What does the advanced Pure content in OCR Maths A cover?

It covers trigonometry (the three functions and their graphs, the sine and cosine rules, the area of a triangle and exact values), radian measure with arc length, sector and segment area and the small-angle approximations, the trigonometric identities including the compound and double angle formulae and the R form, exponentials and logarithms including the number e and linearising models, the language of functions with composites and inverses, the modulus function, partial fractions, and vectors in two and three dimensions. These topics build on the foundation algebra and feed directly into calculus.

Which advanced Pure topics carry the most marks in OCR Maths A?

Trigonometry and its identities are the heaviest, because they appear in equations, proofs, integration and modelling across all three papers. Exponentials and logarithms recur in growth and decay models and in linearising data. Vectors and functions are reliable mid-tariff topics. The R form, the double angle formulae and the log laws are the highest-value single techniques, since each unlocks a whole family of questions.

What techniques recur across the OCR advanced Pure content?

Expressing a sine plus b cosine in the R form to find a maximum or solve an equation, using the three forms of cos 2A, the Pythagorean identity to reduce an equation to one function, the log laws to solve equations with an unknown exponent, linearising a model with a log graph, finding a composite or inverse function and stating its domain, splitting a rational expression into partial fractions, and the displacement, magnitude and unit-vector vector results. These are automatic by exam day.

What is the most common advanced Pure mistake in OCR Maths A?

Losing solutions to a trigonometric equation by dividing by a function instead of factorising, or by not widening the interval for a transformed argument such as 2x. Other frequent errors are taking the log of a sum, composing functions in the wrong order, claiming an inverse for a many-to-one function, omitting the second term for a repeated partial-fraction factor, and reversing a vector displacement. Working in degrees when radians are required also breaks small-angle and calculus results.

How should I revise the OCR advanced Pure content?

Work topic by topic against the specification, drilling each technique until it is automatic, then practise mixed problems under timed conditions. Keep a sheet of standard results: the sine and cosine rules, the exact values, the arc-length and sector formulae, the small-angle approximations, the compound and double angle formulae, the R form, the log laws, and the vector magnitude and unit-vector formulae. Always set the calculator to radians for trigonometric calculus and small-angle questions.

EnglandMaths

OCR A-Level Maths A Pure advanced: trigonometry, exponentials, logarithms, functions and vectors

A deep-dive OCR A-Level Mathematics A guide to the advanced Pure content: trigonometry and the sine and cosine rules, radian measure, trigonometric identities and equations, exponentials and logarithms, functions and the modulus function, partial fractions, and vectors, with the techniques OCR repeats across all three papers.

Generated by Claude Opus 4.820 min readH240/1.05-1.10Updated 2026-06-02

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

Jump to a section

What the advanced Pure content demands
Trigonometry and radians
Identities and equations
Exponentials, logarithms and functions
Partial fractions and vectors
How the advanced content is examined
Check your knowledge

What the advanced Pure content demands

The advanced pure topics in OCR A-Level Mathematics A (H240) take the algebra and graph work of the foundation content and extend it into trigonometry, exponentials, logarithms, functions and vectors. Pure content appears on all three papers, so this material is examined again and again, both on its own and woven into calculus and the applied strands. The examiners reward fluent technique with the standard identities and rules, and the judgement to choose and combine them in unfamiliar multi-step problems.

This guide walks through the advanced topics in specification order, then sets out the exam patterns OCR repeats. Each topic has a matching dot-point page with worked exam questions; this overview ties them together.

Trigonometry and radians

Trigonometry (1.05) covers the sine, cosine and tangent functions and their graphs, the sine and cosine rules, the area of a triangle as $\tfrac{1}{2}ab\sin C$ , the ambiguous case, and the exact values for $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$ . Radian measure introduces the radian, arc length $s = r\theta$ , sector area $\tfrac{1}{2}r^2\theta$ , segment area, and the small-angle approximations $\sin\theta \approx \theta$ , $\tan\theta \approx \theta$ and $\cos\theta \approx 1 - \tfrac{1}{2}\theta^2$ . Radians are essential for trigonometric calculus.

Identities and equations

Trigonometric identities and equations develops the Pythagorean identity $\sin^2\theta + \cos^2\theta \equiv 1$ and its $\sec$ and $\csc$ variants, the reciprocal functions, the compound and double angle formulae, and the $R$ form $a\sin\theta + b\cos\theta = R\sin(\theta + \alpha)$ . These let you prove identities and solve equations over an interval, the single richest source of marks in the strand.

Exponentials, logarithms and functions

Exponentials and logarithms (1.06) covers exponential functions, the number $e$ (the base for which $e^x$ is its own derivative), the natural logarithm, the three log laws, solving exponential and logarithmic equations, and linearising growth and power-law models with log graphs. Functions and the modulus function covers domain and range, composite and inverse functions, the one-to-one condition for an inverse, and solving modulus equations and inequalities.

Partial fractions and vectors

Partial fractions splits a proper rational expression into a sum over each denominator factor (including a repeated factor), which makes integration and binomial expansion straightforward. Vectors (1.10) covers vectors in two and three dimensions, magnitude and unit vectors, position vectors and displacements, and geometric applications such as collinearity and dividing a line in a ratio.

How the advanced content is examined

A typical OCR profile for the advanced pure topics:

Technique questions. Solving a triangle, evaluating a sector area, expressing in $R$ form, solving an exponential equation, finding an inverse, or computing a vector magnitude.
Multi-step problems. Combining the cosine rule with the area formula, using the double angle formulae inside an equation, linearising a model and reading the parameters, or proving collinearity.
Proof and reasoning. Proving a trigonometric identity by reducing one side, or justifying a maximum from the $R$ form.
Synoptic links. Trigonometric and exponential functions feeding into differentiation and integration in the calculus papers.

Check your knowledge

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

State the area of a triangle with sides $a$ and $b$ and included angle $C$ . (1 mark)
A sector has radius $6$ cm and angle $0.5$ radians. Find its area. (2 marks)
Express $\cos^2\theta$ in terms of $\cos 2\theta$ . (1 mark)
Solve $2^x = 10$ , giving your answer to three significant figures. (2 marks)
Given $f(x) = 4x - 1$ , find $f^{-1}(x)$ . (2 marks)
Find the magnitude of $2\mathbf{i} - 3\mathbf{j} + 6\mathbf{k}$ . (2 marks)

Solutions

Step 1: State the formula for the area of a triangle

When the height is not given, the area of a triangle can be expressed using two sides and the included angle. The formula works for any triangle, not just right-angled ones.

\tfrac{1}{2}ab\sin C

Step 2: Calculate the area of the sector

The area of a sector with radius $r$ and angle $\theta$ (in radians) is $\tfrac{1}{2}r^2\theta$ . Substituting $r = 6$ and $\theta = 0.5$ gives the result directly.

\tfrac{1}{2}(6)^2(0.5) = \tfrac{1}{2}(36)(0.5) = 9 \text{ cm}^2

Step 3: Express $\cos^2\theta$ using a double-angle identity

Rearranging the double-angle identity $\cos 2\theta = 2\cos^2\theta - 1$ isolates $\cos^2\theta$ on one side. This form is useful for integrating even powers of cosine.

\cos^2\theta = \tfrac{1}{2}(1 + \cos 2\theta)

Step 4: Solve the exponential equation using logarithms

Taking logarithms of both sides converts the exponential equation into a linear one. The change-of-base formula lets you evaluate $\log_2 10$ using base-10 logarithms.

x = \dfrac{\log 10}{\log 2} = \dfrac{1}{0.3010} \approx 3.32

Step 5: Find the inverse function by swapping and solving

To find an inverse function, write $y = f(x)$ , rearrange to make $x$ the subject, then replace $y$ with $x$ in the result. The inverse undoes what the original function does.

y = 4x - 1 \implies x = \dfrac{y + 1}{4}, \quad \text{so } f^{-1}(x) = \dfrac{x + 1}{4}

Step 6: Find the magnitude of a 3D vector

The magnitude of a vector $a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$ is found using the three-dimensional extension of Pythagoras' theorem. Each component is squared, the squares are summed, and the square root is taken.

\sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7

Sources & how we know this

OCR A Level Mathematics A (H240) specification — OCR (2017)