SACE Specialist Mathematics: induction, complex numbers, vectors and differential equations

Six topics, one exam

Specialist Mathematics is normally studied alongside Mathematical Methods. The exam is a 130-minute paper based on the key questions and key concepts in the six topics, and problems can ask you to connect ideas across topics, for example using complex numbers to describe a vector rotation, or integration to solve a differential equation.

Assessment

School assessment is 70%: five or six Skills and Applications Tasks (50%) and one Mathematical Investigation (20%) with a report. The external exam is 30%. A formula sheet is provided with the exam booklet, you may bring two unfolded A4 sheets (four sides) of handwritten notes, and approved electronic technology is allowed. The outline describes a range of problems, some on knowledge and routine skills and others on analysis and interpretation.

Worked examples

Proof by induction (Topic 1)

Prove that 2 + 4 + 6 + … + 2n = n(n + 1) for every positive integer n.

1. Base case: check n = 1. The sum is just 2, and n(n + 1) = 1 × 2 = 2, so the statement is true.
2. Assume the statement is true for some k, so 2 + 4 + … + 2k = k(k + 1).
3. Step: add the next term, 2(k + 1), to both sides: k(k + 1) + 2(k + 1) = (k + 1)(k + 2), which is the statement with k + 1 in place of k.
4. Conclude: the base case holds and each case forces the next, so by induction the statement is true for every positive integer n.

Write the final sentence. It is a mark in most mark schemes.

De Moivre's theorem (Topic 2)

The modulus of 1 + i is √2, its argument is π/4, so the eighth power has modulus 16 and argument 2π, which is a positive real number. Check the answer with (1 + i)² = 2i and then (2i)⁴ = 16.

Integration by parts (Topic 5)

Differentiate the answer to check it: (x − 1)eˣ has derivative eˣ + (x − 1)eˣ = x eˣ. In a calculator-free setting, that check takes ten seconds and catches sign errors.

Complex-number work can be explored in the Complex Number Calculator, and the Calculus course covers integration and rates of change.

Grades, SATAC and your ATAR

The SACE Board reports an A+ to E− grade. SATAC turns your school and exam results into a raw score out of 15, with 10.5 from the 70% and 4.5 from the 30%, applies the subject's scaling, and uses the result in the university aggregate. Specialist Mathematics is a 20-credit subject, so it can be one of the three subjects in the first 60 credits.

Dates

Use the daily papers as a trial

Trials are set by schools in South Australia. Uni-Sine's free daily papers can work as short timed trials of 8 questions worth 25 to 35 marks, marked as you go. They are not 130-minute SACE exams.

• Maths Extension 2 and Specialist Maths fits on days about complex numbers, proof by induction and polar forms. It also visits hyperbolic functions, Maclaurin series and second-order differential equations, which are not in SACE Stage 2 Specialist.
• Specialist Maths: Mechanics practises forces, moments and pulleys. SACE Specialist has no mechanics topic, so use it only for the calculus-based motion days. For background on the mechanics course, see our Specialist Maths Mechanics guide.

Exam answers

Answers and a worked mark scheme come with each daily paper once it has been marked. They are Uni-Sine's own, not official SACE Board answers.

Mistakes in SACE Specialist

• Proving the inductive step without stating the assumption.
• Choosing the wrong quadrant for an argument in polar form.
• Forgetting the arbitrary constant, or the initial condition, in a differential equation.
• Relying on a technology answer where the question says "show that".

Read next

Australia · SACESACE Specialist Maths formula sheet: the one-page SACE sheetThe SACE Board Specialist Mathematics formula sheet for Stage 2: every formula, what is missing, the A4 notes rule, and worked examples.