A-Level Edexcel Core Pure Maths

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
• Division rule
• Mixed examples
• Exam questions

De Moivre's Theorem

• Introduction
• 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
• 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

• Circles
• Perpendicular bisectors
• Half-lines
• Representing regions on an argand diagram
• Exam questions

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

• Solving simultaneous equations with matrices

Matrix Linear Transformations

• Rotations
• Rotation 90 degrees
• Rotation 180 degrees
• Rotation 270 degrees
• Rotation θ degrees
• Reflections
• Reflections in the x-axis
• Reflections in the y-axis
• Reflection in the line y = x
• Reflection in the line y = -x
• 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
• Exam questions

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

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

• Introduction
• Factorial example
• Exam questions

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 series
• Exam questions

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

• Matrix multiplication proof 1
• Matrix multiplication proof 2
• Exam questions

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

• Convergent integrals
• Divergent integrals

Integrals involving Partial Fractions

• Integrating Partial Fraction types
• Exam questions

Differentiating Inverse Trigonometric Functions

• Derivatives of sin^-1(x), cos^-1(x) and tan^-1(x)
• Differentiating inverse sin(x/a)
• Differentiating inverse cos(x/a)
• Differentiating inverse tan(x/a)
• Example 1 | 2 | 3

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)

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

• Exam questions

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

• 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
• Spirals
• Half-lines
• 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)
• Exam questions

Tangents

• Finding tangents parallel to the initial line
• Finding tangents perpendicular to the initial line
• Exam questions

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)

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

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) = 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
• Exam questions