# ImperialX: Further Mathematics Year 13 course 2: Applications of Differential Equations, Momentum, Work, Energy & Power, The Poisson Distribution, The Central Limit Theorem, Chi Squared Tests, Type I and II Errors

Develop your thinking skills, fluency and confidence in the applied mathematics content of A-level further maths and prepare for undergraduate STEM degrees.

7 weeks
2–4 hours per week
Instructor-paced
Instructor-led on a course schedule
Free

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Started Jan 3
Ends Apr 2, 2026

This course by Imperial College London is designed to help you develop the skills you need to succeed in your A-level further maths exams.

You will investigate key topic areas to gain a deeper understanding of the skills and techniques that you can apply throughout your A-level study. These skills include:

* Fluency – selecting and applying correct methods to answer with speed and efficiency

* Confidence – critically assessing mathematical methods and investigating ways to apply them

* Problem solving – analysing the ‘unfamiliar’ and identifying which skills and techniques you require to answer questions

* Constructing mathematical argument – using mathematical tools such as diagrams, graphs, logical deduction, mathematical symbols, mathematical language, construct mathematical argument and present precisely to others

* Deep reasoning – analysing and critiquing mathematical techniques, arguments, formulae and proofs to comprehend how they can be applied

Over eight modules, you will be introduced to

* Simple harmonic motion and damped oscillations.

* Impulse and momentum.

* The work done by a constant and a variable force, kinetic and potential energy (both gravitational and elastic) conservation of energy, the work-energy principle, conservative and dissipative forces, power.

* Oblique impact for elastic and inelastic collision in two dimensions.

* The Poisson distribution, its properties, approximation to a binomial distribution and hypothesis testing.

* The distribution of sample means and the central limit theorem.

* Chi-squared tests, contingency tables, fitting a theoretical distribution and goodness of fit.

* Type I and type II errors in statistical tests.

Your initial skillset will be extended to give a clear understanding of how background knowledge underpins the A -level further mathematics course. You’ll also be encouraged to consider how what you know fits into the wider mathematical world.

### At a glance

• Institution: ImperialX
• Subject: Math
• Level: Intermediate
• Prerequisites:
None
• Language: English
• Video Transcript: English
• Associated skills:Language Construct, Basic Math, Differential Equations, Contingency Tables, Chi-Squared Tests, Problem Solving, Applied Mathematics

# What you'll learn

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How to derive and solve a second order differential equation that models simple harmonic motion.

How to derive a second order differential equation for damped oscillations.

The meaning of underdamping, critical damping and overdamping.

How to solve coupled differential equations.

How to calculate the impulse of one object on another in a collision.

How to use the principle of conservation of momentum to model collisions in one dimension.

How to use Newton’s experimental law to model inelastic collisions in one dimension.

How to calculate the work done by a force and the work done against a resistive force.

How to calculate gravitational potential energy and kinetic energy.

How to calculate elastic potential energy.

How to solve problems in which energy is conserved.

How to solve problems in which some energy is lost through work against a dissipative force.

How to calculate power and solve problems involving power.

How to model elastic collision between bodies in two dimensions.

How to model inelastic collision between two bodies in two dimensions.

How to calculate the energy lost in a collision.

How to calculate probability for a Poisson distribution.

How to use the properties of a Poisson distribution.

How to use a Poisson distribution to model a binomial distribution.

How to use a hypothesis test to test for the mean of a Poisson distribution.

How to estimate a population mean from sample data.

How to estimating population variance using the sample variance. How to calculate and interpret the standard error of the mean.

How and when to apply the Central Limit Theorem to the distribution of sample means.

How to use the Central Limit Theorem in probability calculations, using a continuity correction where appropriate.

How to apply the Central Limit Theorem to the sum of n identically distributed independent random variables.

How to conduct a chi-squared test with the appropriate number of degrees of freedom to test for independence in a contingency table and interpret the results of such a test.

How to fit a theoretical distribution, as prescribed by a given hypothesis involving a given ratio, proportion or discrete uniform distribution, to given data.

How to use a chi-squared test with the appropriate number of degrees of freedom to carry out a goodness of fit test.

How to calculate the probability of making a Type I error from tests based on a Poisson or Binomial distribution.

How to calculate probability of making Type I error from tests based on a normal distribution.

How to calculate P(Type II error) and power for a hypothesis test for tests based on a normal, Binomial or a Poisson distribution (or any other A level distribution).

# Syllabus

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Module 1: Applications of Differential Equations

Using differential equations in modelling in kinematics and in other contexts.

Hooke’s law.

Simple harmonic motion (SHM).

Damped oscillatory motion.

Light, critical and heavy damping.

Coupled differential equations.

Module 2: Momentum and Impulse

Momentum and the principle of conservation of momentum.

Impulse.

Newton’s experimental law (restitution)

Impulse for variable forces.

Module 3: Work, Energy and Power

The work-energy principle.

Conservation of mechanical energy.

Gravitational potential energy and kinetic energy.

Elastic potential energy.

Conservative and dissipative forces.

Power

Module 4: Oblique Impact

Modelling elastic collision in two dimensions.

Modelling inelastic collision in two dimensions.

The kinetic energy lost in a collision.

Module 5: Expectation and Variance and the Poisson Distribution

The Poisson distribution.

Properties of the Poisson distribution.

Approximating the binomial distribution.

Testing for the mean of a Poisson distribution.

Module 6: The Central Limit Theorem

The distribution of a sample mean.

Underlying normal distributions.

The Central Limit Theorem.

Module 7: Chi-Squared Tests

Chi-squared tests and contingency tables.

Fitting a theoretical distribution.

Testing for goodness of fit.

Module 8: Type I and Type II Errors

What are type I and type II errors?

A summary of all probability distributions encountered in A level maths and further maths.