Uni-Sine
UK exams
UK examsA-level

WJEC A-level Further Maths formula booklet: what is on it

Published 5 min read

Paper resets in

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, and the results most often used follow: summations, matrix transformations, Maclaurin series and hyperbolic functions. Check against WJEC's Formula Booklet for the exact wording.

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
Complex numbersresults for complex numbers and the roots of 1
Vectorsthe resolute of one vector on another, a dividing point, the plane and distances between skew lines, a point and a line, and a point and a plane
Matrix transformationsrotations and reflections in 2-D, and rotations in 3-D
Differentiation and integrationinverse trigonometric and hyperbolic results
Area of a sectorin polar coordinates
Mechanicsmotion in a circle and centres of mass of uniform bodies
Probability and statisticsBayes' theorem, expectation and variance, the binomial and Poisson distributions, continuous distributions, sampling, correlation and regression

Not in the WJEC booklet

  • No statistical tables. The booklet is formulae only, so table values come from the paper.
  • No proofs, and no explanation of when a result applies.
  • The standard A-level Maths recall items, such as the quadratic formula, are not printed.

Sharing a booklet with A-level Maths

Because WJEC uses one booklet for AS/A-level Maths and Further Maths, the first pages cover mensuration, series and the binomial expansion that A-level students use too. The Further content starts with summations and complex numbers.

Vectors get a full page here, including the resolute of one vector on another, a dividing point in a given ratio, the equation of a plane and the perpendicular distance between skew lines.

Use the quadratic formula to find both roots.

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

Worked example: the sum of cubes

Find 1³ + 2³ + … + 6³.

\sum_{r=1}^{6}r^3=\frac{1}{4}(6)^2(7)^2=\frac{36\times49}{4}=441

You can check by adding 1 + 8 + 27 + 64 + 125 + 216, which also gives 441.

The WJEC 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
WJEC (this guide)GCE 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

WJEC and CCEA both share a single booklet between Maths and Further Maths. WJEC's has no tables, while CCEA's does. Of the English boards, Edexcel and AQA publish tables with Further Maths.

Exam answers

The answers to our daily paper appear when you mark it, and are our own work, not WJEC's.

Daily Further Maths practice for WJEC

Use the Further Maths daily paper for eight mixed questions. Since WJEC's booklet lacks tables, check that you can still finish questions that would use one on another board's sheet.

WJEC Further slips

  • Expecting a statistical table that is not there.
  • Using a vector distance formula for the wrong pair of objects.
  • Missing the Poisson distribution section when a question mentions rare events.

Common questions

Do you get a formula booklet in WJEC A-level Further Maths?
Yes. WJEC publishes a Formula Booklet for GCE AS/A Level Mathematics and Further Mathematics, from September 2017.
Is the WJEC Further Maths booklet the same as the Maths one?
Yes. It is one booklet for both.
Does the WJEC booklet have statistical tables?
No. It holds formulae only.
What is in the WJEC Further Maths booklet?
Summations, complex numbers, Maclaurin and Taylor series, hyperbolic functions, vectors, matrices, calculus results, mechanics and statistics.
What is not in the WJEC 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.

Where to go next

  • Solve it

    Complex Number Calculator

    Calculate with complex numbers and plot them on an Argand diagram

    Open the tool
  • Learn it

    Algebra

    Master equations, inequalities and the language of abstract mathematics

    Start the course
  • Test it

    A-level Further Maths

    A new exam-style paper every day, with instant marking and a mark scheme

    Start today's paper
Read next
UK · A-levelWJEC A-level Further Maths: units, papers and gradesWJEC A-level Further Maths explained: the AS and A2 units, the choice of statistics or mechanics, calculator rules and UMS grades.