 \$23.55

# Maths for CAPE® Examinations Volume 2

By Dipchand Bahall
US\$ 23.55
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Book Description

• Cover
• Title Page
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• MODULE 2 TESTS
• MODULE 3 COUNTING, MATRICES AND DIFFERENTIAL EQUATIONS
• CHAPTER 14 PERMUTATIONS AND COMBINATIONS
• The counting principles
• Multiplication 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
• 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
• 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
• 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
• MODULE 3 TESTS
• UNIT 2—MULTIPLE CHOICE TESTS
• INDEX
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