Pure Maths A-Level Tutorials

Outlined below are topics covered for the Pure maths A-Level

It is advisable to check the official Pure Maths A-Level specification for your board.
Here are links to the common examination boards for Edexcel, AQA, and OCR.

I am slowly moving all the videos to my YouTube channel. Please use the Playlists to get to the videos in any topic. This work is ongoing.

Contents for Pure Maths A-Level Tutorials

Prior Knowledge

Algebra and Functions 1

Coordinate Geometry

Algebra and Functions 2

Sequences and Series

Trigonometry

Logarithmic and Exponential Functions

Differentiation

Integration

Numerical Methods

Vectors

Proof

Past Papers

Prior Knowledge

Algebra Basics – Playlist

  1. Expanding a single bracket
  2. Expanding two or more brackets
  3. Squaring a bracket
  4. Terms in expressions and equations
  5. Identity or equation – what is the difference?
  6. Linear equations with a positive x-term
  7. Linear equations with a negative x-term
  8. Linear equations with two x-terms
  9. Linear equations with brackets
  10. Fractional linear equations

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Polynomials – Playlist

  1. Polynomials

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Pythagoras’ Theorem – Playlist

  1. Pythagoras’ Theorem

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Trigonometry Introduction – Playlist

  1. Naming the sides of a right-angled triangle
  2. Trig Ratios: sinθ, cosθ and tanθ
  3. Finding the length of a side
  4. Finding an angle
  5. Mixed exercise – Pythagoras’ Theorem and trigonometry

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Algebra and Functions : 1 – Playlist

Indices – Playlist

  1. Introduction to indices (exponents)
  2. Multiplication rule for indices
  3. Division rule for indices
  4. Negative indices
  5. Fractions raised to a negative index
  6. Rational (fractional) indices
  7. Simplifying terms with negative powers
  8. Expressing terms in the form axn 
  9. Equations in which the power has to be found
  10. Summary of indices
  11. Exam Questions: Indices – Playlist

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Surds – Playlist

  1. Introduction and simplifying surds
  2. Addition and subtraction of surds
  3. Multiplying surds
  4. Dividing surds
  5. Rationalising surds
  6. Exam Questions:  Surds – Playlist

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Functions – Playlist

  1. f(x) notation

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Factorising – Playlist

  1. Introduction to factorising
  2. Highest common factor (HCF)
  3. Factorising by grouping
  4. Factorising quadratic expressions
  5. Mixed Exercise – Factorising

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Completing the Square – Playlist

  1. Completing the square
  2. Applications of completing the square – sketching graphs
  3. Exam Questions: Completing the square – Playlist

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Quadratic Equations – Playlist

  1. Solve by factorising
  2. Solve by completing the square
  3. Solve by the quadratic formula
  4. Exam Questions: Quadratic formula – Playlist
  5. Solving quadratic equations in some function of x

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Quadratic Equations – Roots and Discriminant – Playlist

  1. Roots and discriminant of a quadratic equation
  2. Exam Questions: Roots and discriminant – Playlist

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Quadratic Graphs Playlist

  1. Sketching quadratic graphs

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Simultaneous Equations – Playlist

  1. Elimination method for linear equations
  2. Substitution method for linear and non-linear equations
  3. Exam Questions: Simultaneous equations – Playlist

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Inequalities – Playlist

  1. Linear inequalities
  2. Rules for reversing the inequality sign
  3. Solving a linear type
  4. Solving a simultaneous double inequality
  5. Shading regions for a linear inequality
  6. Quadratic inequalities
  7. Exam Questions: Inequalities – Playlist

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Algebraic Long Division – Playlist

  1. Algebraic long division

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Factor Theorem – Playlist

  1. The factor theorem
  2. How to solve a cubic equation
  3. Exam Questions: Factor theorem – Playlist

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Rational Expressions: Simplifying – Playlist

  1. Simplifying rational expressions
  2. Addition and subtraction of algebraic fractions
  3. Exam Questions: Addition and Subtraction of algebraic fractions – Playlist
  4. Multiplication of algebraic fractions
  5. Further simplifying of ‘stacked fractions’
  6. Division of algebraic fractions
  7. Exam Questions: Algebraic long division – Playlist

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Coordinate Geometry: – Playlist

Gradient of Straight Lines – Playlist

  1. Line segment
  2. Parallel lines
  3. Perpendicular lines

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Straight Lines –  Playlist

  1. Equation of a straight line: y=mx+c
  2. Equation of a line given the gradient and point
  3. Distance between two points
  4. Mid-point of a line segment
  5. Equation of a parallel line
  6. Equation of a perpendicular bisector

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Intersection of Graphs – Playlist

  1. Intersection of two straight lines
  2. Intersection of a straight line and a parabola
  3. Intersection of a straight line and a hyperbola
  4. Nature of the intersection

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Exam Questions – Straight Lines

  1. Exam Questions: Straight lines – Playlist

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Circles – Playlist

  1. Equation of a circle
  2. Finding the centre and radius
  3. Circle properties
  4. Equation of a tangent to a circle
  5. Equation of a circle through 3 points
  6. Exam Questions: Circles – Playlist

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Parametric Equations – Playlist

  1. Parametric equations
  2. Converting to Cartesian form
  3. Exam Questions: Parametric to Cartesian equations – Playlist
  4. Sketching parametric graphs
  5. Parametric equations of circle, ellipse, parabola and hyperbola
  6. Finding points of intersection between a parametric and Cartesian equation
  7. Exam Questions: Parametric equations – Playlist

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Algebra and Functions : 2 – Playlist

Sketching Cubic and Reciprocal Curves – Playlist

  1. Sketching cubic curves
  2. Sketching reciprocal curves of the form y=k/x

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Modulus Functions, Equations and Inequalities – Playlist

  1. The modulus function
  2. Graphing y=|f(x)|
  3. Modulus equations
  4. Modulus inequalities
  5. Exam Questions: Modulus equations and inequalities – Playlist

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Working with Functions – Playlist

  1. f(x) notation
  2. Domain and range
  3. Exam Questions: Domain and range – Playlist
  4. Combination of functions
  5. The inverse of a function
  6. Graphical relationship between f(x) and its inverse
  7. Exam Questions: Inverse functions – Playlist

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Graph Transformations – Playlist

  1. Basic graphs used in transformations
  2. Translations of graphs
  3. Reflections of graphs
  4. Stretches of graphs
  5. Exam Questions: Graph transformations – Playlist

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Asymptotes – Playlist

  1. Asymptotes – horizontal and vertical types

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Partial Fractions – Playlist

  1. Partial fractions: What are they?
  2. Denominator contains 2 or 3 linear factors
  3. Denominator contains repeated factors
  4. Exam Questions: Partial Fractions – Playlist

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Sequences and Series: – Playlist

Binomial Expansion – Playlist

  1. The nCr function
  2. The Binomial expansion using the nCr method
  3. Pascal’s triangle method
  4. Binomial expansion for rational powers
  5. Validity
  6. Exam Questions: Binomial expansion – Playlist
  7. Using partial fractions with the binomial expansion

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Working with Sequences and Series – Playlist

  1. Definition and finding the nth term
  2. Recurrence relationships
  3. Exam Questions: Recurrence relationships – Playlist
  4. Sigma notation

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Arithmetic Sequence and Series – Playlist

  1. Arithmetic progressions
  2. Sum of the first n terms
  3. Finding a and d given two terms
  4. Working with consecutive terms
  5. Exam Questions: Arithmetic sequences and series – Playlist
  6. Examsolutions Beastie – Arithmetic progressions

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Geometric Sequence and Series – Playlist

  1. Geometric series
  2. Proof of the sum of the first n terms, Sn
  3. Sum to infinity
  4. Exam Questions: Geometric series – Playlist

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Trigonometry: – Playlist

Trigonometric Ratios – Playlist

  1. Trigonometric ratios for 30°, 45° and 60°

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Trigonometric Graphs and Transformations – Playlist

  1. Trigonometric graphs
  2. Translations
  3. Reflections
  4. Stretches
  5. Combining transformations
  6. Exam Questions: Transformations of trig. graphs – Playlist

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Applications of Trigonometry – Playlist

  1. Area of a triangle – given 2 sides and an included angle
  2. Sine rule
  3. Exam Questions: Sine rule – Playlist
  4. Cosine rule
  5. Radians
  6. Arcs, sectors and segments
  7. Exam Questions: Arcs, sectors and segments – Playlist

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Trigonometric Equations – Playlist

  1. Quadrant rule to solve trig. equations
  2. Trig equations with different ranges
  3. Trig. equations with multiple angles
  4. Trig. equations that factorise

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Trigonometric Identities and Equations – Playlist

  1. Using the identitities: tanθ ≡sinθ/cosθ and sin²θ+cos²θ ≡1
  2. Using the identities: cos(θ) ≡ cos(-θ), sin(θ) ≡ -sin(-θ)
  3. Solving equations using identities
  4. Exam Questions: Trig. equations(1) – Playlist

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Sec θ, Cosec θ and Cot θ – Playlist

  1. Trig functions sec θ, cosec θ and cot θ
  2. Graphs of sec θ, cosec θ and cot θ

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Inverse trigonometric functions – Playlist

  1. Inverse trigonometric functions – arcsin x, arccos x, arctan x
  2. Examples using inverse trigonometric functions

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Identities & Equations – Pythagorean Type – Playlist

  1. sin²x + cos²x ≡1 , 1 + tan²x ≡ sec²x , 1 + cot²x ≡ cosec²x
  2. Examples using Pythagorean identities
  3. Solving equations using Pythagorean identities

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Small-angle Approximations – Playlist

  1. Small-angle approximations

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Identities – Addition type – Playlist

  1. sin(A±B), cos(A±B) and tan(A±B)
  2. Using the Addition formulae to get exact values
  3. Proving identities using the addition formulae
  4. Identities – Addition type – Equations

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Identities & Equations – Double angle type – Playlist

  1. Identities for sin2A, cos2A and tan2A
  2. Examples using double angle identities
  3. Solving equations using double angle identities
  4. Exam Questions: Double Angles – Playlist

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Identities – Half angle type – Playlist

  1. Examples using half angle identities

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Identities – Triple angle type – Playlist

  1. Identity for cos 3θ and sin 3θ

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Identities & Equations – Harmonic Formulae – Playlist

  1. Harmonic Identities Rsin(x ± α), Rcos(x ± α)
  2. Equations using harmonic identities
  3. Max and min values of a harmonic function
  4. Exam Questions: Harmonic identities and equations – Playlist

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Exam Questions – Mixed trigonometry – Playlist

  1. Exam Questions: Mixed trigonometry – Playlist

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Logarithmic and Exponential Functions: – Playlist

Exponential Functions and Logarithms – Playlist

  1. Exponential functions: what they are and their graphs
  2. What do we mean by a log?
  3. Rules of logs
  4. Logarithms – change of base
  5. Simplifying and expanding
  6. Exponential and log equations
  7. Simultaneous equations
  8. Solving inequalities

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The Exponential Function ex and Natural Log Functions – Playlist

  1. The exponential function ex
  2. Sketching exponential graphs based on transformations
  3. The natural logarithmic function, ln x
  4. Exam Questions: Natural log functions – Playlist

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Modelling Curves of the form y=kxn and y=kax – Playlist

  1. Modelling curves – Converting to linear form

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Differentiation: – Playlist

Differentiation (1) – Playlist

  1. The gradient function dy/dx
  2. Differentiation – terms of the form axn
  3. The second derivative
  4. Exam Questions: Differentiation axn types – Playlist

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Tangents and Normals – Playlist

  1. Equations of tangents and normals
  2. Exam Questions: Tangents and normals – Playlist

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Stationary Points – Playlist

  1. Stationary points
  2. Nature of a stationary point
  3. Exam Questions: Stationary points – Playlist
  4. Applications of stationary points
  5. Exam Questions: Applications of stationary points – Playlist

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Increasing and Decreasing functions – Playlist

  1. Increasing and decreasing functions

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Standard Differentials – Playlist

  1. Exponential function ex
  2. The natural log function, ln(x)
  3. The trig functions sin(x), cos(x) and tan(x)

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The Chain Rule – Playlist

  1. Chain rule: Polynomial to a rational power
  2. Chain rule: Exponential Functions
  3. Chain rule: Log Functions
  4. Chain rule: Trigonometric types

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The Product and Quotient Rules – Playlist

  1. The product rule
  2. The quotient rule

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More Standard Differentials – Playlist

  1. The trig functions, sec(x), cosec(x) and cot(x)

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The Reciprocal Function of dy/dx – Playlist

  1. The reciprocal function of dy/dx

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Exam Questions – Differentiation – Playlist

  1. Exam Questions: Chain, product and quotient rules – Playlist
  2. Exam Questions: Exponential rates of change – Playlist

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Exponential Functions – Playlist

  1. Differentiation: Exponential functions of the form y=ax

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Parametric Functions – Playlist

  1. Differentiation: Parametric functions
  2. Tangents and normals

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Implicit Functions – Playlist

  1. Implicit functions
  2. Tangents and normals
  3. Exam Questions: Implicit functions – Playlist

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Connected Rates of Change – Playlist

  1. Connected rates of change
  2. Using three connected rates of change
  3. Connected rates of change cone type problems
  4. Exam Questions: Connected rates of change – Playlist

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Integration: – Playlist

Integration: introduction – Playlist

  1. Integration – Terms of the form axn
  2. Exam Questions: Integration: axn types – Playlist

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Equations of Curves – Playlist

  1. Finding the equation of a curve given the gradient function
  2. Exam Questions: Equations of curves – Playlist

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Definite Integration – Playlist

  1. Definite integration

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Integration – Common Functions – Playlist

  1. Integration:(ax+b)n types
  2. Exam Questions: Integration:(ax+b)n types – Playlist
  3. Integrating exponential functions ex, eax and e(ax+b)
  4. Exam Questions: Integrating exponential functions ex, eax and e(ax+b) – Playlist
  5. Integrating reciprocal functions 1/x and 1/(ax+b)
  6. Exam Questions: Integrating reciprocal functions 1/x and 1/(ax+b ) – Playlist
  7. Integrals of the form : f ‘(x)/f(x)
  8. Integrals of the form : f'(x)ef(x)

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Integrals of Trigonometric Functions – Playlist

  1. Integrals of sin x, cos x, sec² x
  2. Integrals of the form sin(ax+b), cos(ax+b), sec² (ax+b) types
  3. Identity types
  4. sin2x and cos2x types
  5. Exam Questions: Trigonometric types Playlist

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Integrals involving Partial fractions – Playlist

  1. Integrals involving partial fractions
  2. Exam Questions: Integrals involving partial fractions – Playlist

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Integration by Substitution – Playlist

  1. Integration by substitution
  2. Square root types
  3. Integration of trigonometric functions by substitution
  4. Integration of exponential types by substitution
  5. Integration by substitution using limits
  6. Integration of trigonometric functions by substitution with limits
  7. Exam Questions: Integration by substitution – Playlist

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Integrals of the form f[g(x)]g'(x) by inspection – Playlist

  1. Integrating products of the form f[g(x)]g'(x) by inspection

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Integration by Parts – Playlist

  1. Integration by parts
  2. Integration by parts (ln types)
  3. eaxsin(bx) and eaxcos(bx) types
  4. Integration by parts using limits
  5. Proof of the formula – Integration by parts
  6. Exam Questions: Integration by parts – Playlist

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General Methods – Integration – Playlist

  1. Integration – General Methods
  2. Mixed Examples – Integration

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Applications of Integration – Area bound by a curve – Playlist

  1. Area bound by a curve and x-axis
  2. Exam Questions: Area bound by a curve and x-axis – Playlist
  3. Area under a graph : parametric type

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Differential equations – Separating the variables – Playlist

  1. Finding a general solution and a particular solution
  2. Working with constants in log types
  3. Exponential and trig type

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Differential equations – Forming differential equations – Playlist

  1. Direct proportion type
  2. Inverse proportion type
  3. Newton’s law of cooling
  4. Exam Questions: Forming differential equations – Playlist

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Numerical Methods: – Playlist

Solution of Equations by Numerical methods – Playlist

  1. Graphical methods
  2. Change of sign
  3. Iteration
  4. Exam Questions: Iteration – Playlist
  5. Newton-Raphson method for locating a root in a given interva
  6. Exam Questions: Newton-Raphson – Playlist

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Numerical Integration – Playlist

  1. Trapezium rule
  2. Exam Questions: Trapezium rule – Playlist

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Vectors: – Playlist

Vectors – Playlist

  1. What is a vector and a scalar quantity?
  2. Vector notation 2-dimensions
  3. Vector notation
  4. Position vectors 2-dimensions
  5. Position vectors
  6. Equal and negative vectors
  7. Multiplying a vector by a scalar 2-dimensions
  8. Multiplying a vector by a scalar
  9. Addition and subtraction of vectors 2-dimensions
  10. Addition and subtraction of vectors
  11. Magnitude of a 2 dimensional vector
  12. Magnitude of a 3 dimensional vector
  13. Unit vectors

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Proof – Playlist

  1. Proving Identities
  2. Proof by Exhaustion and Deduction
  3. Using a Counter-Example

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Past Papers

For further revision and practice I have past papers, some with worked solutions – Check them out for:

Edexel | AQA | OCR

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Pure Maths A-Level Index

algebra and functions
arc length
arithmetic sequence/series
asymptote
binomial expansion
circles
chain rule
change of sign
completing the square
connected rates of change
coordinate geometry
cosine rule
cubic graph
decreasing functions
differential equations:
solving
forming
differentiation:
chain rule
connected rates of change
implicit
parametric
product rule
quotient rule
differentials:
ax
ex
ln x
sin, cos, tan(x)
sec, cosec, cot(x)
discriminant
domain – functions
exponential functions
ex
factorising
factor theorem
functions
geometric sequence/series
gradient – straight line
graphs:
cubic
intersection
reciprocal
implicit differentiation
increasing functions
indices
inequalities
integration
by parts
common functions
general methods
partial fractions
substitution
trigonometric
definite
integration applications
area under a curve
iteration
intersection – graphs
inverse functions
logarithms
natural logarithms
long division – algebraic
maximum stationary point
minimum stationary point
modelling curves
modulus function
Newton-Raphson method
normals to curves
numerical methods
integration
solving equations
parametric:
differentiating
equations
partial fractions
product rule
proof
quadrant rule
quadratics:
discriminant
equations
graphs
quotient rule
radian
range – functions
rational expressions
reciprocal graphs
recurrence relationships
sector
segment
sequences and series
sigma notation
simultaneous equations
sine rule
stationary points
straight lines
surds
tangents to curves
transformations:
graphs
trapezium rule
trigonometric:
equations
graphs
ratios 30°, 60°, 45°
trigonometry
cosine rule
cosec, sec and cot θ
inverse functions
sine rule
small angles
trigonometry – identities:
addition formulae
double angle formulae
half angle formulae
harmonic formulae
Pythagorean
triple angle formulae
vectors