Syllabus

Mathematics Extension 1

Stage 6 syllabus (2017), the document examined in the 2026 HSC. 91 Year 12 dot points. Official document, SHA-256 f35eaedf4fac.

Every line below is the official wording, read from the NESA file by a parser and checked verbatim against it. Nothing here was typed by hand.

Proof

ME-P1 Proof by Mathematical Induction

understand the nature of inductive proof, including the ‘initial statement’ and the inductive step (ACMSM064)ACMSM064
prove results using mathematical induction
- prove results for sums, for example $1 + 4 + 9 + \dots + n^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$ for any positive integer $n$ (ACMSM065)ACMSM065
- prove divisibility results, for example $3^{2 n} - 1$ is divisible by 8 for any positive integer $n$ (ACMSM066)ACMSM066
identify errors in false ‘proofs by induction’, such as cases where only one of the required two steps of a proof by induction is true, and understand that this means that the statement has not been proved
recognise situations where proof by mathematical induction is not appropriate

Vectors

ME-V1 Introduction to Vectors

define a vector as a quantity having both magnitude and direction, and examine examples of vectors, including displacement and velocity (ACMSM010)ACMSM010
- explain the distinction between a position vector and a displacement (relative) vector
define and use a variety of notations and representations for vectors in two dimensions (ACMSM014)ACMSM014
- use standard notations for vectors, for example: , $A B$ and $a$
- represent vectors graphically in two dimensions as directed line segments
- define unit vectors as vectors of magnitude 1, and the standard two-dimensional perpendicular unit vectors and
- express and use vectors in two dimensions in a variety of forms, including component form, ordered pairs and column vector notation
perform addition and subtraction of vectors and multiplication of a vector by a scalar algebraically and geometrically, and interpret these operations in geometric terms AAM
- graphically represent a scalar multiple of a vector (ACMSM012)ACMSM012
- use the triangle law and the parallelogram law to find the sum and difference of two vectors
- define and use addition and subtraction of vectors in component form (ACMSM017)ACMSM017
- define and use multiplication by a scalar of a vector in component form (ACMSM018)ACMSM018
define, calculate and use the magnitude of a vector in two dimensions and use the notation for the magnitude of a vector
- prove that the magnitude of a vector, , can be found using:
- identify the magnitude of a displacement vector $A B$ as being the distance between the points $A$ and $B$
- convert a non-zero vector into a unit vector by dividing by its length:
define and use the direction of a vector in two dimensions
define, calculate and use the scalar (dot) product of two vectors and AAM
- apply the scalar product, , to vectors expressed in component form, where
- use the expression for the scalar (dot) product, where $θ$ is the angle between vectors and to solve problems
- demonstrate the equivalence, and use this relationship to solve problems
- establish and use the formula
- calculate the angle between two vectors using the scalar (dot) product of two vectors in two dimensions
examine properties of parallel and perpendicular vectors and determine if two vectors are parallel or perpendicular (ACMSM021)ACMSM021
define and use the projection of one vector onto another (ACMSM022)ACMSM022
solve problems involving displacement, force and velocity involving vector concepts in two dimensions (ACMSM023) AAMACMSM023
prove geometric results and construct proofs involving vectors in two dimensions including to proving that: AAM
- the diagonals of a parallelogram meet at right angles if and only if it is a rhombus (ACMSM039)ACMSM039
- the midpoints of the sides of a quadrilateral join to form a parallelogram (ACMSM040)ACMSM040
- the sum of the squares of the lengths of the diagonals of a parallelogram is equal to the sum of the squares of the lengths of the sides (ACMSM041)ACMSM041
understand the concept of projectile motion, and model and analyse a projectile’s path assuming that:
- the projectile is a point
- the force due to air resistance is negligible
- the only force acting on the projectile is the constant force due to gravity, assuming that the projectile is moving close to the Earth’s surface
model the motion of a projectile as a particle moving with constant acceleration due to gravity and derive the equations of motion of a projectile AAM
- represent the motion of a projectile using vectors
- recognise that the horizontal and vertical components of the motion of a projectile can be represented by horizontal and vertical vectors
- derive the horizontal and vertical equations of motion of a projectile
- understand and explain the limitations of this projectile model
use equations for horizontal and vertical components of velocity and displacement to solve problems on projectiles
apply calculus to the equations of motion to solve problems involving projectiles (ACMSM115) AAMACMSM115

Statistical Analysis

ME-S1 The Binomial Distribution

use a Bernoulli random variable as a model for two-outcome situations (ACMMM143)ACMMM143
- identify contexts suitable for modelling by Bernoulli random variables (ACMMM144)ACMMM144
use Bernoulli random variables and their associated probabilities to solve practical problems (ACMMM146) AAMACMMM146
- understand and apply the formulae for the mean, $E (X) = \overline{x} = p$ , and variance, $Var (X) = p (1 - p)$ , of the Bernoulli distribution with parameter $p$ , and $X$ defined as the number of successes (ACMMM145)ACMMM145
understand the concepts of Bernoulli trials and the concept of a binomial random variable as the number of ‘successes’ in $n$ independent Bernoulli trials, with the same probability of success $p$ in each trial (ACMMM147)ACMMM147
- calculate the expected frequencies of the various possible outcomes from a series of Bernoulli trials
use binomial distributions and their associated probabilities to solve practical problems (ACMMM150) AAMACMMM150
- identify contexts suitable for modelling by binomial random variables (ACMMM148)ACMMM148
- identify the binomial parameter $p$ as the probability of success
- understand and use the notation $X Bin (n, p)$ to indicate that the random variable $X$ is distributed binomially with parameters $n$ and $p$
- apply the formulae for probabilities $P (X = r) =^{n} C_{r} p^{r} (1 - p)^{n - r}$ associated with the binomial distribution with parameters $n$ and $p$ and understand the meaning of $^{n} C_{r}$ as the number of ways in which an outcome with $r$ successes can occur
- understand and apply the formulae for the mean, $E (X) = \overline{x} = n p$ , and the variance, $Var (X) = n p (1 - p)$ , of a binomial distribution with parameters $n$ and $p$
use appropriate graphs to explore the behaviour of the sample proportion on collected or supplied data AAM
- understand the concept of the sample proportion $\overset{p}{^}$ as a random variable whose value varies between samples (ACMMM174)ACMMM174
explore the behaviour of the sample proportion using simulated data AAM
- examine the approximate normality of the distribution of $\overset{p}{^}$ for large samples (ACMMM175)ACMMM175
understand and use the normal approximation to the distribution of the sample proportion and its limitations AAM

Year 11 assumed knowledge

Not examined directly in the HSC, but the Year 12 course builds on it. Collapsed so the examinable material above stays in view.

Functions32 items

ME-F1 Further Work with Functions

examine the relationship between the graph of $y = f (x)$ and the graph of $y = \frac{1}{f ( x )}$ and hence sketch the graphs (ACMSM099)ACMSM099
examine the relationship between the graph of $y = f (x)$ and the graphs of $y^{2} = f (x)$ and $y = f (x)$ and hence sketch the graphs
examine the relationship between the graph of $y = f (x)$ and the graphs of $y = ∣ f (x) ∣$ and $y = f (∣ x ∣)$ and hence sketch the graphs (ACMSM099)ACMSM099
examine the relationship between the graphs of $y = f (x)$ and $y = g (x)$ and the graphs of $y = f (x) + g (x)$ and $y = f (x) g (x)$ and hence sketch the graphs
apply knowledge of graphical relationships to solve problems in practical and abstract contexts AAM
solve quadratic inequalities using both algebraic and graphical techniques
solve inequalities involving rational expressions, including those with the unknown in the denominator
solve absolute value inequalities of the form $∣ a x + b ∣ \geq k$ , $∣ a x + b ∣ \leq k$ , $∣ a x + b ∣ < k$ and $∣ a x + b ∣ > k$
define the inverse relation of a function $y = f (x)$ to be the relation obtained by reversing all the ordered pairs of the function
examine and use the reflection property of the graph of a function and the graph of its inverse (ACMSM096)ACMSM096
- understand why the graph of the inverse relation is obtained by reflecting the graph of the function in the line $y = x$
- using the fact that this reflection exchanges horizontal and vertical lines, recognise that the horizontal line test can be used to determine whether the inverse relation of a function is again a function
write the rule or rules for the inverse relation by exchanging $x$ and $y$ in the function rules, including any restrictions, and solve for $y$ , if possible
when the inverse relation is a function, use the notation $f^{- 1} (x)$ and identify the relationships between the domains and ranges of $f (x)$ and $f^{- 1} (x)$
when the inverse relation is not a function, restrict the domain to obtain new functions that are one-to-one, and compare the effectiveness of different restrictions
solve problems based on the relationship between a function and its inverse function using algebraic or graphical techniques AAM
understand the concept of parametric representation and examine lines, parabolas and circles expressed in parametric form
- understand that linear and quadratic functions, and circles can be expressed in either parametric form or Cartesian form
- convert linear and quadratic functions, and circles from parametric form to Cartesian form and vice versa
- sketch linear and quadratic functions, and circles expressed in parametric form

ME-F2 Polynomials

define a general polynomial in one variable, $x$ , of degree $n$ with real coefficients to be the expression: $a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{2} x^{2} + a_{1} x + a_{0}$ , where $a_{n} \neq = 0$
- understand and use terminology relating to polynomials including degree, leading term, leading coefficient and constant term
use division of polynomials to express $P (x)$ in the form $P (x) = A (x) . Q (x) + R (x)$ where $deg R (x) < deg A (x)$ and $A (x)$ is a linear or quadratic divisor, $Q (x)$ the quotient and $R (x)$ the remainder
- review the process of division with remainders for integers
- describe the process of division using the terms: dividend, divisor, quotient, remainder
prove and apply the factor theorem and the remainder theorem for polynomials and hence solve simple polynomial equations (ACMSM089, ACMSM091)ACMSM089 ACMSM091
solve problems using the relationships between the roots and coefficients of quadratic, cubic and quartic equations AAM
- consider quadratic, cubic and quartic equations, and derive formulae as appropriate for the sums and products of roots in terms of the coefficients
determine the multiplicity of a root of a polynomial equation
- prove that if a polynomial equation of the form $P (x) = 0$ has a root of multiplicity $r > 1$ , then $P^{'} (x) = 0$ has a root of multiplicity $r - 1$
graph a variety of polynomials and investigate the link between the root of a polynomial equation and the zero on the graph of the related polynomial function
- examine the sign change of the function and shape of the graph either side of roots of varying multiplicity

Trigonometric Functions31 items

ME-T1 Inverse Trigonometric Functions

define and use the inverse trigonometric functions (ACMSM119)ACMSM119
- understand and use the notation $arcsin x$ and $sin^{- 1} x$ for the inverse function of $sin x$ when $- \frac{π}{2} \leq x \leq \frac{π}{2}$ (and similarly for $cos x$ and $tan x)$ and understand when each notation might be appropriate to avoid confusion with the reciprocal functions
- use the convention of restricting the domain of $sin x$ to $- \frac{π}{2} \leq x \leq \frac{π}{2}$ , so the inverse function exists. The inverse of this restricted sine function is defined by: $y = sin^{- 1} x$ , $- 1 \leq x \leq 1$ and $- \frac{π}{2} \leq y \leq \frac{π}{2}$
- use the convention of restricting the domain of $cos x$ to $0 \leq x \leq π$ , so the inverse function exists. The inverse of this restricted cosine function is defined by: $y = cos^{- 1} x,$ $- 1 \leq x \leq 1$ and $0 \leq y \leq π$
- use the convention of restricting the domain of $tan x$ to $- \frac{π}{2} < x < \frac{π}{2}$ , so the inverse function exists. The inverse of this restricted tangent function is defined by: $y = tan^{- 1} x$ , $x$ is a real number and $- \frac{π}{2} < y < \frac{π}{2}$
- classify inverse trigonometric functions as odd, even or neither odd nor even
sketch graphs of the inverse trigonometric functions
use the relationships $sin (sin^{- 1} x) = x$ and $sin^{- 1} (sin x) = x, cos (cos^{- 1} x) = x$ and
prove and use the properties: $sin^{- 1} (- x) = - sin^{- 1} x$ , $cos^{- 1} (- x) = π - cos^{- 1} x$ ,
solve problems involving inverse trigonometric functions in a variety of abstract and practical situations AAM

ME-T2 Further Trigonometric Identities

derive and use the sum and difference expansions for the trigonometric functions $sin (A \pm B)$ , $cos (A \pm B)$ and $tan (A \pm B)$ (ACMSM044)ACMSM044
- $sin (A \pm B) = sin A cos B \pm cos A sin B$
- $cos (A \pm B) = cos A cos B \mp sin A sin B$
- $tan (A \pm B) = \frac{t a n A \pm t a n B}{1 \mp t a n A t a n B}$
derive and use the double angle formulae for $sin 2 A$ , $cos 2 A$ and $tan 2 A$ (ACMSM044)ACMSM044
- $sin 2 A$ $= 2 sin A cos A$
- $cos 2 A$ $= cos^{2} A - sin^{2} A$ $= 2 cos^{2} A - 1$ $= 1 - 2 sin^{2} A$
- $tan 2 A$ $= \frac{2 t a n A}{1 - t a n ^{2} A}$
derive and use expressions for $sin A$ , $cos A$ and $tan A$ in terms of $t$ where $t = tan \frac{A}{2}$ (the $t$ -formulae)
- $sin A = \frac{2 t}{1 + t ^{2}}$
- $cos A = \frac{1 - t ^{2}}{1 + t ^{2}}$
- $tan A = \frac{2 t}{1 - t ^{2}}$
derive and use the formulae for trigonometric products as sums and differences for $cos A cos B$ , $sin A sin B$ , $sin A cos B$ and $cos A sin B$ (ACMSM047)ACMSM047
- $cos A cos B = \frac{1}{2} [cos (A - B) + cos (A + B)]$
- $sin A sin B = \frac{1}{2} [cos (A - B) - cos (A + B)]$
- $sin A cos B = \frac{1}{2} [sin (A + B) + sin (A - B)]$
- $cos A sin B = \frac{1}{2} [sin (A + B) - sin (A - B)]$

ME-T3 Trigonometric Equations

convert expressions of the form $a cos x + b sin x$ to $R cos (x \pm α)$ or $R sin (x \pm α)$ and apply these to solve equations of the form $a cos x + b sin x = c$ , sketch graphs and solve related problems (ACMSM048)ACMSM048
solve trigonometric equations requiring factorising and/or the application of compound angle, double angle formulae or the $t$ -formulae
prove and apply other trigonometric identities, for example $cos 3 x = 4 cos^{3} x - 3 cos x$ (ACMSM049)ACMSM049
solve trigonometric equations and interpret solutions in context using technology or otherwise

Calculus42 items

ME-C1 Rates of Change

describe the rate of change of a physical quantity with respect to time as a derivative
- investigate examples where the rate of change of some aspect of a given object with respect to time can be modelled using derivatives AAM
- use appropriate language to describe rates of change, for example ‘at rest’, ‘initially’, ‘change of direction’ and ‘increasing at an increasing rate’
find and interpret the derivative $\frac{d Q}{d t}$ , given a function in the form $Q = f (t)$ , for the amount of a physical quantity present at time $t$
describe the rate of change with respect to time of the displacement of a particle moving along the $x$ -axis as a derivative $\frac{d x}{d t}$ or $\overset{x}{˙}$
describe the rate of change with respect to time of the velocity of a particle moving along the $x$ -axis as a derivative $\frac{d ^{2} x}{d t ^{2}}$ or $\overset{x}{¨}$
construct, analyse and manipulate an exponential model of the form $N (t) = A e^{k t}$ to solve a practical growth or decay problem in various contexts (for example population growth, radioactive decay or depreciation) AAM
- establish the simple growth model, $\frac{d N}{d t} = k N$ , where $N$ is the size of the physical quantity, $N = N (t)$ at time $t$ and $k$ is the growth constant
- verify (by substitution) that the function $N (t) = A e^{k t}$ satisfies the relationship $\frac{d N}{d t} = k N$ , with $A$ being the initial value of $N$
- sketch the curve $N (t) = A e^{k t}$ for positive and negative values of $k$
- recognise that this model states that the rate of change of a quantity varies directly with the size of the quantity at any instant
establish the modified exponential model, $\frac{d N}{d t} = k (N - P)$ , for dealing with problems such as ‘Newton’s Law of Cooling’ or an ecosystem with a natural ‘carrying capacity’ AAM
- verify (by substitution) that a solution to the differential equation $\frac{d N}{d t} = k (N - P)$ is
- sketch the curve $N (t) = P + A e^{k t}$ for positive and negative values of $k$
- note that whenever $k < 0$ , the quantity $N$ tends to the limit $P$ as $t \to \infty$ , irrespective of the initial conditions
- recognise that this model states that the rate of change of a quantity varies directly with the difference in the size of the quantity and a fixed quantity at any instant
solve problems involving situations that can be modelled using the exponential model or the modified exponential model and sketch graphs appropriate to such problems AAM
solve problems involving related rates of change as instances of the chain rule (ACMSM129) AAMACMSM129
develop models of contexts where a rate of change of a function can be expressed as a rate of change of a composition of two functions, and to which the chain rule can be applied

ME-C2 Further Calculus Skills

find and evaluate indefinite and definite integrals using the method of integration by substitution, using a given substitution
- change an integrand into an appropriate form using algebra
prove and use the identities $sin^{2} n x = \frac{1}{2} (1 - cos 2 n x)$ and $cos^{2} n x = \frac{1}{2} (1 + cos 2 n x)$ to solve problems
solve problems involving $\int sin^{2} n x d x$ and $\int cos^{2} n x d x$
find derivatives of inverse functions by using the relationship $\frac{d y}{d x} = \frac{1}{\frac{d x}{d y}}$
solve problems involving the derivatives of inverse trigonometric functions
integrate expressions of the form $\frac{1}{a ^{2} - x ^{2}}$ or $\frac{a}{a ^{2} + x ^{2}}$ (ACMSM121)ACMSM121

ME-C3 Applications of Calculus

calculate area of regions between curves determined by functions (ACMSM124)ACMSM124
sketch, with and without the use of technology, the graph of a solid of revolution whose boundary is formed by rotating an arc of a function about the $x$ -axis or $y$ -axis AAM
calculate the volume of a solid of revolution formed by rotating a region in the plane about the
determine the volumes of solids of revolution that are formed by rotating the region between two curves about either the $x$ -axis or $y$ -axis in both real-life and abstract contexts AAM
recognise that an equation involving a derivative is called a differential equation
recognise that solutions to differential equations are functions and that these solutions may not be unique
sketch the graph of a particular solution given a direction field and initial conditions
- form a direction field (slope field) from simple first-order differential equations
- recognise the shape of a direction field from several alternatives given the form of a differential equation, and vice versa
- sketch several possible solution curves on a given direction field
solve simple first-order differential equations (ACMSM130)ACMSM130
- solve differential equations of the form $\frac{d y}{d x} = f (x)$
- solve differential equations of the form $\frac{d y}{d x} = g (y)$
- solve differential equations of the form $\frac{d y}{d x} = f (x) g (y)$ using separation of variables
recognise the features of a first-order linear differential equation and that exponential growth and decay models are first-order linear differential equations, with known solutions
model and solve differential equations including to the logistic equation that will arise in situations where rates are involved, for example in chemistry, biology and economics (ACMSM132) AAMACMSM132

Combinatorics19 items

ME-A1 Working with Combinatorics

list and count the number of ways an event can occur
use the fundamental counting principle (also known as the multiplication principle)
use factorial notation to describe and determine the number of ways $n$ different items can be arranged in a line or a circle
- solve problems involving cases where some items are not distinct (excluding arrangements in a circle)
solve simple problems and prove results using the pigeonhole principle (ACMSM006)ACMSM006
- understand that if there are $n$ pigeonholes and $n + 1$ pigeons to go into them, then at least one pigeonhole must hold 2 or more pigeons
- generalise to: If $n$ pigeons are sitting in $k$ pigeonholes, where $n > k$ , then there is at least one pigeonhole with at least $\frac{n}{k}$ pigeons in it
- prove the pigeonhole principle
understand and use permutations to solve problems (ACMSM001)ACMSM001
- understand and use the notation $^{n} P_{r}$ and the formula $^{n} P_{r} = \frac{n !}{( n - r ) !}$
solve problems involving permutations and restrictions with or without repeated objects (ACMSM004)ACMSM004
understand and use combinations to solve problems (ACMSM007)ACMSM007
- understand and use the notations $(n r)$ and $^{n} C_{r}$ and the formula $^{n} C_{r} = \frac{n !}{r ! ( n - r )!}$ (ACMMM045, ACMSM008)ACMMM045 ACMSM008
solve practical problems involving permutations and combinations, including those involving simple probability situations AAM
expand $(x + y)^{n}$ for small positive integers $n$ (ACMMM046)ACMMM046
- note the pattern formed by the coefficients of $x$ in the expansion of $(1 + x)^{n}$ and recognise links to Pascal’s triangle
- recognise the numbers $(n r)$ (also denoted $^{n} C_{r}$ ) as binomial coefficients (ACMMM047)ACMMM047
derive and use simple identities associated with Pascal’s triangle (ACMSM009)ACMSM009
- establish combinatorial proofs of the Pascal’s triangle relations $^{n} C_{0} = 1,^{n} C_{n} = 1$ ; $^{n} C_{r} =^{n - 1} C_{r - 1} +^{n - 1} C_{r}$ for $1 \leq r \leq n - 1$ ; and $^{n} C_{r} =^{n} C_{n - r}$

Outcomes

Code	A student
ME11-1	uses algebraic and graphical concepts in the modelling and solving of problems involving functions and their inverses ME11-1
ME11-2	manipulates algebraic expressions and graphical functions to solve problems ME11-2
ME11-6	uses appropriate technology to investigate, organise and interpret information to solve problems in a range of contexts ME11-6
ME11-7	communicates making comprehensive use of mathematical language, notation, diagrams and graphs ME11-7
ME11-3	applies concepts and techniques of inverse trigonometric functions and simplifying expressions involving compound angles in the solution of problems ME11-3
ME11-4	applies understanding of the concept of a derivative in the solution of problems, including rates of change, exponential growth and decay and related rates of change ME11-4
ME11-5	uses concepts of permutations and combinations to solve problems involving counting or ordering ME11-5
ME12-1	applies techniques involving proof or calculus to model and solve problems ME12-1
ME12-6	chooses and uses appropriate technology to solve problems in a range of contexts ME12-6
ME12-7	evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms ME12-7
ME12-2	applies concepts and techniques involving vectors and projectiles to solve problems ME12-2
ME12-3	applies advanced concepts and techniques in simplifying expressions involving compound angles and solving trigonometric equations ME12-3
ME12-4	uses calculus in the solution of applied problems, including differential equations and volumes of solids of revolution ME12-4
ME12-5	applies appropriate statistical processes to present, analyse and interpret data ME12-5