Pearson International A Level Mathematics: P2 Pure Mathematics 2

P2 is an externally assessed modular unit used within Pearson Edexcel International A Level Mathematics. The current Mathematics, Further Mathematics and Pure Mathematics specification is Issue 3. This note follows the official P2 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. Proof by exhaustion and disproof by counterexample.
2. Polynomial division, factor and remainder theorems.
3. Circle equations and tangent-radius geometry.
4. Arithmetic and geometric sequences, recurrence relations and sigma notation.
5. Exponentials, logarithms, identities, differentiation, integration and area.

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

• A universal statement can be disproved by one valid counterexample, but a few confirming cases never prove it.
• Use the factor theorem to test a proposed linear factor before completing polynomial division.
• For a finite region between a curve and the axis, locate all intersections first and split the integral if the sign changes.

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

Let f(x) = x^3 - 4x^2 + x + 6. Testing x = 2 gives f(2) = 8 - 16 + 2 + 6 = 0, so (x - 2) is a factor. Division gives x^2 - 2x - 3, which factorises as (x - 3)(x + 1). The three roots are therefore -1, 2 and 3. This route is auditable because the factor theorem justifies the first factor and division accounts for the remaining degree. A graph can then be used as a check: a cubic with positive leading coefficient should cross the axis at those three simple roots and rise from negative to positive overall.