GCE Edexcel GCE in Mathematics Mathematical Formulae and Statistical Tables

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1 GCE Edecel GCE Mthemtcs Mthemtcl Fomule d Sttstcl Tles Fo use Edecel Advced Susd GCE d Advced GCE emtos Coe Mthemtcs C C4 Futhe Pue Mthemtcs FP FP Mechcs M M5 Sttstcs S S4 Fo use fom Ju 009 UA08598

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3 TABLE OF CONTENTS Pge 4 Coe Mthemtcs C 4 Mesuto 4 Athmetc sees 5 Coe Mthemtcs C 5 Cose ule 5 Boml sees 5 Logthms d epoetls 5 Geometc sees 5 Numecl tegto 6 Coe Mthemtcs C 6 Logthms d epoetls 6 Tgoometc dettes 6 Dffeetto 7 Coe Mthemtcs C4 7 Itegto 8 Futhe Pue Mthemtcs FP 8 Summtos 8 Numecl soluto of equtos 8 Coodte geomet 8 Cocs 9 Mt tsfomtos 0 Futhe Pue Mthemtcs FP 0 Ae of secto 0 Mclu s d Tlo s Sees 0 Tlo polomls Futhe Pue Mthemtcs FP Vectos Hpeolcs Cocs Dffeetto Itegto 4 Ac legth 4 Sufce e of evoluto UA08598 Edecel AS/A level Mthemtcs Fomule Lst: C C4, FP FP Cotets Pge Issue Septeme 007

4 6 Mechcs M 6 Thee e o fomule gve fo M ddto to those cddtes e epected to kow. 6 Mechcs M 6 Cetes of mss 6 Mechcs M 6 Moto ccle 6 Cetes of mss 6 Uvesl lw of gvtto 7 Mechcs M4 7 Thee e o fomule gve fo M4 ddto to those cddtes e epected to kow. 7 Mechcs M5 7 Momets of et 7 Momets s vectos 8 Sttstcs S 8 Polt 8 Dscete dstutos 8 Cotuous dstutos 9 Coelto d egesso 0 The Noml dstuto fucto Pecetge pots of the Noml dstuto Sttstcs S Dscete dstutos Cotuous dstutos Boml cumultve dstuto fucto 8 Posso cumultve dstuto fucto 9 Sttstcs S 9 Epectto lge 9 Smplg dstutos 9 Coelto d egesso 9 No-pmetc tests 0 Pecetge pots of the χ dstuto Ctcl vlues fo coelto coeffcets Rdom umes Sttstcs S4 Smplg dstutos 4 Pecetge pots of Studet s t dstuto 5 Pecetge pots of the F dstuto Thee e o fomule povded fo Decso Mthemtcs uts D d D. UA08598 Edecel AS/A level Mthemtcs Fomule Lst: M M5, S S4 Cotets Pge Issue Septeme 007

5 The fomule ths ooklet hve ee ged ccodg to the ut whch the e fst toduced. Thus cddte sttg ut m e equed to use the fomule tht wee toduced pecedg ut e.g. cddtes sttg C mght e epected to use fomule fst toduced C o C. It m lso e the cse tht cddtes sttg Mechcs d Sttstcs uts eed to use fomule toduced ppopte Coe Mthemtcs uts, s outled the specfcto. UA08598 Edecel AS/A level Mthemtcs Fomule Lst Issue Septeme 007

6 Coe Mthemtcs C Mesuto Sufce e of sphee 4π Ae of cuved sufce of coe π slt heght Athmetc sees u d S l [ d] 4 UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Coe Mthemtcs C Issue Septeme 007

7 UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Coe Mthemtcs C Issue Septeme Coe Mthemtcs C Cddtes sttg C m lso eque those fomule lsted ude Coe Mthemtcs C. Cose ule c c cos A Boml sees N whee!!! C <, R Logthms d epoetls log log log Geometc sees u S S fo < Numecl tegto The tpezum ule: d h{ 0... }, whee h

8 Coe Mthemtcs C Cddtes sttg C m lso eque those fomule lsted ude Coe Mthemtcs C d C. Logthms d epoetls e l Tgoometc dettes s A ± B s Acos B ± cos As B cos A ± B cos Acos B s As B t A ± t B t A ± B A ± B k t At B A B A B s A s B s cos A B A B s A s B cos s A B A B cos A cos B cos cos A B A B cos A cos B s s π Dffeetto f t k sec cot cosec f g f k sec k sec t cosec cosec cot f g f g g 6 UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Coe Mthemtcs C Issue Septeme 007

9 Coe Mthemtcs C4 Cddtes sttg C4 m lso eque those fomule lsted ude Coe Mthemtcs C, C d C. Itegto costt f f d sec k t k k t l sec cot l s cosec l cosec cot l t sec l sec t l t 4 π dv du u d uv v d d d UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Coe Mthemtcs C4 Issue Septeme 007 7

10 Futhe Pue Mthemtcs FP Cddtes sttg FP m lso eque those fomule lsted ude Coe Mthemtcs C d C. Summtos 6 4 Numecl soluto of equtos The Newto-Rphso teto fo solvg f 0 : f f Coodte geomet The pepedcul dstce fom h, k to c 0 s The cute gle etwee les wth gdets m d m s ct h k c m m m m Cocs Pol Rectgul Hpeol Stdd Fom 4 c Pmetc Fom t, t ct, c t Foc, 0 Not equed Dectces Not equed 8 UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Futhe Pue Mthemtcs FP Issue Septeme 007

11 Mt tsfomtos Atclockwse otto though θ out O: cosθ sθ sθ cosθ Reflecto the le cos θ s θ tθ : s θ cos θ UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Futhe Pue Mthemtcs FP Issue Septeme 007 9

12 Futhe Pue Mthemtcs FP Cddtes sttg FP m lso eque those fomule lsted ude Futhe Pue Mthemtcs FP d Coe Mthemtcs C C4. Ae of secto A dθ pol coodtes Comple umes θ e cosθ sθ { cosθ sθ } cosθ s θ k e π The oots of z e gve z, fo k 0,,,, Mclu s d Tlo s Sees f f0 f 0 f 0 f 0!! f f f f f!! f f f f f!! e ep!! l fo ll 5 s! 5!! 4 cos! 4!! 5 ct 5 Tlo polomls h f h f h f f eo! < fo ll fo ll h f h f h f f ξ 0 < ξ < h! f f f f eo! f f f f ξ < ξ <! 0 UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Futhe Pue Mthemtcs FP Issue Septeme 007

13 UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Futhe Pue Mthemtcs FP Issue Septeme 007 Futhe Pue Mthemtcs FP Cddtes sttg FP m lso eque those fomule lsted ude Futhe Pue Mthemtcs FP, d Coe Mthemtcs C C4. Vectos The esolved pt of the decto of s. The pot dvdg AB the to μ λ : s μ λ λ μ Vecto poduct: ˆ s k j θ c. c. c. c c c c..c c If A s the pot wth posto vecto k j d the decto vecto s gve k j, the the stght le though A wth decto vecto hs ctes equto λ z The ple though A wth oml vecto k j hs ctes equto. d d z 0 whee The ple though o-colle pots A, B d C hs vecto equto c c μ λ μ λ μ λ The ple though the pot wth posto vecto d pllel to d c hs equto c t s The pepedcul dstce of,, γ β α fom 0 d z s d γ β α.

14 Hpeolc fuctos cosh sh sh sh cosh cosh cosh sh cosh l{ } sh l{ } th l < Cocs Ellpse Pol Hpeol Rectgul Hpeol Stdd Fom 4 c Pmetc Fom cosθ, sθ t, t sec θ, t θ ± cosh θ, sh θ ct, c t Eccetct e < e e e > e e Foc ± e, 0, 0 ± e, 0 ± c, ± c Dectces ± e ± ± c e Asmptotes oe oe ± 0, 0 UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Futhe Pue Mthemtcs FP Issue Septeme 007

15 Dffeetto f f cs ccos ct sh cosh cosh sh th sech sh cosh th Itegto costt; > 0 whee elevt f f d sh cosh cosh sh th l cosh cs ct < cosh l{ } sh l l th l { } > < UA08598 Edecel AS/A level Mthemtcs Fomule Lst Issue Septeme 007

16 4 UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Futhe Pue Mthemtcs FP Issue Septeme 007 Ac legth s d d d ctes coodtes t t t s d d d d d pmetc fom Sufce e of evoluto d d d d d d S s t t t π π

17 BLANK PAGE TURN OVER FOR MECHANICS & STATISTICS FORMULAE UA08598 Edecel AS/A level Mthemtcs Fomule Lst Issue Septeme 007 5

18 Mechcs M Thee e o fomule gve fo M ddto to those cddtes e epected to kow. Cddtes sttg M m lso eque those fomule lsted ude Coe Mthemtcs C. Mechcs M Cddtes sttg M m lso eque those fomule lsted ude Coe Mthemtcs C, C d C. Cetes of mss Fo ufom odes: Tgul lm: log med fom vete sα Ccul c, dus, gle t cete α : fom cete α sα Secto of ccle, dus, gle t cete α : fom cete α Mechcs M Cddtes sttg M m lso eque those fomule lsted ude Mechcs M, d lso those fomule lsted ude Coe Mthemtcs C C4. Moto ccle Tsvese veloct: v θ Tsvese cceleto: v θ v Rdl cceleto: θ Cetes of mss Fo ufom odes: Sold hemsphee, dus : fom cete 8 Hemsphecl shell, dus : fom cete Sold coe o pmd of heght h: h ove the se o the le fom cete of se to vete 4 Cocl shell of heght h: h ove the se o the le fom cete of se to vete Uvesl lw of gvtto Gmm Foce d 6 UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Mechcs M M Issue Septeme 007

19 Mechcs M4 Thee e o fomule gve fo M4 ddto to those cddtes e epected to kow. Cddtes sttg M4 m lso eque those fomule lsted ude Mechcs M d M, d lso those fomule lsted ude Coe Mthemtcs C C4 d Futhe Pue Mthemtcs FP. Mechcs M5 Cddtes sttg M5 m lso eque those fomule lsted ude Mechcs M d M, d lso those fomule lsted ude Coe Mthemtcs C C4 d Futhe Pue Mthemtcs FP. Momets of et Fo ufom odes of mss m: Th od, legth l, out pepedcul s though cete: ml Rectgul lm out s ple sectg edges of legth l: Th od, legth l, out pepedcul s though ed: 4 ml Rectgul lm out edge pepedcul to edges of legth l: Rectgul lm, sdes d, out pepedcul s though cete: m Hoop o cldcl shell of dus out s though cete: Hoop of dus out dmete: m Dsc o sold clde of dus out s though cete: Dsc of dus out dmete: m Sold sphee, dus, out dmete: 4 m Sphecl shell of dus out dmete: 5 m m m ml 4 ml Pllel es theoem: I I Pepedcul es theoem: Momets s vectos A G z mag The momet out O of F ctg t s I I I fo lm the - ple F UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Mechcs M4 M5 Issue Septeme 007 7

20 Sttstcs S Polt P A B P A P B P A B P A B P A P B A P B A P A P A B P B A P A P B A P A Dscete dstutos Fo dscete dom vle X tkg vlues wth poltes PX Epectto me: EX μ Σ PX Vce: VX σ Σ μ PX Σ PX μ Fo fucto gx : EgX Σg PX Cotuous dstutos Stdd cotuous dstuto: Dstuto of X P.D.F. Me Vce Noml N μ, σ μ σ e σ π μ σ 8 UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 007

21 UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme Coelto d egesso Fo set of ps of vlues, S Σ Σ Σ S Σ Σ Σ S Σ Σ Σ Σ The poduct momet coelto coeffcet s Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ S S S } }{ { The egesso coeffcet of o s S S Σ Σ Lest sques egesso le of o s whee

22 THE NORMAL DISTRIBUTION FUNCTION The fucto tulted elow s Φz, defed s Φz z t π e dt. z Φz z Φz z Φz z Φz z Φz UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 007

23 PERCENTAGE POINTS OF THE NORMAL DISTRIBUTION The vlues z the tle e those whch dom vle Z N0, eceeds wth polt p; tht s, PZ > z Φz p. p z p z UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 007

24 Sttstcs S Cddtes sttg S m lso eque those fomule lsted ude Sttstcs S, d lso those lsted ude Coe Mthemtcs C d C. Dscete dstutos Stdd dscete dstutos: Dstuto of X P X Me Vce Boml B, p p p p p p Posso Poλ λ λ e λ λ! Cotuous dstutos Fo cotuous dom vle X hvg polt dest fucto f Epectto me: E X μ f d Vce: V X f d f d σ Fo fucto gx : Cumultve dstuto fucto: Stdd cotuous dstuto: μ Eg X g f d μ 0 F P X 0 0 f t dt Dstuto of X P.D.F. Me Vce Ufom Rectgul o [, ] UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 007

25 BINOMIAL CUMULATIVE DISTRIBUTION FUNCTION The tulted vlue s PX, whee X hs oml dstuto wth de d pmete p. p , , , , , , UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 007

26 p , , , UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 007

27 p , , UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 007 5

28 p , UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 007

29 p , UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 007 7

30 POISSON CUMULATIVE DISTRIBUTION FUNCTION The tulted vlue s PX, whee X hs Posso dstuto wth pmete λ. λ λ UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 007

31 Sttstcs S Cddtes sttg S m lso eque those fomule lsted ude Sttstcs S d S. Epectto lge Fo depedet dom vles X d Y E XY E X E Y, V X ± Y V X V Y Smplg dstutos Fo dom smple X me μ d vce σ, X,, X of depedet osevtos fom dstuto hvg X s used estmto of μ, wth S s used estmto of σ, whee σ V X Fo dom smple of osevtos fom N μ, σ X μ ~ N0, σ / S Σ X X Fo dom smple of osevtos fom N μ, d, depedetl, dom smple of osevtos fom N μ, X Y μ μ ~ N0, σ σ σ σ Coelto d egesso Spem s k coelto coeffcet s s 6Σd No-pmetc tests Goodess-of-ft test d cotgec tles: O E E ~ χ ν UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 007 9

32 PERCENTAGE POINTS OF THE χ DISTRIBUTION The vlues the tle e those whch dom vle wth the χ dstuto o ν degees of feedom eceeds wth the polt show. ν UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 007

33 CRITICAL VALUES FOR CORRELATION COEFFICIENTS These tles coce tests of the hpothess tht populto coelto coeffcet ρ s 0. The vlues the tles e the mmum vlues whch eed to e eched smple coelto coeffcet ode to e sgfct t the level show, o oe-tled test. Poduct Momet Coeffcet Spem s Coeffcet Level Smple Level Level UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 007

34 RANDOM NUMBERS UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 007

35 Sttstcs S4 Cddtes sttg S4 m lso eque those fomule lsted ude Sttstcs S, S d S. Smplg dstutos Fo dom smple of osevtos fom N μ, σ S σ X μ ~ S / ~ χ t lso vld mtched-ps stutos Fo dom smple of osevtos fom N μ, d, depedetl, dom smple of osevtos fom N μ, σ S S / σ / σ ~ F, σ If σ σ σ ukow the X Y μ μ S p ~ t whee S p S S UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S4 Issue Septeme 007

36 PERCENTAGE POINTS OF STUDENT S t DISTRIBUTION The vlues the tle e those whch dom vle wth Studet s t dstuto o ν degees of feedom eceeds wth the polt show. ν UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S4 Issue Septeme 007

37 PERCENTAGE POINTS OF THE F DISTRIBUTION The vlues the tle e those whch dom vle wth the F dstuto o ν d ν degees of feedom eceeds wth polt 0.05 o 0.0. Polt ν /ν If uppe pecetge pot of the F dstuto o ν d ν degees of feedom s f, the the coespodg lowe pecetge pot of the F dstuto o ν d ν degees of feedom s / f. UA08598 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S4 Issue Septeme 007 5

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