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VCE · VCE Units 3 & 4

Specialist Mathematics

Proof, complex numbers, vectors, differential equations and mechanics — tech-free and tech-active exams with fully worked solutions.

20full-length model exams with mark-by-mark answer guides
20detailed note sets — ~120 pages across every topic
64exam-style practice questions with worked solutions
60flashcards for every key term & formula
5official past papers

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Sample revision note

Implications, equivalence and proof structures (direct, cases, contradiction, contrapositive)

Conjectures, implications and 'if and only if'

A conjecture is a statement put forward to be proved or disproved. A single counter-example disproves a universal claim, but no finite list of supporting examples ever proves one — examples only build intuition or motivate a conjecture.

The conditional P ⟹ Q reads “if P then Q”: P is sufficient for Q, and Q is necessary for P. The biconditional P ⟺ Q (“P if and only if Q”) means both P ⟹ Q and Q ⟹ P hold, so P and Q are equivalent. To prove an “iff”, you must prove both directions.

Direct proof and proof by cases

A direct proof chains together implications from the hypothesis to the conclusion. Example: if n is odd then n² is odd. Write n = 2k+1, then n² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which is odd. ∎

A proof by cases splits the domain into exhaustive cases and proves the result in each. Example: n² + n is even for every integer n.

  • Case n even: n = 2k, so n² + n = 2k(2k+1), even.
  • Case n odd: n = 2k+1, so n² + n = (2k+1)(2k+2) = 2(2k+1)(k+1), even.
Every integer falls into one case, so the result holds for all n. ∎

Sample exam question

Prove by mathematical induction that for all integers n ≥ 1, 1·1! + 2·2! + 3·3! + … + n·n! = (n+1)! − 1. (4 marks)

Show the worked answer

Answer: Worked solution

Let P(n) be the statement 1·1! + 2·2! + … + n·n! = (n+1)! − 1. Base step (n = 1): LHS = 1·1! = 1; RHS = 2! − 1 = 2 − 1 = 1. So LHS = RHS and P(1) is true. Inductive step: assume P(k) holds for some integer k ≥ 1, i.e. 1·1! + … + k·k! = (k+1)! − 1. Consider the sum to k+1 terms: 1·1! + … + k·k! + (k+1)(k+1)! = [(k+1)! − 1] + (k+1)(k+1)! (by the assumption) = (k+1)!·[1 + (k+1)] − 1 = (k+1)!·(k+2) − 1 = (k+2)! − 1 = ((k+1)+1)! − 1. This is exactly P(k+1). Hence P(k) ⇒ P(k+1). Since P(1) is true and the implication holds, by the principle of mathematical induction P(n) is true for all integers n ≥ 1. ∎

All 20 practice exams

  1. Exam 1 — Proof & complex numbers (induction, contradiction, De Moivre) — ~13 marks; By-hand calculus (implicit/related rates, anti-differentiation, kinematics, DEs) — ~17 marks; Vectors & rational graphs (dot/cross product, geometry, asymptotes) — ~10 marks
  2. Exam 2 — Section A — 20 multiple-choice across all six areas of study — 20 marks; Functions, complex numbers & vectors (rational graphs, factorisation over C, intersecting paths) — ~30 marks; Calculus & statistical inference (related rates, SHM kinematics, hypothesis test & CI) — ~30 marks
  3. Exam 3 — Technology-free by-hand algebra, complex numbers and proof — ~16 marks; Calculus by hand: differentiation, anti-differentiation, implicit and related rates — ~14 marks; Vectors and vector proof — ~10 marks
  4. Exam 4 — Multiple choice across all six areas of study — 20 marks; Extended response: complex numbers, rational graphs, vectors and kinematics — ~30 marks; Extended response: calculus, differential equations and statistical inference — ~30 marks
  5. Exam 5 — Proof, complex numbers & rational graphs (by hand) — ~13 marks; Calculus: differentiation, integration & differential equations — ~14 marks; Vectors & kinematics (by hand) — ~13 marks
  6. Exam 6 — Section A — 20 multiple-choice across all six areas of study — 20 marks; Complex numbers, rational-function calculus & vectors (extended) — ~30 marks; Differential equations, vector kinematics & statistical inference (extended) — ~30 marks
  7. Exam 7 — Proof, complex numbers and vectors by hand — ~15 marks; By-hand calculus: differentiation, integration and rational graphs — ~15 marks; Differential equations and kinematics without technology — ~10 marks
  8. Exam 8 — Section A multiple-choice across all six areas of study — 20 marks; Section B: rational graphs, complex factorisation and applied calculus — ~30 marks; Section B: vector motion and statistical inference — ~30 marks
  9. Exam 9 — Proof & complex numbers (induction on divisibility, conjugate roots, De Moivre) — ~13 marks; By-hand calculus (inverse-trig derivatives, related rates, integration by parts, separable DE, kinematics) — ~17 marks; Vectors & rational graphs (cross product & planes, vector proof, oblique asymptote) — ~10 marks
  10. Exam 10 — Section A multiple-choice across all six areas of study — ~20 marks; Complex numbers, rational graphs & vector motion (roots, calculus of graphs, paths/collisions) — ~30 marks; Differential equations, kinematics & statistical inference (mixing/logistic, Euler, CI & hypothesis test) — ~30 marks
  11. Exam 11 — Exam 1 (tech-free) — proof, complex numbers, calculus & vectors by hand — 40 marks; Exam 2 Section A — 20 multiple-choice across all six areas — 20 marks; Exam 2 Section B — extended response (graphs, DEs, kinematics, vectors, inference) — 60 marks
  12. Exam 12 — Exam 1 (tech-free) — induction, complex algebra, integration & vector geometry by hand — 40 marks; Exam 2 Section A — 20 multiple-choice across all six areas — 20 marks; Exam 2 Section B — extended response (quotient graphs, related rates, vector motion, inference) — 60 marks
  13. Exam 13 — Logic, proof & complex numbers (contradiction on irrationality, proof by cases, De Moivre identity, factorisation over C) — ~14 marks; By-hand calculus (implicit differentiation, partial-fraction integration, arc length, separable DE) — ~16 marks; Vectors & rational graphs (scalar-resolute proof, vector geometry, quotient-function asymptotes) — ~10 marks
  14. Exam 14 — Section A multiple-choice across all six areas of study — ~20 marks; Complex numbers, rational graphs & vector motion (roots of 1−i, asymptote analysis, paths & collision) — ~30 marks; Logistic DE, volumes/areas of revolution & statistical inference (carrying capacity, inverse-trig integrals, CI & two-tailed test) — ~30 marks
  15. Exam 15 — Logic, proof and complex numbers by hand (proof by cases, induction on a product, De Moivre) — ~13 marks; By-hand calculus (inverse-trig derivative, by-parts integration, separable DE, rectilinear motion) — ~16 marks; Vectors and rational-graph sketching (cross product & plane, vector midpoint proof, oblique asymptote) — ~11 marks
  16. Exam 16 — Section A multiple-choice across all six areas of study — 20 marks; Section B: quotient-graph analysis, complex 4th roots and logistic growth — ~30 marks; Section B: vector path motion, integral applications and statistical inference — ~30 marks
  17. Exam 17 — Proof & complex numbers (induction on an inequality, proof by cases, nth roots of unity, De Moivre) — ~14 marks; By-hand calculus (implicit 2nd derivative, partial-fraction integral, separable DE, vector kinematics) — ~16 marks; Vectors & quotient graphs (scalar resolute, vector proof of concurrency, square-root quotient curve) — ~10 marks
  18. Exam 18 — Section A multiple-choice across all six areas of study — 20 marks; Section B: rational graph, complex factorisation and applied calculus (revolution & related rates) — ~30 marks; Section B: vector kinematics, mixing-tank differential equation and statistical inference — ~30 marks
  19. Exam 19 — Proof & complex numbers (induction on an inequality, proof by cases, nth roots & factorisation over C) — ~14 marks; By-hand calculus (inverse-trig derivative, partial-fraction integral, separable DE, vector-motion kinematics) — ~17 marks; Vectors & rational graphs (cross product & area, dot-product vector proof, quotient-function sketch) — ~9 marks
  20. Exam 20 — Section A — 20 multiple-choice across all six areas of study — 20 marks; Functions, complex numbers & related rates (quotient graph, factorisation over C, searchlight angle) — ~30 marks; Calculus, mixing DE & statistical inference (x-dependent acceleration, fertiliser tank, CI & two-tail test) — ~30 marks

All 20 revision notes

  • Implications, equivalence and proof structures (direct, cases, contradiction, contrapositive)
  • Proof by mathematical induction (sums, divisibility, inequalities)
  • Rational functions: partial fractions and asymptotic behaviour
  • Stationary points, inflection and the second-derivative test for curve sketching
  • Complex numbers: Cartesian and polar form, the Argand plane and complex regions
  • De Moivre's theorem and the nth roots of complex numbers
  • Factorisation over C: the conjugate root theorem and solving polynomial equations
  • Vector algebra, the dot product, resolutes and the angle between vectors
  • The cross product, normals to planes and vector proofs of geometric results
  • Vector, parametric and Cartesian equations of lines and planes
  • Inverse circular derivatives, second derivatives and implicit/related rates
  • Anti-differentiation: recognising inverse-circular and log forms, and the four core techniques
  • Areas, arc length of parametric curves, and solids of revolution
  • Solving dy/dx = f(x), dy/dx = g(y), separation of variables and second-order by antidifferentiation
  • Logistic growth, direction (slope) fields and Euler's method
  • Rectilinear motion: velocity-time graphs, calculus links and the acceleration forms
  • Vector functions of time: paths, meeting/colliding particles and velocity/acceleration vectors
  • Expectation and variance of sums and linear combinations of random variables
  • The sample mean as a random variable and the central limit behaviour
  • Confidence intervals for a mean and one- and two-tailed hypothesis tests