Syllabus
Below is a summary of the topics to be covered in the S6, five periods mathematics class.
| Module / Topic | Key Content |
|---|---|
| Introduction to Complex Numbers | Basic operations with complex numbers. Determine the real and imaginary part, complex conjugate, inverse. Solve quadratic equations with real coefficients and complex solutions. |
| Sequences | Define a sequence recursively. Graphical behaviour of sequences. Increasing/Decreasing sequences. Limit calculations. |
| Introduction to Real Functions | Study of functions and properties (domain/range, graph interpretation, periodicity, parity, zeroes). Combining Functions. Understand the concept of an inverse function, also in relation to composition of functions and graph. Sketch the graph of basic functions. Investigate composition and the effect on the graph. |
| Limits | Understand the concept of finite and infinite limits of a function tending to a point or infinity. Determine one-sided limits and limits at the end point of a domain. |
| Differentiation | Derivative of a function at a point. First and Second derivative functions, rules for differentiating. Tangent to a graph. Behaviour of a function (finding asymptotes, local/global extrema, inflection points, where the function is increasing/decreasing/convex/concave). |
| Exponential and Logarithms | Explore properties of these functions (including domain, limits, asymptotes, tangents, extrema, curvature, and inflection points). Solve exponential and logarithmic equations and inequalities. Sketching the graphs of these functions. |
| Vectors in 2D | Lines in the plane, distance between two points. Equations of circles. Intersection of lines. Angle between two intersecting lines. Parallel and perpendicular lines. Distance between a line and a point or two lines. Perpendicular projection of a point on a line. Applications. Velocity Problems. |
| Combinatorics | Permutations and combinations with and without repetition. Applying easy combinatorial formulas of  . Pascal’s triangle and Newton’s binomial theorem. Sum notation. |
| Probability (basic) | Elementary probability rules, conditional probability, independence. Total probability law and Bayes’ theorem. |
| Discrete Random Variables | Definition, probability density function, cumulative distribution function. Expected value, variance, and standard deviation. Bernoulli trials and Binomial distribution, including calculating probabilities, expectation, variance, and standard deviation. Modelling with binomial distribution. |
How to Navigate the Programme
The topics of complex numbers, sequences, and vectors in 2 dimensions are meant as introductions in preparation for topics in S7. There, you will explore those topics in greater lengths.
The remaining topics will all likely appear in some form or another in your final baccalaureate. You will not revisit these topics in S7 so that this is your only instance to learn them properly.
Below is a small breakdown of the topics included in the table above.
Complex Numbers: Introduction and examples.
Sequences: Introduction, Monotonic Sequences, Limit of a Sequence.
Introduction to Real Functions: Basic Properties, Combining Functions, Basic Models (Revision), Factorising Polynomials.
Limits: Introduction
Differentiation: Introduction, Rules of Differentiation, The First Derivative, Optimisation Problems, The Second Derivative, Tangent Lines, Sketching Graphs of Functions, Final Remarks.
Exponential and Logarithmic Functions: Introduction, Exponential Equations and Inequalities, Logarithmic Equations and Inequalities, Sketching Graphs of Logarithms.
Vectors in 2D: Basic Operations and Equation of the Line, Scalar Product and Angles between Lines, Projection and Distances, Circles, Velocity Vectors.
Combinatorics: Introduction, Permutations, Combinations without repetition, The Binomial Theorem and Combinations with Repetition.
Probability: Introduction and Basic Forms of Representations, Basic Probabilistic Formulas, Independence, Total Probability and Bayes’ Law.
Discrete Random Variables: Introduction, Expectation, Variance, and Standard Deviation, Bernoulli Processes and Binomial Distribution.