AQA A-Level Further Mathematics Core pure: a complete overview of complex numbers, matrices, calculus and proof

What Core pure actually demands

Core pure is the compulsory heart of AQA A-Level Further Mathematics. It extends A-Level Mathematics with new mathematical objects and deeper techniques, and supplies the tools that every applied option draws on. The examiners test two linked skills: fluent, accurate manipulation, and clear logical presentation, because method marks dominate.

This guide walks through all nine Core pure topics, then sets out the exam patterns AQA repeats. Each topic has a matching dot-point page with worked questions; this overview ties them together.

Complex numbers and matrices

Complex numbers introduce $z = x + yi$ with $i^2 = -1$, their arithmetic and conjugates, the Argand diagram, modulus-argument and exponential form, and de Moivre's theorem $(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$. From this come powers, the $n$ roots of unity, multiple-angle identities and loci such as circles and half-lines.

Matrices cover arithmetic, determinants, the inverse of $2 \times 2$ and $3 \times 3$ matrices, matrices as linear transformations, invariant points and lines, and solving systems of linear equations. The determinant decides whether an inverse exists and gives the area or volume scale factor of a transformation.

Algebra, calculus and vectors

Further algebra and functions extends work on series, the relationships between roots and coefficients of polynomials, and the method of differences for summing series.

Further calculus adds improper integrals (evaluated as limits), volumes of revolution, the mean value of a function, arc length $\int\sqrt{1 + (dy/dx)^2}\,dx$, surface area of revolution, integration by partial fractions, and the Maclaurin series of standard functions.

Further vectors treats lines and planes in three dimensions with the scalar and vector products, angles, intersections and shortest distances, including between skew lines.

Polar coordinates and hyperbolic functions

Polar coordinates describe a point as $(r, \theta)$, link to Cartesian coordinates by $x = r\cos\theta$ and $y = r\sin\theta$, allow sketching of cardioids and spirals, and give the enclosed area $\frac{1}{2}\int r^2\,d\theta$.

Hyperbolic functions are defined from exponentials, with $\cosh^2 x - \sinh^2 x = 1$, logarithmic inverse forms, and derivatives that mirror the trig case with sign changes. They produce standard integrals such as $\int \frac{1}{\sqrt{x^2 + 1}}\,dx = \operatorname{arsinh} x + c$.

Differential equations and proof by induction

Differential equations are solved in two strands: first order linear equations with an integrating factor $e^{\int P\,dx}$, and second order constant-coefficient equations using the auxiliary equation, the complementary function and a particular integral, with applications to damped and forced oscillations.

Proof by induction demands a clearly stated base case, an inductive step using the assumption $P(k)$ to prove $P(k+1)$, and a precise conclusion, applied to summation formulae, divisibility, recurrence relations and powers of matrices.

How Core pure is examined

A typical AQA profile for Core pure:

• Short technique questions. Dividing complex numbers, finding a determinant, differentiating a hyperbolic function, or finding an integrating factor.
• Multi-step problems. Roots of unity and loci, $3 \times 3$ inverses and invariant lines, volumes of revolution and arc lengths, and full second order differential equations with boundary conditions.
• Proof. Rigorous induction proofs for sums, divisibility and matrix powers, where the conclusion wording earns marks.
• Synoptic links. Vectors with geometry, complex numbers with trigonometric identities, and calculus feeding the applied options.

Check your knowledge

A mix of technique and reasoning questions across Core pure. Attempt them under timed conditions, then check against the solutions.

1. Evaluate $\dfrac{2 + i}{1 - i}$. (3 marks)
2. State de Moivre's theorem. (1 mark)
3. Find the determinant of $\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$. (2 marks)
4. Evaluate the improper integral $\int_1^{\infty} \frac{1}{x^2}\,dx$. (3 marks)
5. Write down the arc length integral for a curve $y = f(x)$ from $x = a$ to $x = b$. (1 mark)
6. State the identity linking $\cosh^2 x$ and $\sinh^2 x$. (1 mark)
7. Find the integrating factor for $\frac{dy}{dx} + 4y = x$. (2 marks)
8. Verify the base case for $\sum_{r=1}^{n} r = \frac{n(n+1)}{2}$. (2 marks)