan^-1(x)

Examples using inverse trigonometric functions

Example: Type 1

Example: Type 2

Example: Type 3

Identities and Equations: Pythagorean Type

sin²x + cos²x = 1, 1 + tan²x = sec²x, 1 + cot²x = cosec²x

Proving Pythagorean identities

Solving equations using Pythagorean identities

Small-angle Approximations

Small angle approximations

Applying small angle approximations

Identities and Equations: Addition Type

sin(A + B), cos(A + B) and tan(A + B) addition formulae

Finding exact trig. ratios

Exact values of sin(A + B) etc

Proving identities using the Addition Formulae

Equations that use the Addition Formulae

Identities and Equations: Double Angle Type

Identities for sin(2A), cos(2A) and tan(2A)

Proving identities using the Double Angle identities

Solving equations using Double Angle identities

Proving identities using Half Angle identities

Identities and Equations: Triple Angle Type

Showing cos(3θ) in terms of cos(θ)

Showing sin(3θ) in terms of sin(θ)

Proving identities using the Factor Formulae

Identities and Equations: Harmonic Formulae

Asin(x) + Bcos(x) = Rsin(x + a)

Asin(x) - Bcos(x) = Rsin(x - a)

Acos(x) + Bsin(x) = Rcos(x - a)

Acos(x) - Bsin(x) = Rcos(x + a)

Solving equations using Harmonic identities

Max and min values

Max and min values - fractional type

Logarithmic and Exponential Functions

Logarithmic and Exponential Functions

What is an exponential function?

Logarithm Rules

Proofs of these rules

Equations

Logarithms - change of base

Logarithms - change of base (examples)

Simplifying and expanding equations

Solving equations where x is in the power

Solving equations that contain log terms

Solving equations that contain logs with different bases

Solving equations that contain exponential functions

Simultaneous equations 1

Simultaneous equations 2

Simultaneous equations 3

Solving inequalities

The Exponential Function e^x and Natural Log Functions

Exponential Functions

Transformations of exponential graphs

The natural log function: ln(x)

Modelling Curves of the form y=kxⁿ and y=ka^x

Modelling curves - converting to linear form

Modelling curves - converting to linear form - example 1

Modelling curves - converting to linear form - example 2

Differentiation

Introduction

The gradient function dy/dx

Differentiation from 1st principles

Terms of the form axⁿ

Extending to root types

Extending to fractional types

The second derivative

Stationary Points

What are stationary points?

An example of finding a stationary point

Nature of a stationary point using 1st differential

Nature of a stationary point using 2nd differential

An example of finding stationary points and their nature

Applications of stationary points

Increasing and decreasing functions

Standard Differentials

Exponential function: e^x

Natural log function: ln(x)

Trig. functions: sin(x), cos(x) and tan(x)

How to differentiate sinθ from first principles

How to differentiate cosθ from first principles

Differentiating with the Chain Rule

Polynomial to a rational power

Exponential types

Natural log types

Trigonometric types

Trigonometric functions to a power

Trigonometric functions to a power (part 2)

Differentiating with the Product Rule

What is the Product Rule?

Summary Exercise

Proof of the Product Rule

Differentiating with the Quotient Rule

What is the Quotient Rule?

Quotient Rule example

Proof of the Quotient Rule

More Standard Differentials

Differentiating sec(x), cosec(x) and cot(x)

Applying Chain Rule to sec(x), cosec(x) and cot(x)

Reciprocal Function of dy/dx

Finding dy/dx from dx/dy

Exponential Functions

Differentiation of y = a^x type

Differentiating y = x^x

Parametric Functions

Differentiating Parametric functions

Second Differential

Tangents and normals

Finding stationary points

Implicit Functions

Differentiating Implicit equations

Using the Product Rule with Implicit equations

Tangents to Implicit curves

Finding stationary points

Nature of stationary points

Connected Rates of Change

Connected Rates of Change Example

Three connected rates of change (Best Method)

Three connected rates of change (Alternative Method)

Example of Cone type problem

Integration

Introduction

Terms of the form axⁿ

Fractional and root type terms

Equation of Curves

Finding the equation of a curve given dy/dx

Using f'(x) notation

Definite Integration

What is a definite integral?

Common Functions

(ax + b)ⁿ type functions

Summary Exercise: (ax + b)ⁿ type functions

Exponential functions: e^x, e^ax and e^{(ax + b)}

Reciprocal functions 1/x and 1/(ax + b)

Integrals of the form: f'(x)/f(x)

Integrals of the form: f'(x)/f(x) - Example

Why the modulus sign?

Why the modulus sign? - example

Integrals of the form: f'(x)e^f(x)

Trigonometric Functions

Integrals of sin(x), cos(x) and sec²(x)

Integrals of the form sin(ax + b), cos(ax + b) and sec²(ax + b)

Identity types 1

Identity types 2

Identity types 3

Partial Fractions

Integrating Partial Fraction types

Integration by Substitution

Square root types (Method 1)

Square root types (Method 2)

Integrating trig. functions

Integrating exponential types

Integration by substitution with limits

Integrating trig. functions with limits

Integrating Products of the form f[g(x)]g'(x) by inspection

Integrals of the form f[g(x)]g'(x)

Summary exercise

Integrals of the form Af[g(x)]g'(x)

Summary exercise

Integration by Parts

Applying integration by parts twice over

Natural Log types: ln(x)

e^axsin(bx) and e^axcos(bx) types

How to integrate e^axsin(bx) and e^axcos(bx)

Example - Integrate e^xcos(x)

Example - Integrate e^2xsin(3x)

Example - Integrate e^-3xcos(5x)

Integration by parts with limits

Proof of the formula

General Methods of Integration

Step 1 - Is it a standard Integral?

Step 2a - Is it a product of the form f[g(x)]g'(x)?

Step 2b - If not, try integration by parts or substitution

Step 3 - Is it a fraction?

Area Bound by a Curve

Area under a curve

Area below the x-axis

Area above and below the x-axis

Area bounded by two curves

Differential Equations - Separating the Variables

What do we mean by separating the variables?

Where does the Constant C go?

Handling the constant in log types

Exponential and trig. types

Differential Equations - Forming Differential Equations

Direct proportion type

Inverse proportion type

Newton's Law of Cooling

Numerical Methods

Solving Equations by Numerical Methods

Graphical method of finding roots

Change of sign method

How does Iteration work?

Newton-Raphson method

Trapezium Rule

What is the Trapezium Rule?

Underestimate or overestimate

Vectors

Vectors

What is a vector and a scalar quantity?

2D vector notation

Vector notation

2D position vectors

Position vectors

Equal and negative vectors

2D multiplying a vector by a scalar

Multiplying a vector by a scalar

2D addition and subtraction

2D addition and subtraction of vectors

Addition and subtraction of vectors

Magnitude of a 2 dimensional vector

Distance between 2 points (2D)

Magnitude of a 3 dimensional vector

Proof

Proof identities

Proof by exhaustion and deduction

Proof by exhaustion and deduction - example

Proof using a counter example