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.
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Australian Years 7-12Exercise and test modeParent-managed access
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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 12Mathematics Extension 1Mathematical inductionhardtext
1. To prove 1 + 3 + 5 + ... + (2n - 1) = n² by induction, which line correctly uses the k-case to prove the (k + 1)-case?
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 12Mathematics Extension 1Vectorshardvector 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|?
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 12Mathematics Extension 1Binomial theoremhardtext
3. In the expansion of (1 - 2x)⁷, what is the coefficient of x⁴?
Choices
560
-560
280
35
Explanation:
The x⁴ term is C(7, 4)(-2x)⁴ = 35 × 16x⁴ = 560x⁴, so the coefficient is 560.
HSC Year 12Mathematics Extension 1Trigonometric equationshardtext
4. For 0 <= x < 2π, how many solutions does 2cos² x + cos x - 1 = 0 have?
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 12Mathematics Extension 1Integral calculushardtext
5. For y = x and y = x² between x = 0 and x = 1, what is the exact area between the curves?
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 12Mathematics Extension 1Probabilityhardtext
6. Let X be binomial with n = 8 and p = 0.35. What is E(X), and which interpretation is correct?
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 × 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.
From preview to regular practice
Continue with HSC Year 12 Mathematics Extension 1 in 3 steps
Public samples are free to review before sign-in. Regular full subject access is subscription-based
through a parent-managed student account. Choose the required monthly Practice Plan items on the
Pricing page, subscribe with a parent account, then create the student's separate login from the parent dashboard.
Exam packs, where available, are separate one-off PDF purchases and are not included in a practice subscription.
1
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2
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3
Start student practice
Create the student's separate login from the parent dashboard, then start exercise or test mode.
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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.
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 account and choose the required monthly Practice Plan items.
Create the student's separate login from the parent dashboard after subscribing.
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 requires the relevant monthly Practice Plan items, managed through a parent account with a separate student login. 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.
