# Maths for CAPE® Examinations Volume 1

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
- MATHEMATICAL MODELLING
- MODULE 1 BASIC ALGEBRA AND FUNCTIONS
- CHAPTER 1 REASONING AND LOGIC
- Notation
- Simple statement
- Negation

- Truth tables
- Compound statements
- Connectives
- Conjunction
- Disjunction (‘or’)

- Conditional statements
- Interpretation of p → q
- The contrapositive
- Converse
- Inverse

- Equivalent propositions
- Biconditional statements
- Tautology and contradiction
- Algebra of propositions

- CHAPTER 2 THE REAL NUMBER SYSTEM
- Subsets of rational numbers
- Real numbers
- Operations
- Binary operations
- Closure
- Commutativity
- Associativity
- Distributivity
- Identity
- Inverse

- Constructing simple proofs in mathematics
- Proof by exhaustion
- Direct proof
- Proof by contradiction
- Proof by counter example

- CHAPTER 3 PRINCIPLE OF MATHEMATICAL INDUCTION
- Sequences and series
- Finding the general term of a series
- Sigma notation
- Expansion of a series
- Standard results
- Summation results

- Mathematical induction
- Divisibility tests and mathematical induction

- Sequences and series
- CHAPTER 4 POLYNOMIALS
- Review of polynomials
- Degree or order of polynomials
- Algebra of polynomials
- Evaluating polynomials
- Rational expressions

- Comparing polynomials
- Remainder theorem
- The factor theorem
- Factorising polynomials and solving equations
- Factorising xn - yn

- Review of polynomials
- CHAPTER 5 INDICES, SURDS AND LOGARITHMS
- Indices
- Laws of indices

- Surds
- Rules of surds
- Simplifying surds
- Conjugate surds
- Rationalising the denominator

- Exponential functions
- Graphs of exponential functions
- The number e
- Exponential equations

- Logarithmic functions
- Converting exponential expressions to logarithmic expressions
- Changing logarithms to exponents using the definition of logarithm

- Properties of logarithms
- Solving logarithmic equations
- Equations involving exponents
- Change of base formula (change to base b from base a)
- Logarithms and exponents in simultaneous equations

- Application problems
- Compound interest
- Continuous compound interest

- Indices
- CHAPTER 6 FUNCTIONS
- Relations and functions
- Describing a function
- The vertical line test
- One-to-one function (injective function)
- Onto function (surjective function)
- Bijective functions

- Inverse functions
- Graphs of inverse functions

- Odd and even functions
- Odd functions
- Even functions

- Periodic functions
- The modulus function
- Graph of the modulus function

- Composite functions
- Relationship between inverse functions

- Increasing and decreasing functions
- Increasing functions
- Decreasing functions

- Transformations of graphs
- Vertical translation
- Horizontal translation
- Horizontal stretch
- Vertical stretch
- Reflection in the x-axis
- Reflection in the y-axis
- Graphs of simple rational functions

- Piecewise defined functions
- CHAPTER 7 CUBIC POLYNOMIALS
- Review: Roots of a quadratic and the coefficient of the quadratic
- Cubic equations
- Notation
- Finding a3 + ß3 + y3, using a formula
- Finding a cubic equation, given the roots of the equation

- CHAPTER 8 INEQUALITIES AND THE MODULUS FUNCTION
- Theorems of inequalities
- Quadratic inequalities
- Sign table
- Rational functions and inequalities
- General results about the absolute value function
- Square root of x2

- The triangle inequality
- Applications problems for inequalities

- MODULE 1 TESTS

- CHAPTER 1 REASONING AND LOGIC
- MODULE 2 TRIGONOMETRY AND PLANE GEOMETRY
- CHAPTER 9 TRIGONOMETRY
- Inverse trigonometric functions and graphs
- Inverse sine function
- Inverse cosine function
- Inverse tangent function

- Solving simple trigonometric equations
- Graphical solution of sin x = k
- Graphical solution of cos x = k
- Graphical solution of tan x = k

- Trigonometrical identities
- Reciprocal identities
- Pythagorean identities
- Proving identities

- Solving trigonometric equations
- Further trigonometrical identities
- Expansion of sin (A ± B)
- Expansion of cos (A ± B)
- Expansion of tan (A + B)
- Double-angle formulae
- Half-angle formulae
- Proving identities using the addition theorems and the double-angle formulae

- The form a cos Ɵ + b sin Ɵ
- Solving equations of the form a cos Ɵ + b sin Ɵ = c
- Equations involving double-angle or half-angle formulae
- Products as sums and differences
- Converting sums and differences to products
- Solving equations using the sums and differences as products

- Inverse trigonometric functions and graphs
- CHAPTER 10 COORDINATE GEOMETRY
- Review of coordinate geometry
- The equation of a circle
- Equation of a circle with centre (a, b) and radius r
- General equation of the circle

- Intersection of a line and a circle
- Intersection of two circles
- Intersection of two curves
- Parametric representation of a curve
- Cartesian equation of a curve given its parametric form
- Parametric equations in trigonometric form

- Parametric equations of a circle
- Conic sections
- Ellipses
- Equation of an ellipse
- Equation of an ellipse with centre (h, k)
- Focus–directrix property of an ellipse
- Parametric equations of ellipses
- Equations of tangents and normals to an ellipse

- Parabolas
- Equation of a parabola
- Parametric equations of parabolas
- Equations of tangents and normals to a parabola

- CHAPTER 11 VECTORS IN THREE DIMENSIONS (R3)
- Vectors in 3D
- Plotting a point in three dimensions

- Algebra of vectors
- Addition of vectors
- Subtraction of vectors
- Multiplication by a scalar

- Equality of vectors
- Magnitude of a vector
- Displacement vectors
- Unit vectors
- Special unit vectors
- Scalar product or dot product
- Properties of the scalar product

- Angle between two vectors
- Perpendicular and parallel vectors
- Perpendicular vectors
- Parallel vectors

- Equation of a line
- Finding the equation of a line given a point on a line and the direction of the line
- Finding the equation of a line given two points on the line
- Vector equation of a line
- Parametric equation of a line
- Cartesian equation of a line
- Finding the angle between two lines, given the equations of the lines

- Skew lines
- Equation of a plane
- Equation of a plane, given the distance from the origin to the plane and a unit vector perpendicular to the plane
- Equation of a plane, given a point on the plane and a normal to the plane
- Cartesian equation of a plane

- Vectors in 3D
- MODULE 2 TESTS

- CHAPTER 9 TRIGONOMETRY
- MODULE 3 CALCULUS I
- CHAPTER 12 LIMITS AND CONTINUITY
- Limits
- The existence of a limit
- Limit laws

- Evaluating limits
- Direct substitution
- Factorising method
- Conjugate method

- Tending to infinity
- Limits at infinity
- Special limits

- Continuity
- Types of discontinuity
- Infinite discontinuity
- Point discontinuity
- Jump discontinuity
- Removable and non-removable discontinuity

- Limits
- CHAPTER 13 DIFFERENTIATION 1
- Differentiation
- The difference quotient
- Existence of a derivative
- Notation for derivatives
- Interpretations of derivatives

- Finding derivatives using first principles
- Differentiation of ag(x) where a is a constant

- Differentiation of sums and differences of functions
- First principle and sums and differences of functions of x
- Rate of change
- Chain rule
- Product rule
- Quotient rule
- Differentiation of trigonometric functions
- Higher derivatives

- Differentiation
- CHAPTER 14 APPLICATIONS OF DIFFERENTIATION
- Tangents and normals
- Equations of tangents and normals

- Increasing and decreasing functions
- Stationary points/second derivatives
- Maximum and minimum values
- Stationary points

- Classification of turning points
- First derivative test
- Second derivative test

- Inflexion points
- Practical maximum and minimum problems
- Parametric differentiation

- Rate of change
- Curve sketching
- Polynomials, rational functions, trigonometric functions

- Graph of a polynomial
- Graphs of functions of the form f(x) = xn where n is an even integer
- Graphs of functions of the form f(x) = xn where n is an odd integer greater than 1
- Graphs of polynomials
- Zeros of a polynomial

- Graphing functions
- Graphing functions with a table of values
- Solving simultaneous equations graphically
- Solving inequalities graphically

- Review of trigonometry
- Sine, cosine and tangent of 45°, 30° and 60°
- Graph of cosec x
- Graph of sec x
- Graph of cot x
- Properties and graphs of trigonometric functions

- Transformations of trigonometric functions
- y = a sin (bx) + c and y = a cos (bx) + c
- y = a tan (bx) + c

- Graphs of rational functions
- Vertical asymptotes
- Horizontal asymptotes
- Sketching graphs of rational functions

- Shape of a curve for large values of the independent variable

- Tangents and normals
- CHAPTER 15 INTEGRATION
- Anti-derivatives (integrations)
- The constant of integration
- Integrals of the form axn
- Integration theorems
- Integration of polynomial functions

- Integration of a function involving a linear factor
- Integration of trigonometric functions
- Integration of more trigonometric functions
- Integrating sin2 x and cos2 x
- Integration of products of sines and cosines

- The definite integral
- Integration by substitution
- Substituting with limits

- The equation of a curve

- Anti-derivatives (integrations)
- CHAPTER 16 APPLICATIONS OF INTEGRATION
- Approximating the area under a curve, using rectangles
- Estimating the area under a curve using n rectangles of equal width

- Using integration to find the area under a curve
- Area between two curves
- Area below the x-axis
- Area between the curve and the y-axis
- Volume of solids of revolution
- Rotation about the x-axis
- Rotation about the y-axis
- Volume generated by the region bounded by two curves

- CHAPTER 17 DIFFERENTIAL EQUATIONS
- Families of curves
- Classifying differential equations
- Linear versus non-linear differential equations

- Practical applications of differential equations
- First order differential equations
- Solutions of variable-separable differential equations
- Modelling problems
- Second order differential equations

- CHAPTER 12 LIMITS AND CONTINUITY
- MODULE 3 TESTS
- UNIT 1—MULTIPLE CHOICE TESTS
- INDEX