§-Specialist Mathematics syllabus
QLD · QCAA← Specialist Mathematics
Specialist Mathematics syllabus, dot point by dot point
Every dot point in the QLD Specialist Mathematics syllabus, 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: Mathematical induction, and further vectors, matrices and complex numbers
Unit overview →Topic 1: Proof by mathematical induction and further proof methods
Topic 1: Proof by mathematical induction
Topic 2: Vectors in two and three dimensions
Topic 2: Vectors and matrices
Determine vector, parametric and Cartesian equations of a line in three dimensions, convert between these forms, and find the point of intersection of two lines or establish that they are parallel or skew
Use matrices beyond order two, including the determinant and inverse of a three by three matrix, to represent and solve systems of linear equations, and interpret unique, infinite and no-solution cases geometrically
Determine the vector equation and the Cartesian equation of a plane from a point and a normal or from three points, find the distance from a point to a plane, and find the line of intersection or angle between planes
Topic 3: Complex numbers
Identify and sketch subsets of the complex plane determined by relations involving modulus, argument, distance and inequalities, including lines, circles, perpendicular bisectors, rays and regions
Factorise polynomials with real coefficients over the complex field, apply the conjugate root theorem and the fundamental theorem of algebra, and find all roots of polynomial equations including those with complex coefficients
Topic 3: Further matrices
Unit 4: Further calculus, and statistical inference
Unit overview →Topic 1: Integration and applications of integration
Apply integration techniques including substitution, integration by partial fractions, and trigonometric integrals, and use them to evaluate definite integrals and compute areas and volumes of solids of revolution
Determine areas between curves and volumes of solids of revolution generated by rotating a region about the x-axis or the y-axis, setting up the correct definite integral and evaluating it
Use Simpson's rule to approximate the value of a definite integral or an area, applying the rule with an even number of subintervals and recognising when a numerical method is required
Evaluate integrals using integration by parts and integrate trigonometric expressions using identities such as double-angle and Pythagorean identities and products of sines and cosines
Topic 2: Rates of change and differential equations
Differentiate implicitly defined relations and solve related rates problems using the chain rule, including contexts involving volumes and surface areas of cones, spheres and cylinders
Construct and interpret slope (direction) fields of a first-order differential equation, sketch solution curves through given points, and relate the qualitative behaviour of solutions to the field
Model and solve practical situations with first-order differential equations, including exponential growth and decay, Newton's law of cooling and the logistic equation, and interpret the long-term behaviour of solutions
Formulate and solve first-order differential equations using separation of variables, including growth and decay and the logistic model, and interpret solutions in applied rates-of-change contexts
Topic 3: Statistical inference
Construct and interpret confidence intervals for a population mean using the sample mean and standard error, choosing the appropriate confidence level, and understand the meaning of the confidence level in repeated sampling
Understand the distribution of the sample mean, apply the central limit theorem to describe its shape, mean and standard deviation, and use these to compute probabilities for sample means drawn from a population