Intro
The HSC Inequalities booklet is a free Mathematics Extension 2 resource with 55 worked problems covering AM-GM, Cauchy-Schwarz, triangle inequality, integration-based inequalities, and induction proofs for bounds. It is written for Extension 2 students wanting difficult inequality problems and proof techniques and designed for structured HSC revision on Vu's Maths Hub.
This deep-dive introduces HSC Inequalities — browser-readable, aligned with the NESA syllabus.
Summary
The HSC Inequalities booklet offers a fundamentals review, 55 tiered problems with solutions, appendices, and a conclusion that distils exam habits. Open HSC Inequalities — no account required. Use this post to plan how to work through a 75-page booklet efficiently.
What is this booklet?
This booklet is written for Extension 2 students wanting difficult inequality problems and proof techniques.
Focus: AM-GM, Cauchy-Schwarz, triangle inequality, integration-based inequalities, and induction proofs for bounds.
Topics covered:
- AM-GM inequality
- Cauchy-Schwarz inequality
- Triangle inequality
- Integration-based inequalities
- Inequalities via induction
- Basic inequality properties
How to use it:
- Read fundamentals and worked examples first
- Part 1 without hints; study Takeaways
- Part 2 hints only after attempt
- Practise each inequality type separately before mixing
Approximately 75 pages, CC BY 4.0, readable at HSC Inequalities.
Key fundamental reviews
Core ideas are embedded in the introduction and early problems. Before Part 1, ensure you can handle the following without notes:
- Basic inequality properties and sign rules
- Arithmetic–geometric mean (AM-GM)
- Cauchy-Schwarz in sum and integral forms
- Triangle inequality
- Integration-based bounds
- Induction for inequality chains
Why fundamentals matter
Basic inequality properties and sign rules — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Arithmetic–geometric mean — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Cauchy-Schwarz in sum and integral forms — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Triangle inequality — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Integration-based bounds — 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 Inequalities fundamentals as a closed-book quiz first.
Problems and how to use them
The HSC Inequalities booklet packs 55 practice problems into roughly 75 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
Part 1 — detailed solutions: basic, medium, and advanced inequality proofs with detailed solutions.
Part 2 — hint-based fluency: matching tiers with upside-down hints.
Use Part 1 to learn how complete NSW HSC working is written. Use Part 2 in the fortnight before trials — hints are upside-down so you attempt first.
Part 1 (16 problems)
Advanced (6 problems)
This tier contains 6 problems aimed at extension and synthesis. Representative work includes "Exponential Bounds on Factorials" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.
- Exponential Bounds on Factorials
- Sphere Inequalities via Vector Methods
- Logarithmic Inequalities and the Limit Definition of e
- Homogeneous Inequality via Substitution and AM-GM
- Bernoulli's Inequality and Sequence Monotonicity
- The Cauchy-Schwarz Inequality for Definite Integrals
Basic (5 problems)
This tier contains 5 problems aimed at foundational fluency. Representative work includes "Arithmetic Mean-Geometric Mean Inequality" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.
- Arithmetic Mean-Geometric Mean Inequality
- AM-GM with Non-Negative Reals
- Logarithmic Inequalities and Euler's Number
- Squared Terms Inequality
- Cauchy-Schwarz Inequality Application
Medium (5 problems)
This tier contains 5 problems aimed at exam-standard reasoning. Representative work includes "Arithmetic Sequence of Reciprocals" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.
- Arithmetic Sequence of Reciprocals
- Cascading AM-GM Applications
- Inductive Sum of Squared Reciprocals
- Power Mean Inequality via QM-RMS
- Calculus and Induction for Harmonic Inequality
Part 2 (39 problems)
Advanced (17 problems)
This tier contains 17 problems aimed at extension and synthesis. Representative work includes "Bernoulli's Inequality - Weighted AM-GM" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.
- Bernoulli's Inequality - Weighted AM-GM
- Convexity and Product Constraints (Jensen)
- Cauchy-Schwarz and Sums
- Power Mean, Young's, and AM-GM
- Inductive Proof of AM-GM
- Reciprocal Polynomial with AM-GM
- Nested AM-GM Application
- Triangle Inequality - Quadratic Forms
- Complex Triangle Inequality
- Complex Modulus with Constraint
- Harmonic-Arithmetic Mean Inequality
- Logarithmic Inequality with Factorial
- Cauchy-Schwarz with Homogenization
- Bernoulli's Inequality - Power Form
- Strict Bernoulli via Induction
- Summation Inequality via Induction
- Asymmetric Cubic Inequality
Basic (9 problems)
This tier contains 9 problems aimed at foundational fluency. Representative work includes "Induction with Exponential Growth" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.
- Induction with Exponential Growth
- Vector Cauchy-Schwarz Inequality
- Algebraic Factorization Method
- Multi-Part AM-GM Application
- Substitution with Constrained Variables
- Triangle Inequality for Complex Polynomials
- Problem 21: Constrained AM-GM with Reciprocals
- Problem 22: Central Binomial Coefficient Bound
- Problem 23: AM-GM for Prism Volume Optimization
Medium (13 problems)
This tier contains 13 problems aimed at exam-standard reasoning. Representative work includes "AM-GM with Harmonic Constraint" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 2 marking expectations.
- AM-GM with Harmonic Constraint
- Refining the Wallis Product Bound
- Recurrence, Fixed Points, and Contraction Mapping
- Binomial Inequality via Induction
- Surface Area to Volume Optimization
- Cubic Sum Inequality
- Product of Sums via AM-GM
- Nested Inequalities
- Cauchy-Schwarz Bound
- Cauchy-Schwarz with Constraint
- Bernoulli's Inequality Application
- Exponential Inequality via Induction
- Reciprocal Sum Inequality via AM-HM
Common patterns across the booklet
- Complex numbers: 3 problems — e.g. "Complex Triangle Inequality"
- Integration: 5 problems — e.g. "Homogeneous Inequality via Substitution and AM-GM"
- Vectors & geometry: 2 problems — e.g. "Sphere Inequalities via Vector Methods"
- Mechanics: 3 problems — e.g. "Calculus and Induction for Harmonic Inequality"
- Inequalities: 35 problems — e.g. "Exponential Bounds on Factorials"
- Proof & logic: 2 problems — e.g. "Strict Bernoulli via Induction"
Standout and less-seen problem types
These go beyond routine drills — expect unfamiliar wording or multi-topic synthesis:
- Exponential Bounds on Factorials: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- Sphere Inequalities via Vector Methods: Synthesises two or more Extension 2 topics — typical of harder trial papers and Q16-style investigation work.
- Logarithmic Inequalities and the Limit Definition of e: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- Homogeneous Inequality via Substitution and AM-GM: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- Bernoulli's Inequality and Sequence Monotonicity: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- The Cauchy-Schwarz Inequality for Definite Integrals: Synthesises two or more Extension 2 topics — typical of harder trial papers and Q16-style investigation work.
Working through a large booklet
- Read fundamentals and worked examples first
- Part 1 without hints; study Takeaways
- Part 2 hints only after attempt
- Practise each inequality type separately before mixing
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 Inequalities and work steadily — 55 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
Inequalities appear across algebra, calculus, and complex numbers. Recognise when to use AM-GM, Cauchy-Schwarz, triangle inequality, or induction — pattern recognition matters as much as algebra.
The booklet's closing section reinforces these habits:
- Read fundamentals and worked examples first
- Part 1 without hints; study Takeaways
- Part 2 hints only after attempt
- Practise each inequality type separately before mixing
Revisit HSC Inequalities in the fortnight before trials and redo problems you missed on first pass.
How to study with this booklet
One technique per week; revisit Last Resorts booklet for Q16-style inequality synthesis
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 Inequalities booklet for?
Extension 2 students wanting difficult inequality problems and proof techniques 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 Inequalities on Vu's Maths Hub — free, no account required.
How many problems are in the booklet?
Roughly 55 practice problems across 75 pages, each with worked solutions.
Is this aligned with NESA?
Topics match Mathematics Extension 2 outcomes for AM-GM, Cauchy-Schwarz, triangle inequality, integration-based inequalities, and induction proofs for bounds. Confirm scope with your teacher and current NESA documentation.
Common mistakes to avoid
- Applying AM-GM without non-negativity conditions
- Cauchy-Schwarz with wrong vector pairing — choose terms to collapse
- Triangle inequality direction errors
- Dropping equality conditions when the question asks "when does equality hold?"
- Rushing to advanced tiers before basic fluency — build foundations first.
Practice on Vu's Maths Hub
Open the free HSC Inequalities on Vu's Maths Hub — 55 problems with full worked solutions.
Related resources:
- How to use Vu's Maths Hub — Challenge booklet timing
- HSC Last Resorts — Q16 inequality synthesis
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 Induction — Induction for bounds
- HSC Integrals — Integration-based inequalities
AM-GM inequality — exam context
In NSW Mathematics Extension 2 examinations, am-gm inequality routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Inequalities booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Inequalities and locate items that stress am-gm inequality; 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.
Cauchy-Schwarz inequality — exam context
In NSW Mathematics Extension 2 examinations, cauchy-schwarz inequality routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Inequalities booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Inequalities and locate items that stress cauchy-schwarz inequality; 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.
Triangle inequality — exam context
In NSW Mathematics Extension 2 examinations, triangle inequality routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Inequalities booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Inequalities and locate items that stress triangle inequality; 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-based inequalities — exam context
In NSW Mathematics Extension 2 examinations, integration-based inequalities routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Inequalities booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Inequalities and locate items that stress integration-based inequalities; 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 via induction — exam context
In NSW Mathematics Extension 2 examinations, inequalities via induction routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Inequalities booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Inequalities and locate items that stress inequalities via induction; 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.
Basic inequality properties — exam context
In NSW Mathematics Extension 2 examinations, basic inequality properties routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Inequalities booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Inequalities and locate items that stress basic inequality properties; 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 75 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 Inequalities — 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 Integration-based inequalities
Return to HSC Inequalities and filter mentally for integration-based inequalities. 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 via induction
Return to HSC Inequalities and filter mentally for inequalities via induction. 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 Basic inequality properties
Return to HSC Inequalities and filter mentally for basic inequality properties. 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 AM-GM inequality
Return to HSC Inequalities and filter mentally for am-gm inequality. 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 Cauchy-Schwarz inequality
Return to HSC Inequalities and filter mentally for cauchy-schwarz inequality. 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 Triangle inequality
Return to HSC Inequalities and filter mentally for triangle inequality. 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-based inequalities
Return to HSC Inequalities and filter mentally for integration-based inequalities. 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 via induction
Return to HSC Inequalities and filter mentally for inequalities via induction. 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.
No comments:
Post a Comment