# Maths for CAPE® Examinations Volume 2

The two books in this series provide complete coverage of Units I and II of the new CAPE Pure Mathematics syllabus. They offer a sound platform for students pursuing courses at tertiary institutions throughout the Caribbean. Each topic is covered in depth with additional material in areas that students find most challenging.Key features:• Objectives at the beginning of each chapter aid planning, focus learning and confirm syllabus coverage• Key terms are highlighted to develop students’ vocabulary throughout the course• A wide variety of exercises develops students’ knowledge, application and ability in all areas of the syllabus• Worked solutions throughout the text provide students with easy-to-follow examples of new concepts• Graded exercises at the end of each section can be used to check students’ understanding and monitor progress• Checklists and at-a-glance summaries at the end of each chapter encourage students to review their understanding and go back over areas of weakness• Examination-style questions at the end of each module give students plenty of practice in the types of questions they’ll meet in the examinationsAbout the authorDipchand Bahall has over 20 years’ experience teaching Advanced Level Mathematics at Presentation College, Chaguanas, and St Joseph’s Convent, St Joseph, in Trinidad and Tobago, and at Cayman Prep and High School. He holds a Masters Degree in Statistics, a Diploma in Education (Teaching of Mathematics) and a BSc in Mathematics. Dipchand is presently a Senior Instructor at The University of Trinidad and Tobago, Point Lisas Campus.Please note that this is an eBook version of this title and can NOT be printed. For more information about eBooks, including how to download the software you’ll need, see our FAQs page.

- Cover
- Title Page
- Copyright
- Contents
- INTRODUCTION
- MODULE 1 COMPLEX NUMBERS AND CALCULUS II
- CHAPTER 1 COMPLEX NUMBERS
- Complex numbers as an extension to the real numbers
- Powers of i

- Algebra of complex numbers
- Addition of complex numbers
- Subtraction of complex numbers
- Multiplication of a complex number by a real number
- Multiplication of complex numbers
- Equality of complex numbers
- Conjugate of a complex number
- Division of complex numbers
- Square root of a complex number

- Roots of a polynomial
- Quadratic equations
- Other polynomials

- The Argand diagram
- Addition and subtraction on the Argand diagram
- Multiplication by i
- Modulus (length) of a complex number
- Argument of a complex number

- Trigonometric or polar form of a complex number
- Exponential form of a complex number
- De Moivre’s theorem
- Locus of a complex number
- Circles
- Perpendicular bisector of a line segment
- Half-line
- Straight line
- Inequalities
- Intersecting loci
- Cartesian form of loci

- Complex numbers as an extension to the real numbers
- CHAPTER 2 DIFFERENTIATION
- Standard differentials
- Differentiation of ln x
- Differentiation of ex

- Chain rule (function of a function rule)
- Differentiating exponential functions of the form y = ax
- Differentiating logarithms of the form y = loga x
- Differentiation of combinations of functions
- Differentiation of combinations involving trigonometric functions

- Tangents and normals
- Gradients of tangents and normals
- Equations of tangents and normals

- Implicit differentiation
- Differentiation of inverse trigonometric functions
- Differentiation of y = sin–1x
- Differentiation of y = tan–1x

- Second derivatives
- Parametric differentiation
- First derivative of parametric equations
- Second derivative of parametric equations

- Partial derivatives
- First order partial derivatives
- Second order partial derivatives
- Applications of partial derivatives

- Standard differentials
- CHAPTER 3 PARTIAL FRACTIONS
- Rational fractions
- Proper fractions: Unrepeated linear factors
- Proper fractions: Repeated linear factors
- Proper fractions: Unrepeated quadratic factors
- Proper fractions: Repeated quadratic factors
- Improper fractions

- Rational fractions
- CHAPTER 4 INTEGRATION
- Integration by recognition
- When the numerator is the differential of the denominator
- The form ʃf'(x)[f(x)]n dx, n ǂ –1
- The form ʃf'(x)ef(x)dx

- Integration by substitution
- Integration by parts
- Integration using partial fractions
- Integration of trigonometric functions
- Integrating sin2 x and cos2 x
- Integrating sin3 x and cos3 x
- Integrating powers of tan x
- Integrating products of sines and cosines
- Finding integrals using the standard forms

- CHAPTER 5 REDUCTION FORMULAE
- Reduction formula for ʃsinn x dx
- Reduction formula for ʃcosn x dx
- Reduction formula for ʃtann x dx
- Other reduction formulae

- CHAPTER 6 TRAPEZOIDAL RULE (TRAPEZIUM RULE)
- The area under a curve

- CHAPTER 1 COMPLEX NUMBERS
- MODULE 1 TESTS
- MODULE 2 SEQUENCES, SERIES AND APPROXIMATIONS
- CHAPTER 7 SEQUENCES
- Types of sequence
- Convergent sequences
- Divergent sequences
- Oscillating sequences
- Periodic sequences
- Alternating sequences

- The terms of a sequence
- Finding the general term of a sequence by identifying a pattern
- A sequence defined as a recurrence relation
- Convergence of a sequence

- Types of sequence
- CHAPTER 8 SERIES
- Writing a series in sigma notation (∑)
- Sum of a series
- Sum of a series in terms of n

- Method of differences
- Convergence of a series
- Tests for convergence of a series

- CHAPTER 9 PRINCIPLE OF MATHEMATICAL INDUCTION (PMI): SEQUENCES AND SERIES
- PMI and sequences
- PMI and series

- CHAPTER 10 BINOMIAL THEOREM
- Pascal’s triangle
- Factorial notation
- Combinations
- General formula for nCr

- Binomial theorem for any positive integer n
- The term independent of x in an expansion

- Extension of the binomial expansion
- Approximations and the binomial expansion
- Partial fractions and the binomial expansion

- CHAPTER 11 ARITHMETIC AND GEOMETRIC PROGRESSIONS
- Arithmetic progressions
- Sum of the first n terms of an AP
- Proving that a sequence is an AP

- Geometric progressions
- Sum of the first n terms of a GP (Sn)
- Sum to infinity
- Proving that a sequence is a GP
- Convergence of a geometric series

- Arithmetic progressions
- CHAPTER 12 NUMERICAL TECHNIQUES
- The intermediate value theorem (IMVT)
- Finding the roots of an equation
- Graphical solution of equations
- Interval bisection

- Linear interpolation
- Newton–Raphson method for finding the roots of an equation

- CHAPTER 13 POWER SERIES
- Power series and functions
- Taylor expansion
- The Maclaurin expansion
- Maclaurin expansions of some common functions

- CHAPTER 7 SEQUENCES
- MODULE 2 TESTS
- MODULE 3 COUNTING, MATRICES AND DIFFERENTIAL EQUATIONS
- CHAPTER 14 PERMUTATIONS AND COMBINATIONS
- The counting principles
- Multiplication rule
- Addition rule

- Permutations
- Permutations of n distinct objects
- Permutation of r out of n distinct objects
- Permutations with repeated objects
- Permutations with restrictions
- Permutations with restrictions and repetition

- Combinations
- Combinations with repetition

- The counting principles
- CHAPTER 15 PROBABILITY
- Sample space and sample points
- Events: mutually exclusive; equally likely
- Probability
- Rules of probability
- Conditional probability

- Tree diagrams
- Probability and permutations
- Probability and combinations

- CHAPTER 16 MATRICES
- Matrices: elements and order
- Square matrices
- Equal matrices
- Zero matrix

- Addition and subtraction of matrices
- Multiplication of a matrix by a scalar
- Properties of matrix addition

- Matrix multiplication
- Properties of matrix multiplication
- Identity matrix
- Multiplication of square matrices

- Transpose of a matrix
- Properties of the transpose of a matrix

- Determinant of a square matrix
- Determinant of a 2 × 2 matrix
- Determinant of a 3 × 3 matrix
- Properties of determinants

- Singular and non-singular matrices
- Solving equations using determinants (Cramer’s rule)
- Using Cramer’s rule to solve three equations in three unknowns

- Inverse of a matrix
- Inverse of a 2 × 2 matrix
- Cofactors of a 3 × 3 matrix
- Inverse of a 3 × 3 matrix
- Properties of inverses

- Systems of linear equations
- Row reduction to echelon form
- Finding the inverse of a matrix by row reduction
- Solving simultaneous equations using row reduction

- Systems of linear equations with two unknowns
- Intersecting lines
- Parallel lines
- Lines that coincide

- Systems of linear equations with three unknowns
- Unique solution
- No solutions
- Infinite set of solutions
- Solution of linear equations in three unknowns: geometrical interpretation

- Applications of matrices

- Matrices: elements and order
- CHAPTER 17 DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELLING
- First order linear differential equations
- Practical applications

- Second order differential equations
- When the roots of the AQE are real and equal
- When the roots of the AQE are real and distinct
- When the roots of the AQE are complex

- Non-homogeneous second order differential equations
- When f(x) is a polynomial of degree n
- When f(x) is a trigonometric function
- When f(x) is an exponential function

- Equations reducible to a recognisable form
- Mathematical modelling

- First order linear differential equations

- CHAPTER 14 PERMUTATIONS AND COMBINATIONS
- MODULE 3 TESTS
- UNIT 2—MULTIPLE CHOICE TESTS
- INDEX