HSC Maths Ext 2 study guide
HSC Maths Ext 2 · Work through your exam, one topic at a time.
Mathematics Extension 2 is the most demanding mathematics course in the NSW Higher School Certificate, taken alongside Mathematics Extension 1 and typically chosen by students aiming at mathematics-heavy university degrees or who simply enjoy hard problems. It is assessed by a written examination in which full working must be shown, and marks are awarded for reasoning and proof, not just final answers. The HSC Maths Ext 2 study guide has 8 sections and about 2,459 words.
01Proof
Extension 2 proof questions rest on a small toolkit: direct proof, proof by contradiction, proof by contraposition, disproof by counterexample, and equivalence proofs (proving both directions separately). You must also handle inequalities rigorously - establishing a chain of implications from known truths to the target statement, never assuming the conclusion.
314 words, about 2 min02Vectors
You need fluency with vectors in two and three dimensions: addition, scalar multiplication, position vectors, displacement, magnitude, unit vectors, the dot product and its geometric meaning, and the cross product including its magnitude as an area and its direction by the right-hand rule. You must move confidently between component form and geometric reasoning, and use vectors to prove geometric facts (parallelism, collinearity, perpendicularity, concurrency) and to find lines and planes, distances and angles.
329 words, about 2 min03Complex Numbers
You must handle complex numbers in Cartesian, polar (modulus–argument) and exponential forms, converting fluently. Core skills: arithmetic, conjugates, modulus and argument properties, De Moivre's theorem for powers and roots, sketching loci and regions in the Argand plane, and using complex numbers to factor real polynomials and prove trigonometric identities.
323 words, about 2 min04Calculus: Further Differentiation, Integration and Applications
This domain covers techniques and their applications: implicit and parametric differentiation, logarithmic differentiation, higher derivatives, integration by parts, substitution, partial fractions, trigonometric integrals, and the calculus of inverse trigonometric functions. Applications include curve sketching, maxima and minima, rates of change, volumes of solids of revolution, arc length, and simple differential equations solved by separation of variables.
273 words, about 1 min05Mechanics
Extension 2 mechanics centres on motion with variable acceleration and resisted motion, expressed through differential equations. You must set up equations of motion from Newton's second law, use a = dv/dt = vdv/dx = d^2x/dt^2 appropriately, solve separable differential equations for velocity and displacement, handle resisted motion (proportional to velocity or its square), and analyse projectile motion with or without resistance.
322 words, about 2 min06Sequences and Series
You need arithmetic and geometric sequences, sums to n terms, limiting sums of infinite geometric series, and the behaviour of sequences as n . Extension 2 adds telescoping series, partial fractions to sum series, the method of differences, and convergence tests in an informal sense (comparison, ratio, recognising divergence).
290 words, about 1 min07The Binomial Theorem
You must state and apply the binomial theorem for positive integer powers, use the general term n{r}a^{n-r}b^r to extract a particular term or coefficient, and extend to negative and fractional indices using the expansion of (1+x)^n valid for x <1 . You should also handle identities obtained by substituting values into expansions, and use binomial coefficients in counting and probability contexts.
304 words, about 2 min08Mathematical Induction
Induction proves statements about all positive integers. The structure is fixed: state the proposition P(n) , prove the base case, assume P(k) true for some k , then prove P(k+1) using that assumption, and conclude.
304 words, about 2 min