Intro
The HSC Proofs booklet is a free Mathematics Extension 1 and Extension 2 resource with 38 worked problems covering direct proof, proof by contradiction, cases, divisibility, irrationality, modular arithmetic, and logical quantifiers. It is written for Extension 2 students mastering proof techniques; teachers and tutors and designed for structured HSC revision on Vu's Maths Hub.
This deep-dive introduces HSC Proofs — browser-readable, aligned with the NESA syllabus.
Summary
The HSC Proofs booklet offers a fundamentals review, 38 tiered problems with solutions, appendices, and a conclusion that distils exam habits. Open HSC Proofs — no account required. Use this post to plan how to work through a 76-page booklet efficiently.
What is this booklet?
This booklet is written for Extension 2 students mastering proof techniques; teachers and tutors.
Focus: direct proof, proof by contradiction, cases, divisibility, irrationality, modular arithmetic, and logical quantifiers.
Topics covered:
- Direct proof
- Proof by contradiction
- Mathematical induction crossover
- Proof by cases
- Divisibility
- Irrationality
- Modular arithmetic
- Parity and logical reasoning
How to use it:
- Read the Proof Primer before Part 1
- Attempt Part 1 before reading solutions; study Takeaways
- Part 2: hints after genuine attempt
- Tutors: Part 1 as worked examples, Part 2 as homework
Approximately 76 pages, CC BY 4.0, readable at HSC Proofs.
Key fundamental reviews
Core ideas are embedded in the introduction and early problems. Before Part 1, ensure you can handle the following without notes:
- Proof primer: statements, implications, and quantifiers
- Direct proof structure
- Proof by contradiction
- Proof by cases and exhaustion
- Divisibility and parity arguments
- Modular arithmetic in proofs
Why fundamentals matter
Proof primer — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Direct proof structure — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Proof by contradiction — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Proof by cases and exhaustion — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Divisibility and parity arguments — 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 Proofs fundamentals as a closed-book quiz first.
Problems and how to use them
The HSC Proofs booklet packs 38 practice problems into roughly 76 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: 21 numbered problems with detailed solutions (no difficulty tiers).
Part 2 — hint-based fluency: 17 numbered problems with hints and solution sketches.
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 (21 problems)
Problem 01 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Comparing Radical Sums by Contradiction" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Comparing Radical Sums by Contradiction
Problem 02 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "No Positive Integer Difference of One" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- No Positive Integer Difference of One
Problem 03 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Irrationality of sqrt(23)" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Irrationality of sqrt(23)
Problem 03a (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Irrationality of Non-Perfect-Square Roots" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Irrationality of Non-Perfect-Square Roots
Problem 04 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Contrapositive of Multiples of 6" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Contrapositive of Multiples of 6
Problem 05 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Divisibility of n Squared Minus One by Three" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Divisibility of n Squared Minus One by Three
Problem 06 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Odd Squares and Divisibility by 8" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Odd Squares and Divisibility by 8
Problem 07 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Quadratic Residues Modulo 5" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Quadratic Residues Modulo 5
Problem 08 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Contradiction with a + b <= 5" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Contradiction with a + b <= 5
Problem 09 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Last-Two-Digits Test for 4" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Last-Two-Digits Test for 4
Problem 10 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Parity of a Difference of Squares" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Parity of a Difference of Squares
Problem 11 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Impossible Exponential Diophantine Equation" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Impossible Exponential Diophantine Equation
Problem 12 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Nested Radicals and a Cosine Formula" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Nested Radicals and a Cosine Formula
Problem 12a (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Continued Fraction for 1 + sqrt(2)" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Continued Fraction for 1 + sqrt(2)
Problem 13 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Irrationality of Log Base n of n Plus One" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Irrationality of Log Base n of n Plus One
Problem 14 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Power Difference Divisible by x Minus y" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Power Difference Divisible by x Minus y
Problem 15 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Divisibility by 6: Criterion and Consequence" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Divisibility by 6: Criterion and Consequence
Problem 15a (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Density of Q in R" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Density of Q in R
Problem 15b (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Irrationals Between Any Two Rationals" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Irrationals Between Any Two Rationals
Problem 16 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Rationality of Cube Roots" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Rationality of Cube Roots
Problem 17 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Irrationality of 2 + 3 + 5" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Irrationality of 2 + 3 + 5
Part 2 (17 problems)
Problem 01 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Parity of a Cubed Minus a Plus One" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Parity of a Cubed Minus a Plus One
Problem 02 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Rational Plus Irrational Is Irrational" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Rational Plus Irrational Is Irrational
Problem 03 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "An Irrational Power That Is Rational" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- An Irrational Power That Is Rational
Problem 04 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Consecutive Divisibility Is Impossible" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Consecutive Divisibility Is Impossible
Problem 05 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Mersenne Prime Exponent Cannot Be Even" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Mersenne Prime Exponent Cannot Be Even
Problem 06 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Divisibility of a Shifted Power Expression" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Divisibility of a Shifted Power Expression
Problem 07 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Difference of Squares Cannot Equal 1" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Difference of Squares Cannot Equal 1
Problem 08 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Divisibility of a Product" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Divisibility of a Product
Problem 09 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Extreme Arguments of a Complex Disk" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Extreme Arguments of a Complex Disk
Problem 10 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Evenness Equivalence for x and x Squared" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Evenness Equivalence for x and x Squared
Problem 11 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Product of Two Irrationals: Counterexample" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Product of Two Irrationals: Counterexample
Problem 12 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Parity Obstruction in Pythagorean Triples" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Parity Obstruction in Pythagorean Triples
Problem 13 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Exponential Comparison of Three Four and Five" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Exponential Comparison of Three Four and Five
Problem 14 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Cubic Sums Bound a Quartic Term" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Cubic Sums Bound a Quartic Term
Problem 15 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Induction for a Divisibility Pattern Modulo Ten" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Induction for a Divisibility Pattern Modulo Ten
Problem 16 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Leading Digits of Powers of Two" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Leading Digits of Powers of Two
Problem 17 (1 problems)
This tier contains 1 problem aimed at extension and synthesis. Representative work includes "Divisibility by 9 via Digit Sum" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Divisibility by 9 via Digit Sum
Common patterns across the booklet
- Complex numbers: 1 problem — e.g. "Extreme Arguments of a Complex Disk"
- Inequalities: 1 problem — e.g. "Cubic Sums Bound a Quartic Term"
- Proof & logic: 11 problems — e.g. "Comparing Radical Sums by Contradiction"
- Polynomials: 1 problem — e.g. "Rationality of Cube Roots"
- Trigonometry: 1 problem — e.g. "Nested Radicals and a Cosine Formula"
- Sequences & series: 1 problem — e.g. "Divisibility by 6: Criterion and Consequence"
Standout and less-seen problem types
These go beyond routine drills — expect unfamiliar wording or multi-topic synthesis:
- Comparing Radical Sums by Contradiction: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- No Positive Integer Difference of One: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- Irrationality of sqrt(23): A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- Irrationality of Non-Perfect-Square Roots: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- Contrapositive of Multiples of 6: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- Divisibility of n Squared Minus One by Three: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
Working through a large booklet
- Read the Proof Primer before Part 1
- Attempt Part 1 before reading solutions; study Takeaways
- Part 2: hints after genuine attempt
- Tutors: Part 1 as worked examples, Part 2 as homework
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 Proofs and work steadily — 38 problems is a marathon, not a sprint.
Key appendices
De Morgan's laws — Logical equivalences for negating compound statements. Use for quick reference or enrichment beyond routine exam questions.
Key conclusion
Proof writing is a skill built through repetition. Read the primer, study Takeaways in Part 1, and use Part 2 to practise starting proofs without reading full solutions first.
The booklet's closing section reinforces these habits:
- Read the Proof Primer before Part 1
- Attempt Part 1 before reading solutions; study Takeaways
- Part 2: hints after genuine attempt
- Tutors: Part 1 as worked examples, Part 2 as homework
Revisit HSC Proofs in the fortnight before trials and redo problems you missed on first pass.
How to study with this booklet
Two problems per day from Part 1; rotate techniques weekly; pair with Induction booklet for overlap questions
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 Proofs booklet for?
Extension 2 students mastering proof techniques; teachers and tutors studying Mathematics Extension 1 and 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 Proofs on Vu's Maths Hub — free, no account required.
How many problems are in the booklet?
Roughly 38 practice problems across 76 pages, each with worked solutions.
Is this aligned with NESA?
Topics match Mathematics Extension 1 and Extension 2 outcomes for direct proof, proof by contradiction, cases, divisibility, irrationality, modular arithmetic, and logical quantifiers. Confirm scope with your teacher and current NESA documentation.
Common mistakes to avoid
- Starting induction when direct proof is simpler
- Incomplete cases — missing a branch in proof by cases
- Confusing contrapositive with contradiction
- Quantifier errors: "for all" vs "there exists"
- Rushing to advanced tiers before basic fluency — build foundations first.
Practice on Vu's Maths Hub
Open the free HSC Proofs on Vu's Maths Hub — 38 problems with full worked solutions.
Related resources:
- How to use Vu's Maths Hub — Proof topics on the hub
- HSC Induction — Induction as a core proof method
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 Inequalities — Inequality proofs
- HSC Last Resorts — Harder proof-style questions
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