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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.
The AQA Mathematical Formulae and Statistical Tables booklet is provided to every candidate in all three exam papers. It contains:
You will receive a clean copy — you cannot take your own annotated version into the exam.
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.
Surface area of a sphere = 4πr²
Area of the curved surface of a cone = πrl (where l is slant height)
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 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
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
sin 2A = 2 sin A cos A
cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A
tan 2A = 2 tan A / (1 − tan²A)
sin θ ≈ θ
cos θ ≈ 1 − θ²/2
tan θ ≈ θ
(where θ is in radians)
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)
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
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)
The following results are not in the formula book. You must know them from memory.
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.
f'(x) = lim(h→0) [f(x + h) − f(x)] / h
f(x) f'(x)
xⁿ nxⁿ⁻¹
sin x cos x
cos x −sin x
eˣ eˣ
ln x 1/x
sin kx k cos kx
cos kx −k sin kx
eᵏˣ keᵏˣ
aˣ aˣ ln a
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)
∫ 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
(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.
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: 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).
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.
x = [−b ± √(b² − 4ac)] / 2a
b² − 4ac > 0 → two distinct real roots
b² − 4ac = 0 → one repeated root
b² − 4ac < 0 → no real roots
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).
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.
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.
W = mg
where g = 9.8 m/s² (or as specified in the question)
F ≤ μR (friction ≤ coefficient of friction × normal reaction)
F = μR (at the point of sliding / when friction is limiting)
Moment = Force × Perpendicular distance from the pivot
For equilibrium: sum of clockwise moments = sum of anticlockwise moments
On an inclined plane at angle θ:
Component parallel to the plane: mg sin θ
Component perpendicular to the plane: mg cos θ
Horizontal: x = V cos θ × t (constant velocity, no acceleration)
Vertical: y = V sin θ × t − ½gt² (acceleration = −g)
Time of flight: T = 2V sin θ / g
Maximum height: H = V² sin²θ / (2g)
Range: R = V² sin 2θ / g
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
X ~ B(n, p) if:
- Fixed number of trials (n)
- Two outcomes (success/failure)
- Constant probability of success (p)
- Independent trials
P(X = r) = (n choose r) × pʳ × (1 − p)ⁿ⁻ʳ
Mean: E(X) = np
Variance: Var(X) = np(1 − p)
X ~ N(μ, σ²)
Standardisation: Z = (X − μ) / σ where Z ~ N(0, 1)
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
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.
This is a frequent source of confusion, so here is a clear summary:
| 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 |
You must be able to perform the following integrations from memory, without reference to the formula book:
The formula book provides integrals for sec², tan, cot, cosec, and sec — you do not need to memorise these.
| 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) |
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.
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.
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.
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