Mathematics Extension 1 Scaling HSC 2026: Does It Scale Up or Down?

What Mathematics Extension 1 actually asks of you

Extension 1 is examined in one written paper each year, released by NESA with marking guidelines and, more recently, marking feedback covering Section II. In the questions mapped in our bank the paper opens with one-mark multiple-choice items running to Q10, then moves to multi-part questions numbered from Q11 in which individual parts have carried anywhere from one to seven marks. Those parts usually escalate within a question, starting from a routine technique and ending in an unfamiliar application or a proof.

The Mathematics Extension 1 exam is Fri 23 Oct 2026, 1:50 pm (2 hours 10 minutes (1.50 pm – 4.00 pm block, includes reading time)). Source: HSC timetable.

The 7 areas of study you are examined on

From the Mathematics Extension 1 Stage 6 Syllabus (2017), examined from 2020 to 2026.

• Functions
  This topic pushes the Advanced work on functions into territory where you reason about a graph you have never plotted. You work with reciprocals of functions, square roots of functions and absolute value graphs, deducing asymptotes, turning points and where a transformed curve must sit from the behaviour of the original. Inequalities are treated properly here: rational inequalities with a variable in the denominator, inequalities involving absolute values, and inequalities solved by graphing two functions and reading off the region. Parametric representation of curves appears, including converting between parametric and Cartesian form. The polynomials strand covers division, the remainder and factor theorems, multiplicity of roots and their graphical meaning, and the relationships between the coefficients of a polynomial and the sums and products of its roots. Those root relationships turn up repeatedly in later questions, so the algebra here is worth being fast at.
  In the exam: Expect short items that ask you to solve an inequality exactly or sketch a transformed curve with asymptotes and intercepts labelled, and polynomial parts that use root-coefficient relationships rather than solving. Mapped 2025 questions use the roots of a cubic to evaluate a symmetric expression and, in a harder part, to pin down an angle sum from roots written as tangents.
  Where marks go missing: Multiplying both sides of a rational inequality by the denominator without knowing its sign. Multiply by the square of the denominator, or test intervals, and always exclude the value that makes the denominator zero from the final solution set.
• Trigonometric Functions
  The inverse trigonometric functions are defined here as genuine functions, which means committing their restricted domains and ranges to memory and being able to sketch the graphs of arcsin, arccos and arctan, including their symmetry properties and derivatives. Alongside that sits the identity work: sums and differences of angles, double and triple angle results, products expressed as sums, the t-formulae, and rewriting a cos x + b sin x as a single sine or cosine with an auxiliary angle. Those tools then feed the equation solving, where you solve trigonometric equations exactly over a restricted domain, or give general solutions covering every period. Applications include modelling periodic situations, so you also need to move between an angle in a diagram and a time in a model.
  In the exam: Questions run from one-mark multiple-choice items on general solutions to three-mark parts that solve an equation using a double-angle identity over a stated domain, differentiate a composite of inverse and ordinary trigonometric functions on a restricted domain, or model a rotating system such as clock hands and find the first two times a condition is met.
  Where marks go missing: Losing solutions when you cancel. Dividing an equation by cos x or sin x deletes the roots where that factor is zero, and a restricted-domain question is marked on the complete solution set, not just the first value your calculator returns.
• Calculus
  Three strands sit here. Rates of change covers related rates, where two quantities are linked by a geometric or physical relationship and you differentiate that relationship with respect to time, and exponential models of the form where a quantity approaches a limiting value. Further calculus skills add integration by substitution, integrals producing inverse trigonometric functions, integrating squared sine and cosine using double-angle identities, and differentiating inverse functions. Further applications brings differential equations: sketching and interpreting direction fields, and solving separable equations, including logistic-style models, with an initial condition to fix the constant. Projectile motion is treated in this course as an application of calculus and vectors, so you derive the equations of motion rather than quoting them, and volumes of revolution round out the integration work.
  In the exam: Related-rates parts are typically worth three to six marks and require the relationship written down before differentiating; differential equation parts ask you to separate, integrate and apply the initial condition. Mapped questions include a sector with constant area, a leaking conical container, a reaction-rate model, and a two-particle projectile problem set up as a 'show that'.
  Where marks go missing: Substituting the given numerical value before differentiating in a related-rates question. The radius or height has to stay a variable until after the derivative is formed; substitute the instantaneous value only at the final step, or the derivative collapses.
• Combinatorics
  Counting is done from principles rather than formulas here. You use the multiplication principle for arrangements, then handle the standard complications: arrangements with restrictions, objects that must or must not be adjacent, identical objects, and arrangements in a circle where one position is fixed to remove rotational duplicates. Combinations follow, with selections subject to composition constraints and the use of complementary counting when a condition reads 'at least one'. The pigeonhole principle appears as a proof tool. The binomial theorem strand covers expanding a binomial power, finding a particular term or coefficient, and the identities that come out of Pascal's triangle, including the recurrence linking adjacent coefficients. Proving identities by manipulating those coefficients, or by differentiating or substituting into the expansion, is a regular feature.
  In the exam: Short parts ask for a count under a restriction, such as circular seating where two guests must be separated. Harder parts ask you to prove a binomial-coefficient identity, either by telescoping with the Pascal's triangle recurrence or by deriving two standard identities from the expansion and combining them. Multiple-choice items test at-least-one selections.
  Where marks go missing: Adding overlapping cases when the condition is 'at least one'. Counting the arrangements containing exactly one, then exactly two, and summing them double-counts unless the cases are genuinely disjoint; subtracting from the unrestricted total is usually safer.
• Proof
  Mathematical induction is the named subtopic, and the course expects the full formal structure every time: verify the base case, state the assumption for n equal to k clearly, then prove the statement for k plus one using that assumption, and close with a concluding statement invoking the principle of induction. Induction is applied to summation identities, divisibility results, inequalities and expressions involving factorials or products. The wider proof work in the course also asks you to argue directly rather than compute, using algebraic manipulation, counter-examples, and results about divisibility and factors of integers. Because a proof question tells you what to prove, no marks are available for a correct answer alone; every mark sits in the visible steps, particularly in the line where the assumption is actually used.
  In the exam: A three-mark induction part is standard, often on a summation identity involving factorials, and sometimes preceded by an algebraic step such as factorising a cubic that is then used in the inductive step. Divisibility proofs appear as shorter parts. Marks are attached to stating the assumption and to the point where it is substituted.
  Where marks go missing: Working the k plus one case forward without ever invoking the assumption. If your working would be valid with the assumption deleted, you have verified an identity rather than proved it, and the mark for using the assumption is lost.
• Vectors
  Vectors are new to this course and are examined heavily. You represent vectors in component and column form, add and subtract them, multiply by scalars, find magnitude and direction, and use unit vectors. The scalar (dot) product gives the angle between two vectors, tests for perpendicularity, and produces the projection of one vector onto another, which is the single most-used construction in the topic. Position vectors let you express geometric conditions algebraically, so collinearity, parallelism and ratios of division along a line all become vector equations, and standard plane geometry results can be proved this way. The further work applies calculus to vectors: differentiating a position vector to get velocity and acceleration, and analysing motion, including projectile motion, in vector form rather than as separate horizontal and vertical equations.
  In the exam: Multiple-choice items ask for a projection, for the condition satisfied by three collinear position vectors, or for an angle deduced from a vector sum equal to zero. Longer parts give a vector-valued position function and ask for the time at which velocity and acceleration meet a stated angle, which requires differentiating twice and then using the dot product.
  Where marks go missing: Confusing the scalar projection with the vector projection, and dividing by the wrong magnitude. The projection of a onto b is scaled by the magnitude of b, not of a; getting that the wrong way round produces a plausible number that earns nothing.
• Statistical Analysis
  This topic formalises randomness. Bernoulli trials come first, with a single trial's mean and variance, then the binomial distribution: the probability of a given number of successes, the mean np, the variance np times one minus p, and the shape of the distribution as n and p change. You then move to the normal approximation to the binomial, using continuity where appropriate and standardising to z-scores to estimate probabilities that would be impractical to sum. The sampling distribution of the sample mean follows: how the mean of samples drawn from a population is itself distributed, why its spread is smaller than the population's, and how the standard deviation of that distribution depends on sample size. The practical outcome is being able to reason about how likely a sample result is under an assumed model.
  In the exam: A one-mark item may ask for the mean and variance of a Bernoulli or binomial variable straight from a given probability function. A four-mark part is more typical of the harder work: using a normal approximation with a stated mean and a given tail probability to solve backwards for an unknown threshold value.
  Where marks go missing: Using the population standard deviation where the standard deviation of the sample mean is required. The spread of a sample mean shrinks with sample size, so standardising with the unscaled figure gives a z-score that is too small and a wildly wrong probability.

Full Mathematics Extension 1 study-design guide →

How scaling works in New South Wales

In New South Wales, NESA reports an HSC mark for each course, but the ATAR is not built from those marks. UAC takes the raw examination and assessment marks and scales each course separately, so that a mark means the same thing no matter which course it came from. A course whose students perform strongly across everything else they study is scaled up; a course whose students perform less strongly elsewhere is scaled down. UAC then adds your best 10 units of scaled marks: the best two units of English, which are compulsory, plus the best eight remaining units. That aggregate is ranked statewide and reported as an ATAR. Scaled marks are usually lower than HSC marks, and the statewide average scaled mark is close to 25 out of 50.

Source: official UAC scaling report (PDF). Last checked 2026-08-18.

What scaling is not

Scaling is not a difficulty rating and it is not a bonus. It compares how the students in one subject performed across every other subject they took, so a subject scales up because of its cohort, not because of the paper. The consequence is practical: you cannot scale your way out of a weak result. The only lever you control is the raw mark, and the fastest way to move that is full-length timed practice against the real exam format.

Questions

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