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AQA A-level Maths formula booklet (7357): what is on it and what is not

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Formula sheet

The card below opens today's A-level Maths paper with this sheet in the drawer, so you practise the way you will sit the exam. The contents follow, grouped like the real booklet in our own layout and words. Check against AQA's Formulae for A-level Mathematics before you rely on anything here.

Formula sheet

A-level Maths: Algebra & Functions

Open formula sheet

Binomial series

(a+b)^n=a^n+\binom{n}{1}a^{n-1}b+\binom{n}{2}a^{n-2}b^2+\cdots+\binom{n}{r}a^{n-r}b^r+\cdots+b^nbinomial theorem for a positive whole number n
\binom{n}{r}=\frac{n!}{r!(n-r)!}the binomial coefficient
(1+x)^n=1+nx+\frac{n(n-1)}{1\times2}x^2+\cdots+\frac{n(n-1)\cdots(n-r+1)}{1\times2\times\cdots\times r}x^r+\cdotsbinomial series for any n, when |x| < 1

Arithmetic series

S_n=\frac{1}{2}n(a+l)=\frac{1}{2}n[2a+(n-1)d]arithmetic series, sum of the first n terms

Geometric series

S_n=\frac{a(1-r^n)}{1-r}geometric series, sum of the first n terms
S_\infty=\frac{a}{1-r}geometric series, sum to infinity (for |r| < 1)

Trigonometry: small angles

\sin\theta\approx\thetasine for a small angle (θ in radians)
\cos\theta\approx1-\frac{\theta^2}{2}cosine for a small angle (θ in radians)
\tan\theta\approx\thetatangent for a small angle (θ in radians)

Trigonometric identities

\sin(A\pm B)=\sin A\cos B\pm\cos A\sin Bsine of a sum or difference
\cos(A\pm B)=\cos A\cos B\mp\sin A\sin Bcosine of a sum or difference
\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}tangent of a sum or difference (A ± B not an odd multiple of π/2)

Differentiation

\frac{d}{dx}(\tan x)=\sec^2 xderivative of tan x
\frac{d}{dx}(\sec x)=\sec x\tan xderivative of sec x
\frac{d}{dx}(\cot x)=-\csc^2 xderivative of cot x
\frac{d}{dx}(\csc x)=-\csc x\cot xderivative of cosec x
\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{f'(x)g(x)-f(x)g'(x)}{(g(x))^2}quotient rule
f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}differentiation from first principles

Integration

\int u\frac{dv}{dx}dx=uv-\int v\frac{du}{dx}dxintegration by parts
\int\frac{f'(x)}{f(x)}dx=\ln|f(x)|+cintegral of f′(x)/f(x)
\int\tan x\ dx=\ln|\sec x|+cintegral of tan x
\int\cot x\ dx=\ln|\sin x|+cintegral of cot x

Numerical solution of equations

x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}Newton-Raphson iteration for solving f(x) = 0

Numerical integration

\int_a^b y\ dx\approx\frac{1}{2}h[(y_0+y_n)+2(y_1+y_2+\cdots+y_{n-1})]the trapezium rule (h = (b − a)/n)

Mechanics: constant acceleration

v=u+atfinal velocity
s=ut+\frac{1}{2}at^2displacement from the start velocity
s=vt-\frac{1}{2}at^2displacement from the final velocity
s=\frac{1}{2}(u+v)tdisplacement from the average velocity
v^2=u^2+2asvelocity without time

Probability

P(A\cup B)=P(A)+P(B)-P(A\cap B)addition rule
P(A\cap B)=P(A)\times P(B|A)multiplication rule

Standard deviation

\sigma=\sqrt{\frac{\sum x^2}{n}-\bar{x}^2}standard deviation from the data

Discrete distributions

P(X=x)=\binom{n}{x}p^x(1-p)^{n-x},\quad \mu=np,\quad \sigma^2=np(1-p)binomial distribution B(n, p): probability, mean, variance

Sampling distributions

\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\sim N(0,1)sample mean from a normal population (sample of n from N(μ, σ²))

The larger AQA booklet

AQA also publishes Formulae and statistical tables for A-level Mathematics and A-level Further Mathematics (version 1.5, February 2018). It is required for Further Maths, and AQA says students may also use it in Maths exams. It holds more than the Maths exams need, so a Maths candidate usually finds the shorter booklet quicker to search.

Gaps in the AQA booklet

  • There is no mensuration section. The surface area of a sphere and the curved area of a cone, which Edexcel and WJEC print, are for you to know.
  • The trigonometric identities stop at the addition formulae for sin, cos and tan. Sum to product identities are not given, and neither are the double angle formulae.
  • The short booklet has no statistical tables. Where a question needs a cumulative probability you either get the value in the paper or use the calculator.
  • Formulae that every board leaves to memory are also missing here: the quadratic formula, indices and logarithm laws, the sine and cosine rules, and standard derivatives such as xⁿ and eˣ.

One page, three parts

The booklet is divided into pure mathematics, mechanics, and probability and statistics, and it ends with the line End of formulae. That layout is a gift in a time-pressured paper: the whole thing is short enough to know by heart where each entry sits.

Notice the mechanics section lists the constant acceleration equations in two forms: each of s = ut + ½at² and s = vt − ½at² is printed, even though the second is the first rewritten for the final velocity. Pick the one that omits the unknown you do not have.

Use the quadratic formula to find both roots.

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

Worked example: the trapezium rule on 1/x

Estimate the integral of 1/x from 1 to 3 using four strips.

With four strips h = 0.5, and the y-values at x = 1, 1.5, 2, 2.5, 3 are 1, 0.6667, 0.5, 0.4 and 0.3333.

\int_1^3\frac{1}{x}\ dx\approx\frac{1}{2}(0.5)[(1+0.3333)+2(0.6667+0.5+0.4)]=0.25\times4.4667=1.117

The exact value is ln 3, about 1.099, so the rule over-estimates, as it does for any curve that bends upwards. Say so if the question asks you to comment.

The AQA booklet and the other boards

BoardBookletWhat it adds or leaves out
EdexcelMathematical Formulae and Statistical TablesMensuration, logarithm change of base, sum to product identities, and the statistical tables, all in one booklet for AS, A-level Maths and Further Maths
AQA (this guide)Formulae for A-level Mathematics (7357)A shorter booklet with no statistical tables, no mensuration and no sum to product identities
OCR AFormulae, A Level Mathematics A (H240)Short, with a hypothesis test line, a normal percentage points table and kinematics in two dimensions
OCR MEIFormulae, A Level Mathematics B (MEI) (H640)Close to OCR A, plus the sample variance formula
WJECGCE AS/A Level Mathematics and Further Mathematics Formula BookletOne booklet for both subjects, formulae only, including sum to product identities
CCEAMathematical Formulae and TablesOne booklet for both subjects, with statistical tables inside

AQA offers two booklets, and which one you are handed depends on the paper. The short one is for AS and A-level Maths. The larger booklet with statistical tables belongs to Further Maths, and AQA says Maths candidates may use it too. If a topic looks missing from the short booklet, check whether you are meant to have the bigger one.

Exam answers

Mark a paper on this site and the worked answers appear straight afterwards, showing the step where the booklet formula comes in. They are our own solutions, not AQA's, and there is no past paper content behind them.

Timed practice with the AQA layout in mind

Try the Calculus and Statistics daily papers with the sheet open. They are 8-question papers worth 25 to 35 marks, shorter than the real 7357 papers, but long enough to test whether you find the right entry quickly. The other two tracks, Algebra and Functions and Mechanics, work the same way.

Mistakes the short AQA booklet invites

  • Assuming a missing formula is not needed. If it is not printed, it is on the recall list.
  • Using the small angle approximation for cos with θ in degrees.
  • Using the sampling result for σ unknown: the booklet gives only the known-σ case in N(0, 1) form.
  • Mixing up the two alternative displacement equations and substituting the start velocity into the one that needs the final.

Common questions

Do you get a formula booklet in AQA A-level Maths?
Yes. AQA says its booklet of formulae is required for all AS and A-level Mathematics exams.
Is there a statistical tables booklet for AQA A-level Maths?
Not in the short booklet. AQA's larger booklet of formulae and statistical tables is for Further Maths, and AQA says students may also use it in Maths exams.
What is not in the AQA A-level Maths formula booklet?
The quadratic formula, the sine and cosine rules, laws of indices and logarithms, mensuration and the basic derivatives and integrals. It lists the less common results.
Is the AQA booklet the same as the Edexcel one?
No. They share most of the pure content, but Edexcel's adds mensuration, sum to product identities and statistical tables.
Which version of the AQA formula booklet is current?
The version on AQA's website is v1.2, first issued August 2017, for the specifications first taught from September 2017.

Revision cards

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

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