Intro
The HSC Collections booklet is a free Mathematics Extension 2 resource with 36 worked problems covering curated hard problems spanning complex numbers, integration, vectors, mechanics, and inequalities — multi-topic synthesis at exam difficulty. It is written for Extension 2 students, tutors, and educators preparing for challenging examinations and designed for structured HSC revision on Vu's Maths Hub.
This deep-dive introduces HSC Collections — browser-readable, aligned with the NESA syllabus.
Summary
The HSC Collections booklet offers a fundamentals review, 36 tiered problems with solutions, appendices, and a conclusion that distils exam habits. Open HSC Collections — no account required. Use this post to plan how to work through a 57-page booklet efficiently.
What is this booklet?
This booklet is written for Extension 2 students, tutors, and educators preparing for challenging examinations.
Focus: curated hard problems spanning complex numbers, integration, vectors, mechanics, and inequalities — multi-topic synthesis at exam difficulty.
Topics covered:
- Complex numbers and geometry
- Integration techniques
- Vector geometry in 3D
- Mechanics and particle motion
- Inequalities and optimisation
How to use it:
- Use after completing topic booklets — not as a first resource
- Attempt problems under timed conditions when possible
- Log which topic each problem draws from for targeted revision
- Tutors: select problems matching current class topics
Approximately 57 pages, CC BY 4.0, readable at HSC Collections.
Key fundamental reviews
Core ideas are embedded in the introduction and early problems. Before Part 1, ensure you can handle the following without notes:
- No separate fundamentals chapter — each problem assumes topic fluency
- Complex numbers and geometric interpretations
- Integration techniques and applications
- Vector geometry in three dimensions
- Mechanics and particle motion
- Inequalities and optimisation
Why fundamentals matter
No separate fundamentals chapter — each problem assumes topic fluency — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Complex numbers and geometric interpretations — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Integration techniques and applications — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Vector geometry in three dimensions — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Mechanics and particle motion — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Students who skip this section often repeat the same algebra errors in Part 2. Treat HSC Collections fundamentals as a closed-book quiz first.
Problems and how to use them
The HSC Collections booklet packs 36 practice problems into roughly 57 pages — well beyond a single textbook chapter. Each item includes worked solutions; many include Takeaways that highlight the method to reuse in exams.
Overall structure
Problems: numbered hard problems with hints, detailed solutions, and takeaways.
There is no separate Part 2; schedule timed past-paper questions once you finish each block.
Full problem catalogue (36 items)
Numbered hard problems — each from a different Extension 2 area. Thematic index of every problem title in the booklet:
Complex numbers (10 problems)
- Complex Square with Equilateral Triangle
- Complex 7th Root of Unity
- Complex Triangle Inequality
- Complex Numbers with Argument Condition
- Vectors and Complex Numbers
- Integer Equation with Large Exponents
- Complex Numbers and Equilateral Triangles
- Complex Numbers and Region Sketching
- Complex Numbers with Three Conditions
- Complex Numbers and Equilateral Triangle
This cluster tests complex numbers under exam pressure — after each solution, note which single-topic booklet to reopen if the method felt unfamiliar.
Integration (4 problems)
- Integral with Inverse Sine
- Integral with Cotangent
- Integration with Recurrence Relations
- Integration with Recurrence and Factorial Inequality
This cluster tests integration under exam pressure — after each solution, note which single-topic booklet to reopen if the method felt unfamiliar.
Vectors & geometry (5 problems)
- 3D Vectors and Distance
- Three Unit Vectors Optimization
- A curve C spirals 3 times around the sphere centred at the origin and with radius 3, as shown below.…
- Curve on a Sphere
- Vectors and Ratios
This cluster tests vectors & geometry under exam pressure — after each solution, note which single-topic booklet to reopen if the method felt unfamiliar.
Mechanics (7 problems)
- Two Particles in Resisting Medium
- Projectile with Quadratic Resistance
- Mechanics with Ropes and Forces
- Simple Harmonic Motion
- Projectile with Linear Resistance
- Particle Falling with Resistance
- Two Masses with Pulley
This cluster tests mechanics under exam pressure — after each solution, note which single-topic booklet to reopen if the method felt unfamiliar.
Inequalities (2 problems)
- Triangle Inequality and Rectangular Prism
- Inequalities with Exponentials and Factorials
This cluster tests inequalities under exam pressure — after each solution, note which single-topic booklet to reopen if the method felt unfamiliar.
Proof & logic (3 problems)
- Induction Proof for Series
- Irrationality of Logarithm
- Proposition Logic
This cluster tests proof & logic under exam pressure — after each solution, note which single-topic booklet to reopen if the method felt unfamiliar.
Polynomials (1 problems)
- Cube Roots of Unity and Trigonometric Products
This cluster tests polynomials under exam pressure — after each solution, note which single-topic booklet to reopen if the method felt unfamiliar.
Trigonometry (1 problems)
- Circle and Cosine Function
This cluster tests trigonometry under exam pressure — after each solution, note which single-topic booklet to reopen if the method felt unfamiliar.
Other synthesis (3 problems)
- Bar Magnet and Falling Object
- Parallelogram Geometry
- Basel problem, solved by Leonhard Euler in 1734
This cluster tests other synthesis under exam pressure — after each solution, note which single-topic booklet to reopen if the method felt unfamiliar.
What makes Collections different
Unlike single-topic booklets, Collections forces topic switching — the skill tested in late Extension 2 papers when Q14–Q16 draw from multiple syllabus areas. After each problem, note which topic booklet to revisit if the solution felt unfamiliar.
Common patterns across the booklet
- Complex numbers: 10 problems — e.g. "Complex Square with Equilateral Triangle"
- Integration: 4 problems — e.g. "Integral with Inverse Sine"
- Vectors & geometry: 5 problems — e.g. "3D Vectors and Distance"
- Mechanics: 7 problems — e.g. "Two Particles in Resisting Medium"
- Inequalities: 2 problems — e.g. "Triangle Inequality and Rectangular Prism"
- Proof & logic: 3 problems — e.g. "Induction Proof for Series"
Standout and less-seen problem types
These go beyond routine drills — expect unfamiliar wording or multi-topic synthesis:
- Two Particles in Resisting Medium: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- Complex Numbers with Argument Condition: Synthesises two or more Extension 2 topics — typical of harder trial papers and Q16-style investigation work.
- Vectors and Complex Numbers: Synthesises two or more Extension 2 topics — typical of harder trial papers and Q16-style investigation work.
- Bar Magnet and Falling Object: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- Circle and Cosine Function: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- Cube Roots of Unity and Trigonometric Products: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
Working through a large booklet
- Use after completing topic booklets — not as a first resource
- Attempt problems under timed conditions when possible
- Log which topic each problem draws from for targeted revision
- Tutors: select problems matching current class topics
Timed practice: allow about one minute per mark; write legible structure even when practising alone. Error log: tag mistakes as concept, algebra, or misreading. Rotation: do not camp on advanced tier if basics still slip.
Open HSC Collections and work steadily — 36 problems is a marathon, not a sprint.
Key appendices
This booklet has no separate appendix files — formulas and takeaways are embedded in solutions and the conclusion.
Use the conclusion section as a pre-exam checklist and keep your class formula sheet nearby while practising.
Key conclusion
Mixed practice builds exam stamina and topic-switching fluency. Attempt each problem fully before reading solutions; use takeaways to identify which booklet to revisit.
The booklet's closing section reinforces these habits:
- Use after completing topic booklets — not as a first resource
- Attempt problems under timed conditions when possible
- Log which topic each problem draws from for targeted revision
- Tutors: select problems matching current class topics
Revisit HSC Collections in the fortnight before trials and redo problems you missed on first pass.
How to study with this booklet
Fortnight before trials: three timed problems every second day; map weak topics back to dedicated booklets on vumaths.com
General principles:
- Closed-book first: attempt without notes, then check fundamentals.
- Error log: record concept vs algebra vs reading errors.
- Spaced repetition: redo missed questions after 3 and 7 days.
- Past papers last: fix weak topics here, then sit full papers timed.
Mini-FAQ
Who is the HSC Collections booklet for?
Extension 2 students, tutors, and educators preparing for challenging examinations studying Mathematics Extension 2 under the NSW HSC.
Should I read solutions before attempting problems?
Attempt Part 1 first. Use Part 2 hints only after a genuine try or partial working.
Where can I read the booklet online?
Open HSC Collections on Vu's Maths Hub — free, no account required.
How many problems are in the booklet?
Roughly 36 practice problems across 57 pages, each with worked solutions.
Is this aligned with NESA?
Topics match Mathematics Extension 2 outcomes for curated hard problems spanning complex numbers, integration, vectors, mechanics, and inequalities — multi-topic synthesis at exam difficulty. Confirm scope with your teacher and current NESA documentation.
Common mistakes to avoid
- Starting Collections before single-topic booklets — build foundations first
- Reading solutions too early on hard problems
- Not timing mixed sets — exam conditions matter
- Skipping reflection on which syllabus area each problem tests
- Rushing to advanced tiers before basic fluency — build foundations first.
Practice on Vu's Maths Hub
Open the free HSC Collections on Vu's Maths Hub — 36 problems with full worked solutions.
Related resources:
- How to use Vu's Maths Hub — Mixed revision with all booklets
- HSC Last Resorts — Another mixed challenge set
More on Vu's Maths Hub
All booklets are free for personal and school use under the CC BY 4.0 licence.
Related resources:
- HSC Complex Numbers — Topic depth after mixed practice
- HSC Integrals — Topic depth after mixed practice
Complex numbers and geometry — exam context
In NSW Mathematics Extension 2 examinations, complex numbers and geometry routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Collections booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Collections and locate items that stress complex numbers and geometry; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.
Integration techniques — exam context
In NSW Mathematics Extension 2 examinations, integration techniques routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Collections booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Collections and locate items that stress integration techniques; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.
Vector geometry in 3D — exam context
In NSW Mathematics Extension 2 examinations, vector geometry in 3d routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Collections booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Collections and locate items that stress vector geometry in 3d; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.
Mechanics and particle motion — exam context
In NSW Mathematics Extension 2 examinations, mechanics and particle motion routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Collections booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Collections and locate items that stress mechanics and particle motion; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.
Inequalities and optimisation — exam context
In NSW Mathematics Extension 2 examinations, inequalities and optimisation routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Collections booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Collections and locate items that stress inequalities and optimisation; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.
Syllabus alignment
This booklet supports Mathematics Extension 2 under the NESA syllabus. It supplements school instruction with 57 pages of extra exam-style practice — not a replacement for class teaching.
Additional exam advice
When sitting Mathematics Extension 2 exams, allocate time proportional to marks. Practise concise justification in HSC Collections — NSW markers reward clear communication. Reread the booklet conclusion the night before for a habit checklist.
Why Vu's Maths Hub
Vu's Maths Hub hosts every HSC booklet in a continuous, mobile-friendly viewer — zoom, search, no download required. Maintained by Vu Hung Nguyen; CC BY 4.0 for personal and school use.
More on Mechanics and particle motion
Return to HSC Collections and filter mentally for mechanics and particle motion. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.
More on Inequalities and optimisation
Return to HSC Collections and filter mentally for inequalities and optimisation. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.
More on Complex numbers and geometry
Return to HSC Collections and filter mentally for complex numbers and geometry. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.
More on Integration techniques
Return to HSC Collections and filter mentally for integration techniques. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.
More on Vector geometry in 3D
Return to HSC Collections and filter mentally for vector geometry in 3d. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.
More on Mechanics and particle motion
Return to HSC Collections and filter mentally for mechanics and particle motion. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.
More on Inequalities and optimisation
Return to HSC Collections and filter mentally for inequalities and optimisation. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.
More on Complex numbers and geometry
Return to HSC Collections and filter mentally for complex numbers and geometry. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.
More on Integration techniques
Return to HSC Collections and filter mentally for integration techniques. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.
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