18/06/2026

Year 12 Further Maths - CHAPTER 4: Graphs _ Relations - ANSWERS
Year 12 Further Maths - CHAPTER 4: Graphs _ Relations - QUESTIONS
HELM Stats

07/05/2026

Please complete the HW question.
Note: please look at the 260507.png in the Whiteboard sessions to check your answer. Sequences & Series, Financial Applications, Trigonometric Graphs & Applications.

14/05/2026

Complete schedule
Note: Scroll through each of the HCS exams listed below, complete each question before your test on the 21st.

Create your own exams based on the topics you want to be assessed on.

HSC Paper	Sequences & Series	Financial Applications	Trigonometric Graphs & Applications
2020	Question 12: Sum of an arithmetic series [1].	Question 26: Recurrence relations and account withdrawals [2-10].	Question 6: Range of a cosine function [11]. Question 15: Bearings and the cosine rule [12-14]. Question 19: Trigonometric identity proof [15]. Question 31: Modelling populations with sine functions [16-22].
2021	Question 14: Common difference of an arithmetic sequence [23].	Question 25: Future value interest factors for annuities [24-27]. Question 29: Compound interest and birthday deposits/withdrawals [28-32].	Question 1: Trigonometric identities [33]. Question 12: Right-angled triangles in semicircles [34]. Question 18: Sine rule and obtuse angles [35]. Question 20: Intersection of lines and trigonometric graphs [36]. Question 27: Solar power model using sine functions [37-43].
2022	Question 17: Arithmetic progression in a "house of cards" [44, 45]. Question 29: Limiting sum of a geometric series [46].	Question 21: Future value factors and investment options [47, 48]. Question 32: Reducing-balance loans and monthly repayments [49-51].	Question 3: Angle of elevation [52]. Question 14: Identifying values in a sine graph [53]. Question 23: Tides modelled by cosine equations [54-56]. Question 25: Solving trigonometric derivatives/equations [57]. Question 28: Circular sectors and shaded area [58].
2023	Question 11: Finding terms in an arithmetic sequence [59]. Question 21: Geometric sequence common ratios [60].	Question 15: Future value interest factors and contributions [61-63]. Question 25: Recurrence relations for account balances [64-69].	Question 16: Arc length and perimeter of circular shapes [70, 71]. Question 20: Solving trigonometric equations [72]. Question 22: 3D trigonometry in a rectangular prism [73, 74]. Question 26: Rate of change in a swing motion model [75, 76]. Question 30: Sketching trigonometric functions with stationary points [77-81].
2024	Question 12: Sum of terms in an arithmetic series [82]. Question 30: Range of values for a limiting sum [83].	Question 24: Savings accounts and future value factors [84-87]. Question 26: Monthly withdrawals and present value factors [88, 89].	Question 20: 3D trigonometry and tower elevation [90-92]. Question 28: Ferris wheel height model [93-96]. Question 31: Perimeter and area of regions subtended by angles [97-101].
2025	Question 13: Solving for terms in a geometric sequence [102].	Question 17: Balance owing on reducing-balance loans [103-106]. Question 20: Comparing lump sums and annuities [107, 108].	Question 15: Sound wave amplitude and period models [109-112]. Question 22: Proving complex trigonometric identities [113]. Question 28: Segment area and arc length of circular paddocks [114-117]. Question 29: Bearings and mountain height elevation [118-125]. Question 31: Number of solutions for trigonometric equations [126].

📄 HSC Mathematics Advanced Past Papers

• 2025 HSC Mathematics Advanced Exam Paper
• 2025 HSC Mathematics Advanced Marking Guidelines
• 2024 HSC Mathematics Advanced Exam Paper
• 2024 HSC Mathematics Advanced Marking Guidelines
• 2023 HSC Mathematics Advanced Exam Paper
• 2023 HSC Mathematics Advanced Marking Guidelines
• 2022 HSC Mathematics Advanced Exam Paper
• 2022 HSC Mathematics Advanced Marking Guidelines
• 2021 HSC Mathematics Advanced Exam Paper
• 2021 HSC Mathematics Advanced Marking Guidelines
• 2020 HSC Mathematics Advanced Exam Paper
• 2020 HSC Mathematics Advanced Marking Guidelines

Physics

HSC Paper	Theory of Special Relativity	Relativistic Calculations	Evidence for Special Relativity	Consequences of Special Relativity
2021	Question 4: Constant speed of light pulses between frames.	Question 16: Calculating mass decrease from Sun's energy output using E=mc². Question 28a: Distance calculation based on Earth time and dilated astronaut time. Question 35a,b: Calculating energy release from alpha decay using mass-energy equivalence.		Question 28b: Limitation on the maximum velocity of a spaceship imposed by special relativity.
2022	Question 30a: Qualitative description of how constant speed of light leads to prediction of time dilation.	Question 30b: Calculation of time dilation for light pulses on a moving train at 0.96c.		Question 20: Describing relativistic length contraction at different points on a rolling wheel.
2023		Question 22: Time dilation calculation for light pulses from a spacecraft travelling at 0.9c.
2024		Question 27b: Calculating energy released per pion in a proton-antiproton reaction using mass-energy equivalence.	Question 26: Qualitative explanation of muon survival using frames of reference (evidence for time dilation/length contraction).	Question 12: Relationship between rod speed and measured length contraction. Question 20: Comparing tick rates of atomic clocks in different satellite orbits. Question 27c: Resolving classical predictions of velocities exceeding the speed of light.
2025	Question 34a,b: Foundational theory involving measurements and determination of the speed of light.	Question 19: Calculating mass decrease in a system releasing chemical energy.		Question 20: Plausibility of particle production based on energy-to-mass conversion in accelerators. Question 32: Comprehensive analysis of the consequences for length, time, and motion with evidence.

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Additional Analysis

1. Frequency Count for Topic Areas

• Theory of Special Relativity: 3 questions
• Relativistic Calculations: 7 questions
• Evidence for Special Relativity: 1 question
• Consequences of Special Relativity: 7 questions

2. Most Commonly Assessed Subtopics

• Time Dilation: Appears consistently across multiple years (2021, 2022, 2023) as the primary calculation task.
• Mass-Energy Equivalence (E=mc²): Frequently assessed through nuclear decay or star energy output calculations (2021, 2024, 2025).
• Length Contraction: Often assessed conceptually or graphically rather than through complex arithmetic (2022, 2024, 2025).

3. Noticeable Trends Across Years

• Shift from Calculation to Analysis: Early papers (2021-2023) focused heavily on direct application of the time dilation formula. Recent papers (2024-2025) include more high-mark extended responses requiring analysis of the consequences and evidence of the theory.
• Integration with Other Topics: Relativity is increasingly integrated with Module 8 (Nuclear/Atomic Physics), specifically using E=mc² to explain particle interactions and stability.
• Graphical Interpretation: The 2024 and 2025 papers show a trend toward interpreting relativistic effects via graphs (e.g., length vs. speed) rather than simple text.

4. Predicted High-Frequency Areas

• Relativistic Momentum: While calculations for time and length are frequent, relativistic momentum is present on the formula sheet but has seen lower frequency in recent extended responses. This is a likely candidate for a future multi-mark calculation.
• Muon Evidence: As one of the few named experimental evidences in the syllabus, the muon decay experiment remains a high-probability topic for mid-range (3-4 mark) explanation questions. 04/06/2026 Particle Physics Meets Nuclear Physics. Merged HSC Trial Papers: CTRL+F to find what questions you want..
• Mass-Energy in Accelerators: Given the focus on the Standard Model in recent papers, questions regarding how energy is converted to mass in particle accelerators (using relativistic principles) are highly likely.

Nature of Light Checklist

Electromagnetic Waves

All electromagnetic radiation travels at constant speed (c) in a vacuum How electromagnetic waves are produced and propagated Prediction of speed of light using permittivity and permeability Hertz experiment (spark-gap oscillator and loop antenna) Measuring speed of electromagnetic waves experimentally

Wave Model of Light

Diffraction and interference in double-slit experiment Why small slit gaps are used Interpreting intensity graphs Constructive and destructive interference Factors affecting interference patterns Producing interference patterns experimentally Solving interference problems Polarised vs unpolarised light How light becomes polarised Solving polarisation problems Evidence for wave model of light

Quantum Model of Light

Photon model of light Energy-frequency-wavelength relationships Temperature and particle energy distribution Black body radiation concept Black bodies as perfect absorbers/emitters Black-body radiation curves Solving black-body problems Classical vs experimental black-body curves Quantised energy explanation Quantum relationships (frequency, intensity, photons) Photoelectric effect Energy and kinetic energy of photoelectrons Photoelectric and thermionic emission data analysis Solving photoelectric effect problems Graphs of photoelectric experiments Applications (e.g. solar cells) Evidence for quantum model of light

Light and Special Relativity

Inertial vs non-inertial frames Einstein’s postulates Simultaneity thought experiment Time dilation thought experiment Time dilation, length contraction, relativistic momentum Solving relativity problems Why mass objects cannot reach light speed Evidence for relativity (muons, atomic clocks) >