QLD · QCAASyllabus
Specialist Mathematics syllabus, dot point by dot point
Every dot point in the QLD Specialist Mathematicssyllabus, with a focused answer for each one. Click any dot point for a worked explainer, past exam questions, and links to related dot points. Written by Claude Opus 4.7, Anthropic's latest AI, published by Better Tuition Academy.
Unit 3: Mathematical induction, and further vectors, matrices and complex numbers
Module overview →- Topic 4: Further complex numbersRepresent complex numbers in Cartesian, polar (modulus-argument) and exponential form, convert between forms, and apply de Moivre's theorem to compute powers and roots, locating results on the complex plane6 min answer →
- Topic 2: Vectors and matricesDetermine 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 planes6 min answer →
- Topic 3: Complex numbersFactorise 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 coefficients6 min answer →
- Topic 1: Proof by mathematical inductionUnderstand and use the principle of mathematical induction to prove results involving sums of series, divisibility statements and inequalities for all positive integers, structuring the base step and the inductive step correctly6 min answer →
- Topic 3: Further matricesUse 2x2 matrices to represent linear transformations of the plane, compute determinants and inverses, interpret the determinant as a scaling factor, and combine transformations through matrix multiplication6 min answer →
- Topic 3: Complex numbersIdentify and sketch subsets of the complex plane determined by relations involving modulus, argument, distance and inequalities, including lines, circles, perpendicular bisectors, rays and regions6 min answer →
- Topic 2: Vectors and matricesUse 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 geometrically6 min answer →
- Topic 1: Proof by mathematical induction and further proof methodsConstruct trigonometric proofs and apply direct proof, proof by contrapositive and proof by contradiction, choosing the appropriate method and writing each step with correct logical structure and justification6 min answer →
- Topic 2: Vectors and matricesDetermine 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 skew6 min answer →
- Topic 2: Vectors in two and three dimensionsUse vectors in three dimensions, compute scalar (dot) and vector (cross) products, find angles between vectors, scalar and vector projections, and apply these to geometric problems and the equations of lines and planes6 min answer →
Unit 4: Further calculus, and statistical inference
Module overview →- Topic 1: Integration and applications of integrationDetermine 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 it6 min answer →
- Topic 3: Statistical inferenceConstruct 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 sampling6 min answer →
- Topic 2: Rates of change and differential equationsFormulate 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 contexts6 min answer →
- Topic 2: Rates of change and differential equationsModel 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 solutions7 min answer →
- Topic 2: Rates of change and differential equationsDifferentiate implicitly defined relations and solve related rates problems using the chain rule, including contexts involving volumes and surface areas of cones, spheres and cylinders6 min answer →
- Topic 1: Integration and applications of integrationEvaluate integrals using integration by parts and integrate trigonometric expressions using identities such as double-angle and Pythagorean identities and products of sines and cosines6 min answer →
- Topic 1: Integration and applications of integrationApply 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 revolution6 min answer →
- Topic 3: Statistical inferenceUnderstand 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 population6 min answer →
- Topic 1: Integration and applications of integrationUse 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 required6 min answer →
- Topic 2: Rates of change and differential equationsConstruct 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 field6 min answer →