GCE Edexcel GCE in Mathematics Mathematical Formulae and Statistical Tables For use in Edexcel Advanced Subsidiary GCE and Advanced GCE examinations = Core Mathematics C1 – C4 Further Pure Mathematics FP1 – FP3 Mechanics M1 – M5 Statistics S1 – S4 For use from June 2009 This copy is the property of Edexcel. It is not to be removed from the examination room or marked in any way. = Edexcel AS/A level Mathematics Formulae List: C1 – C4, FP1 – FP3 – Contents Page – Issue 1 – September 2009 1 TABLE OF CONTENTS Page 4 Core Mathematics C1 4 Mensuration 4 Arithmetic series 5 Core Mathematics C2 5 Cosine rule 5 Binomial series 5 Logarithms and exponentials 5 Geometric series 5 Numerical integration 6 Core Mathematics C3 6 Logarithms and exponentials 6 Trigonometric identities 6 Differentiation 7 Core Mathematics C4 7 Integration 8 Further Pure Mathematics FP1 8 Summations 8 Numerical solution of equations 8 Conics 8 Matrix transformations 9 Further Pure Mathematics FP2 9 Area of sector 9 Maclaurin’s and Taylor’s Series 10 Further Pure Mathematics FP3 10 Vectors 11 Hyperbolics 12 Differentiation 12 Integration 13 Arc length 13 Surface area of revolution 2 Edexcel AS/A level Mathematics Formulae List: M1–M5, S1–S4 Contents Page – Issue 1 – September 2009 14 Mechanics M1 14 There are no formulae given for M1 in addition to those candidates are expected to know. 14 Mechanics M2 14 Centres of mass 14 Mechanics M3 14 Motion in a circle 14 Centres of mass 14 Universal law of gravitation 15 Mechanics M4 15 There are no formulae given for M4 in addition to those candidates are expected to know. 15 Mechanics M5 15 Moments of inertia 15 Moments as vectors 16 Statistics S1 16 Probability 16 Discrete distributions 16 Continuous distributions 17 Correlation and regression 18 The Normal distribution function 19 Percentage points of the Normal distribution 20 Statistics S2 20 Discrete distributions 20 Continuous distributions 21 Binomial cumulative distribution function 26 Poisson cumulative distribution function 27 Statistics S3 27 Expectation algebra 27 Sampling distributions 27 Correlation and regression 27 Non-parametric tests 28 Percentage points of the χ 2 distribution 29 Critical values for correlation coefficients 30 Random numbers 31 Statistics S4 31 Sampling distributions 32 Percentage points of Student’s t distribution 33 Percentage points of the F distribution There are no formulae provided for Decision Mathematics units D1 and D2. Edexcel AS/A level Mathematics Formulae List – Issue 1- September 2009 3 The formulae in this booklet have been arranged according to the unit in which they are first introduced. Thus a candidate sitting a unit may be required to use the formulae that were introduced in a preceding unit (e.g. candidates sitting C3 might be expected to use formulae first introduced in C1 or C2). It may also be the case that candidates sitting Mechanics and Statistics units need to use formulae introduced in appropriate Core Mathematics units, as outlined in the specification. 4 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C1 – Issue 1 – September 2009 Core Mathematics C1 Mensuration Surface area of sphere = 4 π r 2 Area of curved surface of cone = π r × slant height Arithmetic series u n = a + (n – 1)d S n = 2 1 n(a + l) = 2 1 n[2a + (n − 1)d] Edexcel AS/A level Mathematics Formulae List: Core Mathematics C2 – Issue 1 – September 2009 5 Core Mathematics C2 Candidates sitting C2 may also require those formulae listed under Core Mathematics C1. Cosine rule a 2 = b 2 + c 2 – 2bc cos A Binomial series 2 1 )( 221 nrrnnnnn bba r n ba n ba n aba ++ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ++ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +=+ −−− KK (n ∈ ℕ) where )!(! ! C rnr n r n r n − == ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∈<+ ××× +−− ++ × − ++=+ nxx r rnnn x nn nxx rn ,1( 21 )1()1( 21 )1( 1)1( 2 K K K K ℝ) Logarithms and exponentials a x x b b a log log log = Geometric series u n = ar n − 1 S n = r ra n − − 1 )1( S ∞ = r a −1 for ⏐r⏐ < 1 Numerical integration The trapezium rule: ⎮ ⌡ ⌠ b a xy d ≈ 2 1 h{(y 0 + y n ) + 2(y 1 + y 2 + + y n – 1 )}, where n ab h − = 6 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C3 – Issue 1 – September 2009 Core Mathematics C3 Candidates sitting C3 may also require those formulae listed under Core Mathematics C1 and C2. Logarithms and exponentials xax a= ln e Trigonometric identities BABABA sincoscossin)(sin ±=± BABABA sinsincoscos)(cos m=± ))(( tantan1 tantan )(tan 2 1 π +≠± ± =± kBA BA BA BA m 2 cos 2 sin2sinsin BABA BA −+ =+ 2 sin 2 cos2sinsin BABA BA −+ =− 2 cos 2 cos2coscos BABA BA −+ =+ 2 sin 2 sin2coscos BABA BA −+ −=− Differentiation f(x) f ′(x) tan kx k sec 2 kx sec x sec x tan x cot x –cosec 2 x cosec x –cosec x cot x )g( )f( x x ) )(g( )(g)f( )g()(f 2 x xxxx ′ − ′ Edexcel AS/A level Mathematics Formulae List: Core Mathematics C4 – Issue 1 – September 2009 7 Core Mathematics C4 Candidates sitting C4 may also require those formulae listed under Core Mathematics C1, C2 and C3. Integration (+ constant) f(x) ⎮ ⌡ ⌠ xx d)f( sec 2 kx k 1 tan kx x tan xsecln x co t xsinln x cosec )tan(ln,cotcosecln 2 1 xxx +− x sec )tan(ln,tansecln 4 1 2 1 π ++ xxx ⎮ ⌡ ⌠ ⎮ ⌡ ⌠ −= x x u vuvx x v u d d d d d d 8 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP1 – Issue 1 – September 2009 Further Pure Mathematics FP1 Candidates sitting FP1 may also require those formulae listed under Core Mathematics C1 and C2. Summations )12)(1( 6 1 1 2 ++= ∑ = nnnr n r 22 4 1 1 3 )1( += ∑ = nnr n r Numerical solution of equations The Newton-Raphson iteration for solving 0)f( =x : )(f )f( 1 n n nn x x xx ′ −= + Conics Parabola Rectangular Hyperbola Standard Form axy 4 2 = xy = c 2 Parametric Form (at 2 , 2at) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ t c ct, Foci )0 ,(a Not required Directrices a x −= Not required Matrix transformations Anticlockwise rotation through θ about O: ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − θθ θθ cos sin sincos Reflection in the line xy )(tan θ = : ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − θθ θθ 2cos2sin 2sin 2cos In FP1, θ will be a multiple of 45°. [...]... formulae listed under Mechanics M2 and M3, and also those formulae listed under Core Mathematics C1–C4 and Further Pure Mathematics FP3 Mechanics M5 Candidates sitting M5 may also require those formulae listed under Mechanics M2 and M3, and also those formulae listed under Core Mathematics C1–C4 and Further Pure Mathematics FP3 Moments of inertia For uniform bodies of mass m: Thin rod, length 2l, about perpendicular... dt ⎠ Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1–September 2009 13 Mechanics M1 There are no formulae given for M1 in addition to those candidates are expected to know Candidates sitting M1 may also require those formulae listed under Core Mathematics C1 Mechanics M2 Candidates sitting M2 may also require those formulae listed under Core Mathematics C1, C2 and. ..Further Pure Mathematics FP2 Candidates sitting FP2 may also require those formulae listed under Further Pure Mathematics FP1 and Core Mathematics C1–C4 Area of a sector A= 1⌠ 2 ⎮ r dθ 2⌡ (polar coordinates) Complex numbers e i θ = cosθ + i sin θ {r (cosθ + i sin θ )}n = r n (cos nθ + i sin nθ ) The roots of z n = 1 are given by z = e 2πk i n , for k = 0, 1, 2, K , n − 1 Maclaurin’s and Taylor’s Series... 0.9993 0.9997 Edexcel AS/A level Mathematics Formulae List: Statistics S2 – Issue 1 – September 2009 Statistics S3 Candidates sitting S3 may also require those formulae listed under Statistics S1 and S2 Expectation algebra For independent random variables X and Y E( XY ) = E( X ) E(Y ) , Var(aX ± bY ) = a 2 Var( X ) + b 2 Var(Y ) Sampling distributions For a random sample X 1 , X 2 , K, X n of n independent... none none 2 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1–September 2009 11 Differentiation f ′(x) f(x) 1 arcsin x 1− x 2 − arccos x 1 1− x 2 1 1 + x2 cosh x sinh x arctan x sinh x cosh x tanh x arsinh x arcosh x sech 2 x 1 1+ x2 1 x2 −1 1 1− x2 artanh x Integration (+ constant; a > 0 where relevant) f(x) ⌠ ⎮ f( x ) dx ⌡ sinh x cosh x tanh x cosh x sinh x ln cosh... 1.9600 2.3263 2.5758 3.0902 3.2905 Edexcel AS/A level Mathematics Formulae List: Statistics S1 – Issue 1 – September 2009 19 Statistics S2 Candidates sitting S2 may also require those formulae listed under Statistics S1, and also those listed under Core Mathematics C1 and C2 Discrete distributions Standard discrete distributions: Distribution of X P( X = x) Mean Variance Binomial B(n, p ) ⎛ n⎞ x ⎜ ⎟ p (1... base on the line from centre of base to vertex h above the base on the line from centre of base to vertex Universal law of gravitation Force = 14 Gm1 m 2 d2 Edexcel AS/A level Mathematics Formulae List: Mechanics M1 – M3 – Issue 1 – September 2009 Mechanics M4 There are no formulae given for M4 in addition to those candidates are expected to know Candidates sitting M4 may also require those formulae listed... 32 74 84 16 61 15 Edexcel AS/A level Mathematics Formulae List: Statistics S3 – Issue 1 – September 2009 Statistics S4 Candidates sitting S4 may also require those formulae listed under Statistics S1, S2 and S3 Sampling distributions For a random sample of n observations from N( μ , σ 2 ) (n − 1) S 2 σ2 X −μ S/ n 2 ~ χ n −1 ~ t n −1 (also valid in matched-pairs situations) 2 For a random sample of n... For uniform bodies: Triangular lamina: 2 3 along median from vertex Circular arc, radius r, angle at centre 2α : r sin α α Sector of circle, radius r, angle at centre 2α : from centre 2r sin α from centre 3α Mechanics M3 Candidates sitting M3 may also require those formulae listed under Mechanics M2, and also those formulae listed under Core Mathematics C1–C4 Motion in a circle Transverse velocity:... ~ N(0, 1) ny Correlation and regression 6Σd 2 Spearman’s rank correlation coefficient is rs = 1 − n(n 2 − 1) Non-parametric tests Goodness-of-fit test and contingency tables: ∑ (Oi − Ei ) 2 Ei Edexcel AS/A level Mathematics Formulae List: Statistics S3 – Issue 1 – September 2009 ~ χ ν2 27 PERCENTAGE POINTS OF THE χ 2 DISTRIBUTION The values in the table are those which a random variable with the χ . GCE Edexcel GCE in Mathematics Mathematical Formulae and Statistical Tables For use in Edexcel Advanced Subsidiary GCE and Advanced GCE. candidates sitting Mechanics and Statistics units need to use formulae introduced in appropriate Core Mathematics units, as outlined in the specification. 4 Edexcel AS/A level Mathematics Formulae. use the formulae that were introduced in a preceding unit (e.g. candidates sitting C3 might be expected to use formulae first introduced in C1 or C2). It may also be the case that candidates