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Further Pure Mathematics 1 (MEI) – Sample chapter

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Chapters:
  FP1 (MEI): Chapter 1: Matrices Working with matrices, Transformations, Multiplying matrices, Properties of matrix multiplication, Composition of transformations, Inverse matrices, Using the determinant of a 2 X 2 matrix, Matrices and simultaneous equations, Invariant points, Invariant lines
  FP1 (MEI): Chapter 2: Complex numbers The growth of the number system, Working with complex numbers, Representing complex numbers geometrically, Sets of points in an Argand diagram, The modulus-argument form of complex numbers, Sets of points using the modulus-argument form, Complex numbers and equations
Sections

Click the links in the menu to access multiple choice tests with fully worked solutions.

Click the links in the Active learning resources menu to access activities to support students' learning.

A puzzle for students to do in small groups/pairs. Motivates manipulation of complex numbers.
Complex Numbers Puzzle
Group/pair activities for learning about complex numbers.
Complex Numbers Activities
This is a Microsoft Publisher file giving a numbered version of the puzzle so that you can replace the numbered pairs with your own questions!
Complex Numbers Activities

Introduction to Complex Numbers

Before you start...
  • You need to be able to use the quadratic formula to solve quadratic equations.
When you have finished you should...
  • Be able to solve any quadratic equation with real coefficients.
  • Understand what is meant by the real part and the imaginary part of a complex number.
  • Understand what is meant by the complex conjugate of a complex number.
  • Be able to add, subtract, multiply and divide complex numbers given in the form x + yj.
  • Know that a complex number is zero if and only if both the real and imaginary parts are zero.
Teach yourself
Addition and subtraction of complex numbers
Teach yourself
Multiplying complex numbers
Teach yourself
Dividing complex numbers
Teach yourself
Questions involving complex conjugates
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Working with Complex Numbers
Complex roots and the graph of a quadratic equation
Complex roots
TestTest Questions
Submit answers Complex numbers 1

Argand Diagrams

Before you start...
  • You need to have completed section 1.
When you have finished you should...
  • Know what is meant by the modulus of a complex number.
  • Know how to represent complex numbers and their conjugates on an Argand diagram.
  • Know what is meant by the real axis and the imaginary axis in an Argand diagram.
  • Be able to represent the sum and difference of two complex numbers on an Argand diagram.
  • Be able to represent simple sets of complex numbers as loci in the Argand diagram, such as circles in the form | z - a | = r.
Spreadsheet to investigate loci
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Flash investigation of loci
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The Argand Diagram
TestTest Questions
Submit answers Complex numbers 2

Modulus-Argument Form

Before you start...
  • You need to have completed sections 1 and 2.
  • You need to be able to work with angles in radians, and know about angles in all four quadrants. This is covered in Core 2, but if you haven't yet done this work, you will find additional help in the 'Notes and examples2, or in chapter 10 of the AS Pure Mathematics textbook.
When you have finished you should...
  • Know what is meant by the argument of a complex number.
  • Be able to represent a complex number in modulus-argument form and be able to convert between the forms z = x + yj and z = r(cos θ + j sin θ).
  • Be able to represent simple sets of complex numbers as loci in the Argand diagram such as half lines of the form arg(z - a - bj) = θ.
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Multiplication and Division of the Argand
TestTest Questions
Submit answers Complex numbers 3

Equations

Before you start...
  • You need to have completed sections 1, 2 and 3.
  • You need to be able to use factor theorem, and to factorise polynomials such as cubics and quartics if you know one factor. This is covered in Core 1, but if you haven't yet done this work you will find additional help in the 'Notes and examples' or in chapter 3 of the AS Pure Mathematics textbook.
When you have finished you should...
  • Know that the complex roots of real polynomial equations with real coefficients occur in conjugate pairs.
  • Be able to solve equations of higher degree with real coefficients in simple cases.
Factorising cubics
Multiplying polynomials
This is a puzzle to test your manipulation skills of complex numbers. Either do it yourself or, better, work with a partner or in a small group.
Complex Numbers puzzle
Multiplying three brackets
Factorising quartics
TestTest Questions
Submit answers Complex numbers 4
Please complete this once you have finished all four sections. Give it to your teacher / tutor for marking.
Complex numbers chapter assessment
This glossary covers all of this chapter.
Glossary
  FP1 (MEI): Chapter 3: Graphs and inequalities Graphs of rational functions, Inequalities, The range of values taken by a function
  FP1 (MEI): Chapter 4: Algebra: identities and roots of equations Identities, Properties of the roots of polynomial equations
  FP1 (MEI): Chapter 5: Induction and series Induction in mathematics, Proof by induction, More proofs by induction, Summation of finite series