Specialist Mathematics (Version 8.4)

solve problems involving permutations  (ACMSM001)

use the multiplication principle (ACMSM002)

use factorial notation (ACMSM003)

solve problems involving permutations and restrictions with or without repeated objects (ACMSM004)

determine and use the formulas for finding the number of elements in the union of two and the union of three sets.  (ACMSM005)

solve problems and prove results using the pigeon-hole principle. (ACMSM006)

solve problems involving combinations (ACMSM007)

use the notation \(\begin{pmatrix}n\\r\end{pmatrix}\) or \({}^nC_r\) (ACMSM008)

derive and use simple identities associated with Pascal’s triangle. (ACMSM009)

examine examples of vectors including displacement and velocity (ACMSM010)

define and use the magnitude and direction of a vector (ACMSM011)

represent a scalar multiple of a vector  (ACMSM012)

use the triangle rule to find the sum and difference of two vectors. (ACMSM013)

use ordered pair notation and column vector notation to represent a vector (ACMSM014)

define and use unit vectors and the perpendicular unit vectors i and j (ACMSM015)

express a vector in component form using the unit vectors i and j (ACMSM016)

examine and use addition and subtraction of vectors in component form (ACMSM017)

define and use multiplication by a scalar of a vector in component form (ACMSM018)

define and use scalar (dot) product  (ACMSM019)

apply the scalar product to vectors expressed in component form (ACMSM020)

examine properties of parallel and perpendicular vectors and determine if two vectors are parallel or perpendicular (ACMSM021)

define and use projections of vectors  (ACMSM022)

solve problems involving displacement, force and velocity involving the above concepts.  (ACMSM023)

use implication, converse, equivalence, negation, contrapositive (ACMSM024)

use proof by contradiction (ACMSM025)

use the symbols for implication (\(\Rightarrow\)), equivalence (\(\Longleftrightarrow\)), and equality (\(=\)) (ACMSM026)

use the quantifiers ‘for all’ and ‘there exists’  (ACMSM027)

use examples and counter-examples.  (ACMSM028)

An angle in a semicircle is a right angle (ACMSM029)

The angle at the centre subtended by an arc of a circle is twice the angle at the circumference subtended by the same arc (ACMSM030)

Angles at the circumference of a circle subtended by the same arc are equal (ACMSM031)

The opposite angles of a cyclic quadrilateral are supplementary  (ACMSM032)

Chords of equal length subtend equal angles at the centre and conversely chords subtending equal angles at the centre of a circle have the same length (ACMSM033)

The alternate segment theorem (ACMSM034)

When two chords of a circle intersect, the product of the lengths of the intervals on one chord equals the product of the lengths of the intervals on the other chord (ACMSM035)

When a secant (meeting the circle at \(A\) and \(B\)) and a tangent (meeting the circle at \(T\)) are drawn to a circle from an external point \(M\), the square of the length of the tangent equals the product of the lengths to the circle on the secant. \((AM\times BM=TM^2)\).  (ACMSM036)

Suitable converses of some of the above results  (ACMSM037)

Solve problems finding unknown angles and lengths and prove further results using the results listed above. (ACMSM038)

The diagonals of a parallelogram meet at right angles if and only if it is a rhombus (ACMSM039)

Midpoints of the sides of a quadrilateral join to form a parallelogram (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)

find all solutions of \(\mathrm f\left(\mathrm a\left(\mathrm x-\mathrm b\right)\right)=\mathrm c\) where \(f\) is one of \(\sin\), \(\cos\) or \(\tan\) (ACMSM042)

graph functions with rules of the form \(\mathrm y=\mathrm f(\mathrm a\left(\mathrm x-\mathrm b\right))\) where \(f\) is one of \(\sin\), \(\cos\) or \(\tan\) (ACMSM043)

prove and apply the angle sum, difference and double angle identities. (ACMSM044)

define the reciprocal trigonometric functions, sketch their graphs, and graph simple transformations of them. (ACMSM045)

prove and apply the Pythagorean identities  (ACMSM046)

prove and apply the identities for products of sines and cosines expressed as sums and differences (ACMSM047)

convert sums \(\mathrm a\cos\mathrm x+\mathrm b\;\sin\mathrm x\) to \(\mathrm R\;\cos{(\mathrm x\pm\mathrm\alpha)}\) or \(\mathrm R\sin{(\mathrm x\pm\mathrm\alpha)}\) and apply these to sketch graphs, solve equations of the form \(\mathrm a\cos\mathrm x+\mathrm b\sin\mathrm x=\mathrm c\) and solve problems (ACMSM048)

prove and apply other trigonometric identities such as \(\cos3\mathrm x=4\;\mathrm c\mathrm o\mathrm s^{3\;}\mathrm x-3\cos\mathrm x\) (ACMSM049)

model periodic motion using sine and cosine functions and understand the relevance of the period and amplitude of these functions in the model. (ACMSM050)

understand the matrix definition and notation  (ACMSM051)

define and use addition and subtraction of matrices, scalar multiplication, matrix multiplication, multiplicative identity and inverse  (ACMSM052)

calculate the determinant and inverse of 2x2 matrices and solve matrix equations of the form AX=B , where A is a 2x2 matrix and X and B are column vectors.  (ACMSM053)

translations and their representation as column vectors (ACMSM054)

define and use basic linear transformations: dilations of the form \((\mathrm x,\mathrm y)\longrightarrow({\mathrm\lambda}_1\mathrm x,{\mathrm\lambda}_2\mathrm y)\) , rotations about the origin and reflection in a line which passes through the origin, and the representations of these transformations by 2x2 matrices (ACMSM055)

apply these transformations to points in the plane and geometric objects (ACMSM056)

define and use composition of linear transformations and the corresponding matrix products (ACMSM057)

define and use inverses of linear transformations and the relationship with the matrix inverse (ACMSM058)

examine the relationship between the determinant and the effect of a linear transformation on area (ACMSM059)

establish geometric results by matrix multiplications; for example, show that the combined effect of two reflections in lines through the origin is a rotation. (ACMSM060)

prove simple results involving numbers.  (ACMSM061)

express rational numbers as terminating or eventually recurring decimals and vice versa (ACMSM062)

prove irrationality by contradiction for numbers such as \(\sqrt[{}]2\) and \(\log_25\) (ACMSM063)

understand the nature of inductive proof including the ‘initial statement’ and inductive step  (ACMSM064)

prove results for sums, such as \(1+4+9\dots+n^2=\frac{n(n+1)(2n+1)}6\) for any positive integer n (ACMSM065)

prove divisibility results, such as \(3^{2n+4}-2^{2n}\)  is divisible by 5 for any positive integer n. 

define the imaginary number i as a root of the equation \(x^2=-1\) (ACMSM067)

use complex numbers in the form a+bi where a and b are the real and imaginary parts  (ACMSM068)

determine and use complex conjugates (ACMSM069)

perform complex-number arithmetic: addition, subtraction, multiplication and division.  (ACMSM070)

consider complex numbers as points in a plane with real and imaginary parts as Cartesian coordinates (ACMSM071)

examine addition of complex numbers as vector addition in the complex plane (ACMSM072)

understand and use location of complex conjugates in the complex plane. (ACMSM073)

use the general solution of real quadratic equations (ACMSM074)

determine complex conjugate solutions of real quadratic equations  (ACMSM075)

determine linear factors of real quadratic polynomials. (ACMSM076)

review real and imaginary parts \(Re\left(z\right)\;\) and \(Im(z)\) of a complex number \(z\) (ACMSM077)

review Cartesian form (ACMSM078)

review complex arithmetic using Cartesian forms. (ACMSM079)

use the modulus \(\left|z\right|\) of a complex number z and the argument \(Arg\;(z)\) of a non-zero complex number \(z\) and prove basic identities involving modulus and argument (ACMSM080)

convert between Cartesian and polar form  (ACMSM081)

define and use multiplication, division, and powers of complex numbers in polar form and the geometric interpretation of these (ACMSM082)

prove and use De Moivre’s theorem for integral powers. (ACMSM083)

examine and use addition of complex numbers as vector addition in the complex plane (ACMSM084)

examine and use multiplication as a linear transformation in the complex plane (ACMSM085)

identify subsets of the complex plane determined by relations such as \(\left|z-3i\right|\leq4\) \(\frac\pi4\leq Arg(z)\leq\frac{3\pi}4\), \(Re\left(z\right)>Im(z)\) and \(\left|z-1\right|=2\vert z-i\vert\) (ACMSM086)

determine and examine the \(n^{th}\) roots of unity and their location on the unit circle (ACMSM087)

determine and examine the \(n^{th}\) roots of complex numbers and their location in the complex plane. (ACMSM088)

prove and apply the factor theorem and the remainder theorem for polynomials (ACMSM089)

consider conjugate roots for polynomials with real coefficients (ACMSM090)

solve simple polynomial equations. (ACMSM091)

determine when the composition of two functions is defined  (ACMSM092)

find the composition of two functions (ACMSM093)

determine if a function is one-to-one (ACMSM094)

consider inverses of one-to-one function  (ACMSM095)

examine the reflection property of the graph of a function and the graph of its inverse.  (ACMSM096)

use and apply the notation \(\left|x\right|\) for the absolute value for the real number \(x\) and the graph of \(y=\left|x\right|\) (ACMSM098)

examine the relationship between the graph of \(y=f(x)\) and the graphs of \(y=\frac1{f(x)}\), \(y=\vert f\left(x\right)\vert\) and \(y=f(\left|x\right|)\) (ACMSM099)

sketch the graphs of simple rational functions where the numerator and denominator are polynomials of low degree. (ACMSM100)

review the concepts of vectors from Unit 1 and extend to three dimensions including introducing the unit vectors i, j and k. (ACMSM101)

prove geometric results in the plane and construct simple proofs in three-dimensions.  (ACMSM102)

introduce Cartesian coordinates for three-dimensional space, including plotting points and the equations of spheres  (ACMSM103)

use vector equations of curves in two or three dimensions involving a parameter, and determine a ‘corresponding’ Cartesian equation in the two-dimensional case (ACMSM104)

determine a vector equation of a straight line and straight-line segment, given the position of two points, or equivalent information, in both two and three dimensions (ACMSM105)

examine the position of two particles each described as a vector function of time, and determine if their paths cross or if the particles meet  (ACMSM106)

use the cross product to determine a vector normal to a given plane (ACMSM107)

determine vector and Cartesian equations of a plane and of regions in a plane. (ACMSM108)

recognise the general form of a system of linear equations in several variables, and use elementary techniques of elimination to solve a system of linear equations  (ACMSM109)

examine the three cases for solutions of systems of equations – a unique solution, no solution, and infinitely many solutions – and the geometric interpretation of a solution of a system of equations with three variables. (ACMSM110)

consider position of vectors as a function of time (ACMSM111)

derive the Cartesian equation of a path given as a vector equation in two dimensions including ellipses and hyperbolas (ACMSM112)

differentiate and integrate a vector function with respect to time (ACMSM113)

determine equations of motion of a particle travelling in a straight line with both constant and variable acceleration  (ACMSM114)

apply vector calculus to motion in a plane including projectile and circular motion. (ACMSM115)