HSC Maths Extension 1 Checklist for 2026-2027 Syllabuses

A syllabus-aware checklist for NSW HSC Mathematics Extension 1, including the 2026 legacy course, the 2027 syllabus, and key Advanced prerequisites.

September 15, 2026 12 min read

You open two “HSC Maths Extension 1 checklists” and immediately find a contradiction. One puts inverse trigonometric functions in Year 11. The other puts them in Year 12. One has rates of change in Year 11. The other doesn’t.

Which checklist is wrong?

At the moment, neither necessarily is. NSW is changing Mathematics Extension 1 syllabuses, so in 2026 there are two different courses running at the same time. If you revise from the wrong checklist, you can end up learning perfectly good mathematics for the wrong HSC.

This page maps the Mathematics Extension 1 11-12 Syllabus (2024), first examined in the 2027 HSC, and the Mathematics Advanced knowledge that supports it. It also gives a separate checklist for students sitting the 2026 HSC under the 2017 syllabus. It does not try to reteach every technique. Each item tells you what you should be able to do, why it matters, and where students commonly come unstuck.

01First, make sure you are using the right syllabus

As at 15 September 2026:

If you are…	Syllabus you should use
In Year 11 during 2026	Mathematics Extension 1 11-12 Syllabus (2024)
Sitting the HSC in 2026	Mathematics Extension 1 Stage 6 Syllabus (2017)
Beginning Year 12 in Term 4, 2026	Mathematics Extension 1 11-12 Syllabus (2024)
Sitting the HSC from 2027	Mathematics Extension 1 11-12 Syllabus (2024)

NESA began teaching the new syllabus to Year 11 in Term 1, 2026. Year 12 changes over from Term 4, 2026, and the first HSC examination based on the new syllabus is in 2027. (NSW Curriculum)

That change matters because the content has genuinely moved around. Under the new syllabus, Extension 1 is organised into seven areas: functions, proof, vectors, trigonometric functions, combinatorics, calculus, and statistical analysis. (NSW Curriculum)

Sitting the 2026 HSC? Use this legacy checklist

The 2017 syllabus has the following Extension 1 structure. (NSW Government)

Year 11 assumed Extension 1 content

[ ] Further work with functions
[ ] Polynomials
[ ] Inverse trigonometric functions
[ ] Further trigonometric identities
[ ] Rates of change
[ ] Working with combinatorics

Year 12 Extension 1 content

[ ] Proof by mathematical induction
[ ] Introduction to vectors
[ ] Trigonometric equations
[ ] Further calculus skills
[ ] Applications of calculus
[ ] The binomial distribution

If you are a 2026 HSC student, do not replace this with the rest of this article’s 2024 syllabus checklist.

For everyone studying towards the 2027 HSC or later, continue below.

02The Advanced maths you need underneath Extension 1

Extension 1 is not a separate mathematical island. Think of Advanced as the floor of the house and Extension 1 as the extra storey. You can spend a lot of time decorating upstairs, but if your algebra downstairs collapses, you’re still going through the floor.

Under the 2024 syllabus, Year 11 Extension 1 is studied alongside Mathematics Advanced. Mathematics Advanced Year 11 is also a prerequisite for entering Year 12 Extension 1, while Mathematics Advanced Year 12 is studied as a corequisite. (Scribd)

You therefore need more than an “Extension 1 formula list”. These Advanced skills need to become automatic.

Functions and graphs

Before Extension 1 functions feel comfortable, make sure you can:

[ ] use function notation such as \(f(x)\)
[ ] identify domain and range
[ ] distinguish a function from a general relation
[ ] solve equations algebraically and graphically
[ ] recognise linear, quadratic, cubic, reciprocal, exponential, and logarithmic behaviour
[ ] apply translations, reflections, stretches, and compressions
[ ] work confidently with composite functions
[ ] read intersections, intercepts, turning points, and asymptotes from graphs

Why it matters: Extension 1 starts asking how functions relate to their reciprocals, inverses, parametric forms, and polynomial structure. Weak graph knowledge turns these into memorisation exercises.

Common trap: treating \(f^{-1}(x)\) as \(\frac{1}{f(x)}\). They are completely different ideas. \(f^{-1}\) means an inverse function. \(\frac{1}{f}\) means a reciprocal.

Trigonometry

You should already be comfortable with:

[ ] radians and exact trigonometric values
[ ] the unit circle
[ ] sine, cosine, and tangent graphs
[ ] periodicity
[ ] solving basic trigonometric equations
[ ] standard identities
[ ] the sine and cosine rules
[ ] bearings and geometric trigonometry

Extension 1 then asks you to push these ideas into three dimensions and more complicated identities.

Calculus

From Advanced, keep these skills active:

[ ] understand derivative as instantaneous rate of change and gradient
[ ] differentiate standard functions
[ ] use product, quotient, and chain rules when they are introduced
[ ] find stationary points and analyse curves
[ ] integrate standard functions
[ ] interpret definite integrals as signed area
[ ] connect displacement, velocity, and acceleration

The new Mathematics Advanced course covers functions, trigonometric functions, calculus, exponential and logarithmic functions, probability and data in Year 11, followed by further modelling, sequences and series, more calculus, random variables, and financial mathematics in Year 12. (NSW Curriculum)

Extension questions often hide an Advanced technique inside a harder setting. If you recognise the sophisticated part but make an algebra error solving \(2x^2-5x-3=0\), you still lose the question.

03Year 11 Extension 1 checklist

The new Year 11 course is unusually concentrated. It contains five focus areas: further work with functions, polynomials, further trigonometry, permutations and combinations, and the binomial theorem. (NSW Curriculum)

Further work with functions

You should be able to:

[ ] solve inequalities and represent their solutions correctly
[ ] understand absolute value graphically and algebraically
[ ] connect \(y=f(x)\) with reciprocal graphs such as \(y=\frac{1}{f(x)}\)
[ ] sketch and analyse \(\sec x\), \(\csc x\), and \(\cot x\)
[ ] determine when a function has an inverse
[ ] find and graph inverse functions
[ ] understand reflection of a function and its inverse in \(y=x\)
[ ] work with parametric equations and eliminate the parameter
[ ] interpret the Cartesian curve represented by parametric equations

Why it matters: this is where a graph stops being just a picture and becomes a way of predicting algebra.

Common trap: finding an algebraic expression for an inverse without checking that the original function is one-to-one on the stated domain.

Polynomials

You should be able to:

[ ] use polynomial terminology including degree, leading coefficient, zeroes, roots, and multiplicity
[ ] predict end behaviour from degree and leading coefficient
[ ] divide polynomials
[ ] apply the remainder theorem
[ ] apply the factor theorem
[ ] find unknown coefficients using known factors or roots
[ ] connect repeated roots with the shape of a graph
[ ] use sums and products of roots for quadratic, cubic, and quartic polynomials

A polynomial is a good example of why Extension 1 rewards connections. A single fact such as \(P(2)=0\) can simultaneously tell you that \(x=2\) is a root, \(x-2\) is a factor, and \((2,0)\) lies on the graph.

Further trigonometry

You should be able to:

[ ] interpret three-dimensional diagrams
[ ] solve 3D length and angle problems using trigonometry
[ ] prove and use compound-angle identities
[ ] simplify more complicated trigonometric expressions
[ ] transform combinations such as \(a\sin x+b\cos x\) into a more useful form
[ ] use identities to solve trigonometric equations
[ ] model periodic situations

Common trap: assuming a visible angle in a 3D drawing is a right angle. A perspective sketch is not proof. Work out which triangle actually contains the required angle.

Permutations and combinations

The key decision is simple:

Does order matter?

If Alice being captain and Ben being vice-captain is different from Ben being captain and Alice being vice-captain, order matters. Think permutation.

If you are merely choosing Alice and Ben for a committee, order does not matter. Think combination.

You should be able to:

[ ] use factorial notation, including \(0!=1\)
[ ] apply the multiplication principle
[ ] calculate permutations
[ ] calculate combinations
[ ] handle restrictions and repeated objects where required
[ ] split a counting problem into cases
[ ] use counting methods in probability

Worked example: choosing students for different roles

Six students are available. A captain and vice-captain are chosen, followed by two ordinary representatives from the remaining four students. How many different leadership groups are possible?

Step 1Choose the two ordered positions

Captain and vice-captain are different jobs, so order matters:

\[
{}^6P_2=6\times5=30
\]

Step 2Choose the two ordinary representatives

The remaining representatives have identical roles, so order does not matter:

\[
{}^4C_2=\frac{4!}{2!2!}=6
\]

The multiplication principle gives:

\[
30\times6=180
\]

Answer: there are 180 possible leadership groups.

The mathematics is not really about remembering which calculator button says \(nPr\) and which says \(nCr\). It is about deciding whether swapping two people creates a genuinely different outcome.

The binomial theorem

You should be able to:

[ ] recognise binomial coefficients in Pascal’s triangle
[ ] use \(\binom nr\)
[ ] expand \((a+b)^n\)
[ ] find a specified term or coefficient without expanding everything
[ ] find a constant term
[ ] use symmetry and identities involving binomial coefficients
[ ] prove identities by comparing coefficients or through counting arguments

This topic is where combinatorics and algebra meet. The coefficient \(\binom nr\) appears because expanding \((a+b)^n\) is secretly a counting problem: you are choosing which \(r\) of the \(n\) factors contribute a \(b\).

04Year 12 Extension 1 checklist

Year 12 expands the course into proof, vectors, inverse trigonometric functions, calculus, and statistics. NESA’s six Year 12 outcomes cover induction, vectors and two-dimensional motion, inverse trigonometric functions, further differentiation and integration, applications of calculus, and binomial and sampling distributions. (NSW Curriculum)

Proof by mathematical induction

You should be able to:

[ ] identify the proposition being proved
[ ] prove the base case
[ ] state an induction hypothesis clearly
[ ] complete the inductive step
[ ] prove summation identities
[ ] prove divisibility results
[ ] detect invalid induction arguments

Induction works a little like an infinite row of dominoes. Showing the first domino falls is not enough. Showing that one falling would knock over the next is also not enough. You need both.

That analogy has a limit: a proof does not involve an actual sequence of physical events. The point is the logical structure.

Introduction to vectors

You should be able to:

[ ] represent vectors in 2D and 3D
[ ] distinguish vectors from scalars
[ ] calculate vector magnitude
[ ] use unit vectors and component notation
[ ] add, subtract, and multiply vectors by scalars
[ ] test whether vectors are parallel
[ ] use the dot product
[ ] find angles between vectors
[ ] test perpendicularity using the dot product
[ ] use vector projections
[ ] model position, velocity, and acceleration
[ ] solve two-dimensional motion and projectile problems

A tempting prediction is that a projectile has zero velocity at its highest point.

It doesn’t. Only its vertical velocity is zero. Unless the projectile has somehow stopped moving sideways as well, it still has horizontal velocity.

Inverse trigonometric functions

You should be able to:

[ ] explain why sine, cosine, and tangent need domain restrictions before inverses can be defined
[ ] state the domains and ranges of \(\sin^{-1}x\), \(\cos^{-1}x\), and \(\tan^{-1}x\)
[ ] sketch inverse trigonometric graphs
[ ] simplify compositions involving trig and inverse trig functions, with domain restrictions in mind
[ ] solve equations involving inverse trigonometric functions

Common trap: assuming \(\sin^{-1}(\sin x)=x\) for every real \(x\). It only returns \(x\) directly when \(x\) lies in the principal range chosen for \(\sin^{-1}\).

Further calculus skills

Your checklist here is:

[ ] differentiate parametrically defined functions
[ ] differentiate inverse functions
[ ] differentiate inverse trigonometric functions
[ ] combine product, quotient, and chain rules efficiently
[ ] integrate expressions connected to inverse trigonometric forms
[ ] use specified substitutions
[ ] integrate expressions involving \(\sin^2x\) and \(\cos^2x\)

Further applications of calculus

You should be able to:

[ ] connect root multiplicity with derivatives and polynomial graph behaviour
[ ] solve related-rates problems
[ ] model quantities approaching limiting values
[ ] calculate areas between curves
[ ] calculate volumes of solids of revolution
[ ] understand what a differential equation represents
[ ] interpret slope fields
[ ] solve suitable first-order differential equations
[ ] use initial conditions
[ ] model growth, decay, cooling, carrying capacity, and logistic behaviour

Worked example: vectors meet calculus

A projectile is launched from ground level. Its position after \(t\) seconds is

\[
\mathbf r(t)=20t\,\mathbf i+(30t-5t^2)\,\mathbf j
\]

where distances are measured in metres. Find its maximum height and horizontal range.

Step 1Differentiate position to find velocity

\[
\mathbf v(t)=\frac{d\mathbf r}{dt}
=20\,\mathbf i+(30-10t)\,\mathbf j
\]

At maximum height, the vertical component of velocity is zero:

\[
30-10t=0
\]

\[
t=3\text{ s}
\]

Step 2Find the height at \(t=3\)

The vertical position is \(y=30t-5t^2\):

\[
y(3)=30(3)-5(3)^2
=90-45
=45\text{ m}
\]

So the maximum height is 45 m.

Step 3Find when the projectile returns to the ground

Set the vertical position equal to zero:

\[
30t-5t^2=0
\]

\[
5t(6-t)=0
\]

Ignoring \(t=0\), which is the launch time:

\[
t=6\text{ s}
\]

Step 4Find the horizontal position after 6 seconds

\[
x=20t=20(6)=120\text{ m}
\]

Answer: the projectile reaches a maximum height of 45 m and has a horizontal range of 120 m.

This is exactly the kind of connection that matters in Extension 1. The vector describes the motion, while calculus extracts the velocity and identifies the maximum.

Binomial distribution and sampling distribution of the mean

You should be able to:

[ ] recognise a Bernoulli trial
[ ] calculate expectation and variance for a Bernoulli variable
[ ] decide when a situation can be modelled by a binomial distribution
[ ] use \(X\sim\operatorname{Bin}(n,p)\)
[ ] calculate exact binomial probabilities
[ ] calculate expected value and variance of a binomial random variable
[ ] distinguish a population from a sample
[ ] understand the sample mean as a random variable
[ ] find the mean and variance of the sampling distribution of the mean
[ ] understand the effect of increasing sample size
[ ] apply the central limit theorem
[ ] calculate probabilities involving sample means

This last part is an important change in the 2024 syllabus. The Year 12 outcome now explicitly includes the sampling distribution of the mean and the central limit theorem. (NSW Curriculum)

05The small equation set worth knowing cold

A checklist should not become a four-page formula dump. These relationships are more useful because several topics grow directly from them.

Idea	Relationship	What it tells you
Permutations	\({}^nP_r=\frac{n!}{(n-r)!}\)	ordered selections
Combinations	\(\binom nr=\frac{n!}{r!(n-r)!}\)	unordered selections
Binomial probability	\(P(X=r)=\binom nr p^r(1-p)^{n-r}\)	exactly \(r\) successes in \(n\) trials
Vector magnitude	(	\mathbf a	=\sqrt{\mathbf a\cdot\mathbf a})	size of a vector
Dot product	(\mathbf a\cdot\mathbf b=	\mathbf a		\mathbf b	\cos\theta)	angle and perpendicularity
Sampling mean	\(E(\bar X)=\mu,\quad \operatorname{Var}(\bar X)=\frac{\sigma^2}{n}\)	how sample means behave

Here \(n\) is a number of objects or trials depending on context, \(r\) is the number selected or the number of successes, \(p\) is the probability of success, \(\theta\) is the angle between two vectors, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation, and \(n\) in the final row is the sample size.

Notice the recurring pattern. Extension 1 is less a collection of unrelated tricks than it first appears. Combinations reappear inside the binomial theorem and binomial probability. Functions lead into inverses, inverse trig, and calculus. Parametric equations return in motion. Polynomial roots return when calculus explains graph shape.

06A better order to revise the course

If you’re finding Extension 1 messy, revising in syllabus order is not always the best fix. Follow the dependencies instead.

Start with algebra, functions, graph transformations, and basic trigonometry from Advanced. Then consolidate Extension 1 functions and polynomials, because they sharpen the algebra used almost everywhere else. Follow with further trigonometry.

Next, learn permutations and combinations before the binomial theorem. The notation becomes much less mysterious when \(\binom nr\) already means something concrete.

For Year 12, build vectors alongside Advanced calculus, then move into the harder Extension 1 calculus applications. Study inverse trigonometric functions before their derivatives and related integrals. Finish the probability chain by moving from Advanced random variables to Bernoulli variables, binomial distributions, sampling means, and the central limit theorem.

Keep proof running through all of it. “Working mathematically” is deliberately embedded throughout the new syllabus rather than being a separate chapter. NESA describes it in terms of understanding and fluency, problem solving, reasoning, and communicating mathematical thinking clearly. (NSW Curriculum)

That gives you a useful final test for each checkbox. Don’t mark a topic complete because you have seen the formula. Mark it complete when you can recognise when it applies, carry out the mathematics accurately, and explain why your method works.