What does the CCEA A2 1 Pure Mathematics unit cover?

A2 1 Pure Mathematics is the A2 pure unit of CCEA Further Mathematics. It covers de Moivre's theorem with the exponential form, trigonometric identities and nth roots, eigenvalues and eigenvectors of a 2x2 matrix with invariant lines, the hyperbolic functions with their identities, calculus and inverses, further calculus including Maclaurin series, improper integrals, arc length and surfaces of revolution, polar coordinates with curve sketching and area, and first-order and second-order linear differential equations with modelling.

How is CCEA A2 1 Pure Mathematics assessed?

A2 1 Pure Mathematics is assessed by a written examination of 2 hours 15 minutes worth 150 raw marks, counting for 30 percent of the full A-Level. The paper is mainly pure mathematics, and it builds directly on the AS 1 Pure unit, so complex numbers, matrices and calculus from AS are assumed knowledge. A formula and statistical tables booklet is provided.

What is de Moivre's theorem used for?

De Moivre's theorem states that the complex number cos theta plus i sin theta raised to the power n equals cos n theta plus i sin n theta. It is used to raise complex numbers to powers, to derive multiple-angle trigonometric identities by expanding with the binomial theorem, and to find the nth roots of a complex number, which lie equally spaced on a circle in the Argand diagram.

How do you solve a second-order linear differential equation?

Form the auxiliary equation a m squared plus b m plus c equals zero. Its roots give the complementary function: distinct real roots give a sum of exponentials, a repeated root gives (A plus B x) times an exponential, and complex roots give an exponential times a sine-and-cosine combination. Then find a particular integral matching the right-hand side, add it to the complementary function for the general solution, and use any conditions to fix the constants.

How should I revise the CCEA A2 1 Pure unit?

Work topic by topic on top of the AS 1 Pure content. Drill de Moivre for powers, identities and nth roots, the characteristic equation for eigenvalues and eigenvectors, the hyperbolic definitions, identities, derivatives and logarithmic inverse forms, Maclaurin series and the limit method for improper integrals, polar conversions, sketches and the half r squared area, and both the integrating-factor and auxiliary-equation methods for differential equations. Practise full method, since the A2 paper rewards structured solutions.

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CCEA A-Level Further Mathematics A2 1 Pure Mathematics: a complete overview

A deep-dive CCEA A2 Further Maths guide to the A2 1 Pure Mathematics unit: de Moivre's theorem and complex roots, eigenvalues and eigenvectors, hyperbolic functions, further calculus with Maclaurin series and improper integrals, polar coordinates, and first- and second-order differential equations.

Generated by Claude Opus 4.817 min readCCEAUpdated 2026-06-16

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this unit demands
De Moivre and complex roots
Eigenvalues and eigenvectors
Hyperbolic functions and further calculus
Polar coordinates and differential equations
How this unit is examined
Check your knowledge

What this unit demands

A2 1 Pure Mathematics is the most advanced pure unit of the course, building directly on AS 1 Pure. CCEA tests six strands: de Moivre's theorem and complex roots, the eigentheory of $2 \times 2$ matrices, the hyperbolic functions and their calculus, further calculus (series, improper integrals and lengths), polar coordinates, and the solution of differential equations. Questions are longer and reward complete, well-structured methods.

This guide walks through the six dot points, then sets out the exam patterns CCEA repeats. Each topic has a matching dot-point page with practice questions; this overview ties them together.

De Moivre and complex roots

De Moivre's theorem $(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$ , and the exponential form $z = re^{i\theta}$ , give powers, multiple-angle identities (by binomial expansion), and the $n$ th roots of a complex number, equally spaced on a circle.

Eigenvalues and eigenvectors

An eigenvector satisfies $\mathbf{A}\mathbf{v} = \lambda\mathbf{v}$ ; the eigenvalues solve the characteristic equation $\det(\mathbf{A} - \lambda\mathbf{I}) = 0$ . Each eigenvector direction is an invariant line of the transformation, and an eigenvalue of $1$ gives a line of invariant points.

Hyperbolic functions and further calculus

The hyperbolic functions come from $e^x$ , satisfy $\cosh^2 x - \sinh^2 x = 1$ , differentiate into each other (no minus sign on $\cosh$ ), and have logarithmic inverses that integrate surds. Further calculus adds Maclaurin series, improper integrals evaluated by limits, arc length $\int\sqrt{1 + (y')^2}\,dx$ and the surface area of revolution.

Polar coordinates and differential equations

Polar coordinates $(r, \theta)$ convert via $x = r\cos\theta$ , $y = r\sin\theta$ ; the area enclosed is $\frac{1}{2}\int r^2\,d\theta$ . Differential equations are solved by the integrating factor (first order) and the auxiliary equation with a particular integral (second order), and model growth, cooling and oscillations.

How this unit is examined

A typical CCEA profile for A2 1 Pure:

De Moivre. Powers, deriving an identity, or finding $n$ th roots.
Eigentheory. The characteristic equation, eigenvectors and invariant lines.
Hyperbolic and calculus. A proof or inverse derivation, a Maclaurin series, or an improper integral.
Polar. A conversion, a sketch, and an area.
Differential equations. A first-order integrating-factor problem and a second-order auxiliary-equation problem, often in a modelling context.

Check your knowledge

A mix of recall and technique questions covering the unit. Attempt them under timed conditions, then check against the solutions.

Use de Moivre to simplify $\left(\cos\frac{\pi}{8} + i\sin\frac{\pi}{8}\right)^4$ . (2 marks)
Write the characteristic equation of $\begin{pmatrix} 1 & 2 \\ 0 & 4 \end{pmatrix}$ and state its eigenvalues. (2 marks)
State the identity for $\cosh^2 x - \sinh^2 x$ . (1 mark)
Find the Maclaurin series for $\cos x$ up to the $x^4$ term. (2 marks)
Evaluate $\displaystyle\int_1^\infty x^{-2}\,dx$ . (2 marks)
State the polar area formula. (1 mark)
Solve $\dfrac{dy}{dx} + y = 0$ . (2 marks)

Solutions

Step 1: Apply de Moivre's theorem

De Moivre's theorem states that $(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$ . Multiplying the angle by the exponent simplifies the expression to a single trigonometric pair, and many special angles then reduce to a simple form.

\left(\cos\frac{\pi}{8} + i\sin\frac{\pi}{8}\right)^4 = \cos\frac{4\pi}{8} + i\sin\frac{4\pi}{8} = \cos\frac{\pi}{2} + i\sin\frac{\pi}{2} = i

Final answer: $i$

Step 2: Read eigenvalues from the characteristic equation

For an upper triangular matrix the diagonal entries are the eigenvalues directly, because the characteristic equation factorises as a product of linear terms $(d_{11} - \lambda)(d_{22} - \lambda) = 0$ . Setting each factor to zero gives the eigenvalues immediately without expanding a full determinant.

(1 - \lambda)(4 - \lambda) = 0

Final answer: $\lambda = 1$ or $\lambda = 4$

Step 3: Recall the fundamental hyperbolic identity

The hyperbolic identity $\cosh^2 x - \sinh^2 x = 1$ is the direct analogue of the circular identity $\cos^2 x + \sin^2 x = 1$ . It follows from substituting the exponential definitions of $\cosh x$ and $\sinh x$ and simplifying.

\cosh^2 x - \sinh^2 x = 1

Final answer: $\cosh^2 x - \sinh^2 x = 1$

Step 4: Construct the Maclaurin series for cos x

A Maclaurin series expands a function about $x = 0$ using the pattern $f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots$ . For $\cos x$ each successive derivative cycles through $\cos x, -\sin x, -\cos x, \sin x$ , so only the even powers survive and the signs alternate.

\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots

Final answer: $1 - \dfrac{x^2}{2} + \dfrac{x^4}{24} - \cdots$

Step 5: Evaluate the improper integral using a limit

Because the upper limit is infinity, the integral is improper and must be evaluated by replacing $\infty$ with a finite bound and taking the limit. Integrating $x^{-2}$ gives $-x^{-1}$ , and the limit as the upper bound grows without bound sends $-\frac{1}{x}$ to zero.

\int_1^\infty x^{-2}\,dx = \left[-\frac{1}{x}\right]_1^\infty = 0 + 1 = 1

Final answer: $1$

Step 6: State the polar area formula

The area enclosed between two radii is built from thin sectors, each approximated by a triangle of area $\frac{1}{2}r^2\,d\theta$ . Integrating over the angular range $\alpha$ to $\beta$ gives the total enclosed area.

A = \frac{1}{2}\int_{\alpha}^{\beta} r^2\,d\theta

Final answer: $A = \dfrac{1}{2}\displaystyle\int_{\alpha}^{\beta} r^2\,d\theta$

Step 7: Solve the first-order linear differential equation

The equation $\frac{dy}{dx} + y = 0$ can be solved either by separating variables ( $\frac{dy}{y} = -dx$ , then integrating) or by the integrating factor method. Both routes lead to an exponential solution, with an arbitrary constant $A$ absorbing the constant of integration.

y = Ae^{-x}

Final answer: $y = Ae^{-x}$

Sources & how we know this

CCEA GCE Further Mathematics specification — CCEA (2018)