§-Specialist Mathematics syllabus

Specialist Mathematics syllabus, dot point by dot point

Every dot point in the TAS Specialist Mathematics course document, in syllabus order, with a focused answer for each. Open any dot point for the answer, exam-style questions and links to related points.

Unit 3

Section overview →

How do equations and inequalities in z describe lines, circles and regions on the Argand diagram?

Sketch curves and regions in the complex plane defined by modulus and argument conditions.

9 min read →

How do we sketch rational functions by finding their intercepts and asymptotes?

Sketch rational functions, identifying vertical, horizontal and oblique asymptotes.

9 min read →

How do translations, dilations and reflections build complex graphs from simple ones?

Sketch graphs using transformations and the addition of ordinates.

8 min read →

How do complex numbers extend the real number system and let us solve every polynomial?

Represent complex numbers in Cartesian and polar form, perform arithmetic, and apply De Moivre's theorem.

9 min read →

How does the graph of one over f of x relate to the graph of f?

Sketch the reciprocal of a function, relating its features to those of the original function.

8 min read →

How do we sketch and solve problems with the absolute value function?

Sketch graphs involving absolute value and solve absolute value equations and inequalities.

8 min read →

How does plotting complex numbers on the Argand diagram reveal the geometry of arithmetic?

Represent complex numbers and their operations geometrically on the Argand diagram.

8 min read →

How do we combine functions and reverse them while keeping domain and range correct?

Form composite and inverse functions, determining the correct domain and range of each.

8 min read →

How do we describe position, direction, lines and planes in three-dimensional space using vectors?

Use 3D vectors with dot and cross products to find angles, projections, lines and planes.

9 min read →

How do we add, multiply, divide and conjugate complex numbers reliably in Cartesian form?

Perform complex arithmetic in Cartesian form and use conjugate properties to divide and simplify.

8 min read →

How does allowing complex roots let us factorise every polynomial completely?

Factorise polynomials over the complex field using the conjugate root theorem and the fundamental theorem of algebra.

9 min read →

What are the nth roots of unity and why do they lie equally spaced on the unit circle?

Find the nth roots of unity and use their symmetry and algebraic properties.

8 min read →

How can we prove a statement is true for every positive integer using induction?

Prove statements for all positive integers using the principle of mathematical induction.

9 min read →

How do we differentiate relations that are not given as explicit functions of x?

Apply implicit and parametric differentiation and use related rates to solve problems.

9 min read →

Unit 4

Section overview →

How do we solve equations that relate a function to its own rate of change?

Solve separable and linear first order differential equations and apply them to models.

9 min read →

How can we draw reliable conclusions about a population from a single sample?

Use the distribution of the sample mean and the central limit theorem to construct confidence intervals.

9 min read →

Which techniques let us integrate functions that the basic rules cannot handle?

Integrate using substitution, integration by parts and partial fractions, and find volumes of revolution.

9 min read →