Calculus | HSC Extension 1

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

Calculus 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

2 marksext1-c2-2

Logarithm of an inverse tangent Differentiate

y=ln(atan(2x))

Part a · 2 marks

Find $dif y/dif x$ .

Worked solution

Answer: 2 / (atan((2 x))(1 + (4 x^2)))

Inverse Tangent Derivative: $2/(1+4x^2)$ .
Chain Rule: $2/(atan(2x)(1+4x^2))$ .

Catalog reference: ext1_jr25_log_inverse_tangent_derivative · seed 41001

Worked example 2

2 marksext1-c2-2

Inverse-tangent integral Evaluate

integral_0^(3) (1)/(3^2+x^2) dif x

Part a · 2 marks

Give the exact value.

Worked solution

Answer: pi / 12

Inverse Trig Integration: 1/3
Definite Integral: pi / 12

Catalog reference: ext1_cssa25_inverse_tangent_definite_integral · seed 41001

Worked example 3

2 marksext1-c3-2

A separable differential equation Solve

dif y/dif x=6sqrt(y)

given

y=1

when

x=0

Part a · 2 marks

Find y as a function of x and state the branch domain $x>=-1/3$ .

Worked solution

Answer: ((3 x) + 1)^2

Separate Variables: $sqrt(y)=3x+C$ .
Apply Initial Condition: $y=(3x+1)^2$ on $x>=-1/3$ .

Catalog reference: ext1_ind_2025_q11b_separable_equation · seed 41001

Content mapping and common errors

Calculus content identifiers

ME-C1

Rates of Change

Solve related rates problems; model and interpret motion using calculus.

Common student mistakes

·Forgetting the chain rule in related rates (dV/dt = dV/dr × dr/dt)
·Incorrect sign for displacement vs distance

ME-C2

Further Integration

Integrate using substitution and partial fractions; apply integration by parts.

Common student mistakes

·Forgetting to adjust the differential when making a substitution (du = f'(x)dx)
·Incorrect partial fraction decomposition setup

ME-C3

Differential Equations

Solve separable differential equations; model logistic growth; interpret slope fields.

Common student mistakes

·Not separating variables correctly before integrating
·Confusing logistic growth curve shape with exponential growth

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

Common mistakes

Common Mistakes in HSC Differential Calculus

Question design

What Makes an HSC Band 6 Maths Question?

Marking

NESA HSC Maths Marking Guidelines Explained

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What these examples cover

Calculus questions with worked solutions

Worked example 1

Worked example 2

Worked example 3

Calculus content identifiers

Rates of Change

Further Integration

Differential Equations

Source and methodology

Other Extension 1 topics

From the blog

Generate a Calculus worksheet in 30 seconds