A-level Further Mathematics for Year 13 - Course 1: Differential Equations, Further Integration, Curve Sketching, Complex Numbers, the Vector Product and Further Matrices from edX

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, the 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

Analytical and numerical methods for solving first-order differential equations
The nth roots of unity, the nth roots of any complex number, geometrical applications of complex numbers.
Coordinate systems and curve sketching.
Improper integrals, integration using partial fractions and reduction formulae
The area enclosed by a curve defined by parametric equations or polar equations, arc length and the surface area of revolution.
Solving second-order differential equations
The vector product and its applications
Eigenvalues, eigenvectors, diagonalization and the Cayley-Hamilton Theorem.

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.

What you'll learn

How to find the general or particular solution to a first-order differential equation by inspection or by using an integrating factor.

How to find a numerical solution to a differential equation using the Euler method or an improved Euler method..

How to find the nth roots of unity

How to find the nth roots of a complex number in the form

How to use complex roots of unity to solve geometrical problems.

How to identify the features of parabolas, rectangular hyperbolae, ellipses and hyperbolae defined by Cartesian and parametric equations.

How to identify features of graphs defined by rational functions.

How to define a parabola, ellipse or hyperbola using focus-directrix properties and eccentricity.

How to evaluate improper integrals.

How to integrate using partial fractions

How to derive and use reduction formulae

How to find areas enclosed by curves that are defined parametrically.

How to find the area enclosed by a polar curve.

How to calculate arc length.

How to calculate the surface area of revolution.

How to find the auxiliary equation for a second order differential equation.

What's inside

Syllabus

Module 1: First Order Differential Equations

Solving first order differential equations by inspection

Solving first order differential equations using an integrating factor

Traffic lights

Read about what's good

what should give you pause

and possible dealbreakers

Focuses on strengthening an existing foundation for intermediate learners, developing skills to comprehend A level further mathematics

Instructors have extensive knowledge and expertise in further mathematics

Teaches essential skills for A level further mathematics exams, focusing on fluency, confidence, and critical thinking

Builds a strong foundation for advanced mathematical topics, expanding knowledge beyond A level further mathematics

Requires learners to come in with a background in mathematics, potentially creating a barrier for some learners

Course materials may not align with the latest industry practices and versions of software

Reviews summary

Comprehensive a-level further maths prep

According to students, this course provides a strong foundation and is highly relevant for A-level Further Mathematics exams. Many learners praised the clarity and expertise of the instructors from Imperial College London, noting their ability to explain complex mathematical concepts effectively. The course is described as well-structured, breaking down advanced topics like differential equations, complex numbers, and matrices into manageable modules. While generally very positive, some students indicated a need for more varied practice problems or suggested that certain sections could benefit from additional supplementary materials to fully solidify understanding, especially for those requiring more in-depth practice beyond the core content.

Pacing suits most, but some topics may need extra time.

"The course moves at a good pace for someone familiar with the basics, but might be fast for others."

"I sometimes needed to pause and re-watch sections to fully grasp the more intricate proofs."

"It’s a great standalone resource, but for true mastery, I'd suggest supplementing with a textbook for deeper dives into certain areas."

Wide range of advanced mathematical topics covered.

"The breadth of topics, from vector products to eigenvalues, was impressive and thorough."

"I appreciated the depth provided on areas like further integration and curve sketching."

"The coverage of complex numbers and matrices was comprehensive for the A-level standard."

Content directly aligns with A-level Further Maths syllabus.

"This course perfectly covers the A-level Further Maths syllabus, preparing me well for my exams."

"I felt much more confident tackling exam-style questions after completing these modules."

"The topics align precisely with what's expected for Year 13 further mathematics."

Instructors deliver complex topics with exceptional clarity.

"The instructor explains difficult concepts with such clarity that I finally understood topics I struggled with before."

"I found the lectures incredibly well-presented, making even the most abstract ideas accessible."

"The way they broke down differential equations was fantastic; I really appreciated the step-by-step approach."

Some learners desired more practice problems for mastery.

"While the examples were good, I felt there weren't quite enough practice problems to truly master every concept."

"I sometimes had to seek out additional exercises from textbooks to get enough practice."

"More practice questions, especially challenging ones, would elevate this course even further."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in A-level Further Mathematics for Year 13 - Course 1: Differential Equations, Further Integration, Curve Sketching, Complex Numbers, the Vector Product and Further Matrices with these activities:

Organize and review class materials

Show steps

Review the lecture notes, assignments, and other materials from the course to reinforce your understanding of the concepts covered and identify areas where you need additional support.

Show steps

Gather all the materials from the course.
Organize the materials into a logical order.
Review the materials and make notes.

Review 'Further Pure Mathematics 2' by Frank Wood

Show steps

Review 'Further Pure Mathematics 2' by Frank Wood to supplement your learning. This book covers advanced topics in further mathematics, providing additional insights and examples to reinforce your understanding of the concepts covered in this course.

Show steps

Read the relevant chapters.
Take notes and highlight important concepts.
Solve practice problems.

Practice first order differential equations

Show steps

Practice solving first order differential equations by inspection and using an integrating factor to strengthen your understanding of the techniques covered in Module 1.

Browse courses on First Order Differential Equations

Show steps

Find the general solution of a first-order differential equation by inspection.
Solve a first-order differential equation using an integrating factor.
Find the particular solution of a first-order differential equation given initial conditions.

One other activity

Expand to see all activities and additional details

Show all four activities

Calculate arc length and surface area of revolution

Show steps

Practice calculating arc length and surface area of revolution using integration techniques to reinforce your understanding of the concepts covered in Module 5.

Show steps

Set up the integral for arc length.
Evaluate the integral to find the arc length.
Set up the integral for surface area of revolution.
Evaluate the integral to find the surface area of revolution.

Career center

Learners who complete A-level Further Mathematics for Year 13 - Course 1: Differential Equations, Further Integration, Curve Sketching, Complex Numbers, the Vector Product and Further Matrices will develop knowledge and skills that may be useful to these careers:

College Mathematics Teacher

A College Mathematics Teacher at a community college or university educates students about foundational and advanced mathematical concepts. Instructing students on these concepts will be a lot easier after taking this comprehensive A-Level Further Mathematics course. The course covers a wide range of mathematical topics, including differential equations, complex numbers, vectors, and matrices. Those enrolling in this course will greatly benefit from the focus on analytical and numerical methods for solving complex mathematical problems, which will be invaluable as an educator of college-level mathematics students.

See salaries and explore the career path for College Mathematics Teacher

Data Scientist

A Data Scientist uses their expertise in statistics, machine learning, and data analysis to solve complex problems. This course in A-Level Further Mathematics provides a strong foundation in the mathematical and statistical concepts that underpin data science. The course's focus on problem solving and critical thinking will also be beneficial for a Data Scientist, who must be able to identify trends and patterns in data.

See salaries and explore the career path for Data Scientist

Operations Research Analyst

An Operations Research Analyst uses mathematical and statistical techniques to help organizations improve their efficiency. This course in A-Level Further Mathematics provides a strong foundation in the mathematical and statistical concepts that underpin operations research. The course's focus on problem solving and critical thinking will also be beneficial for an Operations Research Analyst, who must be able to identify trends and patterns in data.

See salaries and explore the career path for Operations Research Analyst

Financial Analyst

A Financial Analyst uses their knowledge of finance and economics to make investment recommendations. This course in A-Level Further Mathematics provides a strong foundation in the mathematical and statistical concepts that underpin financial analysis. The course's focus on problem solving and critical thinking will also be beneficial for a Financial Analyst, who must be able to identify trends and patterns in data.

See salaries and explore the career path for Financial Analyst

Investment Banker

An Investment Banker provides financial advice to corporations and governments. This course in A-Level Further Mathematics provides a strong foundation in the mathematical and statistical concepts that underpin investment banking. The course's focus on problem solving and critical thinking will also be beneficial for an Investment Banker, who must be able to identify trends and patterns in data.

See salaries and explore the career path for Investment Banker

Software Engineer

A Software Engineer designs, develops, and maintains software applications. This course in A-Level Further Mathematics provides a strong foundation in the mathematical and computational concepts that underpin software engineering. The course's focus on problem solving and critical thinking will also be beneficial for a Software Engineer, who must be able to identify and solve complex problems.

See salaries and explore the career path for Software Engineer

Data Analyst

A Data Analyst collects, cleans, and interprets data to help organizations make informed decisions. This course in A-Level Further Mathematics provides a strong foundation in statistical and mathematical techniques commonly used in the field of data analysis. The course's focus on problem solving and critical thinking will also be beneficial for a Data Analyst who must be able to identify trends and patterns in data.

See salaries and explore the career path for Data Analyst

Market Research Analyst

A Market Research Analyst collects and analyzes data to help businesses understand their target market. This course in A-Level Further Mathematics provides a strong foundation in the statistical and mathematical techniques commonly used in market research. The course's focus on problem solving and critical thinking will also be beneficial for a Market Research Analyst, who must be able to identify trends and patterns in data.

See salaries and explore the career path for Market Research Analyst

Statistician

A Statistician collects, analyzes, and interprets data to help organizations make informed decisions. This course in A-Level Further Mathematics provides a strong foundation in the statistical and mathematical techniques commonly used in statistics. The course's focus on problem solving and critical thinking will also be beneficial for a Statistician, who must be able to identify trends and patterns in data.

See salaries and explore the career path for Statistician

Teacher

A Teacher educates students in a variety of subjects, including mathematics. This course in A-Level Further Mathematics provides a strong foundation in the mathematical concepts that underpin teaching. The course's focus on problem solving and critical thinking will also be beneficial for a Teacher, who must be able to identify and solve complex problems.

See salaries and explore the career path for Teacher

Consultant

A Consultant provides expert advice to businesses and organizations. This course in A-Level Further Mathematics provides a strong foundation in the analytical and problem-solving skills that are essential for a Consultant. The course's focus on critical thinking and communication will also be beneficial for a Consultant, who must be able to communicate complex ideas clearly and effectively.

See salaries and explore the career path for Consultant

Economist

An Economist studies the production, distribution, and consumption of goods and services. This course in A-Level Further Mathematics provides a strong foundation in the mathematical and statistical concepts that underpin economics. The course's focus on problem solving and critical thinking will also be beneficial for an Economist, who must be able to identify and solve complex problems.

See salaries and explore the career path for Economist

Financial Planner

A Financial Planner helps individuals and families plan for their financial future. This course in A-Level Further Mathematics provides a strong foundation in the mathematical and financial concepts that underpin financial planning. The course's focus on problem solving and critical thinking will also be beneficial for a Financial Planner, who must be able to identify and solve complex problems.

See salaries and explore the career path for Financial Planner

Market Researcher

A Market Researcher collects and analyzes data to help businesses understand their target market. This course in A-Level Further Mathematics provides a strong foundation in the statistical and mathematical techniques commonly used in market research. The course's focus on problem solving and critical thinking will also be beneficial for a Market Researcher, who must be able to identify and solve complex problems.

See salaries and explore the career path for Market Researcher

Risk Manager

A Risk Manager identifies and assesses risks to an organization. This course in A-Level Further Mathematics provides a strong foundation in the mathematical and statistical concepts that underpin risk management. The course's focus on problem solving and critical thinking will also be beneficial for a Risk Manager, who must be able to identify and solve complex problems.

See salaries and explore the career path for Risk Manager