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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 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.

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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 k2. 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
  • 13√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 = 13√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 x2. 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

Ready to continue? Use the normal Skill Align pages below to preview questions, check full curriculum coverage, or compare pricing before deciding whether to sign up.

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What this practice 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.

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 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.

Common search wording
HSC Year 12 Mathematics Extension 1 practice questionsHSC Mathematics Extension 1 practice questionsMathematics Extension 1 online practiceHSC Maths - Mathematics Extension 1 practice questions
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No. Skill Align is designed for structured practice after students have learned topics at school or with a teacher.

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Yes. Skill Align uses subject-based access, so families can start with the year level and subject the student needs now.

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