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.
Australian Years 7-12Exercise and test modeParent-managed access
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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 inductionhardsequence 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?
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 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
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 12Mathematics Extension 1Binomial theoremhardweighted score chart
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 x 16x4 = 560x4, so the coefficient is 560.
HSC Year 12Mathematics Extension 1Trigonometric equationshardunit circle
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 calculushardtangent 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?
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 12Mathematics Extension 1Probabilityhardnormal curve
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 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.
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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.
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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. Families may search for either wording; Skill Align uses one curriculum-aligned practice page for both practice questions and exercise questions.
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