Pearson International A Level Pure Mathematics: FP1 Further Pure Mathematics 1

FP1 is an externally assessed modular unit used within Pearson Edexcel International A Level Pure Mathematics. The current Mathematics, Further Mathematics and Pure Mathematics specification is Issue 3. This note follows the official FP1 order and keeps qualification cash-in choices separate from the mathematical content: your centre must still confirm that the unit combination is eligible for the award you intend to claim.

Official unit scope

1. Complex numbers, Argand diagrams and polynomial roots.
2. Roots of quadratic equations and relationships between roots and coefficients.
3. Numerical solution of equations.
4. Coordinate systems and rectangular hyperbolas.
5. Matrix algebra and transformations.
6. Proof by induction and summation of series.

The specification assumes prerequisite knowledge stated for the unit, so later-unit questions may combine earlier methods without re-teaching them. Treat the list above as an integrated toolkit. A question can begin in one topic and finish in another, such as using algebra to form a model, calculus to optimise it and a graph to interpret the result.

Core reasoning and methods

• Represent a complex number algebraically for calculation and geometrically for modulus, argument and loci.
• Use a matrix determinant to test invertibility before attempting an inverse.
• For induction, identify the proposition precisely, establish the base case, assume one arbitrary case, then prove the next.

Write mathematical arguments so another reader can reproduce every transition. Define symbols that are introduced, preserve exact values until a decimal is requested, and place restrictions beside the step that creates them. Calculator use can support arithmetic, graph exploration and checking, but it does not replace a proof, derivation or required chain of working.

When a model is used, state the simplifying assumptions and interpret the answer in the original setting. A mathematically valid root may be inadmissible because it lies outside a time interval, represents a negative length, violates a probability range or conflicts with a geometric domain. The final check is therefore both algebraic and contextual.

Worked example

Solve z^2 - 4z + 13 = 0. The quadratic formula gives z = (4 plus or minus the square root of -36)/2 = 2 plus or minus 3i. On an Argand diagram the roots are the points (2, 3) and (2, -3), symmetric about the real axis because the polynomial has real coefficients. Each has modulus square root 13. Substitution checks the result directly. The diagram also makes the conjugate-root structure visible, while the algebra preserves exact values. If the question asks for an argument, use the correct quadrant rather than applying inverse tangent without a sign check.