A-Level Further Mathematics for Year 12 - Course 2: 3 x 3 Matrices, Mathematical Induction, Calculus Methods and Applications, Maclaurin Series, Complex Numbers and Polar Coordinates 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, 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:

The determinant and inverse of a 3 x 3 matrix
Mathematical induction
Differentiation and integration methods and some of their applications
Maclaurin series
DeMoivre’s Theorem for complex numbers and their applications
Polar coordinates and sketching polar curves
Hyperbolic functions

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 determinant of a complex number without using a calculator and interpret the result geometrically.
How to use properties of matrix determinants to simplify finding a determinant and to factorise determinants.
How to use a 3 x 3 matrix to apply a transformation in three dimensions
How to find the inverse of a 3 x 3 matrix without using a calculator.
How to prove series results using mathematical induction.
How to prove divisibility by mathematical induction.
How to prove matrix results by using mathematical induction.
How to use the chain, product and quotient rules for differentiation.
How to differentiate and integrate reciprocal and inverse trigonometric functions.
How to integrate by inspection.
How to use trigonometric identities to integrate.
How to use integration methods to find volumes of revolution.
How to use integration methods to find the mean of a function.
How to express functions as polynomial series.
How to find a Maclaurin series.
How to use standard Maclaurin series to define related series.
How to use De Moivre’s Theorem.
How to use polar coordinates to define a position in two dimensional space.
How to sketch the graphs of functions using polar coordinates.
How to define the hyperbolic sine and cosine of a value.
How to sketch graphs of hyperbolic functions.
How to differentiate and integrate hyperbolic functions.

What's inside

Syllabus

Module 1: Matrices - The determinant and inverse of a 3 x 3 matrix

Moving in to three dimensions

Conventions for matrices in 3D

The determinant of a 3 x 3 matrix and its geometrical interpretation

Determinant properties

Factorising a determinant

Transformations using 3 x 3 matrices

The inverse of a 3 x 3 matrix

Module 2: Mathematical induction

The principle behind mathematical induction and the structure of proof by induction

Mathematical induction and series

Proving divisibility by induction

Proving matrix results by induction

Module 3: Further differentiation and integration

The chain rule

The product rule and the quotient rule

Differentiation of reciprocal and inverse trigonometric functions

Integrating trigonometric functions

Integrating functions that lead to inverse trigonometric integrals

Integration by inspection

Integration using trigonometric identities

Module 4: Applications of Integration

Volumes of revolution

The mean of a function

Module 5: An Introduction to Maclaurin series

Expressing functions as polynomial series from first principles

Maclaurin series

Adapting standard Maclaurin series

Module 6: Complex Numbers: De Moivre's Theorem and exponential form

De Moivre's theorem and it's proof

Using de Moivre’s Theorem to establish trigonometrical results

De Moivre’s Theorem and complex exponents

Module 7: An introduction to polar coordinates

Defining position using polar coordinates

Sketching polar curves

Cartesian to polar form and polar to Cartesian form

Module 8: Hyperbolic functions

Defining hyperbolic functions

Graphs of hyperbolic functions

Calculations with hyperbolic functions

Inverse hyperbolic functions

* Differentiating and integrating hyperbolic functions

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Designed by Imperial College London, this course is specifically tailored to A-level students preparing for their further mathematics exams

Strengthens the skills required for A-level further mathematics, including fluency, confidence, problem-solving, constructing mathematical arguments, and deep reasoning

Covers key topic areas such as matrices, mathematical induction, differentiation, integration, series, complex numbers, polar coordinates, and hyperbolic functions

Guides students in exploring deeper mathematical concepts and techniques applicable to their A-level studies

Encourages students to connect their knowledge to the broader mathematical world

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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 12 - Course 2: 3 x 3 Matrices, Mathematical Induction, Calculus Methods and Applications, Maclaurin Series, Complex Numbers and Polar Coordinates with these activities:

Review the basics of differentiation and integration

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Refreshing these foundational concepts will strengthen the base for understanding more advanced topics in the course.

Browse courses on Differentiation

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Revisit notes or textbooks on differentiation.
Practice solving basic differentiation problems.
Review the methods of integration.
Solve simple integration problems to reinforce understanding.

Attend a workshop on mathematical induction

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Workshops offer opportunities to interact with experts and peers, reinforcing concepts through practical exercises.

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Locate a workshop on mathematical induction.
Register for and attend the workshop.
Actively participate in discussions and exercises.
Take notes and ask questions for a deeper understanding.
Follow up after the workshop by reviewing notes and practicing techniques.

Derive the determinant of a 3x3 matrix using the Laplace expansion method

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Laplace expansion provides an alternative way of finding the determinant of a matrix that complements the methods taught in the course.

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Review the Laplace expansion formula.
Choose a row or column to expand along.
Calculate the determinants of the resulting 2x2 matrices.
Alternate the signs of the determinants based on their positions.
Sum the products of the determinants and the corresponding elements in the chosen row or column.

Four other activities

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Mentor a student struggling with matrix operations

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Mentoring can help solidify concepts and develop confidence in working with matrices.

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Identify a student who could benefit from your help.
Schedule regular mentoring sessions.
Review the basics of matrix operations.
Work through practice problems together.
Provide feedback and encouragement.

Solve integration problems using trigonometric identities

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Trigonometric identities can simplify complex integrals, making them easier to solve.

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Review the trigonometric identities provided in the course.
Identify the trigonometric functions present in the integral.
Apply appropriate trigonometric identities to simplify the integrand.
Integrate the simplified integrand.
Verify the result by differentiating the answer.

Compile a collection of solved problems on hyperbolic functions

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Creating a compilation of solved problems provides a valuable resource for practicing and reinforcing concepts.

Browse courses on Hyperbolic Functions

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Gather solved problems from textbooks, online resources, or class notes.
Organize the problems by topic or difficulty level.
Review the solutions to identify common patterns and techniques.
Attempt to solve similar problems on your own.
Add your own solutions to the compilation.

Create a visual representation of De Moivre's Theorem

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Visualizing De Moivre's Theorem can deepen understanding of the relationship between complex numbers and trigonometry.

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Choose a complex number and its corresponding angle on the complex plane.
Plot the complex number on the plane.
Use De Moivre's Theorem to calculate the complex number raised to different powers.
Plot the resulting complex numbers on the plane.
Connect the points to visualize the path of the complex number as it is raised to different powers.

Career center

Learners who complete A-Level Further Mathematics for Year 12 - Course 2: 3 x 3 Matrices, Mathematical Induction, Calculus Methods and Applications, Maclaurin Series, Complex Numbers and Polar Coordinates will develop knowledge and skills that may be useful to these careers:

Data Scientist

Data Scientists use mathematical and statistical techniques to extract insights from data. This course will give you a good foundation in the mathematical and statistical skills needed to succeed in this field. The module on calculus methods and applications will help you develop the skills needed to model and analyze complex systems, and the module on Maclaurin series will help you develop the skills needed to solve problems using advanced mathematical techniques.

See salaries and explore the career path for Data Scientist

Operations Research Analyst

As an Operations Research Analyst, you will use mathematical and analytical techniques to solve problems in a variety of industries. This course will give you a good foundation in the mathematical and analytical skills needed to succeed in this field. The module on calculus methods and applications will help you develop the skills needed to model and analyze complex systems, and the module on Maclaurin series will help you develop the skills needed to solve problems using advanced mathematical techniques.

See salaries and explore the career path for Operations Research Analyst

Computational Physicist

Computational Physicists use mathematical and computational techniques to simulate physical systems. This course will give you a good foundation in the mathematical and computational techniques needed to succeed in this field. The module on calculus methods and applications will help you develop the skills needed to model and analyze complex systems, and the module on Maclaurin series will help you develop the skills needed to solve problems using advanced mathematical techniques.

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Quantitative Analyst

Quantitative Analysts use mathematical and statistical techniques to analyze financial markets. This course will give you a good foundation in the mathematical and statistical skills needed to succeed in this field. The module on calculus methods and applications will help you develop the skills needed to model and analyze complex systems, and the module on Maclaurin series will help you develop the skills needed to solve problems using advanced mathematical techniques.

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Actuary

Actuaries use mathematical and statistical techniques to assess risk and uncertainty. This course will give you a good foundation in the mathematical and statistical skills needed to succeed in this field. The module on calculus methods and applications will help you develop the skills needed to model and analyze complex systems, and the module on Maclaurin series will help you develop the skills needed to solve problems using advanced mathematical techniques.

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Astronomer

Astronomers use mathematical and computational techniques to study the universe. This course will give you a good foundation in the mathematical and computational techniques needed to succeed in this field. The module on calculus methods and applications will help you develop the skills needed to model and analyze complex systems, and the module on Maclaurin series will help you develop the skills needed to solve problems using advanced mathematical techniques.

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Simulation Engineer

Simulation Engineers use mathematical and computational techniques to simulate complex systems. This course will give you a good foundation in the mathematical and computational techniques needed to succeed in this field. The module on calculus methods and applications will help you develop the skills needed to model and analyze complex systems, and the module on Maclaurin series will help you develop the skills needed to solve problems using advanced mathematical techniques.

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Financial Analyst

Financial Analysts use mathematical techniques to make investment recommendations. This course will help you build a strong foundation in the mathematical concepts that underpin financial analysis, such as matrix determinants and inverses, calculus methods and applications, and Maclaurin series. You will also gain valuable experience in problem-solving and critical thinking, which are essential skills for success in this field. The course will also give you a good understanding of the theoretical concepts behind complex numbers and their applications. This will help you develop the skills needed to work with complex data sets and make sound financial decisions.

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Market Research Analyst

In this role, you will use mathematical and statistical techniques to understand consumer behavior and trends. This course will give you a good foundation in the mathematical and statistical skills needed to succeed in this field. The module on calculus methods and applications will help you develop the skills needed to model and analyze complex systems, and the module on Maclaurin series will help you develop the skills needed to solve problems using advanced mathematical techniques.

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Game Developer

Game Developers use mathematical and logical reasoning to design and develop video games. This course will help you develop the mathematical and logical reasoning skills needed to succeed in this field. The module on matrix determinants and inverses can help you to understand the mathematics behind 3D graphics, and the module on calculus methods and applications can help you to understand the mathematics behind game physics.

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Data Analyst

As a Data Analyst, you will use mathematical models to understand data and compile reports for your team. While this course's syllabus does not contain any specific data analysis techniques, it provides a solid foundation in numerical problem-solving and mathematical concepts. For example, the module on matrix determinants and inverses can help you understand the impact of variables on data, while mathematical induction is essential for understanding how to develop accurate and reliable models and how to forecast future trends.

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Systems Analyst

Systems Analysts use mathematical and logical reasoning to design and develop complex systems. This course will help you develop the mathematical and logical reasoning skills needed to succeed in this field. The module on mathematical induction will help you develop the skills needed to prove the correctness of algorithms and the module on complex numbers will help you develop the skills needed to understand the mathematical underpinnings of computer systems.

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Software Developer

Software Developers use mathematical and logical reasoning to design and develop software systems. This course will help you develop the mathematical and logical reasoning skills needed to succeed in this field. The module on mathematical induction will help you develop the skills needed to prove the correctness of algorithms, and the module on complex numbers will help you develop the skills needed to understand the mathematical underpinnings of software systems.

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Software Engineer

In this role, you will use mathematical and logical reasoning to design and develop software systems. The skills you acquire in this course will be useful in this role as they will help you to understand the mathematical underpinnings of software development. For example, the module on matrix determinants and inverses can help you to understand how to represent and manipulate data in software, and the module on mathematical induction can help you to develop algorithms and data structures.

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Computer Scientist

Computer Scientists use mathematical and logical reasoning to design and develop computer systems. This course will help you develop the mathematical and logical reasoning skills needed to succeed in this field. The module on mathematical induction will help you develop the skills needed to prove the correctness of algorithms, and the module on complex numbers will help you develop the skills needed to understand the mathematical underpinnings of computer systems.

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