Pure Mathematics
What Pure Mathematics tests
Cambridge IAL Pure Mathematics papers assess algebraic manipulation, functions, coordinate geometry, trigonometry, sequences and series, calculus and vectors. P1 builds fluency in quadratics, circular measure and basic differentiation and integration; P3 extends this to modulus functions, complex numbers, implicit and parametric differentiation, integration by parts and substitution, differential equations and numerical methods. Questions are often chained, so an error in an early part can propagate; keep exact forms until the final step.
Core rules to have automatic
- For ax²+bx+c=0, the discriminant b²−4ac determines the number of real roots; completing the square gives the vertex and range.
- A function has an inverse only if it is one-to-one on its stated domain; swap x and y, solve, then state the inverse domain as the original range.
- In calculus, angles must be in radians. d/dx(sin x)=cos x and d/dx(ln x)=1/x only in radian measure.
- For definite integrals by substitution, change the limits; for integration by parts, ∫u dv = uv − ∫v du, choosing u so that du is simpler.
- A line in vector form is r = a + λb; the direction b is not the position vector of a point on the line.
Worked example
Find ∫ x ln x dx. Let u = ln x and dv/dx = x. Then du/dx = 1/x and v = x²/2. ∫ x ln x dx = (x²/2)ln x − ∫(x²/2)(1/x) dx = (x²/2)ln x − ∫ x/2 dx = (x²/2)ln x − x²/4 + C. Differentiating the answer gives x ln x + x/2 − x/2 = x ln x, which checks the result.
Marks that go missing
- Dropping the ± when solving x² = k, or rejecting a root without checking the domain.
- Integrating 1/x to ln x without the modulus when the domain includes negative x.
- Forgetting +C in an indefinite integral, or not changing limits after a substitution.
- Rounding intermediate values in a “show that” or exact-value question.
- Mixing degrees and radians in trigonometric calculus.