Syllabus

Mathematics Extension 2

Stage 6 syllabus (2017), the document examined in the 2026 HSC. 134 Year 12 dot points. Official document, SHA-256 d0175d23f4de.

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Proof

MEX-P1 The Nature of Proof

use the formal language of proof, including the terms statement, implication, converse, negation and contrapositive (ACMSM024)ACMSM024
- use the symbols for implication $(\Rightarrow)$ , equivalence $(\Leftrightarrow)$ and equality $(=)$ , demonstrating a clear understanding of the difference between them (ACMSM026)ACMSM026
- use the phrases ‘for all’ $(\forall)$ , ‘if and only if’ $(iff)$ and ‘there exists’ $(\exists)$ (ACMSM027)ACMSM027
- understand that a statement is equivalent to its contrapositive but that the converse of a true statement may not be true
prove simple results involving numbers (ACMSM061)ACMSM061
use proof by contradiction including proving the irrationality for numbers such as $2$ and $lo g_{2} 5$ (ACMSM025, ACMSM063)ACMSM025 ACMSM063
use examples and counter-examples (ACMSM028)ACMSM028
prove results involving inequalities. For example:
- prove inequalities by using the definition of $a > b$ for real $a$ and $b$
- prove inequalities by using the property that squares of real numbers are non-negative
- prove and use the triangle inequality $∣ x ∣ + ∣ y ∣ \geq ∣ x + y ∣$ and interpret the inequality geometrically
- establish and use the relationship between the arithmetic mean and geometric mean for two non-negative numbers
prove further results involving inequalities by logical use of previously obtained inequalities

MEX-P2 Further Proof by Mathematical Induction

prove results using mathematical induction where the initial value of $n$ is greater than 1, and/or $n$ does not increase strictly by 1, for example prove that $n^{2} + 2 n$ is a multiple of 8 if $n$ is an even positive integer
understand and use sigma notation to prove results for sums, for example: $\sum_{n = 1}^{N} \frac{1}{( 2 n + 1 ) ( 2 n - 1 )} = \frac{N}{2 N + 1}$
understand and prove results using mathematical induction, including inequalities and results in algebra, calculus, probability and geometry. For example:
- prove inequality results, eg $2^{n} > n^{2}$ , for positive integers $n > 4$
- prove divisibility results, eg $3^{2 n + 4} - 2^{2 n}$ is divisible by 5 for any positive integer $n$
- prove results in calculus, eg prove that for any positive integer $n$ , $\frac{d}{d x} (x^{n}) = n x^{n - 1}$
- prove results related to probability, eg the binomial theorem: $(x + a)^{n} = \sum_{r = 0}^{n}^{n} C_{r} x^{n - r} a^{r}$
- prove geometric results, eg prove that the sum of the exterior angles of an $n$ -sided plane convex polygon is 360°
use mathematical induction to prove first-order recursive formulae

Vectors

MEX-V1 Further Work with Vectors

understand and use a variety of notations and representations for vectors in three dimensions
- define the standard unit vectors, and
- express and use a vector in three dimensions in a variety of forms, including component form, ordered triples and column vector notation
perform addition and subtraction of three-dimensional vectors and multiplication of three-dimensional vectors by a scalar algebraically and geometrically, and interpret these operations in geometric terms
define, calculate and use the magnitude of a vector in three dimensions
- establish that the magnitude of a vector in three dimensions can be found using:
- convert a non-zero vector into a unit vector by dividing by its length:
define and use the scalar (dot) product of two vectors in three dimensions AAM
- define and apply the scalar product to vectors expressed in component form, where , and
- extend the formula for three dimensions and use it to solve problems
prove geometric results in the plane and construct proofs in three dimensions (ACMSM102)ACMSM102
use Cartesian coordinates in two and three-dimensional space
recognise and find the equations of spheres
use vector equations of curves in two or three dimensions involving a parameter, and determine a corresponding Cartesian equation in the two-dimensional case, where possible (ACMSM104) AAMACMSM104
understand and use the vector equation of a straight line through points $A$ and $B$ where $R$ is a point on $A B$ , , , $λ$ is a parameter and
make connections in two dimensions between the equation and $y = m x + c$
determine a vector equation of a straight line or straight-line segment, given the position of two points or equivalent information, in two and three dimensions (ACMSM105)ACMSM105
determine when two lines in vector form are parallel
determine when intersecting lines are perpendicular in a plane or three dimensions
determine when a given point lies on a given line in vector form

Complex Numbers

MEX-N1 Introduction to Complex Numbers

use the complex number system
- develop an understanding of the classification of numbers and their associated properties, symbols and representations
- define the number, $i$ , as a root of the equation $x^{2} = - 1$ (ACMSM067)ACMSM067
- use the symbol $i$ to solve quadratic equations that do not have real roots
represent and use complex numbers in Cartesian form AAM
- use complex numbers in the form $z = a + ib$ , where $a$ and $b$ are real numbers and $a$ is the real part $Re (z)$ and $b$ is the imaginary part $Im (z)$ of the complex number (ACMSM068, ACMSM077)ACMSM068 ACMSM077
- identify the condition for $z_{1} = a + ib$ and $z_{2} = c + i d$ to be equal
- define and perform complex number addition, subtraction and multiplication (ACMSM070)ACMSM070
- define, find and use complex conjugates, and denote the complex conjugate of $z$ as $\overline{z}$
- divide one complex number by another complex number and give the result in the form $a + ib$
- find the reciprocal and two square roots of complex numbers in the form $z = a + ib$
represent and use complex numbers in the complex plane (ACMSM071)ACMSM071
- use the fact that there exists a one-to-one correspondence between the complex number $z = a + ib$ and the ordered pair $(a, b)$
- plot the point corresponding to $z = a + ib$
represent and use complex numbers in polar or modulus-argument form, $z = r (cos θ + i sin θ)$ , where $r$ is the modulus of $z$ and $θ$ is the argument of $z$ AAM
- define and calculate the modulus of a complex number $z = a + ib$ as $∣ z ∣ = a^{2} + b^{2}$
- define and calculate the argument of a non-zero complex number $z = a + ib$ as $arg (z) = θ$ , where $tan θ = \frac{b}{a}$
- define, calculate and use the principal argument $Arg (z)$ of a non-zero complex number $z$ as the unique value of the argument in the interval $(- π, π]$
prove and use the basic identities involving modulus and argument (ACMSM080) AAMACMSM080
- $∣ z_{1} z_{2} ∣ = ∣ z_{1} ∣∣ z_{2} ∣$ and $arg (z_{1} z_{2}) = arg z_{1} + arg z_{2}$
- $\frac{z _{1}}{z _{2}} = \frac{∣ z _{1} ∣}{∣ z _{2} ∣}$ and $arg (\frac{z _{1}}{z _{2}}) = arg z_{1} - arg z_{2}$ , $z_{2} \neq = 0$
- $∣ z^{n} ∣ = ∣ z ∣^{n}$ and $arg (z^{n}) = n arg z$
- $\frac{1}{z ^{n}} = \frac{1}{∣ z ∣ ^{n}}$ and $arg (\frac{1}{z ^{n}}) = - n arg z$ , $z \neq = 0$
- $\overline{z_{1}} + \overline{z_{2}} = \overline{z_{1} + z_{2}}$
- $\overline{z_{1}} \overline{z_{2}} = \overline{z_{1} z_{2}}$
- $z \overline{z} = ∣ z ∣^{2}$
- $z + \overline{z} = 2 Re (z)$
- $z - \overline{z} = 2 i Im (z)$
understand Euler’s formula, $e^{i x} = cos x + i sin x,$ for real $x$
represent and use complex numbers in exponential form, $z = r e^{i θ}$ , where $r$ is the modulus of $z$ and $θ$ is the argument of $z$ AAM
use Euler’s formula to link polar form and exponential form
convert between Cartesian, polar and exponential forms of complex numbers
find powers of complex numbers using exponential form
use multiplication, division and powers of complex numbers in polar form and interpret these geometrically (ACMSM082) AAMACMSM082
solve problems involving complex numbers in a variety of forms AAM

MEX-N2 Using Complex Numbers

use De Moivre’s theorem with complex numbers in both polar and exponential form AAM
- prove De Moivre’s theorem for integral powers using proof by induction (ACMSM083)ACMSM083
- use De Moivre’s theorem to derive trigonometric identities such as $sin 3 θ = 3 cos^{2} θ sin θ - sin^{3} θ$
determine the solutions of real quadratic equations
define and determine complex conjugate solutions of real quadratic equations (ACMSM075) AAMACMSM075
determine conjugate roots for polynomials with real coefficients (ACMSM090) AAMACMSM090
solve problems involving real polynomials with conjugate roots
solve quadratic equations of the form $a x^{2} + b x + c = 0$ , where $a, b, c$ are complex numbers AAM
examine and use addition and subtraction of complex numbers as vectors in the complex plane (ACMSM084) AAMACMSM084
- given the points representing $z_{1}$ and $z_{2}$ , find the position of the points representing $z_{1} + z_{2}$ and $z_{1} - z_{2}$
- describe the vector representing $z_{1} + z_{2}$ or $z_{1} - z_{2}$ as corresponding to the relevant diagonal of a parallelogram with vectors representing $z_{1}$ and $z_{2}$ as adjacent sides
examine and use the geometric interpretation of multiplying complex numbers, including rotation and dilation in the complex plane
recognise and use the geometrical relationship between the point representing a complex number $z = a + ib$ , and the points representing $\overline{z}$ , $cz$ (where $c$ is real) and $i z$
determine and examine the $n$ th roots of unity and their location on the unit circle (ACMSM087)ACMSM087
determine and examine the $n$ th roots of complex numbers and their location in the complex plane (ACMSM088)ACMSM088
solve problems using $n$ th roots of complex numbers AAM
identify subsets of the complex plane determined by relations, for example $∣ z - 3 i ∣ \leq 4$ , $\frac{π}{4} \leq Arg (z) \leq \frac{3 π}{4}$ , $Re (z) > Im (z)$ and $∣ z - 1∣ = 2∣ z - i ∣$ (ACMSM086)ACMSM086

Calculus

MEX-C1 Further Integration

find and evaluate indefinite and definite integrals using the method of integration by substitution, where the substitution may or may not be given
integrate rational functions involving a quadratic denominator by completing the square or otherwise
decompose rational functions whose denominators have simple linear or quadratic factors, or a combination of both, into partial fractions
use partial fractions to integrate functions
evaluate integrals using the method of integration by parts (ACMSM123)ACMSM123
- develop the method for integration by parts, expressed as $\int u v^{'} d x = uv - \int v u^{'} d x$ or
derive and use recurrence relationships
apply these techniques of integration to practical and theoretical situations AAM

Mechanics

MEX-M1 Applications of Calculus to Mechanics

derive equations for displacement, velocity and acceleration in terms of time, given that a motion is simple harmonic and describe the motion modelled by these equations AAM
- establish that simple harmonic motion is modelled by equations of the form:
- establish that when a particle moves in simple harmonic motion about $c$ , the central point of motion, then $\overset{x}{¨} = - n^{2} (x - c)$
prove that motion is simple harmonic when given an equation of motion for acceleration, velocity or displacement and describe the resulting motion
sketch graphs of $x, \overset{x}{˙}$ and $\overset{x}{¨}$ as functions of $t$ and interpret and describe features of the motion
prove that motion is simple harmonic when given graphs of motion for acceleration, velocity or displacement and determine equations for the motion and describe the resulting motion
derive $v^{2} = g (x)$ and the equations for velocity and displacement in terms of time when given $\overset{x}{¨} = f (x)$ and initial conditions, and describe the resulting motion
use relevant formulae and graphs to solve problems involving simple harmonic motion AAM
examine force, acceleration, action and reaction under constant and non-constant force (ACMSM133, ACMSM134) AAMACMSM133 ACMSM134
examine motion of a body under concurrent forces (ACMSM135) AAMACMSM135
consider and solve problems involving motion in a straight line with both constant and non-constant acceleration and derive and use the expressions $\frac{d v}{d t}$ , $v \frac{d v}{d x}$ and $\frac{d}{d x} (\frac{1}{2} v^{2})$ for acceleration (ACMSM136) AAMACMSM136
use Newton’s laws to obtain equations of motion in situations involving motion other than projectile motion or simple harmonic motion AAM
- use $F = m \overset{x}{¨}$ where $F$ is the force acting on a mass, $m$ , with acceleration $\overset{x}{¨}$
describe mathematically the motion of particles in situations other than projectile motion and simple harmonic motion AAM
- interpret graphs of displacement-time and velocity-time to describe the motion of a particle, including the possible direction of a force which acts on the particle
derive and use the equations of motion of a particle travelling in a straight line with both constant and variable acceleration (ACMSM114) AAMACMSM114
solve problems involving resisted motion of a particle moving along a horizontal line AAM
- derive, from Newton’s laws of motion, the equation of motion of a particle moving in a single direction under a resistance proportional to a power of the speed
- derive an expression for velocity as a function of time
- derive an expression for velocity as a function of displacement
- derive an expression for displacement as a function of time
- solve problems involving resisted motion along a horizontal line
solve problems involving the motion of a particle moving vertically (upwards or downwards) in a resisting medium and under the influence of gravity AAM
- derive, from Newton’s laws of motion, the equation of motion of a particle moving vertically in a medium, with a resistance $R$ proportional to the first or second power of its speed
- derive an expression for velocity as a function of time and for velocity as a function of displacement (or vice versa)
- derive an expression for displacement as a function of time
- determine the terminal velocity of a falling particle from its equation of motion
- solve problems by using the expressions derived for acceleration, velocity and displacement including obtaining the maximum height reached by a particle, and the time taken to reach this maximum height and obtaining the time taken for a particle to reach ground level when falling
solve problems involving projectiles in a variety of contexts AAM
- use parametric equations of a projectile to determine a corresponding Cartesian equation for the projectile
- use the Cartesian equation of the trajectory of a projectile, including problems in which the initial speed and/or angle of projection may be unknown
solve problems involving projectile motion in a resisting medium and under the influence of gravity which include consideration of the complete motion of a particle projected vertically upwards or at an angle to the horizontal AAM

Outcomes

Code	A student
MEX12-1	understands and uses different representations of numbers and functions to model, prove results and find solutions to problems in a variety of contexts MEX12-1
MEX12-2	chooses appropriate strategies to construct arguments and proofs in both practical and abstract settings MEX12-2
MEX12-7	applies various mathematical techniques and concepts to model and solve structured, unstructured and multi-step problems MEX12-7
MEX12-8	communicates and justifies abstract ideas and relationships using appropriate language, notation and logical argument MEX12-8
MEX12-3	uses vectors to model and solve problems in two and three dimensions MEX12-3
MEX12-4	uses the relationship between algebraic and geometric representations of complex numbers and complex number techniques to model and solve problems MEX12-4
MEX12-5	applies techniques of integration to structured and unstructured problems MEX12-5
MEX12-6	uses mechanics to model and solve practical problems MEX12-6