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HSC Year 12 Mathematics Extension 1 practice questions

HSC Year 12 Mathematics Extension 1 practice questions

Use Skill Align for HSC Year 12 Mathematics Extension 1 practice questions and exercise questions after the relevant topic has been taught. Students can start with the pathway demo, then practise by topic and mode.

30 practice skills

HSC Year 12 Mathematics Extension 1 includes 30 practice skills across Advanced functions and proof, Calculus and applications, Vectors and complex numbers, and Extension problem solving.

Try public sample questions
Australian Years 7-12 Exercise and test mode Parent-managed access

What is a practice skill?

A practice skill is a focused topic or question type designed to help students practise one curriculum-aligned concept with instant feedback and explanations. Skill Align uses practice skills to organise questions by year level, subject, strand, and curriculum focus.

Sample HSC Year 12 Mathematics Extension 1 questions

These sample questions are visible on the page before login. They show HSC Year 12 Mathematics Extension 1 induction, vectors, binomial, trigonometry, calculus, and probability explanations before opening the demo.

HSC Year 12 Mathematics Extension 1 Mathematical induction hard sequence graph

1. To prove 1 + 3 + 5 + ... + (2n - 1) = n² by induction, which line correctly uses the k-case to prove the (k + 1)-case?

Induction step for odd-number sums Pₖ₊₁ n S
Choices
  • k² + (2k + 1) = (k + 1)²
  • k² + 2k = (k + 1)²
  • k² + (k + 1) = (k + 1)²
  • 2k + 1 = (k + 1)²
Explanation:

Assume the sum of the first k odd numbers is k^2. The next odd number is 2k + 1, so the next sum is k² + 2k + 1 = (k + 1)^2.

HSC Year 12 Mathematics Extension 1 Vectors hard vector diagram

2. In the vector diagram, a = (3, 1), b = (2, 5), and p denotes the component of b perpendicular to a. What is |p|?

Perpendicular component in a vector diagram a b p
Choices
  • 13sqrt(10) / 10
  • 11 / 10
  • √10
  • 13 / 10
Explanation:

Since b . a = 11 and a . a = 10, proj_a b = (11 / 10)a. Hence p = b - proj_a b = (-13 / 10, 39 / 10), with magnitude √1690 / 10 = 13sqrt(10) / 10.

HSC Year 12 Mathematics Extension 1 Binomial theorem hard weighted score chart

3. In the expansion of (1 - 2x)⁷, what is the coefficient of x⁴?

Highlighted binomial coefficient x³ x⁴ 560
Choices
  • 560
  • -560
  • 280
  • 35
Explanation:

The x⁴ term is C(7, 4)(-2x)⁴ = 35 x 16x⁴ = 560x⁴, so the coefficient is 560.

HSC Year 12 Mathematics Extension 1 Trigonometric equations hard unit circle

4. For 0 <= x < 2π, how many solutions does 2cos² x + cos x - 1 = 0 have?

Unit-circle solutions for cosine 1 1/2 θ
Choices
  • 3
  • 2
  • 4
  • 5
Explanation:

Factor the equation as (2cos x - 1)(cos x + 1) = 0. Thus cos x = 1 / 2 or cos x = -1. On 0 <= x < 2π, these occur at π / 3, 5π / 3 and π, giving three solutions.

HSC Year 12 Mathematics Extension 1 Integral calculus hard tangent and area graph

5. The graph shows y = x and y = x² between x = 0 and x = 1. What is the exact area between the curves?

Area between two curves P T R
Choices
  • 1 / 6
  • 1 / 3
  • 1 / 2
  • 2 / 3
Explanation:

On 0 <= x <= 1, x is above x^2. The area is the integral from 0 to 1 of x - x², which is 1 / 2 - 1 / 3 = 1 / 6.

HSC Year 12 Mathematics Extension 1 Probability hard normal curve

6. Let X be binomial with n = 8 and p = 0.35. What is E(X), and which interpretation is correct?

Binomial distribution centre A B μ
Choices
  • 2.8 expected successes
  • 0.35 expected successes
  • 8 expected successes
  • 5.2 expected failures
Explanation:

For a binomial random variable, E(X) = np. Here E(X) = 8 x 0.35 = 2.8, meaning the long-run average number of successes is 2.8 per eight trials.

For parents comparing HSC Year 12 Mathematics Extension 1 support

HSC Year 12 Mathematics Extension 1 practice should give strong students enough structure to revise induction, vectors, binomial theorem, trigonometry, calculus, and probability. These examples preview that style before the no-login Extension 1 demo.

Continue with Skill Align

Public samples are free to review before sign-in. Regular full subject access is subscription-based through a parent-managed student account, with included access to start and optional monthly subject access explained on the Pricing page. Exam packs, where available, are separate one-off PDF purchases and are not included in a practice subscription.

Open School Learning Browse Curriculum Coverage View Monthly Practice Pricing Browse HSC Mathematics Extension 1 Year 12 Exam Packs
What this practice and exercise page covers

Mathematics Extension 1 practice sits inside NSW Year 12 HSC Mathematics course structures, with Year 11 Preliminary context available through the full curriculum coverage, with Skill Align keeping the route focused on the selected maths pathway.

Senior practice is organised by pathway, unit, topic, and mode so students can revise targeted areas rather than sitting a full-paper workflow every time. Skill Align treats practice questions and exercise questions as the same learning workflow: students answer curriculum-aligned questions, review explanations, and move between exercise mode and test mode.

Start with the public sample questions to check the question style, then use the curriculum coverage page to choose a topic that matches the student's current classwork.

Preview question styles
  • Advanced functions and proof: Students practise advanced functions and proof through short targeted questions, explanations, and mode-specific feedback.
  • Calculus and applications: Students practise calculus and applications through short targeted questions, explanations, and mode-specific feedback.
  • Vectors and complex numbers: Students practise vectors and complex numbers through short targeted questions, explanations, and mode-specific feedback.
Suggested first practice steps
  • Preview the public sample practice and exercise questions before creating a saved student session.
  • Choose one focus area that has already been introduced at school.
  • Use exercise mode for immediate explanations, then test mode when the student is ready for delayed feedback.
Example focus areas
View full curriculum coverage

These examples are not the full topic list. Use the curriculum coverage page for the complete mapped pathway.

  • Advanced functions and proof
  • Calculus and applications
  • Vectors and complex numbers
  • Extension problem solving
Who it is for

NSW students studying Mathematics Extension 1.

How access works
  • Review public sample questions without signing in.
  • Create a parent-managed student account for saved practice.
  • Use included access first, then add an optional monthly subject subscription when more practice is needed.
Questions parents ask
Can students try hsc year 12 mathematics extension 1 practice questions before subscribing?

Yes. Public sample pages let visitors preview curated Skill Align questions without creating a saved student test record.

Does Skill Align replace school lessons or tutoring?

No. Skill Align is designed for structured practice after students have learned topics at school or with a teacher.

Are practice questions and exercise questions the same on Skill Align?

Yes. Families may search for either wording; Skill Align uses one curriculum-aligned practice page for both practice questions and exercise questions.

How does regular hsc year 12 mathematics extension 1 practice questions access work?

Public samples are free before sign-in. Regular full subject access uses a parent-managed account with included access to start and an optional monthly subject subscription when more practice is needed. Exam packs are separate one-off PDF purchases.

Skill Align independently prepares practice pathways aligned to publicly available curriculum and syllabus information. Official requirements should always be checked with the relevant authority.