100+ Free AS-Level Mathematics Practice Questions

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Question 1

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Differentiate f(x) = x^2 from first principles. Which is the correct limit expression?

lim h->0 [(x+h)^2 - x^2] / h^2

lim h->0 [(x+h)^2 - x^2] / h = 2x

lim x->0 [(x+h)^2 - x^2] / h

lim h->0 [x^2 + h^2 - x^2] / h

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Key Facts: AS-Level Mathematics Exam

A-E

Grading scale (no A*)

Ofqual

2 papers

1 hour 30 minutes each, 100 marks each

AQA, Edexcel, OCR specifications

~50%

Pure Mathematics weighting

AQA 7356 / Edexcel 8MA0 / OCR H230

100

Free practice questions here

OpenExamPrep

AS-Level Mathematics (AQA 7356 / Edexcel 8MA0 / OCR H230) is the Year 12 standalone qualification covering Pure Mathematics, Statistics and Mechanics. Two 1h 30m calculator papers; graded A-E (no A*) on 2026 specifications.

Sample AS-Level Mathematics Practice Questions

Try these sample questions to test your AS-Level Mathematics exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Simplify (3x^2)^3 * x^4.

A.27x^10

B.9x^10

C.27x^9

D.27x^24

Explanation: (3x^2)^3 = 3^3 * x^(2*3) = 27x^6. Then multiply by x^4: 27x^6 * x^4 = 27x^(6+4) = 27x^10.

2Simplify sqrt(50) + sqrt(18).

A.8 sqrt(2)

B.sqrt(68)

C.5 sqrt(2) + 3 sqrt(2)

D.2 sqrt(17)

Explanation: sqrt(50) = sqrt(25*2) = 5 sqrt(2) and sqrt(18) = sqrt(9*2) = 3 sqrt(2). Adding gives 5 sqrt(2) + 3 sqrt(2) = 8 sqrt(2).

3Rationalise the denominator: 4 / (sqrt(5) + 1).

A.sqrt(5) - 1

B.sqrt(5) + 1

C.(sqrt(5) - 1)/4

D.4(sqrt(5) - 1)

Explanation: Multiply numerator and denominator by the conjugate (sqrt(5) - 1): 4(sqrt(5) - 1) / (5 - 1) = 4(sqrt(5) - 1)/4 = sqrt(5) - 1.

4Solve x^2 - 7x + 12 = 0.

A.x = 3 or x = 4

B.x = -3 or x = -4

C.x = 2 or x = 6

D.x = 1 or x = 12

Explanation: Factorise: (x - 3)(x - 4) = 0, so x = 3 or x = 4. Check: 3 + 4 = 7 (matches middle term) and 3 * 4 = 12 (constant).

5Write x^2 + 8x - 3 in the form (x + a)^2 + b.

A.(x + 4)^2 - 19

B.(x + 4)^2 - 3

C.(x + 8)^2 - 3

D.(x - 4)^2 - 19

Explanation: Half of 8 is 4, so (x + 4)^2 = x^2 + 8x + 16. To keep the original expression, subtract 16: (x + 4)^2 - 16 - 3 = (x + 4)^2 - 19.

6The discriminant of 2x^2 + 5x + k = 0 is zero. Find k.

A.25/8

B.8/25

C.5/2

D.25/4

Explanation: Discriminant = b^2 - 4ac = 25 - 4(2)(k) = 25 - 8k. Set equal to zero: 25 - 8k = 0, so k = 25/8.

7Solve the simultaneous equations 2x + y = 7 and x - y = 2.

A.x = 3, y = 1

B.x = 1, y = 3

C.x = 2, y = 3

D.x = 3, y = -1

Explanation: Add the equations: 3x = 9, so x = 3. Substitute back: 2(3) + y = 7, so y = 1. Check second equation: 3 - 1 = 2. Correct.

8Solve the simultaneous equations y = x + 1 and x^2 + y^2 = 13. Find one pair (x, y).

A.(2, 3)

B.(3, 2)

C.(1, 0)

D.(0, 1)

Explanation: Substitute: x^2 + (x+1)^2 = 13, expand: 2x^2 + 2x + 1 = 13, so x^2 + x - 6 = 0, (x+3)(x-2) = 0. Take x = 2, giving y = 3.

9Solve the inequality x^2 - 5x + 6 > 0.

A.x < 2 or x > 3

B.2 < x < 3

C.x < 3

D.x > 2

Explanation: Factorise: (x - 2)(x - 3) > 0. The parabola opens upwards with roots 2 and 3, so the expression is positive outside the roots: x < 2 or x > 3.

10When f(x) = x^3 - 4x^2 + x + 6 is divided by (x - 2), the remainder is...

A.0

B.4

C.-2

D.6

Explanation: By the remainder theorem, remainder = f(2) = 8 - 16 + 2 + 6 = 0. Therefore (x - 2) is a factor of f(x).

About the AS-Level Mathematics Exam

AS-Level Mathematics is a standalone Year 12 qualification covering the first half of full A-Level Mathematics content. The course is split into Pure Mathematics (about 50%) plus Statistics and Mechanics (about 25% each). Assessment is via two 1 hour 30 minute calculator papers worth 100 marks each.

Questions

100 scored questions

Time Limit

3 hours total (2 x 1h 30m papers)

Passing Score

Grade E is the minimum pass; AS grades A-E (no A*)

Exam Fee

GBP 50-100 per subject (school-set entry fee) (AQA, Edexcel (Pearson), OCR)

AS-Level Mathematics Exam Content Outline

~50%

Pure Mathematics

Algebra (indices, surds, quadratics, simultaneous equations, inequalities, polynomials), coordinate geometry (lines, circles), sequences and series including binomial expansion, trigonometry (sine and cosine rules, identities, equations), exponentials and logarithms, differentiation, integration, 2D vectors, proof

~25%

Statistics

Sampling methods (random, stratified, systematic, opportunity, quota), data presentation (histograms, box plots, cumulative frequency, scatter diagrams), measures of location and spread, correlation and regression, probability, discrete uniform and binomial distributions, hypothesis testing for binomial proportion

~25%

Mechanics

Kinematics for constant acceleration (suvat equations, displacement-time and velocity-time graphs, vertical motion under gravity g = 9.8), forces and Newton's three laws (F = ma, weight, friction, tension), connected particles and pulleys, variable acceleration using calculus, moments and equilibrium of rigid bodies

How to Pass the AS-Level Mathematics Exam

What You Need to Know

Passing score: Grade E is the minimum pass; AS grades A-E (no A*)
Exam length: 100 questions
Time limit: 3 hours total (2 x 1h 30m papers)
Exam fee: GBP 50-100 per subject (school-set entry fee)

Keys to Passing

Complete 500+ practice questions
Score 80%+ consistently before scheduling
Focus on highest-weighted sections
Use our AI tutor for tough concepts

AS-Level Mathematics Study Tips from Top Performers

1Memorise the suvat equations and conditions (constant acceleration only) - they appear on both Mechanics and applied questions

2Practise binomial hypothesis testing carefully: define H0 and H1, compute P(X >= r) under H0, then compare to the significance level

3Use past papers from your specific board (AQA 7356 / Edexcel 8MA0 / OCR H230); question style varies and so does the formula booklet

4Master completing the square - it unlocks quadratic graphs, circle equations and discriminant problems all in one technique

Frequently Asked Questions

What is the difference between AS-Level and A-Level Mathematics?

AS-Level Mathematics covers only the Year 12 content (about half of full A-Level). It is a standalone qualification that does NOT count toward the final A-Level grade since the 2017 reform. Students who sit AS receive an AS certificate graded A-E.

How many papers does AS-Level Mathematics have?

All three boards (AQA 7356, Edexcel 8MA0, OCR H230) use two 1 hour 30 minute papers worth 100 marks each. Paper 1 is Pure Mathematics; Paper 2 is Pure, Statistics and Mechanics (or split Pure + Statistics + Mechanics depending on board). A calculator is permitted on both papers.

Is AS-Level Mathematics graded A*-E?

No. AS-Level grades range from A to E only; there is no A* grade at AS. Full A-Levels are graded A*-E. AS grades also carry fewer UCAS tariff points than A-Level grades.

What calculator is allowed for AS-Level Mathematics?

A scientific or graphical calculator is permitted on both papers. The specifications require a calculator with iterative function, statistical distribution functions (binomial), and the ability to compute summary statistics. Symbolic algebra (CAS) calculators are NOT allowed.