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100+ Free A-Level Further Mathematics Practice Questions

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Show by induction sum from r = 1 to n of (2r - 1) = n^2. In the inductive step, after adding (2(k+1) - 1) to k^2, you get:

A
B
C
D
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Key Facts: A-Level Further Mathematics Exam

A*-E

Grading scale

Ofqual

May-June

Exam series

AQA, Edexcel, OCR timetable

3 boards

Specifications available

AQA, Edexcel, OCR

100

Free practice questions here

OpenExamPrep

AQA, Edexcel, OCR A-Level Further Mathematics is assessed through linear end-of-course exam papers (Year 13). Coverage spans core pure mathematics, further statistics, further mechanics, and grading uses the A*-E scale on 2026 specifications.

Sample A-Level Further Mathematics Practice Questions

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

1What is the modulus of the complex number z = 3 + 4i?
A.5
B.7
C.25
D.1
Explanation: The modulus of z = a + bi is |z| = sqrt(a^2 + b^2). For z = 3 + 4i, |z| = sqrt(9 + 16) = sqrt(25) = 5.
2What is the argument (in radians, principal value) of z = -1 + i?
A.pi/4
B.3pi/4
C.-pi/4
D.5pi/4
Explanation: z = -1 + i lies in the second quadrant. Reference angle is arctan(1/1) = pi/4, so arg(z) = pi - pi/4 = 3pi/4. The principal argument lies in (-pi, pi].
3Express z = 1 + i sqrt(3) in polar form r*e^(i*theta) with theta in principal range.
A.2 e^(i pi/6)
B.2 e^(i pi/3)
C.sqrt(2) e^(i pi/3)
D.2 e^(i pi/4)
Explanation: |z| = sqrt(1 + 3) = 2. arg(z) = arctan(sqrt(3)/1) = pi/3 (first quadrant). So z = 2 e^(i pi/3).
4Using De Moivre's theorem, what is (cos(pi/6) + i sin(pi/6))^6?
A.-1
B.1
C.i
D.-i
Explanation: By De Moivre, (cos theta + i sin theta)^n = cos(n theta) + i sin(n theta). Here n*theta = 6*(pi/6) = pi, so the result is cos pi + i sin pi = -1 + 0i = -1.
5What are the three cube roots of unity?
A.1, i, -i
B.1, -1, i
C.1, e^(2pi i/3), e^(4pi i/3)
D.1, e^(pi i/3), e^(2pi i/3)
Explanation: The nth roots of unity are e^(2pi i k/n) for k = 0, 1, ..., n-1. For n = 3: e^0 = 1, e^(2pi i/3), and e^(4pi i/3). These are equally spaced around the unit circle.
6If z1 = 2 e^(i pi/4) and z2 = 3 e^(i pi/6), what is z1 * z2 in exponential form?
A.5 e^(i pi/4)
B.6 e^(i 5pi/12)
C.6 e^(i pi/24)
D.5 e^(i 5pi/12)
Explanation: When multiplying in polar form: multiply moduli and add arguments. |z1*z2| = 2*3 = 6 and arg = pi/4 + pi/6 = 3pi/12 + 2pi/12 = 5pi/12.
7The locus |z - 2| = 3 in the Argand diagram represents which shape?
A.A line
B.A circle centred at (2, 0) with radius 3
C.A circle centred at (-2, 0) with radius 3
D.An ellipse
Explanation: |z - a| = r describes the set of points whose distance from a is r — a circle of radius r centred at a. Here a = 2 (i.e., (2, 0)) and r = 3.
8What does the locus arg(z - 1) = pi/4 represent?
A.A full line through (1, 0) with gradient 1
B.A half-line (ray) from (1, 0) at angle pi/4 to the positive real axis, excluding (1, 0)
C.A circle through (1, 0)
D.The vertical line x = 1
Explanation: arg(z - a) = theta defines a half-line (ray) starting at a, going in the direction making angle theta with the positive real axis. The point z = a itself is excluded because arg(0) is undefined.
9Find all solutions to z^4 = 16, giving them in the form a + bi.
A.2, -2, 2i, -2i
B.4, -4, 4i, -4i
C.2, 2i
D.16, -16, 16i, -16i
Explanation: 16 = 16 e^(i*2pi k). The fourth roots have modulus 16^(1/4) = 2 and arguments 2pi k/4 = 0, pi/2, pi, 3pi/2. These give 2, 2i, -2, -2i.
10Using De Moivre's theorem, express cos(3 theta) in terms of cos theta.
A.3 cos theta - 4 cos^3 theta
B.4 cos^3 theta - 3 cos theta
C.cos^3 theta - 3 sin^2 theta cos theta
D.3 cos^3 theta - cos theta
Explanation: Expanding (cos theta + i sin theta)^3 = cos^3 theta + 3 cos^2 theta (i sin theta) + 3 cos theta (i sin theta)^2 + (i sin theta)^3. Taking the real part: cos^3 theta - 3 cos theta sin^2 theta = cos^3 theta - 3 cos theta (1 - cos^2 theta) = 4 cos^3 theta - 3 cos theta.

About the A-Level Further Mathematics Exam

A-Level Further Mathematics is offered by AQA, Edexcel, OCR as part of the UK A-Level qualification framework. The course covers core pure mathematics, further statistics, further mechanics, decision mathematics and is assessed primarily through written exam papers at the end of the two-year course.

Questions

100 scored questions

Time Limit

5-7 hours total across multiple papers

Passing Score

Grade E is the minimum pass, Grades A*-E count as a pass (A*-A-B-C-D-E)

Exam Fee

£75-£130 per subject (school-set entry fee) (AQA, Edexcel, OCR)

A-Level Further Mathematics Exam Content Outline

Core

Core Pure: Complex Numbers and Matrices

Argand diagrams, polar form, De Moivre's theorem, matrix algebra, transformations

Core

Core Pure: Further Algebra and Calculus

Series, polar coordinates, hyperbolic functions, further integration, differential equations

Core

Core Pure: Vectors and Proof

Lines and planes in 3D, scalar product, vector product, induction proofs

Core

Further Statistics

Discrete distributions (Poisson, geometric, negative binomial), continuous distributions, chi-squared, regression

Core

Further Mechanics

Momentum and impulse, collisions, elastic strings and springs, centres of mass, dynamics in 2D

Core

Decision Mathematics

Algorithms (Prim's, Kruskal's, Dijkstra's), graph theory, linear programming, simulation

How to Pass the A-Level Further Mathematics Exam

What You Need to Know

  • Passing score: Grade E is the minimum pass, Grades A*-E count as a pass (A*-A-B-C-D-E)
  • Exam length: 100 questions
  • Time limit: 5-7 hours total across multiple papers
  • Exam fee: £75-£130 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

A-Level Further Mathematics Study Tips from Top Performers

1Use past papers from your specific exam board — questions follow the same style year on year
2Time yourself on full papers to build pacing for the long extended-response questions
3Build a clear understanding of mark schemes — examiners reward specific assessment objectives
4Review examiner reports each summer; common errors repeat

Frequently Asked Questions

What exam boards offer A-Level Further Mathematics?

A-Level Further Mathematics is offered by AQA, Edexcel, OCR. All boards follow Ofqual subject content but vary in the choice of set texts, optional topics, and paper structure.

When is the A-Level Further Mathematics exam taken?

Exams are written in the May-June series at the end of the two-year linear A-Level course. Most students sit the papers in Year 13.

How is A-Level Further Mathematics graded?

A-Levels are graded A*-E. A* is the highest grade and E is the minimum pass. UCAS tariff points are awarded for A-Level grades on most university applications.

How many papers does A-Level Further Mathematics have?

Most A-Level subjects have 3 written papers. The exact number, timing, and weighting depend on the chosen exam board. Some subjects also include a non-examined assessment (NEA) coursework component.