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100+ Free HSC Maths Ext 2 Practice Questions

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2026 Statistics

Key Facts: HSC Maths Ext 2 Exam

100 marks

Total marks for the HSC Mathematics Extension 2 written examination

NESA - Assessment and reporting in Mathematics Extension 2

10 + 90

Section I objective response (10 marks) and Section II extended response (90 marks)

NESA - Assessment and reporting in Mathematics Extension 2

3 hours

Working time plus 10 minutes reading time for the exam

NESA - Assessment and reporting in Mathematics Extension 2

5 topics

Proof, Vectors, Complex Numbers, Calculus and Mechanics

NESA - Mathematics Extension 2 Stage 6 syllabus

1 unit

Mathematics Extension 2 is a 1-unit Year 12 course in the NSW HSC

NESA - Mathematics Extension 2 Stage 6 syllabus

Bands E1–E4

1-unit Extension results reported across four performance bands out of 50

NESA - HSC results and bands

$1,443

Full Fee Paying Overseas Students HSC fee per candidate (2026)

NESA - Overseas students fee schedule

100

Free original MCQ practice questions in this bank

OpenExamPrep

HSC Mathematics Extension 2 is NESA’s 1-unit Year 12 advanced mathematics course. The written exam is 100 marks over 3 hours plus 10 minutes reading time (Section I 10 marks objective response; Section II 90 marks extended response), typically scheduled about 1.50 pm–5.00 pm in 2026. Syllabus topics are Proof, Vectors, Complex Numbers, Further Integration and Mechanics. Results are reported on Extension bands E1–E4 out of 50. Entry is included in HSC enrolment; overseas full-fee HSC candidates pay $1,443. This 100-question bank is a free MCQ study adaptation with worked explanations.

Sample HSC Maths Ext 2 Practice Questions

Try these sample questions to test your HSC Maths Ext 2 exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Which statement is logically equivalent to the implication P ⇒ Q?
A.¬Q ⇒ ¬P
B.Q ⇒ P
C.¬P ⇒ ¬Q
D.P ∧ ¬Q
Explanation: The contrapositive ¬Q ⇒ ¬P is always logically equivalent to P ⇒ Q.
2To prove ∀n ∈ ℕ, P(n) by mathematical induction, which step is required after the base case?
A.Assume P(k+1) and deduce P(k)
B.Assume P(k) for some k ≥ n₀ and prove P(k+1)
C.Verify P(n) for all n ≤ 10
D.Show that P(n) is false for some n
Explanation: After verifying P(n₀), assume P(k) for some integer k ≥ n₀ (inductive hypothesis) and prove P(k+1).
3The negation of “∀x ∈ ℝ, x² ≥ 0” is which statement?
A.∀x ∈ ℝ, x² < 0
B.∃x ∈ ℝ such that x² ≥ 0
C.∃x ∈ ℝ such that x² < 0
D.∀x ∈ ℝ, x² ≤ 0
Explanation: Negating ∀x, P(x) gives ∃x, ¬P(x). Here ¬(x² ≥ 0) is x² < 0.
4Proof by contradiction that √2 is irrational typically begins by assuming which statement?
A.√2 is an integer
B.2 is prime
C.√2 > 1
D.√2 is rational
Explanation: Contradiction proofs of irrationality assume the number is rational and derive an impossibility.
5If a|b and a|c for integers a, b, c with a ≠ 0, which conclusion must follow?
A.a|(b + c)
B.b|c
C.a = 1
D.b + c = 0
Explanation: If b = aq and c = ar, then b + c = a(q + r), so a divides b + c.
6Which is a correct base case for proving ∑_{k=1}^{n} k = n(n+1)/2 for all integers n ≥ 1?
A.For n = 0 only, without checking n = 1
B.For n = 1: 1 = 1·2/2
C.For n = 2 only
D.Assume true for all n
Explanation: With n₀ = 1, the left side is 1 and the right side is 1·2/2 = 1.
7To prove P ⇒ Q by contraposition, one proves which implication?
A.Q ⇒ P
B.¬P ⇒ ¬Q
C.¬Q ⇒ ¬P
D.P ∧ Q
Explanation: Contraposition replaces P ⇒ Q with the equivalent ¬Q ⇒ ¬P.
8Suppose one proves: if n² is even then n is even, for integer n. Which method is most natural?
A.Directly expanding (2k)²
B.Induction on n
C.Assuming n is even
D.Contraposition: if n is odd then n² is odd
Explanation: The contrapositive “n odd ⇒ n² odd” is a clean direct argument and proves the claim.
9In an inductive proof that 3 divides 4^n − 1 for all integers n ≥ 1, the inductive step typically uses which identity?
A.4^{k+1} − 1 = 4(4^k − 1) + 3
B.4^{k+1} − 1 = (4^k − 1)^2
C.4^{k+1} − 1 = 4^k − 4
D.4^{k+1} − 1 = 3^k
Explanation: 4^{k+1} − 1 = 4·4^k − 1 = 4(4^k − 1) + 3. If 3|(4^k − 1), then 3 divides the right-hand side.
10Which argument correctly shows that √3 is irrational by contradiction, assuming √3 = p/q in lowest terms?
A.Then p² + q² = 3
B.Then 3q² = p², so 3|p² hence 3|p; write p = 3m, deduce 3|q, contradiction
C.Then p = 3 and q = 1 only
D.Then 3 is even
Explanation: From 3q² = p², 3 divides p² so 3 divides p; substituting yields 3 divides q, contradicting lowest terms.

About the HSC Maths Ext 2 Exam

HSC Mathematics Extension 2 is the highest-level 1-unit Year 12 mathematics course of the NSW Higher School Certificate, examined by NESA under the Mathematics Extension 2 Stage 6 Syllabus (2017). It must be studied with Mathematics Extension 1, and candidates also sit the Extension 1 paper. The Year 12 course covers Proof, Vectors, Complex Numbers, Calculus (further integration) and Mechanics. The written HSC examination runs for 3 hours plus 10 minutes reading time and is worth 100 marks: Section I has 10 marks of objective-response questions and Section II has extended response. A shared NESA mathematics reference sheet is supplied and approved calculators are permitted. Course results are reported on Extension bands E1–E4 (mark out of 50). This bank provides original multiple-choice practice adapted from the syllabus.

Assessment

The written HSC examination totals 100 marks over 3 hours plus 10 minutes reading time. Section I has objective-response questions worth 10 marks; Section II has short-answer and extended-response questions worth 90 marks (typically 37–42 items, with at least two items worth 4 or 5 marks). School assessment and the external exam each contribute to the course result. Note: this 100-question bank is an MCQ study adaptation of syllabus topics, not a replica of the official paper format.

Time Limit

3 hours of working time plus 10 minutes reading time (2026 timetable typically about 1.50 pm–5.00 pm).

Passing Score

There is no fixed pass mark. As a 1-unit Extension course, results are reported as an HSC mark out of 50 across Bands E1 to E4 (Band E4 highest, typically 45–50/50). ATAR scaling is handled separately by UAC.

Exam Fee

Included in NSW HSC enrolment for domestic students. Full Fee Paying Overseas Students (FFPOS) pay the 2026 NESA HSC fee of $1,443 AUD; there is no separate per-subject Mathematics Extension 2 fee. (NSW Education Standards Authority (NESA))

HSC Maths Ext 2 Exam Content Outline

18%

Proof

Logical connectives, implication and quantifiers, proof by contradiction and contraposition, and further mathematical induction including inequalities and divisibility results.

20%

Vectors

Vector algebra in R² and R³, magnitude and unit vectors, scalar product and projections, equations of lines and planes, and geometric applications such as angles and perpendicularity.

22%

Complex Numbers

Cartesian and polar forms, Argand geometry, conjugates and moduli, De Moivre’s theorem, nth roots and roots of unity, and sketching loci and regions in the complex plane.

22%

Calculus

Further integration: substitution, parts, partial fractions, products of trigonometric functions, inverse trigonometric integrals, and applications to areas and volumes of solids of revolution.

18%

Mechanics

Simple harmonic motion, projectiles, resisted motion under forces, and using calculus and differential equations to model displacement, velocity and acceleration.

How to Pass the HSC Maths Ext 2 Exam

What You Need to Know

  • Passing score: There is no fixed pass mark. As a 1-unit Extension course, results are reported as an HSC mark out of 50 across Bands E1 to E4 (Band E4 highest, typically 45–50/50). ATAR scaling is handled separately by UAC.
  • Assessment: The written HSC examination totals 100 marks over 3 hours plus 10 minutes reading time. Section I has objective-response questions worth 10 marks; Section II has short-answer and extended-response questions worth 90 marks (typically 37–42 items, with at least two items worth 4 or 5 marks). School assessment and the external exam each contribute to the course result. Note: this 100-question bank is an MCQ study adaptation of syllabus topics, not a replica of the official paper format.
  • Time limit: 3 hours of working time plus 10 minutes reading time (2026 timetable typically about 1.50 pm–5.00 pm).
  • Exam fee: Included in NSW HSC enrolment for domestic students. Full Fee Paying Overseas Students (FFPOS) pay the 2026 NESA HSC fee of $1,443 AUD; there is no separate per-subject Mathematics Extension 2 fee.

Keys to Passing

  • Complete 500+ practice questions
  • Score 80%+ consistently before scheduling
  • Focus on highest-weighted sections
  • Use our AI tutor for tough concepts

HSC Maths Ext 2 Study Tips from Top Performers

1Practise past HSC Mathematics Extension 2 papers under timed conditions, then use this MCQ bank to drill weak syllabus topics between full papers.
2Memorise what is on the shared NESA reference sheet versus what you must recall, especially complex-number identities and integration techniques.
3For Complex Numbers, switch fluently between Cartesian and polar form, and sketch loci before algebraically solving modulus–argument equations.
4For Calculus, drill substitution, parts and partial fractions until technique choice is automatic, then apply to volumes of solids of revolution.
5For Mechanics, always define a clear positive direction and write F = ma or a = dv/dt before substituting the force law.
6For Proof and induction, write the base case and inductive hypothesis explicitly; examiners award method marks for clear logical structure.
7Show full working in Section II even when practising MCQs mentally, because the real paper rewards method and communication marks.

Frequently Asked Questions

How is the HSC Mathematics Extension 2 examination structured?

The written paper is worth 100 marks. Section I has objective-response questions worth 10 marks; Section II has short-answer and extended-response questions worth 90 marks, typically with 37–42 items and at least two items worth 4 or 5 marks.

How long is the HSC Mathematics Extension 2 exam?

Students have 3 hours of working time plus 10 minutes reading time. In the 2026 HSC timetable the paper is typically scheduled in the afternoon window around 1.50 pm to 5.00 pm.

Is there a pass mark for HSC Mathematics Extension 2?

No. As a 1-unit Extension course, NESA reports an HSC mark out of 50 and a performance band from E1 to E4. There is no simple pass or fail mark, and the ATAR is calculated separately by UAC.

What topics are examined in HSC Mathematics Extension 2?

The Year 12 course covers Proof, Vectors, Complex Numbers, Calculus (further integration) and Mechanics. Mathematics Advanced and Extension 1 content is assumed knowledge.

Do I also sit the Extension 1 paper?

Yes. Students studying Mathematics Extension 2 must also complete the Mathematics Extension 1 HSC examination in addition to the Extension 2 paper.

Can I use a calculator in the exam?

Yes. NESA-approved calculators are permitted, and students receive the shared Mathematics Advanced / Extension 1 / Extension 2 reference sheet.

What does the exam cost?

Domestic NSW candidates do not pay a separate per-subject fee; Mathematics Extension 2 is included in HSC enrolment. Full Fee Paying Overseas Students entered for the HSC pay AUD $1,443 per student under the 2026 NESA overseas fee schedule.

Are these official NESA HSC questions?

No. These are original OpenExamPrep multiple-choice practice questions adapted from the Mathematics Extension 2 syllabus. The official exam mixes objective-response and extended-response items; NESA publishes past papers separately.