A-Level Maths: Algebra and Functions
Algebra and functions is the largest single section of the A-Level Mathematics prescribed content, and its techniques are assumed knowledge in every other topic — calculus, trigonometry and series all break down without fluent algebraic manipulation. This quiz covers the section's full span.
The manipulation questions start with algebraic fractions: simplifying (x² − 4)/(x² + 4x + 4) by factorising top and bottom, with the domain restriction that must accompany the cancellation. Indices and surds follow — evaluating (8x⁶)^(1/3) through fractional index laws, and combining √72 + √18 into a single surd, with the equivalent form √162 tested alongside 9√2 to check genuine understanding rather than one memorised route.
The factor theorem gets a full workout: identifying which of five candidate linear factors divide the cubic x³ − 2x² − 5x + 6 (each requiring an evaluation of f at the candidate root), and polynomial division of the perfect cube x³ − 3x² + 3x − 1 by (x − 1), where recognising the binomial structure shortcuts the working.
The functions questions cover the A2 material: composite functions with the order-of-application trap (fg means g first), inverse functions found by rearrangement, and the elegant self-inverse question — which functions satisfy f(f(x)) = x — where 1/x and 5 − x both qualify. The modulus function appears twice: solving |2x − 5| = 3 with its two cases, and reading properties (intercepts, symmetry) from f(x) = |x| − 2.
Each explanation shows the full manipulation, and the distractors encode the classic slips: cancelled terms that were never factors, composite functions applied in the wrong order, and sign errors in the two-case modulus split.
- Simplify algebraic fractions by factorising, stating domain restrictions
- Apply fractional and negative index laws and combine surds
- Use the factor theorem and polynomial division on cubics
- Form composite and inverse functions, recognising self-inverse cases
- Solve modulus equations with the two-case method
Topic scope follows section B (Algebra and functions) of the DfE's prescribed AS and A level mathematics subject content — 100% common across all exam boards: indices, surds, quadratics, polynomial division and the factor theorem, composite and inverse functions, and the modulus function.