Year 12 Maths Advanced study notes and practice questions

For Year 11 and 12 students sitting the HSC, VCE, QCE, SACE, WACE and more. Each unit has short study notes, flashcards and practice questions, and the app builds timed practice exams from the whole course.

• Further differentiationThe chain rule; Product and quotient rules; Exponential and logarithmic functions; Trigonometric functions
• Curve sketching and optimisationIncreasing, decreasing and stationary points; The second derivative and concavity; Curve sketching; Optimisation and global extrema
• IntegrationAntiderivatives; Exponential, reciprocal and trigonometric integrals; Definite integrals; Areas and the trapezoidal rule
• Kinematics and rates of changeDisplacement, velocity and acceleration; Interpreting motion; Integration in kinematics; Rates of change and exponential models
• Sequences, series and financial mathsArithmetic sequences and series; Geometric sequences and series; Compound interest and annuities; Reducing-balance loans
• Statistical analysisBivariate data and regression; Continuous random variables; The normal distribution and z-scores; The empirical rule and comparing scores

4 multiple choice and 2 written questions. Try each one before you open the answer. Written answers show the model answer and what a full-marks response needs.

1. Question 1. Further differentiationMultiple choice

   d/dx ln(x^³) = …

   1. A3/x
   2. B3x^²
   3. C1/x^³
   4. D3 ln x
   Show answer

   A. 3/x

   ln(x^³) = 3 ln x.

2. Question 2. Curve sketching and optimisationMultiple choice

   The point of inflection of y = x^³ − 6x^² is at x = …

   1. A2
   2. B0
   3. C6
   4. D4
   Show answer

   A. 2

   y″ = 6x − 12.

3. Question 3. Curve sketching and optimisation: the second derivative and concavity2 marks

   Explain why f″(a) = 0 does not guarantee a point of inflection, using y = x^⁴ as an example.

   Show model answer and marking criteria

   Model answer. A point of inflection needs the concavity to change. For y = x^⁴, y″ = 12x^², which is 0 at x = 0 but positive on both sides, so the concavity does not change. In fact (0, 0) is a minimum turning point.

   A full-marks answer:

   • For y = x^⁴, y″ = 12x^² = 0 at x = 0
   • But y″ ≥ 0 on both sides (no change of concavity), so (0, 0) is a minimum, not an inflection
4. Question 4. IntegrationMultiple choice

   The area between y = x^² − 1 and the x-axis from x = 0 to x = 1 is…

   1. A0
   2. B2/3
   3. C−2/3
   4. D1/3
   Show answer

   B. 2/3

   The integral is −2/3; area is positive.

5. Question 5. Integration: exponential, reciprocal and trigonometric integrals2 marks

   Show that ∫₀^^π/2 (sin x + cos x) dx = 2.

   Show model answer and marking criteria

   Model answer. ∫ (sin x + cos x) dx = −cos x + sin x. From 0 to π/2: (−cos(π/2) + sin(π/2)) − (−cos 0 + sin 0) = (0 + 1) − (−1 + 0) = 2.

   A full-marks answer:

   • Antiderivative −cos x + sin x
   • (−0 + 1) − (−1 + 0) = 2
6. Question 6. Kinematics and rates of changeMultiple choice

   x = t^³. The acceleration at t = 2 is…

   1. A12
   2. B8
   3. C6
   4. D3
   Show answer

   A. 12

   a = 6t.

A few of the 91 Year 12 Maths Advanced flashcards. In the app they use spaced repetition: rate each card Again, Hard, Good or Easy and it comes back just before you would forget it.

• Chain ruledy/dx = dy/du × du/dx
• f′(x) > 0Increasing
• ∫ x^ⁿ dxx^^{n + 1}/(n + 1) + C, n ≠ −1
• vdx/dt

Set out every step. Senior Maths papers give most of the marks for method, so a right answer with no working can score little on a multi-mark question. Learn the standard derivatives, integrals and identities until they are automatic, then spend your time on mixed questions where you have to choose the method yourself.

Press Study for me and the app builds a 15 to 20 minute session from the cards that are due, the mistakes that are due and questions on your weakest topics. Practice questions come in three levels, and anything you get wrong comes back 1, 3 and 7 days later until you get it right. When an exam is close, sit a timed practice exam with AI marking.

Your state, your exam

Maths Advanced goes by different names around Australia. In the app the course shows under your state’s name: NSW: Maths Advanced; VIC, QLD, SA, ACT, NT: Mathematical Methods; WA, TAS: Mathematics Methods.

When you set up Year 12, you pick your state (NSW, VIC, QLD, SA, WA, TAS, ACT or NT) and your subjects. The AI then pitches explanations and marking at your certificate’s level. Courses differ between states, so compare the units above with your course outline, and turn your class notes into extra decks for anything that’s different.

Maths Advanced in Year 11

Year 11 Maths Advanced study notes and practice questions.

More Year 12 courses

Year 12 Maths Standard, Year 12 English, Year 12 Modern History, Year 12 Biology, Year 12 Chemistry, Year 12 Business Studies, Year 12 Legal Studies, Year 12 PDHPE. All Year 11–12 subjects.