§-Specialist Mathematics syllabus
SA · SACE Board← Specialist Mathematics
Specialist Mathematics syllabus, dot point by dot point
Every dot point in the SA Specialist Mathematics syllabus, with a focused answer for each. Click any dot point for a worked explainer, past exam questions and links to related points.
Topic 1: Mathematical Induction
Module overview →Topic 2: Complex Numbers
Module overview →How do we add, multiply and divide numbers that involve the square root of minus one, and how do we picture them?
Complex numbers extend the reals with i where i squared equals minus one, and are represented as points or vectors on the Argand plane.
How does writing a complex number by its modulus and angle make multiplication, division and powers easy?
Polar form expresses a complex number by its modulus and argument, and De Moivre's theorem raises it to integer powers.
Why does a complex number have exactly n distinct nth roots, and how are they arranged?
Find the nth roots of a complex number using polar form and de Moivre's theorem, and represent them geometrically on the Argand plane.
Topic 3: Functions and Sketching Graphs
Module overview →When can a function be reversed, and how do composition and inversion interact with domain and range?
Form composite functions and find inverse functions, attending to domain and range and the condition for an inverse to exist.
How does taking the absolute value reshape a graph, and how do we solve equations and inequalities involving it?
Sketch graphs involving the modulus (absolute value) function and solve equations and inequalities containing modulus expressions.
How do the zeros of a numerator and denominator dictate the shape of a rational function's graph?
Sketch graphs of rational functions, identifying intercepts, vertical asymptotes, and horizontal or oblique asymptotes.
Topic 4: Vectors in Three Dimensions
Module overview →How do we find a vector perpendicular to two others, and what does its length tell us?
Compute the cross product of two vectors, interpret its direction and magnitude, and use it to find areas and normals.
How do vector and cartesian equations describe lines and planes, and how do we find where they meet?
Write vector, parametric and cartesian equations of lines and planes in three dimensions and find intersections and angles between them.
How does the dot product measure the angle between two vectors and detect perpendicularity?
Represent vectors in three dimensions, compute magnitudes and unit vectors, and use the dot product to find angles and test for perpendicularity.
Topic 5: Integration Techniques
Module overview →How does reversing the product rule let us integrate products such as x times e to the x?
Evaluate integrals using integration by parts, including choosing u and dv and applying the method more than once where needed.
How does reversing the chain rule let us integrate composite functions?
Evaluate integrals using the method of substitution, including changing the limits for definite integrals.
How does splitting a rational function into simpler fractions make it integrable?
Decompose rational functions into partial fractions and use the decomposition to integrate them.
Topic 6: Rates of Change and Differential Equations
Module overview →How do we translate a real situation into a differential equation and interpret its solution?
Formulate and solve differential equations modelling real situations such as growth, cooling and limited growth, and interpret the solutions.
How do we solve a differential equation by separating the two variables onto opposite sides?
Solve first-order separable differential equations and find particular solutions using initial conditions.