List of formulae and statistical tables - Cambride International AS & A Level Mathematics (9709) and Further Mathematics (9231)

Q: What subjects and syllabus scope does the formula list cover?

It covers Cambridge International AS & A Level Mathematics (9709) and Further Mathematics (9231), for use from 2020, with sections spanning Pure Mathematics, Further Pure Mathematics, Mechanics, Further Mechanics, and Probability & Statistics.

Q: Which Pure Mathematics topics are included?

Mensuration, algebra (including quadratic equations and series), trigonometric identities, differentiation, and integration are included as formula summaries.

Q: What probability and statistics formulas are provided?

The list provides summary statistics for ungrouped and grouped data, formulas for expectation and variance of discrete variables, distribution-specific results (binomial, geometric, Poisson), and expectation integrals for continuous variables.

Oct 23, 2026

Uploaded by River Wang

A consolidated reference list of mathematics results for Cambridge International AS & A Level Mathematics (9709) and Further Mathematics (9231), intended for use from 2020 across the stated syllabuses. It compiles key formulae spanning Pure Mathematics topics such as mensuration, algebraic equation solving, arithmetic and geometric series, binomial and Maclaurin series, trigonometric identities, differentiation rules, and integration methods. It also includes Further Pure Mathematics extensions, Mechanics motion and string-spring relationships, and Probability & Statistics summary statistics with discrete and continuous distributions, expectations, variances, and sampling concepts.

LanguageEnglish

CategoryExam

FormatPDF

Pages16

Reading Time40 min

AccessFree to download

Mensuration

Algebra

Trigonometry

Differentiation

Integration

Further Pure Mathematics

Summations and series

Hyperbolic and inverse functions

Mechanics

Uniformly accelerated motion

Projectile motion and springs

Motion in a circle

Centres of mass

Further Mechanics

Probability & Statistics

Summary statistics and distributions

List MF19 List of formulae and statistical tables Cambridge International AS & A Level Mathematics (9709) and Further Mathematics (9231) For use from 2020 in all papers for the above syllabuses. CST319 *2994182781* PURE MATHEMATICS Mensuration Volume of sphere = 43~~ ~~ πr3 Surface area of sphere = 4πr2 Volume of cone or pyramid = 13 × base area × height Area of curved surface of cone = πr × slant height Arc length of circle = r θ (θ in radians) Area of sector of circle = ~~ ~~12~~ ~~ r 2 θ (θ in radians) Algebra For the quadratic equation ax2 + bx + c = 0 : −b ± b2 − 4ac x = ~~ ~~ 2a For an arithmetic series: un = a + (n −1)d , Sn = ~~ ~~12~~ ~~ n(a + l) = ~~ ~~12~~ ~~ n{2a + (n −1)d } For a geometric series: un = arn−1 , Sn = a(1 − rn ) (r ≠ 1) , S∞ =~~ ~~a~~ ~~ ( r < 1 ) 1 − r 1 − r Binomial series: (a + b)n = an + 􀂧􀂩􀂨 n1􀂷􀂹􀂸 an−1b + 􀂧􀂨􀂩 n2􀂷􀂹􀂸 an−2b2 + 􀂧􀂩􀂨 n3􀂷􀂹􀂸 an−3b3 + K + bn , where n is a positive integer and 􀂧􀂨􀂩 nr􀂷􀂹􀂸 = ~~ ~~r!(nn−!~~ ~~r)! (1 + x)n = 1 + nx + ~~ ~~n(n2−!~~ ~~1)~~ ~~ x2 + ~~ ~~n(n~~ ~~−~~ ~~13)!(n~~ ~~−~~ ~~2)~~ ~~ x3 + K , where n is rational and x < 1 2 Trigonometry tan θ ≡ ~~ ~~scθθ cos2 θ + sin 2 θ ≡ 1 , 1 + tan 2 θ ≡ sec2 θ , cot 2 θ +1 ≡ cosec2 θ sin(A ± B) ≡ sin AcosB ± cosA sin B cos(A ± B) ≡ cosAcos B m sinA sin B tan(A ± B) ≡ tan A ± tan B 1 m tan A tan B sin 2A ≡ 2sinAcosA cos 2A ≡ cos2 A − sin2 A ≡ 2cos2 A −1 ≡ 1− 2sin 2 A tan 2 A ≡ 2 tan A 1 − tan2 A Principal values: − ~~ ~~12~~ ~~ π ⩽ sin −1 x ⩽ ~~ ~~12~~ ~~ π , 0 ⩽ cos−1 x ⩽ π , − ~~ ~~12~~ ~~ π < tan −1 x < ~~ ~~12~~ ~~ π Differentiation f( x) xn ln x x esin x cos x tan x sec x cosec x cot x −1 tan x uv u v If x = f(t ) and y = g(t) then dydx = ~~ ~~dydt ÷ ~~ ~~ddxt f ′( x ) n−1 nx 1 x xe cos x −sin x 2 sec x sec x tan x − cosec x cot x 2 − cosec x 1 1 + x2 du dv v dx + u dx du dv v dx − u dx 2 v 3 Integration (Arbitrary constants are omitted; a denotes a positive constant.) f( x) xn 1 x ex sin x cos x sec2 x x2 2 1 + 1 a 2 2 x − a 1 2 2 a − x ∫ f( x ) dx n+1 x n +1 1 a ln x xe − cos x sin x tan x 1 􀂧 x 􀂷 tan − 􀂩􀂨 a 􀂹􀂸 1 ln x − a 2a x + a 1 ln a + x 2a a − x ∫u dv dx dx = uv − ∫v du dx dx ∫ f ′(x) f(x) dx = ln f (x) (n ≠ −1) (x > a) ( x < a) Vectors If a = a1i + a2 j + a3k and b = b1i + b2 j + b3k then a.b = a1b1 + a2b2 + a3b3 = a b cos θ 4 FURTHER PURE MATHEMATICS Algebra Summations: ∑r1 r = ~~ ~~12~~ ~~ n(n + 1) , ∑r1 r 2 = ~~ ~~16~~ ~~ n(n + 1)(2n + 1) , ∑r1 r 3 = ~~ ~~14~~ ~~ n2 (n + 1)2 Maclaurin’s series: f(x) = f(0) + xf ′(0) + ~~ ~~x22! f ′′(0) + K + ~~ ~~xrr!~~ ~~ f (r ) (0) + K ex = exp(x) = 1 + x + ~~ ~~x22! + K + ~~ ~~xrr! + K (all x) 2 3 r ln(1 + x) = x − ~~ ~~x~~ ~~+ ~~ ~~x~~ ~~ − K + (−1)r+1 ~~ ~~x~~ ~~+ K (–1 < x ⩽ 1) 2 3 r 3 5 2r+1 sin x = x − ~~ ~~x3! + ~~ ~~x5! − K + (−1)r ~~ ~~(x2r~~ ~~+~~ ~~1)! + K (all x) 2 4 2r cos x = 1 − ~~ ~~x2! + ~~ ~~x4! − K + (−1)r ~~ ~~(x2r)! + K (all x) 3 5 2r+1 tan −1 x = x − ~~ ~~x~~ ~~ + ~~ ~~x~~ ~~ − K + (−1)r ~~ ~~x~~ ~~ + K (–1 ⩽ x ⩽ 1) 3 5 2r +1 3 5 2r+1 sinh x = x + ~~ ~~x3! + ~~ ~~x5! + K + ~~ ~~(x2r~~ ~~+~~ ~~1)! + K (all x) 2 4 2r cosh x = 1 + ~~ ~~x2! + ~~ ~~x4! + K + ~~ ~~(x2r)! + K (all x) 3 5 2r +1 tanh −1 x = x + ~~ ~~x~~ ~~ + ~~ ~~x~~ ~~ + K + ~~ ~~x~~ ~~ + K (–1 < x < 1) 3 5 2r +1 Trigonometry If t = tan ~~ ~~12~~ ~~ x then: sin x = ~~ ~~1tt~~ ~~2~~ ~~ and cos x = ~~ ~~11~~ ~~tt22 Hyperbolic functions cosh 2 x − sinh2 x ≡ 1 , sinh 2x ≡ 2sinh x cosh x , cosh 2x ≡ cosh 2 x + sinh 2 x sinh −1 x = ln(x + x2 +1) cosh −1 x = ln(x + x2 −1) (x ⩾ 1) tanh −1 x = ~~ ~~12~~ ~~ ln 􀂨􀂧􀂩 ~~ ~~11~~ ~~xx~~ ~~ 􀂷􀂹􀂸 (| x | < 1) 5 Differentiation f( x) sin −1 x −1 cos x sinh xcosh x tanh x sinh −1 xcosh −1 xtanh −1 x f ′( x ) 1 1 − ~~ ~~ 1 − x2 cosh x sinh x sech 2 x

FAQs

What subjects and syllabus scope does the formula list cover?

It covers Cambridge International AS & A Level Mathematics (9709) and Further Mathematics (9231), for use from 2020, with sections spanning Pure Mathematics, Further Pure Mathematics, Mechanics, Further Mechanics, and Probability & Statistics.

Which Pure Mathematics topics are included?

Mensuration, algebra (including quadratic equations and series), trigonometric identities, differentiation, and integration are included as formula summaries.

What probability and statistics formulas are provided?

The list provides summary statistics for ungrouped and grouped data, formulas for expectation and variance of discrete variables, distribution-specific results (binomial, geometric, Poisson), and expectation integrals for continuous variables.

Oct 23, 2026

Uploaded by River Wang