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CCEA A-level Further Maths formulae and tables: 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 all the formulae pages and the cycle indices in our own layout and words, apart from the six statistical tables, and the results most often used follow: summations, matrix transformations, Maclaurin series and hyperbolic functions. Check against CCEA's Mathematical Formulae and Tables 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, except the statistical tables, which the drawer leaves out.

SectionWhat it holds
Complex numbersresults for complex numbers
Vectors and matrix transformationsvector results and 2-D and 3-D transformations
Differentiation and integrationinverse trigonometric and hyperbolic results, and the area of a sector
Mechanicsmotion in a circle, centres of mass and the universal law of gravitation
Probability and statisticsprobability, discrete and continuous distributions, expectation algebra, sampling, correlation and regression, and non-parametric tests
Statistical tablesnormal, binomial, Poisson, chi-squared, Student t and correlation tables
Discrete and decision mathematicscycle indices for 3-D rotational symmetry groups

What CCEA's booklet does not do

  • No proofs. CCEA lists results, such as summations and series, without derivation.
  • The A-level Maths recall list is not repeated: quadratic formula, triangle rules and routine derivatives.
  • No help choosing between the pure content for Further Maths and the decision mathematics page.

A booklet with tables and a decision page

CCEA's Mathematical Formulae and Tables was first used in 2018 and serves A-level Maths and Further Maths. It lists six statistical tables and a discrete and decision mathematics page with cycle indices for 3D rotational symmetry groups.

The contents page lists everything in order, so use it. Pure content runs from mensuration and summations through complex numbers, Maclaurin series, hyperbolic functions, vectors and matrices.

Use the quadratic formula to find both roots.

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

Worked example: a Maclaurin series for ln(1 + x)

Use the first three terms of the series to estimate ln 1.2.

\ln(1.2)\approx0.2-\frac{0.2^2}{2}+\frac{0.2^3}{3}=0.2-0.02+0.00267=0.1827

The calculator gives 0.1823, so three terms are good to three decimal places. The series is valid because |x| = 0.2 is less than 1.

The CCEA 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
AQAFormulae 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
CCEA (this guide)Mathematical Formulae and TablesOne booklet shared with Maths, with statistical tables and a decision mathematics page

CCEA is the one board whose shared booklet includes statistical tables and a discrete mathematics page. WJEC shares a booklet with no tables, and OCR and Edexcel keep them apart.

Exam answers

After you mark the daily paper, our worked answers show. They are written here and are not CCEA mark schemes.

Practising for CCEA Further Maths

Open the Further Maths daily paper with the sheet in the drawer for eight mixed questions. It does not cover the decision mathematics page, so revise that from your CCEA course.

CCEA Further slips

  • Reading the Poisson table for a binomial question.
  • Skipping the contents list and turning pages blindly.
  • Using a series outside its range.

Common questions

Do you get a formula booklet in CCEA A-level Further Maths?
Yes. CCEA's Mathematical Formulae and Tables is for candidates taking AS and A level Mathematics and Further Mathematics.
Does the CCEA Further Maths booklet have statistical tables?
Yes. The contents list normal, binomial, Poisson, chi-squared, Student t and correlation tables.
Is the CCEA Further Maths booklet the same as the Maths one?
Yes. One booklet covers both subjects.
Where can I find the CCEA Further Maths formulae?
On CCEA's GCE Mathematics support page, as the Mathematical Formulae and Tables PDF.
What is not in the CCEA Further Maths booklet?
Proofs, definitions and A-level Maths basics such as the quadratic formula.

Revision cards

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

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