Further Pure 2


Chapter 1 - Complex Numbers (17.2)

1.1 Introduction

1.2 The general complex number

1.3 The modulus and argument of a complex number

1.4 The polar form of a complex number

1.5 Addition, subtraction and multiplication of complex numbers

1.6 The conjugate of a complex number and the division of complex numbers

1.7 Products and quotients of complex numbers in their polar form

1.8 Equating real and imaginary parts

1.9 Further consideration of |z2 - z1| and arg(z2-z1)

1.10 Loci on Argand diagrams

x + iy and r(cosθ + i sinθ ).

The parts of this topic also included in module Further Pure 1 will be

examined only in the context of the content of this module.

For example,

Maximum level of difficulty where a and b are

complex numbers.

This chapter introduces the idea of a complex number.

When you have completed it, you will:

  • know what is meant by a complex number;

  • know what is meant by the modulus and argument of a complex number;

  • know how to add, subtract, multiply and divide complex numbers;

  • know how to solve equations using real and imaginary parts;

  • understand what an Argand diagram is;

  • know how to sketch loci on Argand diagrams.


Chapter 2 - Roots of Polynomial Equations (17.1)

2.1 Introduction

2.2 Quadratic equations

2.3 Cubic equations

2.4 Relationship between the roots of a cubic equation and its coefficients

2.5 Cubic equations with related roots

2.6 An important result

2.7 Polynomial equations of degree n

2.8 Complex roots of polynomial equations with real coefficients

This chapter revises work already covered on roots of equations and extends those ideas.

When you have completed it, you will:

  • know how to solve any quadratic equation;

  • know that there is a relationship between the number of real roots and form of a polynomial equation, and be able to sketch graphs;

  • know the relationship between the roots of a cubic equation and its coefficients;

  • be able to form cubic equations with related roots;

  • know how to extend these results to polynomials of higher degree;

  • know that complex conjugates are roots of polynomials with real coefficients.

Chapter 3 - Finite Series (17.4 & 17.5)

3.1 Introduction

3.2 Summation of series by the method of differences

3.3 Summation of series by the method of induction

3.4 Proof by induction extended to other areas of mathematics


E.g. proving that 7n2+4n+1 is divisible by 6, or (cosθ + isinθ ) = cosnθ+ isinnθ where n is a positive integer.

This chapter extends the idea of summation of simple series, with which you are familiar from earlier studies, to other kinds of series.

When you have completed it, you will:

  • know new methods of summing series;

  • know which method is appropriate for the summation of a particular series;

  • understand an important method known as the method of induction;

  • be able to apply the method of induction in circumstances other than in the summation of series.


Chapter 4 - De Moivre’s Theorem (17.3)

4.1 De Moivre’s theorem

4.2 Using de Moivre’s theorem to evaluate powers of complex numbers

4.3 Application of de Moivre’s theorem in establishing trigonometric identities

4.4 Exponential form of a complex number

4.5 The cube roots of unity

4.6 The nth roots of unity

4.7 The roots of zn =α, where α is a non-real number

Use of z + 1/z = 2cosθ and z−1/z = 2isin θ , leading to, for example, expressing sin5θ in terms of multiple angles and tan 5θ

in term of powers of tanθ .

Applications in evaluating integrals, for example, ∫ sin5θ dθ .

The use, without justification, of the identity eix =cosx +isinx.

To include geometric interpretation and use, for example, in expressing cos 5π/12 in surd form.

This chapter introduces de Moivre’s theorem and many of its applications.

When you have completed it, you will:

  • know the basic theorem;

  • be able to find shorter ways of working out powers of complex numbers;

  • discover alternative methods for establishing some trigonometric identities;

  • know a new way of expressing complex numbers;

  • know how to work out the nth roots of unity and, in particular, the cube roots;

  • be able to solve certain types of polynomial equations.


Chapter 5 - Calculus of Inverse Trig Functions (17.6)

5.1 Introduction and revision

5.2 The derivatives of standard inverse trigonometrical functions

5.3 Application to more complex differentiation

5.4 Standard integrals integrating to inverse trigonometrical functions

5.5 Applications to more complex integrals

Use of the derivatives of sin−1x, cos−1x, tan−1x as given in the formulae booklet.

To include the use of the standard integrals.

given in the formulae booklet

This chapter revises and extends work on inverse trigonometrical functions.

When you have studied it, you will:

  • be able to recognise the derivatives of standard inverse trigonometrical functions;

  • be able to extend techniques already familiar to you to differentiate more complicated expressions;

  • be able to recognise algebraic expressions which integrate to standard integrals;

  • be able to rewrite more complicated expressions in a form that can be reduced to standard integrals.


Chapter 6 - Hyperbolic Functions (17.7)

6.1 Definitions of hyperbolic functions

6.2 Numerical values of hyperbolic functions

6.3 Graphs of hyperbolic functions

6.4 Hyperbolic identities

6.5 Osborne’s rule

6.6 Differentiation of hyperbolic functions

6.7 Integration of hyperbolic functions

6.8 Inverse hyperbolic functions

6.9 Logarithmic form of inverse hyperbolic functions

6.10 Derivatives of inverse hyperbolic functions

6.11 Integrals which integrate to inverse hyperbolic functions

6.12 Solving equations

The proofs mentioned below require expressing hyperbolic functions in terms of exponential functions.

To include solution of equations of the form asinhx+bcoshx=c.

Use of basic definitions in proving simple identities.

Maximum level of difficulty:

sinh(x+y) = sinhxcoshy+coshxsinhy.

The logarithmic forms of the inverse functions, given in the formulae booklet, may be required. Proofs of these results may also be required.

Proofs of the results of differentiation of the hyperbolic functions, given in the formula booklet, are included.

Knowledge, proof and use of:

cosh2x− sinh2x= 1



Familiarity with the graphs of

sinh x , coshx, tanhx, sinh−1x, cosh−1x, tanh−1x.

This chapter introduces you to a wholly new concept.

When you have completed it, you will:

  • know what hyperbolic functions are;

  • be able to sketch them;

  • be able to differentiate and integrate them;

  • have learned some hyperbolic identities;

  • understand what inverse hyperbolic functions are and how they can be expressed in alternative forms;

  • be able to sketch inverse hyperbolic functions;

  • be able to differentiate inverse hyperbolic functions and recognise integrals which integrate to them;

  • be able to solve equations involving hyperbolic functions.


Chapter 7 - Arc length & Surface of revolution (17.8)

7.1 Introduction

7.2 Arc length

7.3 Area of surface of revolution

This chapter introduces formulae which allow calculations concerning curves. 

When you have completed it, you will:

  • know a formula which can be used to evaluate the length of an arc when the equation of the curve is given in Cartesian form;

  • know a formula which can be used to evaluate the length of an arc when the equation of the curve is given in parametric form;

  • know methods of evaluating a curved surface area of revolution when the equation of the curve is given in Cartesian or in parametric form.