Mathematics (UPCSE)

Find out more about the Mathematics foundation subject on the UPCSE. Learn about what you will study, teaching methods, assessments and recommended reading.

Subject description

The Mathematics subject is an optional part of the Undergraduate Preparatory Certificate for Science and Engineering (UPCSE) course.

We’ll provide you with an in-depth introduction into Pure Maths. You’ll develop your skills in Algebra, Trigonometry, Geometry and their related topics.

The subject aims to prepare students thoroughly with the skills needed to study at UK universities. Students typically go on to study degrees such as Mathematics, Engineering, Life Sciences and Physical Sciences.

The syllabus covers the following topics:

Algebra

• Quadratic functions
• Long division
• Factorisation
• Fundamental theorem of algebra
• Remainder and factor theorem
• Partial fractions
• Proof by induction
• Binomial theorem and generalised binomial theorem

Differentiation

• Limits at a point
• Differentiation from first principles
• Rules of differentiation (product rule, quotient rule, chain rule)
• Implicit differentiation
• Tangents and normals
• Finding and classifying stationary points
• Optimisation

Curve Sketching

• x-intercepts, y-intercepts, points of inflection, end behaviour/limits at infinity, vertical asymptotes
• Numerical methods, Newton-Raphson method, bisection method
• Transformations of curves

Integration

• Indefinite integrals
• Integration by parts, integration by substitution
• Definite integration
• Trigonometric substitutions
• Using integration to find area and volumes of revolution

Series

• Arithmetic and geometric progressions/series
• Method of differences/telescoping series
• Sum of squares and cubes
• Maclaurin Series, Taylor Series

Complex numbers

• Complex arithmetic, properties of complex numbers
• Factorisation of polynomials over complex numbers
• Argand diagram
• Polar form, exponential form
• De Moivre’s theorem
• Loci
• Transformation of the complex plane

Vectors

• Vector arithmetic
• Scalar product, cross product, triple product
• Equations of lines and planes
• Finding the angle/distance between planes/lines/points
• Finding the intersection of lines/planes

How we teach Mathematics

You'll have two face-to-face lectures every week. You’ll also have two in-person smaller-group tutorials. During the tutorials, you’ll work on Pure Maths exercises based on the lectures. We expect you to carry out independent study and problem-solving outside of your classes.

Office hours will be available for additional support throughout your foundation year.

Intended learning outcomes

By the end of the subject, you should be able to:

• state and prove the main theoretical results the syllabus is based on
• apply the theory to problem-solving
• present well-written solutions to problems with good attention to detail and a logical structure
• work independently on mathematics exercises, taking responsibility for your own understanding and asking for help when necessary
• confidently discuss mathematical ideas in English, using correct vocabulary

Overview of Summative Assessment

Recommended reading

Before starting the course, you should prepare by revising high school work. You’ll need to be comfortable with exponentials, logarithms and have strong trigonometry knowledge.

You should be able to work with all six trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) and the inverse trigonometric functions. You should also be familiar with standard trigonometric identities and using them in proofs. Any standard high school mathematics text should cover this Pure Maths material (e.g., ‘Mathematics- The Core Course for A-level’ – see below).

Additional costs

• Calculator: Casio fx-83GT X, cost is approximately £20
   
• Textbook: Mathematics: ‘The Core Course for A-level’, by L Bostock and S Chandler, published by Stanley Thornes, ISBN 0859503062. This book costs approximately £35 if bought new. Cheaper second-hand copies are available.
   
• Textbook: ‘Mathematics: Further Pure Mathematics’, by L Bostock, S Chandler and C P Rourke, published by Stanley Thornes, ISBN 9780859501033). This also costs around £35 if bought new. Cheaper second-hand copies are readily available.

Please note the information on all the UPC subject pages, including this one, reflects the subject as it was taught in the 2024-25 academic year (unless otherwise stated). The information may change for the next year of entry.