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Edexcel A-level Further Maths formula booklet (9FM0): 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 of section 4 in our own layout and words, apart from the statistical tables, and the results most often used follow: summations, matrix transformations, Maclaurin series and hyperbolic functions. Check against Pearson's booklet (Issue 1, July 2017) 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

Section 4 of the booklet also holds these, which are in the drawer above but not repeated below.

SectionWhat it holds
Complex numbersresults for complex numbers and the roots of z^n = 1
Vectorsvector formulae and equations of lines and planes
Conicsstandard and parametric forms for the ellipse, parabola, hyperbola and rectangular hyperbola, with eccentricity and asymptotes
Differentiation and integrationresults for inverse trigonometric and hyperbolic functions
Arc length and surface area of revolutioncartesian, parametric and polar forms
Statisticsdiscrete and continuous distributions, correlation and regression, sampling distributions and non-parametric tests
Mechanicscentres of mass
Statistical tablesbinomial, Poisson, chi-squared, correlation, t and F tables, plus normal percentage points and random numbers (none are in the drawer)

What the Edexcel booklet will not give you

  • Proof. Pearson lists results only, so induction, the method of differences and the derivations behind a Maclaurin series are skills you carry in.
  • Anything from the A-level Maths recall list in Appendix 1 of the Maths specification, including the quadratic formula and the sine and cosine rules.
  • A choice of method. A Further Pure question does not say whether the result you need is in the summations, the series or the hyperbolic section.

Section 4 and its neighbours

Pearson arranges the booklet by qualification, so A-level Further Maths students work from section 4 and dip into section 2 for any A-level Maths result a question borrows. Learn that detour; it is where students lose time.

Section 4 runs from summations and matrix transformations to conics, arc length and the surface area of revolution, then statistics with negative binomial and non-parametric tests, then centres of mass, then the tables.

Use the quadratic formula to find both roots.

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

Worked example: summing squares

Find the sum of the squares of the integers from 1 to 10.

\sum_{r=1}^{10}r^2=\frac{1}{6}(10)(11)(21)=385

The last bracket is 2n + 1 = 21. Students often write 11 there, which gives 220.

The Edexcel booklet and the other boards

BoardBookletNotes
Edexcel (this guide)Mathematical 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
CCEAMathematical Formulae and TablesOne booklet shared with Maths, with statistical tables and a decision mathematics page

Edexcel keeps every qualification in one booklet, with statistical tables for binomial, Poisson, chi-squared, correlation, t and F at the back. AQA also bundles tables, OCR A adds discrete mathematics, and MEI adds a technology section.

Exam answers

Mark a Further Maths daily paper and the worked solutions appear, written by us and not Edexcel. They show which section of the booklet each step used.

Further Maths practice

The single Further Maths daily paper is a short 8-question set, so one sitting covers a few topics rather than the whole of 9FM0. Use it to check you can reach the right section of the booklet in seconds.

Slips with the Edexcel Further Maths section

  • Using the Maclaurin series for ln(1 + x) outside −1 < x ≤ 1.
  • Working in degrees for series in sin x or cos x.
  • Reading the matrix transformation as clockwise when it is printed anticlockwise.

Common questions

Do you get a formula booklet in Edexcel A-level Further Maths?
Yes. Pearson provides Mathematical Formulae and Statistical Tables for the Advanced GCE in Further Mathematics (9FM0).
Is the Edexcel Further Maths booklet the same as the Maths one?
It is one booklet arranged by qualification. Further Maths students use section 4 and may also need the formulae in section 2.
What is in the Edexcel Further Maths formula booklet?
Summations, matrix transformations, Maclaurin and Taylor series, hyperbolic functions, conics, differentiation and integration results, statistics, centres of mass and statistical tables.
Are proofs in the Edexcel Further Maths booklet?
No. It gives results, not derivations.
Do I need the A-level Maths section for Further Maths?
Pearson says you may, because Further Maths papers can use formulae introduced in A-level Maths.

Revision cards

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