A Level Maths
Formula Booklet
Pure
Arithmetic Series
𝑆𝑛 =
1
𝑛 𝑎+𝑙
2
1
𝑆𝑛 = 𝑛 2𝑎 + 𝑛 − 1 𝑑
2
Geometric Series
𝑎(1 − 𝑟 𝑛 )
𝑆𝑛 =
1−𝑟
𝑆∞ =
𝑎
for 𝑟 < 1
1−𝑟
Binomial Expansion
(𝑎 + 𝑏)𝑛 = 𝑎𝑛 + 𝑛𝐶1 𝑎𝑛−1 𝑏 + 𝑛𝐶2 𝑎𝑛−2 𝑏 2 + ⋯ + 𝑛𝐶𝑟 𝑎𝑛−𝑟 𝑏 𝑟 + ⋯ + 𝑏 𝑛
where 𝑛 ∈ ℕ and 𝑛𝐶𝑟 =
(1 + 𝑥)𝑛 = 1 + 𝑛𝑥 +
𝑛!
𝑛
=
𝑟
𝑟! 𝑛 − 𝑟 !
𝑛(𝑛 − 1) 2
𝑛 𝑛−1 … 𝑛−𝑟+1
𝑥 + ⋯+
+⋯
1×2
1 × 2 × ⋯× 𝑟
for 𝑥 < 1, 𝑛 ∈ ℝ
Curved Surface Area
Surface area of sphere = 4𝜋𝑟 2
Area of curved surface of cone = 𝜋𝑟 × slant height
Exponentials and Logarithms
log 𝑎 (𝑥)
log 𝑏 (𝑥)
log 𝑏 (𝑎)
𝑒 𝑥ln(𝑎) = 𝑎 𝑥
Trigonometric Identities
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
𝐴±𝐵 ≠ 𝑘+
sin 𝐴 ± sin 𝐵 = 2 sin
𝐴±𝐵
𝐴∓𝐵
cos
2
2
cos 𝐴 + cos 𝐵 = 2 cos
𝐴+𝐵
𝐴−𝐵
cos
2
2
cos 𝐴 − cos 𝐵 = −2 sin
𝐴+𝐵
𝐴−𝐵
sin
2
2
Small Angle Approximations
sin(𝜃) ≈ 𝜃
1
cos 𝜃 ≈ 1 − 𝜃 2
2
tan(𝜃) ≈ 𝜃
Differentiation
𝑓 𝑥 + ℎ − 𝑓(𝑥)
ℎ→0
ℎ
First principles: 𝑓 ′ 𝑥 = lim
𝑑 𝑢 𝑥
Quotient rule:
𝑑𝑥 𝑣 𝑥
=
𝑣
𝑑𝑢
𝑑𝑣
−𝑢
𝑑𝑥
𝑑𝑥
2
𝑣
𝒇(𝒙)
𝒇′(𝒙)
tan(𝑘𝑥)
𝑘sec 2 (𝑘𝑥)
sec(𝑘𝑥)
𝑘sec 𝑘𝑥 tan(𝑘𝑥)
cot(𝑘𝑥)
−𝑘cosec 2 (𝑘𝑥)
cosec(𝑘𝑥)
−𝑘cosec 𝑘𝑥 cot(𝑘𝑥)
1
𝜋
2
Integration
න
𝑓′(𝑥)
𝑑𝑥 = ln 𝑓(𝑥) + 𝑐
𝑓(𝑥)
න 𝑓′ 𝑥 𝑓 𝑥
𝑛
𝑑𝑥 =
1
(𝑓 𝑥 )𝑛+1 + 𝑐
𝑛+1
Integration by parts: න 𝑢
𝑑𝑣
𝑑𝑢
𝑑𝑥 = 𝑢𝑣 − න 𝑣
𝑑𝑥
𝑑𝑥
𝑑𝑥
𝒇(𝒙)
න 𝒇 𝒙 𝒅𝒙
sec 2 (𝑘𝑥)
1
tan 𝑘𝑥 + 𝑐
𝑘
1
ln sec(𝑘𝑥) + 𝑐
𝑘
1
ln sin(𝑘𝑥) + 𝑐
𝑘
tan(𝑘𝑥)
cot(𝑘𝑥)
cosec(𝑘𝑥)
1
− ln cosec 𝑘𝑥 + cot 𝑘𝑥 + 𝑐
𝑘
1
1
ln tan 𝑘𝑥 + 𝑐
𝑘
2
sec(𝑘𝑥)
1
− ln sec 𝑘𝑥 + tan 𝑘𝑥 + 𝑐
𝑘
1
1
𝜋
ln tan 𝑘𝑥 +
+𝑐
𝑘
2
4
Numerical Methods
𝑏
1
Trapezium Rule: ‫ = 𝑥𝑑 𝑦 𝑎׬‬2 ℎ(𝑦0 + 𝑦𝑛 + 2(𝑦1 + 𝑦2 + ⋯ + 𝑦𝑛−1 )) where ℎ =
𝑓(𝑥 )
Newton-Raphson iteration for solving 𝑓 𝑥 = 0 is 𝑥𝑛+1 = 𝑥𝑛 − 𝑓′(𝑥𝑛 )
𝑛
𝑏−𝑎
𝑛
Statistics
Measures of Variation
Interquartile Range = IQR = 𝑄3 − 𝑄1
2
𝑆𝑥𝑥 = ෍(𝑥𝑖 − 𝑥)ҧ =
෍ 𝑥𝑖2
(σ 𝑥𝑖 )
−
𝑛
Standard deviation = variance
𝜎=
𝑆𝑥𝑥
𝑛
𝜎=
σ 𝑥2
− 𝑥ҧ 2
𝑛
𝜎=
σ(𝑥 − 𝑥)ҧ 2
𝑛
𝜎=
σ 𝑓(𝑥 − 𝑥)ҧ 2
σ𝑓
𝜎=
σ 𝑓𝑥 2
− 𝑥ҧ 2
σ𝑓
2
Probability
P A∪B = P A +P B −P A∩B
P A ∩ B = P A P(B|A)
P BA =
P(A ∩ B)
P(A)
P A′ = 1 − P A
P AB =
P B A P(A)
P B A P A + P B A′ P(A′ )
Independent Events
P BA =P B
P A B = P(A)
P A ∩ B = P A P(B)
Binomial Distribution
If 𝑋 ∼ 𝐵(𝑛, 𝑝), then:
P 𝑋=𝑥 =
𝑛 𝑥
𝑝 (1 − 𝑝)𝑛−𝑥
𝑥
𝑋ത = 𝑛𝑝
Var 𝑋 = 𝑛𝑝(1 − 𝑝)
Normal Distribution
If 𝑋 ∼ 𝑁(𝜇, 𝜎 2 ), then:
𝑋ത ∼ 𝑁 𝜇,
𝜎2
𝑛
𝑋ത − 𝜇
𝜎 ∼ 𝑁(0,1)
𝑛
Mechanics
Motion in a Straight Line
𝑣 = 𝑢 + 𝑎𝑡
𝑠=
1
𝑢+𝑣 𝑡
2
1
𝑠 = 𝑢𝑡 + 𝑎𝑡 2
2
1
𝑠 = 𝑢𝑡 + 𝑎𝑡 2
2
𝑣 2 = 𝑢2 + 2𝑎𝑠
Motion in Two Dimensions
𝐯 = 𝐮 + 𝐚𝑡
𝐬=
1
𝐮+𝐯 𝑡
2
1
𝐬 = 𝐮𝑡 + 𝐚𝑡 2
2
1
𝐬 = 𝐯𝑡 − 𝐚𝑡 2
2