What does the CCEA AS 1 Pure Mathematics unit cover?

AS 1 is the pure mathematics foundation of the course. It covers algebra and functions including indices, surds, quadratics and polynomials, coordinate geometry of the straight line and the circle, sequences and series with the binomial expansion for positive integer powers, trigonometry with the sine and cosine rules and the identities, exponentials and logarithms, differentiation from first principles through to stationary points, integration as the reverse of differentiation and the area under a curve, and two-dimensional vectors.

How is the CCEA AS 1 Pure Mathematics unit assessed?

Unit AS 1 is assessed by a written examination of 1 hour 45 minutes worth 100 raw marks, with typically 6 to 10 questions. It is the pure content of AS Mathematics and is compulsory for both AS and the full A level. A calculator is allowed and a formulae sheet is provided in the examination.

What is the discriminant and what does it tell you?

The discriminant is the quantity b squared minus 4ac for a quadratic ax squared plus bx plus c. It counts the real roots: a positive discriminant gives two distinct real roots, a zero discriminant gives one repeated root, and a negative discriminant gives no real roots so the graph does not cross the x-axis. It is the standard tool for deciding how many times a line meets a curve.

What is the most common mistake in AS 1 Pure Mathematics?

Frequent errors include misapplying the laws of indices and the rule for fractional powers, forgetting the constant of integration on an indefinite integral, giving only the principal value when solving a trigonometric equation instead of every solution in the interval, and confusing the gradients of a tangent and its normal. Careful algebra and reading the question avoid most lost marks.

How should I revise the AS 1 Pure Mathematics unit?

Work topic by topic against the specification statements. Master indices, surds and quadratics first, then coordinate geometry, then the binomial expansion, then trigonometry, then exponentials and logarithms, then differentiation and integration, then vectors. Learn each definition and method precisely, drill past-paper questions under timed conditions, and practise carrying algebra through cleanly, because most marks are method marks.

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CCEA A-Level Mathematics AS 1 Pure Mathematics: a complete overview of algebra, trigonometry, calculus and vectors

A deep-dive CCEA A-Level Mathematics guide to the AS 1 Pure Mathematics unit. Covers algebra and functions, coordinate geometry and graphs, sequences and series with the binomial expansion, trigonometry, exponentials and logarithms, differentiation, integration and vectors, with the definitions and methods CCEA examines.

Generated by Claude Opus 4.818 min readCCEAUpdated 2026-06-16

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

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What this unit demands
Algebra, functions and coordinate geometry
Sequences, trigonometry and logarithms
Differentiation, integration and vectors
How this unit is examined
Check your knowledge

What this unit demands

AS 1 Pure Mathematics is the foundation of CCEA A-Level Mathematics. It builds the algebraic, geometric and calculus toolkit that every later unit relies on, from the applied mechanics and statistics of AS 2 to the harder pure content of A2 1. The examiners test two linked skills: precise recall of definitions and methods, and confident, accurate algebra carried through multi-step problems.

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

Algebra, functions and coordinate geometry

The unit opens with algebra and functions: the laws of indices and surds, solving quadratics by factorising, completing the square and the formula, the discriminant for counting roots, simultaneous equations, inequalities, and polynomial division with the factor and remainder theorems. Coordinate geometry then handles the straight line (gradients of parallel and perpendicular lines) and the circle (centre and radius by completing the square, the tangent-radius and chord properties), along with sketching and transforming standard curves.

Sequences, trigonometry and logarithms

Sequences and series introduces the binomial expansion of $(a + b)^n$ for a positive integer $n$ , with coefficients from Pascal's triangle and a general term for picking out a single coefficient. Trigonometry covers the ratios and their graphs, the sine and cosine rules and the area formula, the identities $\sin^2\theta + \cos^2\theta = 1$ and $\tan\theta = \sin\theta/\cos\theta$ , and solving equations over an interval. Exponentials and logarithms introduces $e$ and $\ln$ , the laws of logs, solving $a^x = b$ , and linearising data.

Differentiation, integration and vectors

Differentiation runs from first principles to the rule $\frac{d}{dx}x^n = nx^{n-1}$ , the gradient of a curve, tangents and normals, increasing and decreasing functions, and stationary points classified by the second derivative. Integration reverses this: indefinite integrals with the constant of integration, definite integrals, and the area under a curve including regions below the axis. Vectors closes the unit with two-dimensional components, magnitude and direction, addition and scaling, and position vectors for geometric proofs.

How this unit is examined

A typical CCEA profile for AS 1:

Algebra and manipulation. Surds, the discriminant, factorising cubics, and solving inequalities.
Calculus. Differentiating to find tangents and stationary points, and integrating to find areas and recover functions.
Trigonometry and logs. Solving equations over an interval, the sine and cosine rules, and releasing a variable from an exponent.
Geometry. Circle equations, perpendicular bisectors, and vector collinearity.

Check your knowledge

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

Rationalise the denominator of $\frac{4}{\sqrt{2}}$ . (2 marks)
State what the discriminant tells you when it is negative. (1 mark)
Find the coefficient of $x^2$ in the expansion of $(1 + 2x)^5$ . (3 marks)
Solve $\sin\theta = \tfrac{1}{2}$ for $0^\circ \le \theta \le 360^\circ$ . (2 marks)
Solve $5^x = 30$ , to three significant figures. (3 marks)
Differentiate $y = 2x^3 - 4x + 1$ . (2 marks)
Evaluate $\int_0^2 3x^2\,dx$ . (2 marks)
Find the magnitude of the vector $3\mathbf{i} + 4\mathbf{j}$ . (2 marks)

Solutions

Step 1: Rationalise the denominator by multiplying by the surd

A surd in the denominator is removed by multiplying the numerator and denominator by that surd. This works because $\sqrt{2} \times \sqrt{2} = 2$ , leaving a rational denominator.

\frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2} = 2\sqrt{2}

Final answer: $2\sqrt{2}$

Step 2: Interpret a negative discriminant geometrically

The discriminant $b^2 - 4ac$ determines the nature of the roots of a quadratic. When it is negative, the square root in the quadratic formula is imaginary, so no real solutions exist and the parabola does not touch the $x$ -axis at all.

A negative discriminant means the quadratic has no real roots, so its graph does not cross the $x$ -axis.

Final answer: No real roots; the graph does not cross the $x$ -axis.

Step 3: Identify the $x^2$ term using the binomial theorem

The binomial theorem gives the general term of $(1 + 2x)^5$ as $\binom{5}{r}(2x)^r$ . To find the coefficient of $x^2$ , set $r = 2$ and evaluate.

\binom{5}{2}(2x)^2 = 10 \times 4x^2 = 40x^2

Final answer: The coefficient of $x^2$ is $40$ .

Step 4: Solve a sine equation over a full interval

$\sin\theta = \tfrac{1}{2}$ has a principal value of $30^\circ$ . Because sine is also positive in the second quadrant, the second solution in $[0^\circ, 360^\circ]$ is found using the symmetry $180^\circ - 30^\circ$ .

\theta = 30^\circ \quad \text{and} \quad \theta = 150^\circ

Final answer: $\theta = 30^\circ$ and $\theta = 150^\circ$

Step 5: Solve an exponential equation using logarithms

When the unknown is in the exponent, take logarithms of both sides to bring it down. Using the change-of-base rule, $x = \frac{\log 30}{\log 5}$ , then evaluate with a calculator.

x = \frac{\log 30}{\log 5} = \frac{1.4771}{0.6990} \approx 2.11

Final answer: $x \approx 2.11$

Step 6: Differentiate a polynomial term by term

The power rule states $\frac{d}{dx}x^n = nx^{n-1}$ . Apply it to each term of the polynomial separately, treating the constant $1$ as having zero derivative.

\frac{dy}{dx} = 6x^2 - 4

Final answer: $\dfrac{dy}{dx} = 6x^2 - 4$

Step 7: Evaluate a definite integral using the fundamental theorem

Integrate $3x^2$ to get $x^3$ , then substitute the limits and subtract. The fundamental theorem of calculus converts the antiderivative evaluated at the upper and lower limits into the exact area.

\int_0^2 3x^2\,dx = \left[x^3\right]_0^2 = 8 - 0 = 8

Final answer: $8$

Step 8: Find the magnitude of a two-dimensional vector

The magnitude of a vector $a\mathbf{i} + b\mathbf{j}$ is $\sqrt{a^2 + b^2}$ , which follows from Pythagoras's theorem applied to the component triangle.

\left|3\mathbf{i} + 4\mathbf{j}\right| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5

Final answer: $5$

Sources & how we know this

CCEA GCE Mathematics specification — CCEA (2018)