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(z2z1) 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:

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:

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 7n^{2}+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:

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 nonreal 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 e^{ix} =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:

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:

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: cosh^{2}x− sinh^{2}x= 1 1−tanh^{2}x=sech^{2}x coth^{2}x−1=cosech^{2}x Familiarity with the graphs of sinh x , coshx, tanhx, sinh^{−1}x, cosh^{−1}x, tanh^{−1}x. This chapter introduces you to a wholly new concept. When you have completed it, you will:

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:
