Pure Maths can be thought of as the heart of mathematics. It consists of the core aspects of maths before any application to the real world. There are many extensive topics in Pure Maths but before any student can explore these, they must learn the basics at A-Level. The Pure Maths elements of both the AS and the A-Level in Mathematics consist of the following topic areas:

Proof, Algebra & Functions, Coordinate Geometry, Sequences & Series, Trigonometry, Exponentials & Logarithms, Differentiation, Integration, Vectors

### Proof in AS-Level Pure Maths:

In AS-Level pure maths, students will be expected to be familiar with the following areas of proof:

### Proof in A-Level Pure Maths:

In addition to the above:

- Proof by contradiction.

### Algebra & Functions in AS-Level Pure Maths:

In AS-Level Maths, students will be expected to be familiar with the following area of Algebra & Functions:

- Indices.
- Surds.
- Sketching and solving quadratics., completing the square, Discriminants.
- Simultaneous equations.
- Inequalities.
- Polynomials.
- Curve sketching.
- Transformations.

### Algebra & Functions in A-Level Pure Maths:

In addition to the above:

- Simplifying rational expressions.
- The modulus of a linear function.
- Composite and inverse functions.
- More transformations.
- Partial fractions.
- Modelling.

### Coordinate Geometry in AS-Level Maths:

In AS-Level Maths, students will be expected to be familiar with the following area of Coordinate Geometry:

### Coordinate Geometry in A-Level Pure Maths:

In addition to the above:

- Parametric equations.

### Sequences & Series in AS-Level Pure Maths:

At AS-Level Maths, students will be expected to be familiar with the following areas of Sequences & Series:

- Binomial expansion (including factorial notation)

### Sequences & Series in A-Level Maths:

In addition to the above:

- More binomial expansion, nth term.
- Increasing, decreasing and periodic sequences.
- Sigma notation.
- Arithmetic sequences & series.
- Geometric sequences & series.
- Sequences in modelling.

### Trigonometry in AS-Level Maths:

In AS-Level Maths, students will be expected to be familiar with the following area of Trigonometry:

- The sine and cosine rule, area of a triangle.
- The graphs of sine, cosine and tan.
- trigonometric identities
- trigonometric equations

### Trigonometry in A-Level Maths:

In addition to the above:

- arc length and area of a sector
- small angle approximations
- exact values of sin, cos and tan
- reciprocal and inverse trigonometric functions
- more trigonometric identities
- double angle and compound angle formulae
- trigonometric proof
- problems in context

### Exponential & Logarithmic Functions in AS-Level Maths:

In AS-Level Maths, students will be expected to be familiar with the following area of Exponentials & Logarithms:

- functions of the form a to the x, including e
- the gradient of e to the kx
- the logarithm functions with different bases
- the logarithm rules
- solving exponential equations
- logarithmic graphs
- exponential growth and decay

### Exponential & Logarithmic Functions in A-Level Maths:

- No additional content

### Differentiation in AS-Level Maths:

In AS-Level Maths, students will be expected to be familiar with the following area of Differentiation:

- derivatives, sketching gradient functions, second derivatives, differentiation from first principles
- differentiation of linear combinations of polynomials
- equations of tangents and normals, stationary points, increasing and decreasing functions

### Differentiation in A-Level Maths:

In addition to the above:

- differentiate trigonometric functions from first principles, convex/concave functions
- differentiate trigonometric and exponential functions
- product rule, quotient rule and chain rule
- implicit and parametric differentiation
- construct simple differential equations

### Integration in AS-Level Pure Maths:

In AS-Level Maths, students will be expected to be familiar with the following area of Integration:

### Integration in A-Level Pure Maths:

In addition to the above:

- Integrate linear combinations, exponential and trigonometric functions.
- Finding areas.
- Understand that integration is the limit of a sum.
- Integration by substitution and integration by parts.
- Integrate using partial fractions.
- Separation of variables.
- Interpret the solution of a first order differential equation.

### Vectors in AS-Level Maths:

In AS-Level Maths, students will be expected to be familiar with the following area of Vectors:

- two-dimensional vectors
- magnitude and direction of vectors
- vector addition and scalar multiplication
- position vectors and distance between two points
- using vectors in context

### Vectors in A-Level Pure Maths:

In addition to the above:

- Three-dimensional vectors

### Numerical Methods in AS-Level Maths:

In AS-Level Maths, students will be expected to be familiar with the following area of Numerical Methods:

- Not covered in AS-Level Pure Maths

### Numerical Methods in A-Level Maths:

- Approximate location of roots
- Iterative methods
- Newton-Raphson method
- Numerical integration
- Problems in context

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The Edexcel Specification gives a long list of expected criteria. In general, there is a larger emphasis on the deeper understanding of pure maths and its applications. This means that students will be required to use a high level of reasoning. This means that they must be able to justify with logic and recognise incorrect reasoning. Skills include being able to generalise mathematically and constructing mathematical proof. Students must be able to select a strategy for challenging problems and use diagrams to explore mathematical scenarios where appropriate. Effective communication of interpretation of solution will be required.