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AQA A-level Further Maths formula booklet (7367): what is on it

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Formula sheet

The card below opens today's Further Maths paper with this sheet in the drawer. The drawer holds the whole booklet in our own layout and words, apart from the three statistical tables, and the results most often used follow: summations, matrix transformations, Maclaurin series and hyperbolic functions. Check against AQA's booklet for the statistical tables.

Formula sheet

A-level Further Maths

Open formula sheet

Summations

\sum_{r=1}^{n}r^2=\frac{1}{6}n(n+1)(2n+1)sum of the squares of the first n integers
\sum_{r=1}^{n}r^3=\frac{1}{4}n^2(n+1)^2sum of the cubes of the first n integers

Matrix transformations

anticlockwise rotation through θ about the origin. The 2 by 2 matrix with rows (cos θ, −sin θ) and (sin θ, cos θ).

Maclaurin series

f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+\cdots+\frac{x^r}{r!}f^{(r)}(0)+\cdotsMaclaurin series of a function
e^x=1+x+\frac{x^2}{2!}+\cdots+\frac{x^r}{r!}+\cdotsseries for e^x, for all x
\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots+(-1)^{r+1}\frac{x^r}{r}+\cdotsseries for ln(1 + x) (−1 < x ≤ 1)
\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots+(-1)^r\frac{x^{2r+1}}{(2r+1)!}+\cdotsseries for sin x, for all x
\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots+(-1)^r\frac{x^{2r}}{(2r)!}+\cdotsseries for cos x, for all x

Hyperbolic functions

\cosh^2x-\sinh^2x=1hyperbolic identity
\sinh2x=2\sinh x\cosh xdouble angle for sinh
\cosh2x=\cosh^2x+\sinh^2xdouble angle for cosh
\text{arcosh}\ x=\ln(x+\sqrt{x^2-1})inverse hyperbolic cosine (x ≥ 1)
\text{arsinh}\ x=\ln(x+\sqrt{x^2+1})inverse hyperbolic sine
\text{artanh}\ x=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)inverse hyperbolic tangent (|x| < 1)

What else is on the sheet

The booklet also holds these sections, which are in the drawer above but not repeated below.

SectionWhat it holds
Series and identitiesarithmetic and geometric series, the binomial series and trigonometric identities
Further pure resultscomplex numbers, vectors, differentiation and integration results, area of a sector, arc length and surface area of revolution
Conicsellipse, parabola and hyperbola
Numerical methodsnumerical solution of equations, numerical integration and further numerical integration
Mechanicsconstant acceleration and centres of mass
Probability and statisticsprobability, standard deviation, discrete distributions and sampling distributions
Statistical tablesthree tables: Student's t, chi-squared and product moment correlation (not in the drawer)

What AQA's booklet leaves for you

  • Derivations and proofs. The v1.5 booklet is a list of results with their conditions.
  • The short A-level Maths booklet's recall items are also missing here: quadratic formula, triangle rules and everyday derivatives.
  • No indication of which topics belong to AS Further Maths and which to A-level, so check your specification for what you are examined on.

AQA's one big booklet

AQA gives Further Maths candidates the larger formulae and statistical tables booklet, and says Maths candidates may also use it. If you take both subjects, you can practise from a single document, which is an advantage over boards with separate booklets.

It runs from pure mathematics, through mechanics, to probability and statistics, and then the tables. Notice that area of a sector, arc length and surface area of revolution have their own headings among the pure mathematics, not under geometry.

Use the quadratic formula to find both roots.

Press Mark to have your answer marked in the Algebra course.

Worked example: a series for e<sup>x</sup>

Use the first four terms of the Maclaurin series for eˣ to estimate e^(0.1).

e^{0.1}\approx1+0.1+\frac{0.1^2}{2!}+\frac{0.1^3}{3!}=1+0.1+0.005+0.000167=1.10517

The calculator value is 1.10517, so four terms are good to five decimal places for so small an x.

The AQA booklet and the other boards

BoardBookletNotes
EdexcelMathematical Formulae and Statistical TablesOne booklet arranged by qualification, with statistical tables for binomial, Poisson, chi-squared, correlation, t and F
AQA (this guide)Formulae and statistical tables for A-level Mathematics and A-level Further MathematicsVersion 1.5, February 2018, with statistical tables; required for Further Maths and optional for Maths
OCR AFormulae Booklet, A Level Further Mathematics A (H245)Pure, statistics, mechanics and a discrete mathematics section with network and sorting algorithms
OCR MEIFormulae Booklet, AS and A Level Further Mathematics B (MEI) (H635, H645)Holds the A-level Maths content too, and a Further Pure with Technology section
WJECGCE AS/A Level Mathematics and Further Mathematics Formula BookletOne booklet shared with Maths, formulae only
CCEAMathematical Formulae and TablesOne booklet shared with Maths, with statistical tables and a decision mathematics page

AQA is the board with the clearest split: a short booklet for Maths and this larger one for Further. Edexcel folds everything into a single document, OCR splits by specification, and WJEC and CCEA share one booklet between the two subjects.

Exam answers

On this site, answers arrive after you mark a paper. They are our own worked solutions and do not come from AQA.

Trying AQA-style Further questions

The Further Maths daily paper gives eight questions across the topics in the booklet. Treat it as a quick drill in finding an entry under pressure, not as a replica of a 7367 paper.

Booklet errors AQA candidates make

  • Using the short A-level Maths booklet by mistake and missing the Further results.
  • Forgetting the Maclaurin series are only valid for a range of x.
  • Confusing the surface area of revolution forms.

Common questions

Do you get a formula booklet in AQA A-level Further Maths?
Yes. AQA says its formulae and statistical tables booklet is required for all AS and A-level Further Maths exams.
Is the AQA Further Maths booklet the same as the Maths one?
No. The Maths exams have a shorter booklet with no tables. The larger booklet is for Further Maths, and students may also use it in Maths exams.
Does the AQA Further Maths booklet have statistical tables?
Yes. AQA's booklet is called formulae and statistical tables.
What version is the AQA Further Maths booklet?
Version 1.5, issued in February 2018, for specifications first taught from September 2017.
What is not in the AQA Further Maths booklet?
Proofs, definitions and the basic A-level Maths results such as the quadratic formula.

Revision cards

Say the answer first, then flip the card to check it.

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