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 Jue 009 Ths cop s the popet of Edecel. It s ot to e emoved fom the emto oom o mked w.

Pue Mthemtcs FP FP Mechcs M M5 Sttstcs S S4 Fo use fom Jue 009 Ths

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 Cocs 8 Mt tsfomtos 9 Futhe Pue Mthemtcs FP 9 Ae of secto 9 Mclu s d Tlo s Sees 0 Futhe Pue Mthemtcs FP 0 Vectos Hpeolcs Dffeetto Itegto Ac legth Sufce e of evoluto Edecel AS/A level Mthemtcs Fomule Lst: C C4, FP FP Cotets Pge Issue Septeme 009

Summtos 8 Numecl soluto of equtos 8 Cocs 8 Mt tsfomtos 9 Futhe Pue Mthemtcs FP 9 Ae of secto 9 Mclu s d Tlo s Sees 0 Futhe Pue Mthemtcs

4 4 Mechcs M 4 Thee e o fomule gve fo M ddto to those cddtes e epected to kow. 4 Mechcs M 4 Cetes of mss 4 Mechcs M 4 Moto ccle 4 Cetes of mss 4 Uvesl lw of gvtto 5 Mechcs M4 5 Thee e o fomule gve fo M4 ddto to those cddtes e epected to kow. 5 Mechcs M5 5 Momets of et 5 Momets s vectos 6 Sttstcs S 6 Polt 6 Dscete dstutos 6 Cotuous dstutos 7 Coelto d egesso 8 The Noml dstuto fucto 9 Pecetge pots of the Noml dstuto 0 Sttstcs S 0 Dscete dstutos 0 Cotuous dstutos Boml cumultve dstuto fucto 6 Posso cumultve dstuto fucto 7 Sttstcs S 7 Epectto lge 7 Smplg dstutos 7 Coelto d egesso 7 No-pmetc tests 8 Pecetge pots of the χ dstuto 9 Ctcl vlues fo coelto coeffcets 0 Rdom umes Sttstcs S4 Smplg dstutos Pecetge pots of Studet s t dstuto Pecetge pots of the F dstuto Thee e o fomule povded fo Decso Mthemtcs uts D d D. Edecel AS/A level Mthemtcs Fomule Lst: M M5, S S4 Cotets Pge Issue Septeme 009

5 Mechcs M5 5 Momets of et 5 Momets s vectos 6 Sttstcs S 6 Polt 6 Dscete dstutos 6 Cotuous dstutos 7 Coelto d egesso 8 The Noml dstuto fucto 9 Pecetge pots of the Noml dstuto 0 Sttstcs S 0 Dscete

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. Edecel AS/A level Mthemtcs Fomule Lst Issue - Septeme 009

It m lso e the cse tht cddtes sttg Mechcs d Sttstcs uts eed to use fomule toduced ppopte Coe

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 Edecel AS/A level Mthemtcs Fomule Lst: Coe Mthemtcs C Issue Septeme 009

7 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 K K N whee!!! C <, K K K K R Logthms d epoetls log log log Geometc sees u S S fo < Numecl tegto The tpezum ule: d h{ 0... }, whee h

Cose ule c c cos A Boml sees K K N whee!

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 m s As B t A ± t B t A ± B A ± B k m t A t 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 Edecel AS/A level Mthemtcs Fomule Lst: Coe Mthemtcs C Issue Septeme 009

± B k m t A t 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

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 Edecel AS/A level Mthemtcs Fomule Lst: Coe Mthemtcs C4 Issue Septeme 009 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 Cocs Pol Rectgul Hpeol Stdd Fom 4 c Pmetc Fom t, t ct, c t Foc, 0 Not equed Dectces Not equed Mt tsfomtos Atclockwse otto though θ out O: cosθ sθ sθ cosθ Reflecto the le cos θ s θ tθ : s θ cos θ I FP, θ wll e multple of Edecel AS/A level Mthemtcs Fomule Lst: Futhe Pue Mthemtcs FP Issue Septeme 009

Fom t, t ct, c t Foc, 0 Not equed Dectces Not equed Mt tsfomtos Atclockwse otto though θ out O: cosθ sθ sθ cosθ

11 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 } cos θ s θ k e π The oots of z e gve z, fo k 0,,, K, Mclu s d Tlo s Sees f f0 f 0 f 0 K f 0 K!! f f f f K f K!! f f f f K f K!! e ep! K K! l K fo ll K 5 s K K! 5!! 4 cos K K! 4!! 5 ct K K 5 < fo ll fo ll Edecel AS/A level Mthemtcs Fomule Lst: Futhe Pue Mthemtcs FP Issue Septeme 009 9

K, Mclu s d Tlo s Sees f f0 f 0 f 0 K f 0 K!! f f f f K f K!! f f f f K f K!! e ep! K K! l K fo ll K 5 s K K! 5!! 4 cos K K!

12 0 Edecel AS/A level Mthemtcs Fomule Lst: Futhe Pue Mthemtcs FP Issue Septeme 009 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 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.

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

13 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 Edecel AS/A level Mthemtcs Fomule Lst: Futhe Pue Mthemtcs FP Issue Septeme 009

t Eccetct e < e e e > e e Foc ± e, 0, 0 ± e, 0 ± c, ± c Dectces ± e ± ± c e Asmptotes oe

14 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 l { } th > < Edecel AS/A level Mthemtcs Fomule Lst: Futhe Pue Mthemtcs FP Issue Septeme 009

15 Edecel AS/A level Mthemtcs Fomule Lst: Futhe Pue Mthemtcs FP Issue Septeme 009 Ac legth s d d d ctes coodtes t t t s d d d d d pmetc fom Sufce e of evoluto S ds π d d d π t t t d d d d d π

16 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 4 Edecel AS/A level Mthemtcs Fomule Lst: Mechcs M M Issue Septeme 009

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

17 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 Edecel AS/A level Mthemtcs Fomule Lst: Mechcs M4 M5 Issue Septeme 009 5

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.

18 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 6 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 009

19 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

The poduct momet coelto coeffcet s Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ S S S } }{ {

20 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 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 009

56 0.7.06 0.8554.56 0.9406. 0.980 0.07 0.579 0.57 0.757.07 0.8577.57 0.948.4 0.988 0.08 0.59 0.58 0.790.08 0.8599.58 0.949.6 0.9846 0.09 0.559 0.59 0.74.09 0.86.59 0.944.8 0.9854 0.0 0.598 0.60 0.757.0 0.864.

21 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 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 009 9

22 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 σ μ f d f d Fo fucto gx : Eg X g f d Cumultve dstuto fucto: Stdd cotuous dstuto: F μ 0 P X 0 0 f t d Dstuto of X P.D.F. Me Vce t Ufom Rectgul o [, ] 0 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 009

23 BINOMIAL CUMULATIVE DISTRIBUTION FUNCTION The tulted vlue s PX, whee X hs oml dstuto wth de d pmete p. p , , , , , , Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 009

24 p , , , Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 009

25 p , , Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 009

26 p , Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 009

27 p , Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 009 5

28 POISSON CUMULATIVE DISTRIBUTION FUNCTION The tulted vlue s PX, whee X hs Posso dstuto wth pmete λ. λ λ Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 009

29 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, K, X of depedet osevtos fom dstuto hvg X s used estmto of μ, wth S s used estmto of σ, whee σ V X Σ X X S Fo dom smple of osevtos fom N μ, σ X μ ~ N0, σ / 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 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 009 7

30 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 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 009

31 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 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 009 9

32 RANDOM NUMBERS Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S Issue Septeme 009

33 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 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S4 Issue Septeme 009

34 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 Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S4 Issue Septeme 009

35 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. Edecel AS/A level Mthemtcs Fomule Lst: Sttstcs S4 Issue Septeme 009

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