Proof | HSC Extension 2

Assessment evidence

What these examples cover

These examples show assessable skills, not an official HSC topic weighting or prediction. The 2024 NSW syllabuses have their first HSC examination in 2027; students sitting the 2026 HSC continue under the applicable 2017 syllabus.

Teacher check

Teacher verification note: check the mathematics, notation, permitted technology and syllabus fit for your cohort before classroom or assessment use.

3 deterministic examples

Proof questions with worked solutions

Each question is generated from an existing curriq QDSL archetype at a fixed seed. Marks and catalog content mappings travel with the generated question.

Worked example 1

3 marksext2-proof-1

An exponential inequality Prove that if

x>0

, then

e^(3x)>1+3x

Part a · 3 marks

Give a calculus proof.

Worked solution

Answer: Apply derivatives to $f(x)=e^(3x)-1-3x$ .

Auxiliary Function: Let $f(x)=e^(3x)-1-3x$ .
Derivative Sign: $f'(x)=3(e^(3x)-1)>0$ for $x>0$ .
Inequality: $f(x)>f(0)=0$ .

Catalog reference: ext2_morris_2025_q16a_exponential_inequality · seed 41001

Worked example 2

2 marksext2-proof-1

A chained absolute-value inequality Prove

3(abs(a-b)+abs(c-b)+abs(c-d))>=3(a-d)

Part a · 2 marks

Give a direct proof.

Worked solution

Answer: Apply the triangle inequality to obtain $abs(a-c)+abs(c-d)>=abs(a-d)$ , then use $abs(a-d)>=a-d$ .

Triangle Inequality: $abs(a-b)+abs(b-c)>=abs(a-c)$
Absolute Value Bound: $abs(a-d)>=a-d$

Catalog reference: ext2_nor25_chained_triangle_inequality · seed 41001

Worked example 3

3 marksext2-proof-1

A logarithmic inequality For a>1 and b>1, prove

log_a(a b^(3))+log_b(a^(3) b)>= 8

Part a · 3 marks

Give a complete proof.

Worked solution

Answer: Set $t=log_a(b)>0$ . The left side is $2+3(t+1/t)$ , and $(t-1/t)^2>=0$ gives $t+1/t>=2$ .

Change Of Base: $log_b(a)=1/log_a(b)$ .
Nonnegative Square: $t+1/t>=2$ .
Substitute Bound: 8

Catalog reference: ext2_ind25_reciprocal_log_inequality · seed 41001

Content mapping and common errors

Proof content identifiers

MEX-P1

The Nature of Proof

Construct rigorous proofs using various techniques including contradiction, contrapositive and counterexample.

Common student mistakes

·Assuming the result during a proof by contradiction instead of negating it
·Confusing contrapositive (¬Q → ¬P) with converse (Q → P)
·Not providing a valid counterexample that disproves both the statement and its converse

MEX-P2

Further Proof by Mathematical Induction

Prove results involving inequalities and series using strong and recursive induction.

Common student mistakes

·Weak induction used where strong induction is required
·Inequality direction errors in the inductive step

Source and methodology

Examples are deterministic curriq catalog generations from existing QDSL archetypes at the published fixed seed. The generator copies only student-facing prompts, mark allocations, topic mappings, display tables or options, checked answers and marking-path results into this resource.

Review the official NSW Curriculum source ↗

From the blog

Reasoning

What Makes an HSC Band 6 Maths Question?

Marking

How to Structure an HSC Maths Marking Rubric

Assessment

How to Write an HSC Maths Assessment Task

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What these examples cover

Proof questions with worked solutions

Worked example 1

Worked example 2

Worked example 3

Proof content identifiers

The Nature of Proof

Further Proof by Mathematical Induction

Source and methodology

Other Extension 2 topics

From the blog

Generate a Proof worksheet in 30 seconds