Formula Book & What to Memorise

One of the most important strategic decisions in your A-Level Mathematics preparation is knowing what is in the formula book and what you must memorise. The AQA formula book is provided in the exam and contains many key results — but it does not contain everything. If you rely on the formula book for something that is not there, you will waste precious time and may lose marks.

This lesson provides a comprehensive breakdown of what is given and what must be committed to memory.

Spec Mapping — This lesson develops transferable exam-skill content for AQA 7357 Papers 1, 2 and 3, drawing on the contents of the AQA formula booklet and the must-memorise list. Refer to the official AQA specification document for exact wording.

---

What Is the AQA Formula Book?

The AQA Mathematical Formulae and Statistical Tables booklet is provided to every candidate in all three exam papers. It contains:

• Key formulae for pure mathematics
• Key formulae for mechanics
• Key formulae for statistics
• Statistical tables (binomial cumulative distribution function, percentage points of the normal distribution, etc.)

You will receive a clean copy — you cannot take your own annotated version into the exam.

---

Pure Mathematics: What IS in the Formula Book

The following results are provided in the AQA formula book. You do not need to memorise these, but you must know how to use them.

Mensuration

• Surface area of a sphere: $4\pi r^{2}$4πr2
• Area of the curved surface of a cone: $\pi r l$πrl (where $l$l is slant height)

Binomial Series

The general binomial expansion for (1 + x)ⁿ where n is not a positive integer:

(1 + x)ⁿ = 1 + nx + n(n-1)/2! × x² + n(n-1)(n-2)/3! × x³ + ...   (|x| < 1)

Arithmetic and Geometric Series

Arithmetic series:
  uₙ = a + (n − 1)d
  Sₙ = n/2 × (2a + (n − 1)d) = n/2 × (a + l)

Geometric series:
  uₙ = arⁿ⁻¹
  Sₙ = a(1 − rⁿ)/(1 − r)
  S∞ = a/(1 − r)  for |r| < 1

Trigonometric Identities (given in the formula book)

$$\begin{aligned} sin(A \pm B) = sin A cos B \pm cos A sin B \\ cos(A \pm B) = cos A cos B ∓ sin A sin B \\ tan(A \pm B) = (tan A \pm tan B) / (1 ∓ tan A tan B) \\ sin 2A = 2 sin A cos A \\ cos 2A = cos^{2}A - sin^{2}A = 2cos^{2}A - 1 = 1 - 2sin^{2}A \\ tan 2A = 2 tan A / (1 - tan^{2}A) \end{aligned}$$sin(A±B)=sinAcosB±cosAsinBcos(A±B)=cosAcosB∓sinAsinBtan(A±B)=(tanA±tanB)/(1∓tanAtanB)sin2A=2sinAcosAcos2A=cos2A−sin2A=2cos2A−1=1−2sin2Atan2A=2tanA/(1−tan2A)​

Small Angle Approximations

$$\begin{aligned} \sin \theta &\approx \theta \\ \cos \theta &\approx 1 - \tfrac{\theta^{2}}{2} \\ \tan \theta &\approx \theta \end{aligned}$$sinθcosθtanθ​≈θ≈1−2θ2​≈θ​

(where $\theta$θ is in radians)

Differentiation (given)

f(x)        f'(x)
tan kx      k sec²kx
sec x       sec x tan x
cot x       −cosec²x
cosec x     −cosec x cot x

f(x)·g(x)  →  f'(x)g(x) + f(x)g'(x)    (product rule — given in words but formula given)
f(g(x))    →  f'(g(x)) × g'(x)           (chain rule — given in words but formula given)

Integration (given)

f(x)        ∫f(x) dx
sec²kx      (1/k) tan kx + C
tan x       ln|sec x| + C
cot x       ln|sin x| + C
cosec x     −ln|cosec x + cot x| + C
sec x       ln|sec x + tan x| + C

Numerical Methods

Trapezium rule:
∫ₐᵇ y dx ≈ h/2 [y₀ + 2(y₁ + y₂ + ... + yₙ₋₁) + yₙ]
where h = (b − a)/n

Newton-Raphson:
xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)

---

Pure Mathematics: What You MUST Memorise

The following results are not in the formula book. You must know them from memory.

Laws of Logarithms

log(ab) = log a + log b
log(a/b) = log a − log b
log(aⁿ) = n log a
logₐa = 1
logₐ1 = 0
aˡᵒᵍₐˣ = x
Change of base: logₐb = logₓb / logₓa = ln b / ln a

Warning: These are NOT in the formula book. Many students assume they are. Memorise them thoroughly.

Differentiation from First Principles

f'(x) = lim(h→0) [f(x + h) − f(x)] / h

Standard Derivatives (not given for basic functions)

$$\begin{aligned} f(x) f'(x) \\ x^{n} nx^{n-1} \\ sin x cos x \\ cos x -sin x \\ e^{x} e^{x} \\ ln x 1/x \\ sin kx k cos kx \\ cos kx -k sin kx \\ e^{kx} ke^{kx} \\ a^{x} a^{x} ln a \end{aligned}$$f(x)f′(x)xnnxn−1sinxcosxcosx−sinxexexlnx1/xsinkxkcoskxcoskx−ksinkxekxkekxaxaxlna​

Product Rule, Quotient Rule, Chain Rule

While the formula book mentions these, you must know how to apply them fluently:

Product rule:   d/dx [uv] = u(dv/dx) + v(du/dx)
Quotient rule:  d/dx [u/v] = [v(du/dx) − u(dv/dx)] / v²
Chain rule:     dy/dx = (dy/du) × (du/dx)

Standard Integrals (not given for basic functions)

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C    (n ≠ −1)
∫ 1/x dx = ln|x| + C
∫ eˣ dx = eˣ + C
∫ eᵏˣ dx = (1/k)eᵏˣ + C
∫ sin x dx = −cos x + C
∫ cos x dx = sin x + C
∫ sin kx dx = −(1/k)cos kx + C
∫ cos kx dx = (1/k)sin kx + C

Binomial Expansion (positive integer)

(a + b)ⁿ = Σ (n choose r) aⁿ⁻ʳ bʳ   for r = 0 to n

where (n choose r) = n! / [r!(n − r)!]

(1 + x)ⁿ = 1 + nx + n(n−1)/2! × x² + ...  (for positive integer n, this terminates)

Note: The formula book gives the general binomial expansion (for non-integer n). The basic expansion for positive integer n — and the formula for binomial coefficients — must be memorised.

Equations of Lines and Circles

Straight line:
  y − y₁ = m(x − x₁)
  y = mx + c
  Gradient = (y₂ − y₁) / (x₂ − x₁)
  Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
  Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]

Perpendicular gradients: m₁ × m₂ = −1

Circle:
  (x − a)² + (y − b)² = r²     (centre (a, b), radius r)
  x² + y² + 2gx + 2fy + c = 0  (centre (−g, −f), radius √(g² + f² − c))

Factor Theorem and Remainder Theorem

Factor theorem: If f(a) = 0, then (x − a) is a factor of f(x).
Remainder theorem: When f(x) is divided by (x − a), the remainder is f(a).

Fundamental Trigonometric Identities (must know)

sin²θ + cos²θ ≡ 1
tan θ ≡ sin θ / cos θ
1 + tan²θ ≡ sec²θ
1 + cot²θ ≡ cosec²θ

Key Distinction: The addition formulae (sin(A ± B), cos(A ± B), etc.) and double angle formulae ARE in the formula book. The Pythagorean identities (sin² + cos² = 1 and its variants) are NOT. You must memorise them.

Quadratic Formula

$$x = [-b pm sqrt{b^{2} - 4ac}] / 2a$$x=[−bpmsqrtb2−4ac]/2a

Discriminant

b² − 4ac > 0  →  two distinct real roots
b² − 4ac = 0  →  one repeated root
b² − 4ac < 0  →  no real roots

---

Mechanics: What You MUST Memorise

All mechanics formulae must be memorised — none are given in the AQA formula book for A-Level Maths (they are only in the Further Maths book).

SUVAT Equations

v = u + at
s = ut + ½at²
s = vt − ½at²
v² = u² + 2as
s = ½(u + v)t

Where: s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time.

Exam Tip: You must know all five SUVAT equations. In any kinematics problem, identify the three known quantities and the one unknown, then select the equation that does not contain the fifth (unwanted) quantity.

Newton's Laws of Motion

First law:  A body remains at rest or moves with constant velocity unless acted upon by a resultant force.
Second law: F = ma (resultant force = mass × acceleration)
Third law:  For every action, there is an equal and opposite reaction.

Weight

$W = mg$W=mg where $g = 9.8\,\text{m/s}^{2}$g=9.8m/s2 (or as specified in the question).

Friction

• $F \leq \mu R$F≤μR — friction $\leq$≤ coefficient of friction $\times$× normal reaction
• $F = \mu R$F=μR — at the point of sliding (when friction is limiting)

Moments

• Moment = Force $\times$× Perpendicular distance from the pivot
• For equilibrium: sum of clockwise moments = sum of anticlockwise moments

Resolving Forces

On an inclined plane at angle θ:
  Component parallel to the plane:     mg sin θ
  Component perpendicular to the plane: mg cos θ

Projectiles

• Horizontal: $x = V \cos\theta \times t$x=Vcosθ×t (constant velocity, no acceleration)
• Vertical: $y = V \sin\theta \times t - \tfrac{1}{2}gt^{2}$y=Vsinθ×t−21​gt2 (acceleration $= -g$=−g)
• Time of flight: $T = \dfrac{2V \sin\theta}{g}$T=g2Vsinθ​
• Maximum height: $H = \dfrac{V^{2} \sin^{2}\theta}{2g}$H=2gV2sin2θ​
• Range: $R = \dfrac{V^{2} \sin 2\theta}{g}$R=gV2sin2θ​

---

Statistics: What You MUST Memorise

Probability

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(A | B) = P(A ∩ B) / P(B)
For independent events: P(A ∩ B) = P(A) × P(B)
For mutually exclusive events: P(A ∩ B) = 0

Binomial Distribution

$X \sim B(n, p)$X∼B(n,p) if:

• Fixed number of trials ($n$n)
• Two outcomes (success/failure)
• Constant probability of success ($p$p)
• Independent trials

$P(X = r) = \binom{n}{r} p^{r} (1 - p)^{n-r}$P(X=r)=(rn​)pr(1−p)n−r

• Mean: $E(X) = np$E(X)=np
• Variance: $\mathrm{Var}(X) = np(1 - p)$Var(X)=np(1−p)

Normal Distribution

$X \sim N(\mu, \sigma^{2})$X∼N(μ,σ2)

Standardisation: $Z = \dfrac{X - \mu}{\sigma}$Z=σX−μ​ where $Z \sim N(0, 1)$Z∼N(0,1).

Measures of Central Tendency and Spread

Mean of grouped data: x̄ = Σfx / Σf
Variance: σ² = Σf(x − x̄)² / Σf = Σfx² / Σf − x̄²
Standard deviation: σ = √(variance)

For coded data: if y = (x − a)/b, then x̄ = bȳ + a and σₓ = b × σᵧ

Hypothesis Testing

Binomial test:
  H₀: p = p₀
  H₁: p > p₀ (or p < p₀ or p ≠ p₀)
  Significance level: usually 5% or 1%
  Compare P(X ≥ observed) or P(X ≤ observed) with the significance level

Normal test (for means):
  H₀: μ = μ₀
  Test statistic: Z = (x̄ − μ₀) / (σ/√n)
  Compare with critical value from tables

Exam Tip: When conducting a hypothesis test, always state H₀ and H₁, define the test statistic, find the p-value or critical region, and write a conclusion in context. Saying "reject H₀" without interpretation will lose you marks.

---

Trigonometric Identities: The Full Picture

This is a frequent source of confusion, so here is a clear summary:

Given in the formula book (do NOT memorise — just know where to find them)

• sin(A ± B), cos(A ± B), tan(A ± B)
• sin 2A, cos 2A, tan 2A
• Small angle approximations

NOT in the formula book (MUST memorise)

• sin²θ + cos²θ ≡ 1
• tan θ ≡ sin θ / cos θ
• 1 + tan²θ ≡ sec²θ (derived from dividing sin² + cos² = 1 by cos²θ)
• 1 + cot²θ ≡ cosec²θ (derived from dividing sin² + cos² = 1 by sin²θ)
• Exact values: sin 0°, sin 30°, sin 45°, sin 60°, sin 90° and corresponding cos and tan values

Exact trigonometric values (must know)

Angle	sin	cos	tan
0°	0	1	0
30° (π/6)	1/2	√3/2	1/√3
45° (π/4)	√2/2	√2/2	1
60° (π/3)	√3/2	1/2	√3
90° (π/2)	1	0	undefined

---

Integration Techniques: What Is Expected Without the Formula Book

You must be able to perform the following integrations from memory, without reference to the formula book:

1. Power rule for xⁿ (including negative and fractional powers)
2. eˣ and eᵏˣ
3. sin x, cos x, sin kx, cos kx
4. 1/x (giving ln|x|)
5. Integration by substitution — the method must be known; specific substitutions may be suggested in the question
6. Integration by parts — the formula ∫u dv = uv − ∫v du must be memorised
7. Partial fractions — decomposition technique must be known
8. Using trigonometric identities to rewrite integrands (e.g., using cos 2A = 1 − 2sin²A to integrate sin²x)

The formula book provides integrals for sec², tan, cot, cosec, and sec — you do not need to memorise these.

---

Tips for Using the Formula Book Effectively

Before the exam

1. Get a copy of the AQA formula book and practise using it during revision
2. Know the layout — where are the trig identities? Where are the statistical tables?
3. Practise finding formulae quickly — in the exam, fumbling through pages wastes time
4. Mark up your own revision copy with notes about when each formula is used (you cannot take this into the exam, but it helps you learn the structure)

During the exam

1. Check immediately if you cannot remember a formula — it might be in the book
2. Do not waste time deriving formulae that are provided — use them directly
3. Copy formulae carefully — transcription errors are a common source of lost marks
4. Use the statistical tables for binomial cumulative probabilities and normal distribution critical values rather than trying to calculate them

Common mistakes with the formula book

Mistake	Consequence
Assuming laws of logarithms are given	They are NOT — you must memorise them
Looking for SUVAT equations	They are NOT in the A-Level Maths formula book
Confusing the general binomial expansion with the integer version	The book gives the general version; you must know how to use both
Not using the statistical tables	The cumulative binomial tables save enormous time
Misreading the normal distribution table	Check whether you need P(Z < z) or P(Z > z)

---

Synoptic Links

This topic connects to:

• Trigonometry identities and equations (aqa-alevel-maths-pure-1 / trigonometry) — knowing which trig results are printed versus memorised drives equation-solving speed in the exam.
• Normal distribution and statistical tables (aqa-alevel-maths-statistics / normal-distribution) — the printed $z$z-table and percentage-points table underpin every Paper 3 normal-distribution question.
• Integration by parts and standard integrals (aqa-alevel-maths-calculus-applications / integration-by-parts) — the booklet's integral table sets the boundary between standard recall and memorised technique.

---

Summary

• The AQA formula book provides addition formulae, double angle formulae, small angle approximations, advanced derivatives and integrals, the trapezium rule, Newton-Raphson, and statistical tables.
• You must memorise: laws of logarithms, basic derivatives and integrals (xⁿ, sin, cos, eˣ, 1/x), SUVAT equations, Newton's laws, probability rules, binomial conditions, Pythagorean trig identities, exact trig values, quadratic formula, factor theorem, equations of lines and circles, and integration by parts formula.
• Know the layout of the formula book so you can find things quickly.
• Practise with the formula book during revision so you are comfortable using it under exam conditions.

Exam Tip: Make flash cards for every formula NOT in the book. Test yourself daily in the weeks leading up to the exam. The single biggest reason students lose marks is forgetting formulae they assumed were given.

Deeper Strategy: AQA Formula Book and What to Memorise

The AQA A-Level Mathematics (7357) examination provides every candidate with a printed formula booklet at the start of each paper. Knowing precisely what is and is not in that booklet is one of the highest-leverage pieces of exam preparation available. Candidates who treat the booklet as a safety net for all formulae arrive at the exam under-prepared; candidates who memorise things already printed waste revision time. This deep dive is about drawing the seam exactly.

---

What's in the AQA formula book

The booklet is organised by topic, with separate sections for Pure Mathematics, Mechanics and Statistics. Without reproducing any printed text, the broad coverage is as follows.