Intro
HSC Mathematics Extension 2 Question 16 is typically an investigation-style problem worth 15 marks — it rewards persistence, clear working, and partial results even when you do not reach a full solution. Treat it as a multi-part journey: explore small cases, spot a pattern, then prove or generalise. For NSW Year 12 Extension 2 candidates. Keywords: hardest HSC math questions, Extension 2 Q16, advanced math problem solving.
Summary
Question 16 combines syllabus areas, expects proof-level reasoning, and allocates marks for method. Read all parts before writing, test small values, record lemmas, and move on if stuck — return with fresh eyes. Practise with past HSC and trial papers at full difficulty.
Question 16 is not designed to be finished by every candidate — the mark distribution rewards partial progress across all parts. Schools often use past Q16s as assessment extension tasks; treat those as low-stakes practice before the HSC.
Key Points
- Q16 often links two topics — e.g. complex numbers with geometry, or induction with inequalities.
- Full marks may require a rigorous proof, not only a numeric answer.
- Working counts: definitions, substitutions, and partial cases earn method marks.
- Try n = 1, 2, 3 or special values to guess the form before proving.
- Leave space and return; burning 40 minutes on one part hurts the rest of the paper.
- The HSC Last Resorts booklet targets Q16-level Extension 2 practice.
Worked example
Question (simplified Q16-style). Let S(n) = 1² + 2² + … + n². Part (a) Find S(1), S(2), S(3). Part (b) Conjecture a formula for S(n). Part (c) Prove your conjecture.
Solution.
- (a) S(1) = 1, S(2) = 1 + 4 = 5, S(3) = 1 + 4 + 9 = 14.
- (b) Compare with n(n + 1)(2n + 1)/6: for n = 3, 3 × 4 × 7 / 6 = 14. Conjecture S(n) = n(n + 1)(2n + 1)/6.
- (c) Base n = 1: LHS = 1, RHS = 1 × 2 × 3 / 6 = 1.
- Assume true for n = k; add (k + 1)² to both sides and algebraically rearrange to (k + 1)(k + 2)(2k + 3)/6.
Answer. S(n) = n(n + 1)(2n + 1)/6.
Takeaway. Part (a) exists to help you conjecture — always use early parts before attacking the proof.
Exam Preparation
Build Q16 stamina gradually: start with single-topic hard questions, then mixed investigations. In the exam, allocate 35–45 minutes but do not begin Q16 with less than 50 minutes on the clock unless earlier questions were unusually fast.
- Skim the full paper. Identify Q16 structure and note which earlier parts feed later ones.
- Secure marks elsewhere first. Complete confident questions before committing deep time to Q16.
- Write partial progress. State conjectures, prove base cases, and cite lemmas — method marks add up.
Past Extension 2 papers show Q16 frequently chaining parts: (a) numeric exploration, (b) conjecture, (c) proof. Even if part (c) fails, a correct conjecture from (b) often scores. Write 'From part (a), we observe …' to signal linkage. When stuck, prove a special case of the general claim — partial generalisation earns marks. Schedule at least one full Q16 attempt per fortnight in Term 3 with a 45-minute timer.
Mini-FAQ
Should I attempt Q16 if I am running out of time?
Write any definitions, special cases, and conjectures you can in the last minutes. Blank pages score zero; partial structure often earns marks.
How do I practise at Q16 difficulty?
Use past HSC Extension 2 papers and the Last Resorts booklet. Compare every solution line-by-line with worked answers.
Does Q16 always need induction?
No — but many investigations end with induction, contradiction, or inequality proof. Be ready to switch proof technique.
Common mistakes to avoid
- Starting Q16 before scanning whether parts (b) and (c) hint at part (a).
- Giving up after part (a) when later parts are worth more marks.
- No labelled steps — markers cannot award method marks for unexplained leaps.
- Ignoring domain restrictions hidden in the setup (e.g. n must be a positive integer).
Practice on Vu's Maths Hub
Need more practice on this topic? Open the free HSC Last Resorts booklet on Vu's Maths Hub — worked examples and exam-style questions, readable in your browser with no account required. It is built for Question 16–level Extension 2 practice.
More on Vu's Maths Hub
All booklets are free for personal and school use under the CC BY 4.0 licence.
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