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New Senior Mathematics Extension 2 Worked Solutions Pdf

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This article needs to be updated. Please help update this article to reflect recent events or newly available information. (April 2018)

Mathematics education in Australia varies significantly between states, especially at the upper secondary level.

Secondary [edit]

New South Wales [edit]

Higher School Certificate [edit]

The Higher School Certificate (HSC) in NSW contains a number of mathematics courses catering for a range of abilities. There are four courses offered by NESA (NSW Education Standards Authority ) for HSC Study:

  • Mathematics Standard 1 or 2: A basic mathematics course containing precalculus concepts; the course is heavily based on practical mathematics used in everyday life.[1] While the more advanced courses include statistical topics, this is the only course which introduces normal distributions, standard deviations and z-scores. These topics are alluded to in more advanced courses though not formally considered.
  • Mathematics Advanced: An advanced level calculus-based course with detailed study in probability and statistics, trigonometry, curve sketching, and applications of calculus. It is the highest level non-extension mathematics course.[2] The calculus is only a single variable in all of year 12 mathematics in NSW. Computational methods such as the trapezoidal rule are encountered for evaluating integrals. The course includes a brief foray into series and sequences, including an application to basic finance through the modelling of compound interest. The nature of lines, circles and parabolas as loci are investigated however these properties are not exploited by the plane geometry coursework. Quadratic equations are studied and students learn techniques to reduce special quintic and exponential equations to quadratics.
  • Mathematics Extension 1 (Must be studied concurrently with Mathematics Advanced): A more advanced course building on concepts in calculus, trigonometry, polynomials, basic combinatorics, vectors, and further statistics. [3] Students learn the binomial theorem to extend their knowledge of probability, along with using circle geometry to prove a greater family of statements. The trigonometry component includes double-angle identities and factoring the addition of a sine and cosine function into a single sinusoid. In calculus, students are exposed to a greater variety of integration techniques such as substitution. Parametrization of planar curves is introduced, mainly focusing on lines, circles and parabolas. The plotting of cubic equations and solution of specific cases through polynomial long division and the remainder theorem enable a deeper understanding of polynomials.
  • Mathematics Extension 2 (Must be studied concurrently with Mathematics Advanced and Mathematics Extension 1): A highly advanced mathematics course containing an introduction to complex numbers, advanced calculus, motion, and further work with vectors.[4] While NSW Mathematics curricula does not include matrix theory nor group theory, the geometric properties of complex numbers alludes to both of these. The former is hinted at in the multiplicative properties of complex numbers, as students are required to plot the products, sums and quotients of complex numbers on the Argand plane. While group theory is not explicitly mentioned, roots of unity and cyclic groups are extensively studied. With their newfound familiarity with complex numbers, the fundamental theorem of algebra can now be formally stated. Students are now able to exploit this closure to solve even more polynomial equations. Recursive integral sequences, integration by parts and partial fraction decomposition techniques allow the solution to a wider class of problems. Projectile motion is studied in the kinematics module, which surpasses the depth of study found in HSC physics. This course synergizes with HSC Physics, as students are able to apply this knowledge in their Physics exams to arrive at more elegant and efficient solutions. The parametrisation of lines, circles and parabolas in Mathematics Extension 1 is further developed to the entire family of conics, including degenerate cases. Students are exposed to rectangular hyperbolas, however hyperbolic trigonometric functions are not included. Despite this students are expected to adapt to novel material, such as proving properties of the catenary via its expression in exponential functions.

The defining feature of content progression from Mathematics Advanced through to Extension 2 Mathematics is the level of Mathematical Maturity expected of students. In higher courses, students have exposure to a greater breadth and depth of techniques, and are expected to synthesize knowledge from seemingly disparate topics. In Mathematics Advanced exams students may be asked to apply familiar techniques to unfamiliar contexts, such as being given an identity through which they must solve a problem. Further mathematical maturity is vital to success in Extension 2 exams, as assessment focuses on both conceptual understanding and computational abilities.

The difficulty in HSC final exam questions generally increases throughout the course of the paper. In one Extension 2 HSC examination, the final question provides students with a series of prompts and smaller questions, which culminate in a proof of the Basel Problem.

Victoria [edit]

Victorian Certificate of Education [edit]

The Victorian Certificate of Education (VCE) mathematics subjects are designed to cater for the varying abilities and aptitudes of Victorian students. There are four courses offered for VCE study:[5]

  • Foundation Mathematics (2 units): Very basic, practical maths skills are taught in this strand, which is only run at a Unit 1 + 2 level.
  • Further Mathematics (4 Units): Further Mathematics is considered to be the second-least demanding of the four maths subjects and is the "standard" maths course taken by Victorian students[ citation needed ]. It deals with topics including basic algebra, matrices, trigonometry, straight line geometry, business-related mathematics and other concepts
  • Mathematical Methods (CAS) (4 Units): Mathematical Methods is the most common non-compulsory prerequisite for tertiary study in Victoria. Methods deals with concepts including differential calculus, integral calculus, circular functions, probability and the behaviour of functions with a single real variable, incorporating the use of computer algebra system technology (CAS) throughout the course.
  • Specialist Mathematics (4 Units): Specialist Mathematics is considered the most advanced high school mathematics subject in Victoria. The subject covers concepts including conic sections, complex numbers, differential equations, kinematics, vector calculus and mechanics. The idea behind Specialist Mathematics is to integrate already learnt concepts of calculus into other fields of mathematics, thus giving Specialist Mathematics a far more practical orientation than standard mathematical subjects.

In addition to the regular mathematics courses offered in VCE, a new subject called Algorithmics (Higher Education Scored Study) began its coursework in 2015.[6] Since it is a fully scored subject supported by members in both the University of Melbourne and Monash University, it is different from University Extension studies. Beginning with definitions of abstract data types and algorithms, this subject explores the concepts of pseudocode, sorting, common graph algorithms, recursion, algorithm design patterns, algorithm analysis and computational complexity theory.

Queensland [edit]

In Queensland, the senior (years 11 and 12) curriculum is divided into three parts: Maths A, Maths B, and Maths C.

Mathematics A [edit]

Maths A covers more practical topics than Maths B and C, but it is still OP eligible. There are considerably fewer algebraic concepts in this subject, and it is suitable for students who either struggled with mathematics in Year 10, or who do not require a knowledge of abstract mathematics in the future. Maths A is designed to help students to develop an appreciation of the value of Mathematics to humanity. Students learn how mathematical concepts may be applied to a variety of life situations including business and recreational activities. The skills encountered are relevant to a vast array of careers (trade, technical, business etc.). Assessments in the subject include both formative and summative written tests, assignments and practical work. It is assessed in the categories: Knowledge & Procedures (KAPS); Modelling & Problem Solving (MAPS); Communication & Justification (CAJ). Although Maths A is not a pre-requisite subject, but it is sufficient for entrance to many tertiary courses.[7]

The course is divided into four semesters. The skills learned in each semester are as follows:

Semester 1 (Year 11/Form 5):

  • Data Analysis
  • Managing Money
  • Applied Geometry
  • Linking 2 and 3 Dimensions

Semester 2 (Year 11/Form 5):

  • Land Measurement
  • Applied Geometry
  • Statistics
  • Managing Money

Semester 3 (Year 12/Form 6):

  • Managing Money
  • Land Measurement
  • Data Analysis
  • Operations Research

Semester 4 (Year 12/Form 6):

  • Statistics
  • Land Measurement
  • Navigation
  • and an elective topic on Data

Mathematics B [edit]

Maths B is considerably more theoretical than Maths A, requiring advanced algebra skills to successfully complete. It is a common prerequisite for science and engineering courses at Queensland Universities. Maths B (in some schools) can be studied at the same time with either Maths A or Maths C, but not both. Maths B gives students an understanding of the methods and principles of mathematics and the ability to apply them in everyday situations and in purely mathematical contexts; the capacity to model actual situations and deduce properties from the model; an interest and ability in framing and testing mathematical hypotheses; the ability to express and communicate any results obtained; some knowledge of the history of mathematics; encouragement to think independently and creatively. Assessments are similar as those of Maths A, which includes both formative (Semester 1) and summative (Semesters 2,3 and 4) written tests, assignments and post-assignment tests. It is also assessed in the three categories Knowledge & Procedures (KAP); Modelling & Problem Solving (MAP); Communication & Justification (CAJ). Maths B is a pre-requisite for any tertiary course which deals with or uses math and/or science.[8] According to the Queensland Studies Authority, in 2010, 93% of students who studied Maths B were OP eligible.

The course is divided into four (4) semesters. The skills learned each semester are as follows:

Semester 1 (Year 11/Form 5):

  • Functions (Linear, Quadratic, Absolute Value)
  • Periodic Functions (Trigonometry, Sin/Cosine Functions)
  • Applied Statistics (Mean, Median, Mode, Lie Factor)
  • Applied Statistics 2 (Linear/Quadratic Regression, Residual Plots)

Semester 2 (Year 11/Form 5):

  • Exploring Data / Statistics
  • Indices and Logarithms/ Exponential Functions
  • Limits and Differential Calculus 1

Semester 3 (Year 12/Form 6):

  • Exponential and Log Functions
  • Optimization Using Derivatives
  • Integration
  • Integral Calculus

Semester 4 (Year 12/Form 6):

  • Applied Statistical Analysis
  • Integration
  • Differential Calculus 2
  • Optimisation (Other Methods)

Mathematics C [edit]

Maths C extends the topics taught in Maths B, and covers additional pure-maths topics (including complex numbers, matrices, vectors, further calculus and number theory). Although not necessarily more difficult, it must be studied in conjunction with Maths B. Maths C gives the students an understanding of the methods and principles of mathematics and the ability to apply them in everyday situations and in purely mathematical contexts; the capacity to model actual situations and deduce properties from the model; an interest and ability in framing and testing mathematical hypotheses; the ability to express and communicate any results obtained; some knowledge of the history of mathematics; encouragement to think independently and creatively. Assessments are in the same as the other two courses, formative and summative written tests, assignments and practical work. The student is assessed in the areas of Knowledge & Procedures (KAPS); Modelling & Problem Solving (MAPS); Communication & Justification (CAJ). Maths C can be a pre-requisite to tertiary courses with a heavy maths/science basis. Some skills learned in Maths C would be found in business and economics degrees.[9]

The course is divided into four (4) semesters. The areas learned are in the following:

Semester 1 (Year 11/Form 5):

  • Real and Complex Numbers
  • Matrices
  • Vectors
  • Groups
  • Structures & Patterns

Semester 2 (Year 11/Form 5):

  • Applications of Matrices
  • Vectors
  • Real and Complex Numbers
  • Dynamics
  • Structures and Patterns

Semester 3 (Year 12/Form 6):

  • Structures and Patterns
  • Real and Complex Numbers
  • Matrices
  • Periodic Functions
  • Calculus
  • Option I & II

Semester 4 (Year 12/Form 6):

  • Vectors
  • Calculus
  • Dynamics
  • Vectors
  • Option I & II

Western Australia [edit]

New WACE mathematics courses were introduced for Year 11 students in 2015 to replace previous mathematics courses and being the Western Australian course in line with the Australian Curriculum.

The new WACE mathematics courses consist of four units. Each unit is studied over one semester. Therefore, Unit 1 & 2 is studied in Year 11, and Unit 3 & 4 is studied in Year 12.

The new WACE mathematics courses are:[10]

  • Mathematics Preliminary General[11]
  • Mathematics Foundation General[12] [13]
  • Mathematics Essential General[14] [15]
  • Mathematics Applications ATAR[16] [17]
  • Mathematics Methods ATAR[18] [19]
  • Mathematics Specialist ATAR[20] [21]

ATAR mathematics courses are for university-bound students, whereas general courses are for non-ATAR students.

Syllabus information is available from the School Curriculum and Standards Authority (SCSA) website.

South Australia [edit]

In South Australia the mathematics courses are split into six[ clarification needed ] levels:

  • Numeracy for Work and Community Life (up to and including Stage 1)[22]
  • Essential Mathematics/Mathematical Pathways[23]
  • General Mathematics/Mathematical Applications[24]
  • Mathematical Methods/Mathematical Studies[25]
  • Specialist Mathematics — more advanced topics that complement and are taken concurrently with Mathematical Studies[26]

Tertiary [edit]

University Campus Degree(s)[note 1]
Australian National University Canberra BMathSci,[27] MMathSci (advanced),[28] and PhD[29]
Charles Sturt University Online BSc (mathematics major).[30]
Curtin University Perth BAdvSci (hons), BSc (maths-related majors available),[31] BSc (hons), MSc, MPhil and PhD[32]
Deakin University Geelong, Melbourne BSc (mathematics major),[33] and BSc (hons)[34]
Edith Cowan University Perth BSc (mathematics major),[35] BSc (hons)[36] and MSc[37]
Federation University Ballarat, Churchill, Melbourne, online BMathSci, BSc (mathematics major), BSc (advanced), BSc (hons) and MSc[38]
Flinders University Adelaide BMathSci,[39] BMathSci (honours)[40] and MSc (mathematics)[41]
Griffith University Brisbane, Gold Coast BSc (applied mathematics major),[42] BSc (hons),[43] MSc,[44]
James Cook University Townsville BSc (mathematics major)[45] and BSc (hons)[46]
La Trobe University Melbourne BA, BA (hons), BSc, BSc (hons), MAppSci, MPhil, MSc and PhD[47]
Macquarie University Sydney BSc (mathematics major)[48]
Monash University Melbourne BSc (mathematics major),[49] BSc (hons)[50] and MMaths[51]
Murdoch University Perth, external BSc (mathematics and statistics major),[52] BSc (hons)[53]
Queensland University of Technology Brisbane BMaths,[54] BMaths (hons),[55] MPhil (mathematical sciences)[56] and PhD[57]
RMIT University Melbourne BSc (mathematics and statistics major), BSc (hons),[58] MSc, masters of statistics and operations research and PhD[59]
Southern Cross University Lismore, online BSc (mathematics major)[60]
Swinburne University of Technology Melbourne BSc, BSc (professional), BSc (hons), MSc and PhD[61]
University of Adelaide Adelaide BMathSci,[62] BMathSci (honours),[63] MMathSci,[64] MPhil[65] and PhD[66]
University of Melbourne Melbourne BSc (mathematics and statistics major),[67] BSc (extended, mathematics and statistics major)[68]
University of Newcastle Newcastle BMaths[69] and BMaths (hons)[70]
University of New England Armidale and online BSc (mathematics major),[71] BSc (hons),[72] MPhil,[73] MSc,[74] and PhD[75]
University of New South Wales Sydney BSc (mathematics[76] and statistics[76] majors)
University of Queensland Brisbane BMaths,[77] BMaths (hons),[78] GradCertSci (mathematics),[79] GradDipSci (mathematics)[80] and MSc (mathematics)[81]
University of South Australia Adelaide, online (some degrees only) BAppSci (online available), BAppSci (online available), BMaths (online available), BSc, BSc (honours), MPhil[82] and PhD[83] [84]
University of Southern Queensland Toowoomba, online BSc (mathematics, statistics and combination majors),[85] BSc (hons),[86] MSc[87] and PhD[88] [89]
University of the Sunshine Coast Moreton Bay, Sunshine Coast BSc (mathematics major)[90]
University of Sydney Sydney BSc (many different mathematics-related majors available)[91]
University of Tasmania Hobart BSc (mathematics, and statistics/decision science majors available),[92] [93] BSc (hons)[94] and PhD[95]
University of Technology Sydney Sydney BSc (mathematics and statistics majors available),[96] BSc (hons)[97] and MSc[98]
University of Western Australia Albany and Perth BSc/BPhil (mathematics and statistics major)[99] and BSc (hons)[100]
University of Wollongong Wollongong BMaths,[101] BMaths (honours),[102] MMaths[103] and PhD[104]
Abbreviations
BA = Bachelor of Art. BAdvSci = Bachelor of Advanced Science. BAppSci = Bachelor of Applied Science. BMaths = Bachelor of Mathematics. BMathSci = Bachelor of Mathematical Sciences. BSc = Bachelor of Science. GradCertSci = graduate certificate in science. GradDipSci = Graduate Diploma in Science. Hons = honours. MMaths = Master of Mathematics. MPhil = Master of Philosophy. MSc = Master of Science. PhD = Philosophy of Science.
Notes
  1. ^ Excluding double degrees

References [edit]

  1. ^ NSW Mathematics Standard Syllabus
  2. ^ NSW Mathematics Advanced Syllabus
  3. ^ NSW Mathematics Extension 1 Syllabus
  4. ^ NSW Mathematics Extension 2 Syllabus
  5. ^ VCE Mathematics Study Design Units 1-4: 2006-2009 (PDF)
  6. ^ VCE Algorithmics Study Design Units 3-4: 2015-2016 Archived 2015-11-25 at the Wayback Machine (PDF)
  7. ^ Mathematics A syllabus Archived 2007-12-11 at the Wayback Machine (PDF)
  8. ^ Mathematics B syllabus Archived 2007-12-11 at the Wayback Machine (PDF)
  9. ^ Mathematics C syllabus Archived 2007-11-11 at the Wayback Machine (PDF)
  10. ^ "WACE 15-16 - Mathematics". scsa.wa.edu.au.
  11. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26 . CS1 maint: archived copy as title (link)
  12. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26 . CS1 maint: archived copy as title (link)
  13. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26 . CS1 maint: archived copy as title (link)
  14. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26 . CS1 maint: archived copy as title (link)
  15. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26 . CS1 maint: archived copy as title (link)
  16. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26 . CS1 maint: archived copy as title (link)
  17. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26 . CS1 maint: archived copy as title (link) (PDF)
  18. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26 . CS1 maint: archived copy as title (link)
  19. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26 . CS1 maint: archived copy as title (link)
  20. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26 . CS1 maint: archived copy as title (link)
  21. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-12-26. Retrieved 2014-12-26 . CS1 maint: archived copy as title (link)
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  24. ^ Mathematical Applications syllabus Archived 2007-08-31 at the Wayback Machine (PDF)
  25. ^ Mathematical Methods syllabus Archived 2007-08-31 at the Wayback Machine (PDF)
  26. ^ Specialist Mathematics syllabus Archived 2007-08-31 at the Wayback Machine (PDF)
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New Senior Mathematics Extension 2 Worked Solutions Pdf

Source: https://en.wikipedia.org/wiki/Mathematics_education_in_Australia

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