Mathematics Specialised is designed for learners with a strong interest in mathematics, including those intending to study mathematics, statistics, all sciences and associated fields, economics, or engineering at university
This course provides opportunities, beyond those presented in Mathematics Methods Level 4, to develop rigorous mathematical arguments and proofs, and to use mathematical models more extensively.
Mathematics is the study of order, relation and pattern. From its origins in counting and measuring it has evolved in highly sophisticated and elegant ways to become the language now used to describe much of the modern world. Mathematics is also concerned with collecting, analysing, modelling and interpreting data in order to investigate and understand real-world phenomena and solve problems in context. Mathematics provides a framework for thinking and a means of communication that is powerful, logical, concise and precise. It impacts upon the daily life of people everywhere and helps them to understand the world in which they live and work.
Mathematics Specialised provides opportunities, beyond those presented in Mathematics Methods, to develop rigorous mathematical arguments and proofs, and to use mathematical and statistical models more extensively. Topics are developed systematically and lay the foundations for future studies in quantitative subjects in a coherent and structured fashion. Learners will be able to appreciate the true nature of mathematics, its beauty and its functionality.
This course contains topics in functions, sequences and series, calculus, matrices and complex numbers that build on and deepen the ideas presented in Mathematics Methods and demonstrate their application in many areas. Complex numbers, mathematics in three dimensions and matrices are introduced.
Mathematics Specialised is designed for learners with a strong interest in mathematics, including those intending to study mathematics, statistics, all sciences and associated fields, economics or engineering at the tertiary level.
Mathematics Specialised aims to develop each learner’s:
On successful completion of this course, learners will be able to:
Learners must have access to calculator algebraic system (CAS) calculators and become proficient in their use. These calculators can be used in all aspects of this course in the development of concepts and as a tool for solving problems. Refer to 'What can I take to my exam?' and 'Use of calculator policy guidelines (PDF)' for the current TASC Calculator Policy that applies to Level 3 and 4 courses.
The use of computer software is also recommended as an aid to learning and mathematical development. A range of packages such as, but not limited to: Wolfram Mathematica, Microsoft Excel, Autograph, Efofex Stat, and Graph and Draw are appropriate for this purpose.
This course has complexity level of 4.
In general, courses at this level provide theoretical and practical knowledge and skills for specialised and/or skilled work and/or further learning, requiring:
This Level 4 course has a size value of 15.
This course is made up of four (4) compulsory areas of study:
Each area of study relates to a specific Assessment Criteria (4–8). Assessment Criteria 1–3 apply to all four areas of study.
It is recommended that each area is addressed in the order presented in this document.
It is a requirement of this course that learners study and analyse real-world applications involving the concepts studied in the four major areas of study. This provides learners with mathematical experiences that are much richer than a collection of skills. Learners thereby have the opportunity to observe and make connections between related aspects of the course and the real world and to develop further some important abstract ideas.
SEQUENCES AND SERIES (Criterion 4)
Learners will study a range of sequences and series by learning about their properties, meet some of their important uses and begin to understand the ideas of convergence and divergence and develop some methods of proof.
This area of study will include:
Area Work Requirements
The minimum work requirements for this area of study, assessing Criterion 4, will include learners completing one (1) major and two (2) minor assessments where they will find solutions to mathematical questions/problems that may:
Also see Work Requirements – Applications below.
COMPLEX NUMBERS (Criterion 8)
Learners will be introduced to a different class of numbers and understand that such numbers can be represented in several ways and that their use allows factorisation to be carried out more fully than was previously possible.
This area of study will include:
Area Work Requirements
The minimum work requirements for this area of study, assessing Criterion 8, will include learners completing one (1) major and two (2) minor assessments where they will find solutions to mathematical questions/problems that may:
Also see Work Requirements – Applications below.
MATRICES AND LINEAR ALGEBRA (Criterion 5)
Learners will be introduced to new mathematical structures and appreciate some of the ways in which these structures can be put to use. Trigonometric identities will be used to develop ideas in matrices and linear transformations.
This area of study will include:
Area Work Requirements
The minimum work requirements for this area of study, assessing Criterion 5, will include learners completing one (1) major and two (2) minor assessments where they will find solutions to mathematical questions/problems that may:
Also see Work Requirements – Applications below.
CALCULUS (Criteria 6 and 7)
Learners will extend their existing knowledge and understanding of this vital branch of mathematics by developing a greater capacity for integrating functions and by introducing simple differential equations and their uses.
This section of the course develops and extends the ideas introduced in the Mathematics Methods course.
This area of study will include:
PART 1: (Criterion 6)
PART 2: (Criterion 7)
Area Work Requirements
The minimum work requirements for these areas of study, assessing Criterion 6 and Criterion 7, will include learners completing two (2) major and four (4) minor assessments where they will find solutions to mathematical questions/problems that may:
Also see Work Requirements – Applications below.
APPLICATIONS
In their study of this course, learners will investigate a minimum of two applications. It is recommended that the selected applications relate to different course areas. Such extended problems, investigations and applications of technology provide opportunities to reinforce many of the skills and concepts studied in this course.
Examples include the following:
SUMMARY
In total, the minimum work requirements for this course are:
Criterion-based assessment is a form of outcomes assessment that identifies the extent of learner achievement at an appropriate end-point of study. Although assessment – as part of the learning program – is continuous, much of it is formative, and is done to help learners identify what they need to do to attain the maximum benefit from their study of the course. Therefore, assessment for summative reporting to TASC will focus on what both teacher and learner understand to reflect end-point achievement.
The standard of achievement each learner attains on each criterion is recorded as a rating ‘A’, ‘B’, or ‘C’, according to the outcomes specified in the standards section of the course.
A ‘t’ notation must be used where a learner demonstrates any achievement against a criterion less than the standard specified for the ‘C’ rating.
A ‘z’ notation is to be used where a learner provides no evidence of achievement at all.
Providers offering this course must participate in quality assurance processes specified by TASC to ensure provider validity and comparability of standards across all awards. For further information, see quality assurance and assessment processes.
Internal assessment of all criteria will be made by the provider. Providers will report the learner’s rating for each criterion to TASC.
TASC will supervise the external assessment of designated criteria which will be indicated by an asterisk (*). The ratings obtained from the external assessments will be used in addition to internal ratings from the provider to determine the final award.
The following processes will be facilitated by TASC to ensure there is:
Process – TASC gives course providers feedback about any systematic differences in the relationship of their internal and external assessments and, where appropriate, seeks further evidence through audit and requires corrective action in the future.
The external assessment for this course will comprise:
For further information, see Exams and Assessment for the current external assessment specifications and guidelines for this course.
The assessment for Mathematics Specialised Level 4 will be based on the degree to which the learner can:
Note: * denotes criteria that are both internally and externally assessed
The learner:
Rating A | Rating B | Rating C |
---|---|---|
presents work that conveys a logical line of reasoning that has been followed between question and answer | presents work that conveys a line of reasoning that has been followed between question and answer | presents work that shows some of the mathematical processes that have been followed between question and answer |
consistently uses mathematical conventions and symbols correctly | generally uses mathematical conventions and symbols correctly | uses mathematical conventions and symbols (There may be some errors or omissions in doing so.) |
presents work with the final answer clearly identified, and articulated in terms of the question as required | presents work with the final answer clearly identified | presents work with the final answer apparent |
uses correct units and includes them in an answer for routine and non-routine problems | uses correct units and includes them in an answer for routine problems | uses correct units and includes them in an answer for routine problems |
presents detailed tables, graphs and diagrams that convey accurate meaning and precise information | presents detailed tables, graphs and diagrams that convey clear meaning | presents tables, graphs and diagrams as directed |
ensures a degree of accuracy appropriate to task is maintained – including the use of exact values – communicated throughout a problem | determines and works to a degree of accuracy appropriate to task, including the use of exact values where required | works to a degree of accuracy appropriate to tasks, as directed |
The learner:
Rating A | Rating B | Rating C |
---|---|---|
selects and applies an appropriate strategy (where several may exist) to solve routine and non-routine problems in a variety of contexts | selects and applies an appropriate strategy to solve routine and simple non-routine problems | identifies an appropriate strategy to solve routine problems |
interprets solutions to routine and non-routine problems | interprets solutions to routine and simple non-routine problems | describes solutions to routine problems |
explains the reasonableness of results and solutions to routine and non-routine problems | describes the reasonableness of results and solutions to routine problems | describes the appropriateness of the results of calculations |
identifies and describes limitations of presented models, and as applicable, explores the viability of possible alternative models | identifies and describes limitations of presented models | identifies limitations of simple models |
uses available technological aids in familiar and unfamiliar contexts | chooses to use available technological aids when appropriate to solve routine problems | uses available technological aids to solve routine problems |
explores the use of technology in familiar and unfamiliar contexts | explores the use of technology in familiar contexts | |
constructs and solves problems derived from routine and non-routine scenarios | constructs and solves problems derived from routine scenarios |
The learner:
Rating A | Rating B | Rating C |
---|---|---|
uses planning tools and strategies to achieve and manage activities within proposed times | uses planning tools to achieve objectives within proposed time | uses planning tools, with prompting, to achieve objectives within proposed times |
divides tasks into appropriate sub-tasks in multiple step operations | divides a task into appropriate sub-tasks | divides a task into sub-tasks |
selects strategies and formulae to successfully complete routine and non-routine problems | selects from a range of strategies and formulae to successfully complete routine and non-routine problems | selects from a range of strategies and formulae to complete routine problems |
plans timelines and monitors and analyses progress towards meeting goals, making adjustments as required | plans timelines and monitors progress towards meeting goals | monitors progress towards meeting goals |
addresses all of the required elements of a task with a high degree of accuracy | addresses the elements of required tasks | addresses most elements of required tasks |
plans future actions, effectively adjusting goals and plans where necessary | plans future actions, adjusting goals and plans where necessary | uses prescribed strategies to adjust goals and plans where necessary |
This criterion is both internally and externally assessed.
Rating 'A'
In addition to the standards for a C and a B rating, the learner:
Rating 'B'
In addition to the standards for a C rating, the learner:
Rating 'C'
The learner:
Rating A | Rating B | Rating C |
---|---|---|
determines whether a given number is a term of a given sequence | determines the `n^(th)` term of a sequence | determines the next few terms of a sequence, e.g. `{u_n} = 1/(2 × 3), -5/(3 × 4), 9/(4 × 5), ...` |
determines the sum to `n` terms of arithmetic and geometric progressions in questions of a greater complexity | determines the sum to n terms of recursively defined arithmetic and geometric progressions, e.g. find the sum to `n` terms of `u_1 = xy^2, u_(n+1) = x/y u_n` | determines the sum to n terms of arithmetic and geometric progressions, e.g. find the sum to `n` terms of `{u_n}= 3k, 7k, 11k, ...` |
interprets the conditions required for the infinite sum of geometric series to exist, for series of a greater complexity | interprets the conditions required for the infinite sum of a geometric series to exist | determines the infinite sum of simple geometric series and states the condition required for it to exist |
states the formal definitions of a sequence and requirements for a sequence to converge and diverge to `+-oo` | ||
states and prove convergence or divergence of a sequence where a variety of algebraic techniques are required | states and proves convergence or divergence of a sequence and interprets results given `K` or `epsilon` | states and proves convergence or divergence of a sequence in simple cases |
determines sums of series to a given number or an infinite number of terms using standard results in cases where there may be a number of sub-series | determines sums of series to `n` terms using standard results in cases with a greater complexity | determines sums of series to `n` terms using standard results in simple cases |
uses mathematical induction to show that a series has a given sum to `n` terms, where the `n^(th)` term is given, and interpret the results | uses mathematical induction to show that a series has a given sum to `n` terms, where the `n^(th)` term is given | |
uses a method of differences to sum routine and non-routine series | uses a method of differences to sum routine series | |
determines MacLaurin series for functions of the prescribed type, and integrates, differentiates or substitutes to find other series or approximations | determines MacLaurin series for functions of the prescribed type |
This criterion is both internally and externally assessed.
Rating ‘A’
In addition to the standards for a C and a B rating, the learner:
Rating ‘B’
In addition to the standards for a C rating, the learner:
Rating A | Rating B | Rating C |
---|---|---|
use properties of matrices to determine or demonstrate results, e.g. show `(AB)^(-1) = B^(-1) A^(-1)`, where `A` and `B` are non-singular matrices | recognises and applies properties of matrices in less routine cases | recognises and applies properties of matrices in routine cases, e.g. find `a` and `b` given that `((1,b),(a,1))((2a,a),(a,1)) = ((10,5),(10,5))` |
solves equations involving matrices where the solution is expressed in a general form | solves equations involving matrices that may require the use of an inverse matrix | solves routine equations involving matrices |
uses Gauss-Jordan reduction to solve systems of two equations in two unknowns, or two or three equations in three unknowns where one or more element(s) in the matrix is designated symbolically | uses Gauss-Jordan reduction to solve systems of two equations in two unknowns, or two or three equations in three unknowns where no solutions or an infinite number of solutions may exist | uses Gauss-Jordan reduction to solve systems of two equations in two unknowns, or three equations in three unknowns where, if a solution exists, it is unique |
determines the equation of a plane passing through three points using Gauss-Jordan reduction | interprets solutions to systems of equations as points or lines (in parametric or symmetric form) in 3D space | interprets equations as lines or planes in 3 dimensional space |
solves non-routine problems involving lines and planes in 3 dimensional space | demonstrate whether two or three planes are parallel using Gauss-Jordan reduction | demonstrate whether a line or a point is embedded in a plane |
applies techniques in linear transformations to solve routine and non-routine problems in a variety of contexts, e.g. show that the image of a circle when rotated by `theta` radians is also a circle | uses techniques in composite linear transformations, e.g. find the image of the circle `x^2 + y^2 = 1` when it is dilated by a factor of `4` parallel to the `y`-axis and rotated clockwise through `pi/2` | uses techniques in linear transformations, e.g. find the image of the circle `x^2 + y^2 = 1` when it is dilated by a factor of `4` parallel to the `y`-axis |
applies composite transformations to develop addition theorems and associated results | ||
uses area-related information to make inferences about determinants or matrix elements in order to solve problems | applies determinants to areas of images, e.g. if the linear transformation `T:(x,y) -> (x + y, x - y)` is applied to the unit square, find the resulting area |
This criterion is both internally and externally assessed.
Rating ‘A’
In addition to the standards for a C and a B rating, the learner:
Rating ‘B’
In addition to the standards for a C rating, the learner:
Rating ‘C’
The learner:
Rating A | Rating B | Rating C |
---|---|---|
determines the derivatives of `a^x` and `log_ax`, and of inverse trigonometric functions, including products and compositions of these functions | determines the derivatives of `a^x` and `log_ax`, and of inverse trigonometric functions | |
finds tangents and normals of routine and non-routine explicit or implicit functions | finds tangents and normals of routine explicit or implicit composite functions | applies techniques in derivatives of explicit and implicit functions to find tangents and normals |
finds and classifies stationary points and non-stationary points of inflection of functions of greater complexity and interprets the answer | finds and classifies stationary points and stationary and non-stationary points of inflection of functions including basic trigonometric functions | finds and classifies stationary points and points of inflection of functions, excluding trigonometric functions |
determines areas and volumes in routine and non-routine cases, including compound (or composite) areas and interprets the answer | determines areas and volumes in routine cases including finding the area between a curve and the `y`-axis | applies properties of definite integrals to calculate areas and volumes |
uses the concavity of a function in sketching its curve (where the function has features of a greater complexity) | determines the domain for particular concavity of functions | uses points of inflection to determine the concavity of functions in given domains |
This criterion is both internally and externally assessed.
Rating ‘A’
In addition to the standards for a C and a B rating, the learner:
Rating ‘B’
In addition to the standards for a C rating, the learner:
Rating ‘C’
The learner:
Rating A | Rating B | Rating C |
---|---|---|
resolves expressions into partial fractions and integrates rational functions with non-linear factors | resolves expressions into partial fractions and integrates rational functions with repeating linear factors | resolves expressions into partial fractions and integrates proper rational functions with non-repeating linear factors |
manipulates trigonometric identities to determine integrals in cases of greater complexity, e.g. find `int sin^3 3x dx` | uses trigonometric identities to determine integrals in cases of more complexity, e.g. find `int x^2/(x^2 + 1) dx` | uses trigonometric identities to determine integrals, e.g. find `int 9/(1 + 4x^2) dx` |
determines integrals involving manipulations and substitutions of greater complexity , e.g. find `int x^2/((2x - 3)^3) dx` | determines integrals involving substitutions and linear substitutions of more complexity, e.g. find `int (3x + 1)/(sqrt(4x - 1)) dx` | determines integrals involving simple substitutions and linear substitutions, e.g. find `int x(4x - 1)^3 dx` |
determines integrals involving integration by parts more than once | determines integrals involving integration by parts once | |
solves homogeneous equations | solves differential equations involving separable variables | solves differential equations involving a function of `x` or `y` |
establishes and applies differential equations to solve practical problems | applies given differential equations to solve practical problems |
This criterion is both internally and externally assessed.
Rating ‘A’
In addition to the standards for a C and a B rating, the learner:
Rating ‘B’
In addition to the standards for a C rating, the learner:
Rating ‘C’
The learner:
Rating A | Rating B | Rating C |
---|---|---|
infers equations from real and imaginary parts of a complex equation and solves associated problems | uses the notation associated with complex numbers and performs basic operations | |
converts between rectangular and polar forms when solving problems of a greater complexity, e.g. simplify `(isqrt3 + 1)^2/((-sqrt3 + i)^4(- 1 - i)^4)` | converts between rectangular and polar forms when solving basic problems, e.g. demonstrate that `1 - i = sqrt2 cis(-pi/4) = sqrt2 e^((i, -pi/4)` | converts between rectangular and polar forms |
locates points, lines, curves or regions on the Argand plane defined by one or more conditions including points at intersections of boundaries of regions | locates points, lines, curves or regions on the Argand plane defined by more than one simple condition | locates points, lines, curves or regions on the Argand plane defined by one simple condition |
manipulates and solves simple polynomial equations with complex roots and factorises associated polynomials | solves simple polynomial equations with complex roots | solves simple linear, quadratic or simultaneous equations with complex roots; finds `n^(th)` roots of complex numbers, where `n` is a positive integer |
solves polynomial equations, where the terms are in geometric progression, in non-routine situations | solves polynomial equations, where the terms are in geometric progression, in routine situations | |
applies conjugate root theorem to solve degree 4 polynomials with real coefficients | applies conjugate root theorem to solve degree 3 polynomials with real coefficients |
Mathematics Specialised Level 4 (with the award of):
EXCEPTIONAL ACHIEVEMENT (EA)
HIGH ACHIEVEMENT (HA)
COMMENDABLE ACHIEVEMENT (CA)
SATISFACTORY ACHIEVEMENT (SA)
PRELIMINARY ACHIEVEMENT (PA)
The final award will be determined by the Office of Tasmanian Assessment, Standards and Certification from 13 ratings (8 from the internal assessment, 5 from external assessment).
The minimum requirements for an award in Mathematics Specialised Level 4 are as follows:
EXCEPTIONAL ACHIEVEMENT (EA)
11 ‘A’ ratings, 2 ‘B’ ratings (4 ‘A’ ratings and 1 ‘B’ rating in the external assessment)
HIGH ACHIEVEMENT (HA)
5 ‘A’ ratings, 5 ‘B’ ratings, 3 ‘C’ ratings (2 ‘A’ ratings, 2 ‘B’ ratings and 1 ‘C’ rating in the external assessment)
COMMENDABLE ACHIEVEMENT (CA)
7 ‘B’ ratings, 5 ‘C’ ratings (2 ‘B’ ratings and 2 ‘C’ ratings in the external assessment)
SATISFACTORY ACHIEVEMENT (SA)
11 ‘C’ ratings (3 ‘C’ ratings in the external assessment)
PRELIMINARY ACHIEVEMENT (PA)
6 ‘C’ ratings
A learner who otherwise achieves the ratings for a CA (Commendable Achievement) or SA (Satisfactory Achievement) award but who fails to show any evidence of achievement in one or more criteria (‘z’ notation) will be issued with a PA (Preliminary Achievement) award.
The Department of Education’s Curriculum Services will develop and regularly revise the curriculum. This evaluation will be informed by the experience of the course’s implementation, delivery and assessment.
In addition, stakeholders may request Curriculum Services to review a particular aspect of an accredited course.
Requests for amendments to an accredited course will be forward by Curriculum Services to the Office of TASC for formal consideration.
Such requests for amendment will be considered in terms of the likely improvements to the outcomes for learners, possible consequences for delivery and assessment of the course, and alignment with Australian Curriculum materials.
A course is formally analysed prior to the expiry of its accreditation as part of the process to develop specifications to guide the development of any replacement course.
The statements in this section, taken from documents endorsed by Education Ministers as the agreed and common base for course development, are to be used to define expectations for the meaning (nature, scope and level of demand) of relevant aspects of the sections in this document setting out course requirements, learning outcomes, the course content and standards in the assessment.
For the content areas of Mathematics Methods, the proficiency strands – Understanding; Fluency; Problem Solving; and Reasoning – build on learners’ learning in F-10 Australian Curriculum: Mathematics. Each of these proficiencies is essential, and all are mutually reinforcing. They are still very much applicable and should be inherent in the five areas of study.
Specialist Mathematics
Unit 2 – Topic 1: Trigonometry
The basic trigonometric functions:
Compound angles:
The reciprocal trigonometric functions, secant, cosecant and cotangent:
Trigonometric identities:
Applications of trigonometric functions to model periodic phenomena:
Unit 2 – Topic 2: Matrices
Matrix arithmetic:
Transformations in the plane:
Unit 2 – Topic 3: Real and Complex Numbers
Rational and irrational numbers:
An introduction to proof by mathematical induction:
Complex numbers:
The complex plane:
Roots of equations:
Unit 3 – Topic 1: Complex Numbers
Cartesian forms:
Complex arithmetic using polar form:
The complex plane (the Argand plane):
Roots of complex numbers:
Factorisation of polynomials:
Unit 3 – Topic 2: Functions and sketching graphs
Functions:
Unit 3 – Topic 3: Vectors in three dimensions
Vector and Cartesian equations:
Systems of linear equations:
Unit 4 – Topic 1: Integration and Applications of Integration
Integration techniques:
Applications of integral calculus:
Unit 4 – Topic 2: Rates of Change and Differential Equations
The accreditation period for this course has been renewed from 1 January 2022 until 31 December 2025.
During the accreditation period required amendments can be considered via established processes.
Should outcomes of the Years 9-12 Review process find this course unsuitable for inclusion in the Tasmanian senior secondary curriculum, its accreditation may be cancelled. Any such cancellation would not occur during an academic year.
Version 1 – Accredited on 13 August 2017 for use from 1 January 2018. This course replaces MTS415114 Mathematics Specialised that expired on 31 December 2017.
Accreditation renewed on 22 November 2018 for the period 1 January 2019 until 31 December 2021.
Version 1.a - Renewal of Accreditation on 14 July 2021 for the period 31 December 2021 until 31 December 2025, without amendments.
GLOSSARY
Addition of matrices (See Matrix)
If `A` and `B` are matrices with the same dimensions, and the entries of `A` are `a_(ij)` and the entries of `B` are `b_(ij)`, then the entries of `A + B` are `a_(ij) + b_(ij)`.
For example, if `A = [(2,1),(0,3),(1,4)]` and `B = [(5,1),(2,1),(1,6)]`, then `A + B = [(7,2),(2,4),(2,10)]`.
Angle sum and difference identities
`sin(A + B) = sinAcosB + sinBcosA`
`sin(A - B) = sinAcosB - sinBcosA`
`cos(A + B) = cosAcosB - sinBsinA`
`cos(A - B) = cosAcosB + sinBsinA`
Arithmetic sequence
An arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For instance, the sequence `2, 5, 8, 11, 14, 17, ...` is an arithmetic sequence with common difference 3.
If the initial term of an arithmetic sequence is a and the common difference of successive members is `d`, then the `n^(th)` term, `t_n`, of the sequence is given by `t_n = a + (n - 1)d` where `n >= 1`.
A recursive definition is `U_n = U_(n - 1) + d` where `n >= 1`.
Arithmetic series
An arithmetic series is the sum of an arithmetic sequence `S_n = U_1 + U_2 + U_3 + ... + U_n = sum_(r=1)^nU_r`.
The infinite series is given by `S_(oo) = U_1 + U_2+U_3+ ... = sum_(r=1)^(oo)U_r`. This can be found by evaluating `lim_(n -> oo)S_n`.
Argument (abbreviated Arg)
If a complex number is represented by a point `P` in the complex plane, then the argument of `z`, denoted `Arg(z)`, is the angle theta that `OP` makes with the positive real axis `O_x`, with the angle measured anticlockwise from `O_x`. The principal value of the argument is the one in the interval `(–pi, pi]`.
Complex arithmetic
If `z_1 = x_1 + y_1i` and `z_2 = x_2 + y_2i`, then:
Complex conjugate
For any complex number `z = x + yi`, its conjugate is `barz = x - yi`. The following properties hold:
Complex plane (Argand plane)
The complex plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. The complex plane is sometimes called the Argand plane.
Convergent Sequences
A sequence `{a_n}` converges to a finite number `L` if given `epsilon > 0`, however small, there exists `N` (dependent on `epsilon`) such that `|a_n - L| < epsilon`, provided that `n > N`.
`lim_(n->oo) a_n = L`.
Cosine and Sine functions
Since each angle `theta` measured anticlockwise from the positive x-axis determines a point `P` on the unit circle, we will define:
De Moivre’s Theorem
For all integers `n`, `(costheta + isintheta)^n = cosntheta + isinntheta`
Determinant of a 2 x 2 matrix
If `A = [(a,b),(c,d)]`, the determinant of `A` is denoted by `detA = ad - bc`.
If `detA != 0`,
Dimension (or Size) of a matrix
Two matrices are said to have the same dimensions (or size) if they have the same number of rows and columns.
For example, the matrices `[(1,8,0),(2,5,7)]` and `[(3,4,5),(6,7,8)]` have the same dimensions. They are both `2×3` matrices.
An `m×n` matrix has `m` rows and `n` columns.
Difference Method (or Method of Differences)
The method of differences can be used determine some ‘special series’, when we are not given the sum of the series.
For example, as `n^2 - (n - 1)^2 = 2n - 1`, this will mean that `n^2 = 2 sum_(r=1)^n n^2 - 1`.
Divergent Sequences
A sequence `{a_n}` diverges to `oo`, if given `K > 0`, however great, there exists `N` (dependent on `K` ) such that `a_n > K`, provided `n > N`.
`lim_(n-> oo) a_n = oo`
Double angle formulae
`sin2A = 2sinAcosA`
`cos2A = cos^2A - sin^2A = 2cos^2A - 1 = 1 - 2sin^2A`
`tan2A= (2tanA)/(1 - tan^2A)`
Entries (Elements) of a matrix
The symbol `a_(ij)` represents the `(i, j)` entry which occurs in the `i^(th)` row and the `j^(th)` column.
For example, a general `3×2` matrix is `[(a_(11),a_(12)),(a_(21),a_(22)),(a_(31),a_(32))]`, and `a_(32)` is the entry in the third row and the second column.
Geometric sequence
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. For example, the sequence `6, 12, 24, ...` is a geometric sequence with common ratio `2`.
Similarly, the sequence `40, 20, 10, 5, 2.5, ...` is a geometric sequence with common ratio `1/2`.
If the initial term of a geometric sequence is `a`, and the common ratio of successive members is `r`, then the `n^(th)` term, `t_n`, of the sequence is given by `t_n = ar^(n - 1)` for `n >= 1`.
A recursive definition is `U_n = rU_(n-1)` where `U_1 = a`.
Geometric series
A geometric series is the sum of a geometric sequence `S_n = U_1 + U_2 + U_3 + ... + U_n = sum_(r=1)^n U_r`
`S_n = (a(1 - r^n))/((1 - r))` where `r != 1`.
The infinite series `S_(oo) = U_1 + U_2 + U_3 + ... = sum_(r=1)^(oo) U_r` can be found by evaluating `lim_(n->oo)S_n`.
`S_(oo) = a/(1 - r)`, provided `|r| < 1`.
Imaginary part of a complex number
A complex number `z` may be written as `x + yi`, where `x` and `y` are real, and then `y` is the imaginary part of `z`. It is denoted by `Im(z)`.
Implicit differentiation
When variables `x` and `y` satisfy a single equation, this may define `y` as a function of `x` even though there is no explicit formula for `y` in terms of `x`. Implicit differentiation consists of differentiating each term of the equation as it stands and making use of the chain rule. This can lead to a formula for `dy/dx`. For example, if `x^2 + xy^3 - 2x + 3y = 0`, then `2x + x(3y^2)dy/dx + y^3 - 2 + 3dy/dx = 0`, and so `dy/dx = (2 - 2x - y^3)/(3xy^2 + 3)`.
Inverse trigonometric functions
The inverse sine function, `y = sin^(-1)x`
If the domain for the `"sine"` function is restricted to the interval `[-pi/2,pi/2]`, a one-to-one function is formed and so an inverse function exists.
The inverse of this restricted `"sine"` function is denoted by `sin^(-1)`, and is defined by `sin^(-1):[-1,1] -> R`, `sin^(-1)x = y` where`sin y = x`, `y in [-pi/2,pi/2]`.
`sin^(-1)` is also denoted by `"arcsin"`.
The inverse cosine function, `y = cos^(-1)x`
If the domain of the `"cosine"` function is restricted to `[0,pi]`, a one-to-one function is formed and so the inverse function exists.
`cos^(-1)x`, the inverse function of this restricted `"cosine"` function, is defined by `cos^(-1):[-1,1] -> R`, `cos^(-1)x = y` where `cosy = x`, `y in [0,pi]`.
`cos^(-1)` is also denoted by `"arccos"`.
The inverse tangent function, `y = tan^(-1)x`
If the domain of the `"tangent"` function is restricted to `(-pi/2,pi/2)`, a one-to-one function is formed and so the inverse function exists.
`tan^(-1):R -> R`, `tan^(-1)x = y`, where `tany = x`, `y in (-pi/2,pi/2)`
`tan^(-1)` is also denoted by `"arctan"`.
Leading diagonal
The leading diagonal of a square matrix is the diagonal which runs from the top left corner to the bottom right corner.
Linear Transformation Defined by a 2x2 matrix
The matrix multiplication `[(a,b),(c,d)][(x),(y)] = [(ax + by),(cx + dy)]` defines a transformation `T(x,y) = (ax + by, cx + dy)`.
Linear Transformations in 2-dimensions
A linear transformation in the plane is a mapping of the form `T(x,y) = (ax + by,cx + dy)`.
A transformation `T` is linear if and only if `T(alpha(x_1,y_1) + beta(x_2,y_2)) = alphaT(x_1,y_1) + betaT(x_2,y_2)`.
Linear transformations include:
Translations are not linear transformations.
MacLaurin’s Series
A Maclaurin series is a special power series expansion for `f(x)`.
`f(x) = f(0) + f'(0)x + (f''(0))/(2!)x^2 + (f^(3)(0))/(3!)x^3 + ... + (f^(n)(0))/(n!)x^n + ...`
It equals `f(x)` whenever the series converges. MacLaurin series converge for all real `x`, some for a subset of `x` and others for no values of `x`.
Matrix (matrices)
A matrix is a rectangular array of elements or entries displayed in rows and columns. For example, `A = [(2,1),(0,3),(1,4)]` and `B = [(1,8,0),(2,5,7)]` are both matrices.
Matrix `A` is said to be a `3×2` matrix (three rows and two columns), while `B` is said to be a `2×3` matrix (two rows and three columns).
A square matrix has the same number of rows and columns.
A column matrix (or vector) has only one column.
A row matrix (or vector) has only one row.
Matrix algebra of 2×2 matrices
If `A`, `B` and `C` are `2×2` matrices, `I` the `2×2` (multiplicative) identity matrix, and `0` the `2×2` zero matrix, then:
`A + B = B + A` (commutative law for addition)
`(A + B) + C = A + (B + C)` (associative law for addition)
`A + 0 = A` (additive identity)
`A + (-A) = 0` (additive inverse)
`(AB)C = A(BC)` (associative law for multiplication)
`AI = A = IA` (multiplicative identity)
`A(B + C) = AB + AC` (left distributive law)
`(B + C)A = BA + CA` (right distributive law)
Matrix multiplication
Matrix multiplication is the process of multiplying a matrix by another matrix. The product `AB` of two matrices `A` and `B` with dimensions `m×n` and `p×q` is defined if `n = p`. If it is defined, the product `AB` is an `m×q` matrix and it is computed as shown in the following example.
`[(1,8,0),(2,5,7)][(6,10),(11,3),(12,4)] = [(94,34),(151,63)]`
The entries are computed as shown:
`1 × 6 + 8 × 11 + 0 × 12 = 94`
`1 × 10 + 8 × 3 + 0 × 4 = 34`
`2 × 6 + 5 × 11 + 7 × 12 = 151`
`2 × 10 + 5 × 3 + 7 × 4 = 63`
The entry in row `i` and column `j` of the product `AB` is computed by ‘multiplying’ row `i` of `A` by column `j` of `B` as shown.
If `A = [(a_(11),a_(12)),(a_(21),a_(22)),(a_(31),a_(32))]` and `B = [(b_(11),b_(12),b_(13)),(b_(21),b_(22),b_(23))]`, then
`AB = [(a_(11)b_(11) + a_(12)b_(21),a_(11)b_(12) + a_(12)b_(22),a_(11)b_(13) + a_(12)b_(23)),(a_(21)b_(11) + a_(22)b_(21),a_(21)b_(12) + a_(22)b_(22),a_(21)b_(13) + a_(22)b_(23)),(a_(31)b_(11) + a_(32)b_(21),a_(31)b_(12) + a_(32)b_(22),a_(31)b_(13) + a_(32)b_(23))]`
Modulus (Absolute value) of a complex number
If `z` is a complex number and `z = x + yi`, then the modulus of `z` is the distance of `z` from the origin in the Argand plane. The modulus of `z` is denoted by `|z| = sqrt(x^2 + y^2)`.
Multiplication by a scalar
Let `a` be a non-zero vector and `k` a positive real number (scalar), then the scalar multiple of `a` by `k` is the vector `ka` which has magnitude `|k||a|` and the same direction as `a`. If `k` is a negative real number, then `ka` has magnitude `|k||a|` but is directed in the opposite direction to `a`.
Some properties of scalar multiplication are:
`k(a + b) = ka + kb`
`h(k(a)) = (hk)a`
`Ia = a`
(Multiplicative) identity matrix
A (multiplicative) identity matrix is a square matrix in which all the elements in the leading diagonal are 1s and the remaining elements are 0s. Identity matrices are designated by the letter `I`.
For example, `[(1,0),(0,1)]` and `[(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]` are both identity matrices.
There is an identity matrix for each order of square matrix. When clarity is needed, the order is written with subscript, `I_n`.
Multiplicative inverse of a square matrix
The inverse of a square matrix `A` is written as `A^(-1)` and has the property that `A A^(-1) = I`.
Not all square matrices have an inverse. A matrix that has an inverse is said to be invertible.
Multiplicative inverse of a `2×2` matrix:
The inverse of the matrix `A = [(a,b),(c,d)]` is `A^(-1) = 1/detA [(d,-b),(-c,a)]`, when `detA != 0`.
Scalar multiplication (Matrices)
Scalar multiplication is the process of multiplying a matrix by a scalar (number). For example, forming the product `10[(2,1),(0,3),(1,4)] = [(20,10),(0,30),(10,40)]` is an example of the process of scalar multiplication.
In general, for the matrix `A` with entries `a_(ij)`, the entries of `kA` are `ka_(ij)`.
Polar form of a complex number
For a complex number `z`, let `theta = arg(z)`. Then `z = r(costheta + isintheta)` is the polar form of `z`.
Power Series
A power series is an infinite series `a_0 + a_1x + a_2x^2 + a_3x^3 + ...` where `a_0, a_1, a_2, a_3, ...` are real constants.
Principle of mathematical induction
Let there be associated with each positive integer `n`, a proposition `P(n)`.
If `P(1)` is true, and for all `k`, `P(k)` is true implies `P(k + 1)` is true, then `P(n)` is true for all positive integers `n`.
Products as sums and differences
`cosAcosB = 1/2[cos(A – B) + cos(A + B)]`
`sinAsinB = 1/2[cos(A – B) – cos(A + B)]`
`sinAcosB = 1/2[sin(A + B) + sin(A – B)]`
`cosAsinB = 1/2[sin(A + B) – sin(A – B)]`
Pythagorean identities
`cos^2A + sin^2A = 1`
`tan^2A + 1 = sec^2A`
`cot^2A + 1 = cosec^2A`
Rational function
A rational function is a function such that `f(x) = (g(x))/(h(x))`, where `g(x)` and `h(x)` are polynomials. Usually, `g(x)` and `h(x)` are chosen so as to have no common factor of degree greater than or equal to 1, and the domain of `f` is usually taken to be `{x in R: h(x) != 0}`.
Real part of a complex number
A complex number `z` may be written as `x + yi`, where `x` and `y` are real, and then `x` is the real part of `z`. It is denoted by `Re(z)`.
Reciprocal trigonometric functions
`secA = 1/(cosA)`, `cosA != 0`
`cosecA = 1/(sinA)`, `sinA != 0`
`cotA = cosA/sinA`, `sinA != 0`
Root of unity
Given a complex number `z` such that `z^n = 1`, then the `n^(th)` roots of unity are: `cos((2kpi)/n) + isin((2kpi)/n)`, where `k = 0, 1, 2, ..., n - 1`.
The points in the complex plane representing roots of unity lie on the unit circle.
The cube roots of unity are `z_1 = 1`, `z_2 = 1/2(–1 + isqrt3)`, `z_3 = 1/2(–1 – isqrt3)`.
Note: `z_3 = bar(z_2)`; `z_3 = 1/(z_2)`; and `z_2z_3 = 1`.
Separation of variables
Differential equations of the form `dy/dx = g(x)h(y)` can be rearranged as long as `h(y) != 0` to obtain `1/(h(y))dy/dx = g(x)`.
Sigma Notation Rules
`sum_(r=1)^n f(r) = f(1) + f(2) + f(3) + ... + f(n)`
`sum_(r=1)^n (f(r) + g(r)) = sum_(r=1)^n f(r) + sum_(r=1)^n g(r)`
`sum_(r=1)^n kf(r) = ksum_(r=1)^n f(r)`
Singular matrix
A matrix is singular if `detA = 0`. A singular matrix does not have a multiplicative inverse.
Vector equation of a plane
Let `a` be a position vector of a point `A` in the plane, and `n` a normal vector to the plane, then the plane consists of all points `P` whose position vector `p` satisfies `(p - a)n = 0`. This equation may also be written as `pn = an`, `a` constant.
(If the normal vector `n` is the vector `(l,m,n)` in ordered triple notation and the scalar product `an = k`, this gives the Cartesian equation `lx + my + nz = k` for the plane.)
Vector equation of a straight line
Let `a` be the position vector of a point on a line `l`, and `u` any vector with direction along the line. The line consists of all points `P` whose position vector `p` is given by `p = a + tu` for some real number `t`.
(Given the position vectors of two points on the plane `a` and `b`, the equation can be written as `p = a + t(b - a)` for some real number `t`.)
Zero matrix
A zero matrix is a matrix if all of its entries are zero. For example, `[(0,0,0),(0,0,0)]` and `[(0,0),(0,0)]` are zero matrices.
There is a zero matrix for each size of matrix.
LINE OF SIGHT – Mathematics Specialised Level 4
Learning Outcomes | Separating Out Content From Skills in 4 of the Learning Outcomes | Criteria and Elements | Content | |
utilise concepts and techniques from sequences and series, complex numbers, matrices and linear algebra, function and equation study and calculus |
Sequences and Series | concepts and techniques; problem solving; reasoning skills; interpret and evaluate ; select and use appropriate tools | C4 E1-8 C4 E1-8; C2 E1,4,7 C2 E1,3,4,7 C1 E5 C1 E2,5,6; C2 E5-6 |
Sequences and Series (can include some, non examinable applications) |
solve problems using concepts and techniques drawn from algebraic processes, sequences and series, complex numbers, matrices and linear algebra, function and equation study and calculus | Matrices and Linear Algebra | concepts and techniques; problem solving; reasoning skills; interpret and evaluate; select and use appropriate tools | C5 E1-8 C5 E1-8; C2 E1,4,7 C2 E1,3,4,7 C3 E1; C1 E5 C1 E2,5,6; C2 E5-6 |
Matrices and Linear Algebra (can include some, non examinable applications) |
apply reasoning skills in the contexts of algebraic processes, sequences sand series, complex numbers, matrices and linear algebra, function and equation study and calculus | Calculus – Part 1: implicit differentiation, fundamental theorem,definite integrals | concepts and techniques; problem solving; reasoning skills; interpret and evaluate; select and use appropriate tools | C6 E1-5 C6 E1-5; C2 E1,4,7 C2 E1,3,47 C1 E5 C1 E2,5,6; C2 E5-6 |
Calculus – Part 1 (can include some, non examinable applications) |
interpret and evaluate mathematical information and ascertain the reasonableness of solutions to problems | Calculus – Part 2: techniques of integration, first order differential equations, applications | concepts and techniques; problem solving; reasoning skills; interpret and evaluate; select and use appropriate tools | C7 E1-6 C7 E1-6; C2 E1,4,7 C2 E1,3,47 C3 E1; C1 5 C1 E2,5,6; C2 E5-6 |
Calculus – Part 2 (can include some, non examinable applications) |
Complex Numbers | concepts and techniques; problem solving; reasoning skills; interpret and evaluate; select and use appropriate tools | C8 E1-6 C8 E1-6; C2 E1,4,7 C2 E1,3,4,7 C1 E5 C1 E2,5,6; C2 E5-6 |
Complex Numbers (can include some, non examinable applications) | |
communicate their arguments and strategies when solving problems | C1 E1-7 | All content areas | ||
plan activities and monitor and evaluate progress; use strategies to organise and complete activities, and meet deadlines in context of mathematics | C1 E1,2,5,6 C3 E5-7 |
All content areas | ||
select and use appropriate tools, including computer technology, when solving mathematical problems | C1 E5-7 C2 E5-7 |
All content areas |