VCE Vault

VCE Specialist Mathematics (Year 12)

VCE Specialist Mathematics (Year 12) on VCE Vault: 22 lessons written for Year 12 (VCE, Victorian Certificate of Education), each with worked examples, practice questions and Aari, an AI tutor. The first 4 lessons are free; the whole subject is a one-off A$14.99.

VCE Specialist Mathematics past exams 2023–2025: 247 questions matched to these lessons.

VCE Specialist Mathematics (Year 12) study notes: 111 pages covering 22 topics (PDF, A$9.99).

Lessons

Complex Numbers

  • Complex Numbers — Introduction and the Complex Plane (free) — Define complex numbers, represent them in Cartesian and polar forms, and perform arithmetic operations in both forms.
  • Complex Number Geometry — Loci and Transformations — Describe and sketch loci in the Argand plane defined by conditions on complex numbers.
  • Argand Diagrams and Polar Form — Represent complex numbers on the Argand plane, convert fluently between Cartesian and polar forms, and interpret multiplication and division geometrically.
  • De Moivre's Theorem and Roots of Unity — Apply de Moivre's theorem to compute powers and roots of complex numbers, find all nth roots of unity, and use them to factorise polynomials over the complex numbers.

Calculus

  • Implicit Differentiation and Related Rates (free) — Differentiate implicitly defined functions and solve related rate problems using the chain rule.
  • Integration Techniques — Substitution and Partial Fractions (free) — Apply substitution and partial fractions to evaluate integrals not in standard form.
  • Differential Equations — Separation of Variables — Solve first-order differential equations using separation of variables and apply them to growth and decay models.
  • Techniques of Integration — Apply integration by parts, trigonometric integrals, and trigonometric substitution to evaluate integrals beyond standard forms.
  • Volumes of Revolution and Arc Length — Calculate volumes of solids of revolution using the disc and washer methods, and compute arc lengths of curves using integration.
  • Exam-Style Extended-Response Practice — Solve multi-part extended-response problems that integrate calculus techniques across differentiation, integration, and differential equations.

Kinematics (Calculus and Vectors)

  • Vector Calculus — Kinematics in Two Dimensions (free) — Model 2D motion using vector-valued functions; differentiate and integrate to find velocity, acceleration and displacement.
  • Kinematics with Calculus — Apply differentiation and integration to analyse rectilinear and curvilinear motion, including velocity, acceleration, displacement, and distance travelled.

Vectors

  • Vector Proofs and Applications — Use vector methods including the dot product and cross product to prove geometric results and solve problems involving angles, areas, and projections.
  • Lines and Planes in 3D — Write vector and parametric equations for lines and planes in three dimensions, and find intersections, angles, and distances.

Extension — not in the 2023–2027 study design

  • Dynamics — Newton's Laws with Calculus — Apply Newton's second law F = ma with variable forces to set up and solve differential equations of motion, including problems involving resistance proportional to velocity.

Proof and Logic

  • Proof by Mathematical Induction — Construct rigorous proofs by mathematical induction for summation formulas, divisibility results, and inequalities.
  • Proof Techniques — Contradiction and Contrapositive — Construct proofs using contradiction and contrapositive, and distinguish between direct proof, proof by contradiction, and proof by contrapositive.
  • Specialist Maths Exam Strategy — Develop effective strategies for the VCAA Specialist Mathematics Exams, including time management, common pitfalls, and techniques for maximising marks.

Probability and Statistics

  • Linear Combinations of Random Variables and Sample Means — Find the mean and variance of linear combinations of random variables and describe the distribution of the sample mean.
  • Confidence Intervals and Hypothesis Testing for a Mean — Construct approximate confidence intervals for a population mean and carry out one- and two-tailed hypothesis tests using p-values.

Functions and Relations

  • Reciprocal and Inverse Circular Functions — Compound and Double Angle Formulas — Sketch and use sec, cosec and cot and the inverse functions arcsin, arccos and arctan, and apply the compound and double angle formulas to find exact values, simplify and solve equations.
  • Rational Functions — Asymptotes, Sketching and Partial Fractions — Sketch graphs of rational functions of low degree by finding asymptotes (vertical, horizontal and oblique), intercepts and stationary points, and decompose rational expressions into partial fractions.