Getting Started September 2007

Transcription

1 Gettg Strted Septemer 007 GCE Mthemtcs Edecel Advced Susdr GCE Mthemtcs (837) Edecel Advced Susdr GCE Further Mthemtcs (837) Edecel Advced Susdr GCE Pure Mthemtcs (8373) Edecel Advced Susdr GCE Further Mthemtcs (Addtol) (8374) Frst emto 009 Edecel Advced GCE Mthemtcs (937) Edecel Advced GCE Further Mthemtcs (937) Edecel Advced GCE Pure Mthemtcs (9373) Edecel Advced GCE Further Mthemtcs (Addtol) (9374) Frst emto 009

2 Edecel GCE e-spec Your free e-spec Everthg ou eed oe CD Es-to-use

3 Cotets Gettg strted for techers Itroducto Wht s ew? Course overvews 3 Formule Booklet 4Gettg strted for studets Studet Gude 44 Edecel GCE Mthemtcs Edecel Lmted 007 Gettg Strted

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5 Itroducto Itroducto Ths Gettg Strted ook wll gve ou overvew of the ew Edecel GCE Mthemtcs d wht t mes for ou d our studets. Ke prcples The specfcto hs ee developed wth the followg ke prcples: Focus o choce 8 uts tested full wrtte emto All uts equll weghted, llowg m dfferet comtos of uts d greter flelt Choce of pthws ledg to full Advced Susdr (AS) d Advced (A level) GCE Mthemtcs, Further Mthemtcs, Pure Mthemtcs d Further Mthemtcs (Addtol), so ou c choose the most pproprte pthw for our studets Well supported Pst ppers, specme ppers, emer reports d further support mterls vlle A vret of edorsed electroc support mterl, cludg Em Wzrd, Topc Tutor d Em Tutor Edorsed tetooks d revso ooks, s well s formto o how to mp updted uts to curret tetooks Susttl professol developmet d trg progrmme Strghtforwrd ssessmet Oe wrtte emto per ut Ech emto pper lsts hour 30 mutes Ech emto pper hs 75 mrks Clcultors c e used for ll ut emtos ecept C Edecel GCE Mthemtcs Gettg Strted

6 Wht s ew? Wht s ew? Ths specfcto s ver smlr to the prevous specfcto, ut cludes ew Decso Mthemtcs ut cotet d updted Further Pure Mthemtcs ut cotet. I summr, ths specfcto offers: No chge to Core, Mechcs or Sttstcs ut cotet Updted Decso Mthemtcs d Decso Mthemtcs uts, gvg more lced pproch to the cotet Updted Further Pure Mthemtcs ut for techg the frst er of stud Updted Further Pure Mthemtcs d Further Pure Mthemtcs 3 uts to offer coheret currculum Further Mthemtcs Edecel GCE Mthemtcs Gettg Strted

7 Course overvews Course overvews These course overvews hve ee developed to help ou pl the orgsto d delver of the course. Core mthemtcs: C to C4 progresso C. Alger d fuctos. Coordte geometr the (, ) ple 3. Sequeces d seres 4. Dfferetto 5. Itegrto C 4. Trgoometr. Coordte geometr the (, ) ple. Alger d fuctos 5. Epoetls d logrthms 3. Sequeces d seres 6. Dfferetto 7. Itegrto C3. Alger d fuctos 3. Epoetls d logrthms. Trgoometr 5. Numercl methods 4. Dfferetto C4. Coordte geometr the (, ) ple. Alger d fuctos 4. Dfferetto 6. Vectors 5. Itegrto 3. Sequeces d seres Edecel GCE Mthemtcs Gettg Strted 3

8 Course overvews Mechcs progresso M Fclt lgerc mpulto d the lt to solve ler, qudrtc d smulteous equtos s specfed C s essetl prerequste for ths ut. Prgrph Descrpto Prerequstes Modellg Noe Kemtcs C Prgrph, Alger 3 Vectors C Prgrph, Alger 4 Dmcs Mometum & mpulse 5 Sttcs of prtcle (Dmcs wth = 0) C C Prgrph, Alger Prgrph, Alger 6 Momets C Prgrph, Alger M Kowledge of the M specfcto d the lger, trgoometr, dfferetto d tegrto s specfed C d C re essetl for ths ut. Prgrph Descrpto Prerequstes Projectles M Prgrph 3 Kemtcs wth vrle veloct/ccelerto C Prgrph 4, 5 C Prgrph 6, 7 Cetres of mss M Prgrph 6 3 Work d eerg M Prgrph 4, 5 4 Collsos M Prgrph 4 Mometum & mpulse 5 Sttcs of rgd odes M Prgrph 5, 6 4 Edecel GCE Mthemtcs Gettg Strted

9 Course overvews M3 Kowledge of the M d M specfctos d the dfferetto, tegrto d dfferetl equtos s specfed C, C d C3 re essetl for ths ut. Prgrph Descrpto Prerequstes Elstc strgs d sprgs M Prgrph 3 4 Moto crcle Horzotl crcles Vertcl crcles 5 Cetres of mss of rgd odes Sttcs of rgd odes M3 Prgrph M Prgrph M Prgrph 5 Further kemtcs M Prgrph Soluto of dfferetl equtos s specfed o C, C d C3 3 Vrle force Smple Hrmoc Moto M3 Prgrph M4 Kowledge of the M, M d M3 specfctos d the clculus o FP together wth d + ; d + re essetl for ths ut. Prgrph Descrpto Prerequstes Elstc collsos D M Prgrph 4 3 Further prtcle moto strght le FP Prgrph 5, 6 M3 Prgrph 4 Stlt M Prgrph 3 M3 Prgrph C3 Prgrph 4 Reltve moto M Prgrph C4 Prgrph 6 M5 Kowledge of the M, M, M3 d M4 specfctos, the clculus o FP d the sclr d vector products re essetl for ths ut. Prgrph Descrpto Prerequstes Applcto of vectors FP Prgrph 5, 6 M Prgrph 3 C3 Prgrph 4 FP3 Prgrph 3 3 Momets of ert Itegrto o C to 4 4 Rotto out fed s M3 Prgrph 4 M4 Prgrph 3 Vrle mss M Prgrph 4 (Mometum & mpulse) FP Prgrph 5, 6 Edecel GCE Mthemtcs Gettg Strted 5

10 Course overvews Sttstcs progresso S Prgrph Topc Prerequstes Notes Mthemtcl modellg C e strtg pot ut usull etter looked t towrds the ed, s overvew of the cotet. Represetto d summr of dt Uderstdg of otto from C Sesle strtg pot. There re lks to GCSE e.g. the dgrms, me, med etc. Suggest leve codg utl Prgrph 5. 3 Prolt A good ltertve strtg pot. Ag lks to GCSE lthough stle m e dfferet d P(A B) s ew. 4 Correlto Me d stdrd devto from S Prgrph 4 Regresso S d S etc from S Prgrph 4 d Prgrph Coordte geometr of strght le from C 5 5 5c Dscrete prolt dstrutos Me d vrce of dscrete rdom vrles Dscrete uform dstruto Prolt from S Prgrph 3 Me d vrce from S Prgrph Arthmetc seres des from C c e useful here. 6 Norml dstruto Me d stdrd devto d hstogrms from S Prgrph Prolt from S Prgrph 3 Totl prolt = from S Prgrph 5 Suggest use E(X + ) d Vr(X +) formul to del wth codg. Ths topc s ofte left utl the ed d cddtes fd t chllegg. Possle Routes through S 6: Mtches the specfcto order d some tet ooks ut levg orml dstruto to the ed, some studets m ot grsp ths cotet full., 3, 5, 6, 4, : Correlto d regresso ol deped o the me d stdrd devto mterl d so c e left utl the ed. Prolt followed prolt dstrutos does t sut everoe s tste. 3,, 5, 6, 4, : Ths splts up the prolt d rgs the orml dstruto work lttle erler. There re other possle vrtos. 6 Edecel GCE Mthemtcs Gettg Strted

11 Course overvews S Prgrph Topc Prerequstes Notes Boml dstruto Dscrete prolt dstrutos from S Prgrph 5 Boml theorem from C Posso dstruto Evluto of e o oml for ppromtos from S Prgrph S Prgrph 5 Cotuous rdom vrles Clculus d cocept of re uder curve d m/m from C/C Prllels wth dscrete dstrutos from S Prgrph 5 d orml dstrutos from S Prgrph 6 A good strtg pot provded work from C hs ee covered. A resole ltertve strtg pot. Depeds o lter work C though. M d m s ol eeded for some mode questos. 3 Cotuous dstrutos S Prgrph Rectgulr dstruto should e compred wth dscrete uform dstruto. 3 Norml ppromtos Norml dstruto from S Prgrph 6 d S Prgrph d 4 Hpothess testg Overvew of sttstcs from S Prgrph d S Prgrph d Ths s ke de d pupls eed tme to ssmlte t d prctce the techques. Possle Routes through S Most studets wll hve lred covered C, C d S. Some Further Mthemtcs studets m stll e coverg C/C whe the strt d the frst suggesto s recommeded., 4,, 3: Ths troduces hpothess testg erl d ol requres the oml theorem from C. If C s eg tught t the sme tme, t hs the dvtge of levg prgrph utl lter the course whe the C clculus hs ee covered.,, 3, 4: Ths follows the order of the specfcto d some tet ooks d does hve the dvtge of rek from Boml d Posso efore usg them hpothess tests., 3,, 4, 3: The order of 3 d d of 3 d 4 c e swpped. Ths mkes use of prgrph s ltertve strtg pot. Other comtos re possle. If hpothess testg s covered efore orml ppromtos the cre should e tke to vod questos tht requre the use of orml ppromto to evlute prolt hpothess test. The sc de of hpothess tests c e covered d most emples requre the use of Boml or Posso tles. Edecel GCE Mthemtcs Gettg Strted 7

12 Course overvews S3 Prgrph Topc Prerequstes Notes Comto of rdom vrles Norml dstruto from S Prgrph 6 E(X + ) d Vr(X + ) from S Prgrph 5 Smplg Overvew from S Prgrph Ides of smplg frme, populto, smples from S Prgrph 4 3 Estmto, cofdece tervls d tests 4 Goodess of ft tests Cotgec tle tests 5 Sperm s rk correlto coeffcet d hpothess tests for correlto. Hpothess tests from S Prgrgh 4 Work o comg rdom vrles from S3 Prgrph Boml d Posso from S Prgrph Rectgulr dstruto from S Prgrph 3 Ides of hpothess tests from S Prgrph 4 Prolt for des of depedece from S Prgrph 3 Product momet correlto coeffcet from S Prgrph 4 Ides of hpothess test from S Prgrph 4 A good strtg pot A ltertve strtg pot. Ths depeds clerl o S3 Prgrph d rgul the work o smplg (S3 Prgrph ) provdes some troducto to the estmto topc. A ltertve strtg pot. Aother smple strtg pot. Possle Routes through S3 Some kowledge of S s requred efore S3 c e tught, prtculr the des ehd hpothess testg whch re essetl. Assumg ths hs ee covered, studets c strt S3 o prgrph ecept 3. 5, 4,,, 3: Sperm s rk correlto d the hpothess tests for correlto re possl the smplest topcs S3 d requre lttle ckgroud kowledge. The Ch squred work prgrph 4 c e troduced wthout referece to Boml or Posso dstrutos, lthough these re ke emples tht the wll eed to kow. But prgrphs 5 d 4 provde sutle d frl es strtg pots for S3. Prgrph eeds to come efore prgrph 3 ut prgrph c e ftted lmost where. Ths route provdes getle troducto d covers some ke tests tht mght e useful other sujects such s olog d geogrph t the strt of the course.,, 3, 4, 5: Ths hs the dvtge of followg the order of the specfcto d some tet ooks, d splts the work o comg rdom vrles wth ts chef pplctos prgrph 3 wth the work o smplg whch rgul should come efore the topcs o estmto. There re other possle vrtos. 8 Edecel GCE Mthemtcs Gettg Strted

13 Course overvews S4 Prgrph Topc Prerequstes Notes Qult of tests d Estmtors Oe smple t-test d cofdece tervl Oe smple test for vrce d cofdece tervls Comg rdom vrles from S3 Prgrph Estmtors from S3 Prgrph 3 Ides of hpothess tests from S Prgrph 4 Clculus from C/C S3 pr 3 Cocept of χ v dstruto from S3 Prgrph 4 3 F-test Ide of tests for vrce from S4 Prgrph 3 Test d cofdece tervls for dfferece of two mes usg t-test S3 Prgrph 3 S4 Prgrph 3c Pred t-test S3 Prgrph 3 S4 Prgrph Usull the most demdg secto S4. C e tught depedetl of the other sectos. C e tught logsde the prllel test S3. Ths c e tught logsde the prllel test S3. Ths lks ck to the smple t-test met S4 Prgrph. Possle Routes through S4 The course requres kowledge of prgrph 3 S3 d prts of S4 c deed e tught logsde S3. Tests o S3 tht ssume kowledge of σ c e tught logsde those from S4 ( d 3) tht do t requre ths ssumpto. Prgrph c e tught t pot the S4 course., 3, : The tests re troduced frst d cddtes usull fd ths spect of the course more strghtforwrd th the work prgrph.,, 3: Ths follows the order of the specfcto d some tetooks d covers the more dffcult d theoretcl work frst. Edecel GCE Mthemtcs Gettg Strted 9

14 Course overvews Decso Mthemtcs Progresso Mtchgs (D) Prm s lgorthm (D) Kruskl s lgorthm (D) Trsportto (D) Allocto (D) Trvellg slesm prolem (D) D Prgrph Descrpto Prerequstes Algorthms Noe Prm s d Kruskl s lgorthms Noe 3 Route specto prolem Noe 4 Crtcl pth lss Noe 5 Ler progrmmg Noe 6 Mtchgs Noe D Prgrph Descrpto Prerequstes Trsportto prolems D Prgrph 6 - Mtchgs Allocto (ssgmet) prolems D Prgrph 6 - Mtchgs 3 Trvellg slesm prolem D Prgrph - Prm s d Kruskl s lgorthms 4 Further ler progrmmg Noe 5 Gme theor Noe 6 Flows etworks Noe 7 Dmc progrmmg Noe Most prts of the Decso Mthemtcs specfcto c e tught wthout ssumg pror kowledge; however, some sectos D ssume kowledge from D s show the tle. 0 Edecel GCE Mthemtcs Gettg Strted

15 Formule Booklet Formule Booklet The formule ths ooklet hve ee rrged ccordg to the ut whch the re frst troduced. Thus cddte sttg ut m e requred to use the formule tht were troduced precedg ut (e.g. cddtes sttg C3 mght e epected to use formule frst troduced C or C). It m lso e the cse tht cddtes sttg Mechcs d Sttstcs uts eed to use formule troduced pproprte Core Mthemtcs uts, s outled the specfcto. Edecel GCE Mthemtcs Gettg Strted

16 Formule Booklet Core Mthemtcs C Mesurto Surfce re of sphere = 4πr Are of curved surfce of coe = πr slt heght Arthmetc seres u = + ( )d S = ( + l) = [ + ( )d] 4 UA08598 Edecel AS/A level Mthemtcs Formule Lst: Core Mthemtcs C Issue Septemer 007 Edecel GCE Mthemtcs Gettg Strted

17 Formule Booklet Core Mthemtcs C Cddtes sttg C m lso requre those formule lsted uder Core Mthemtcs C. Cose rule = + c c cos A Boml seres r r ( + ) = r ( )! where = C r = r r!( r)! ( ) ( ) ( r + ) r ( + ) = r ( <, ) Logrthms d epoetls log log = log Geometrc seres u = r S = ( r ) r S = r for r < Numercl tegrto The trpezum rule: d h{( 0 + ) + ( )}, where h = UA08598 Edecel AS/A level Mthemtcs Formule Lst: Core Mthemtcs C Issue Septemer Edecel GCE Mthemtcs Gettg Strted 3

18 Formule Booklet Core Mthemtcs C3 Cddtes sttg C3 m lso requre those formule lsted uder Core Mthemtcs C d C. Logrthms d epoetls e l = Trgoometrc 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 ) Dfferetto 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 Formule Lst: Core Mthemtcs C3 Issue Septemer Edecel GCE Mthemtcs Gettg Strted

19 Formule Booklet Core Mthemtcs C4 Cddtes sttg C4 m lso requre those formule lsted uder Core Mthemtcs C, C d C3. Itegrto (+ costt) f() f( ) d sec k t cot t k k l sec 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 Formule Lst: Core Mthemtcs C4 Issue Septemer Edecel GCE Mthemtcs Gettg Strted 5

20 Formule Booklet Further Pure Mthemtcs FP Cddtes sttg FP m lso requre those formule lsted uder Core Mthemtcs C d C. Summtos r = r= r r 3 = = 6 4 ( + )( + ) ( +) Numercl soluto of equtos The Newto-Rphso terto for solvg f( ) = 0 : + = f( ) f ( ) Coordte geometr The perpedculr dstce from (h, k) to + + c = 0 s h + k + c + m m The cute gle etwee les wth grdets m d m s rct + m m Cocs Prol Rectgulr Hperol Stdrd Form = 4 = c Prmetrc Form (t, t) ct, c t Foc (, 0) Not requred Drectrces = Not requred 8 UA08598 Edecel AS/A level Mthemtcs Formule Lst: Further Pure Mthemtcs FP Issue Septemer Edecel GCE Mthemtcs Gettg Strted

21 Formule Booklet Mtr trsformtos cosθ Atclockwse rotto through θ out O: sθ sθ cosθ Reflecto the le cos θ s θ = (tθ ) : s θ cos θ UA08598 Edecel AS/A level Mthemtcs Formule Lst: Further Pure Mthemtcs FP Issue Septemer Edecel GCE Mthemtcs Gettg Strted 7

22 Formule Booklet Further Pure Mthemtcs FP Cddtes sttg FP m lso requre those formule lsted uder Further Pure Mthemtcs FP d Core Mthemtcs C C4. Are of sector A = r dθ (polr coordtes) Comple umers θ e = cosθ + sθ { r(cosθ + sθ )} = r (cosθ + s θ ) k e π The roots of z = re gve z =, for k = 0,,,, Mclur s d Tlor s Seres r ( r) f( ) = f(0) + f (0) + f (0) + + f (0) +! r! r ( ) ( ) ( r) f( ) = f( ) + ( ) f ( ) + f ( ) + + f ( ) +! r! r ( r) f( + ) = f( ) + f ( ) + f ( ) + + f ( ) +! r! r e = ep( ) = for ll! r! 3 r r+ l ( + ) = + + ( ) + ( < ) 3 r 3 5 r + r s = + + ( ) + for ll 3! 5! (r + )! 4 r r cos = + + ( ) + for ll! 4! (r)! 3 5 r+ r rct = + + ( ) + ( ) 3 5 r + Tlor polomls h f( + h) = f( ) + h f ( ) + f ( ) + error! h f( + h) = f( ) + h f ( ) + f ( + ξ ) (0 < ξ < h)! ( ) f( ) = f( ) + ( ) f ( ) + f ( ) + error! ( ) f( ) = f( ) + ( ) f ( ) + f ( ξ ) ( < ξ < )! 0 UA08598 Edecel AS/A level Mthemtcs Formule Lst: Further Pure Mthemtcs FP Issue Septemer Edecel GCE Mthemtcs Gettg Strted

23 Edecel GCE Mthemtcs Gettg Strted 9 Edecel GCE Mthemtcs Formule Booklet UA08598 Edecel AS/A level Mthemtcs Formule Lst: Further Pure Mthemtcs FP3 Issue Septemer 007 Further Pure Mthemtcs FP3 Cddtes sttg FP3 m lso requre those formule lsted uder Further Pure Mthemtcs FP, d Core Mthemtcs C C4. Vectors The resolved prt of the drecto of s. The pot dvdg AB the rto μ λ : s μ λ λ μ + + Vector product: = = = ˆ s k j θ ) ( ) ( ) ( c. c. c. = = = c c c c..c c ) ( ) ( ) ( = If A s the pot wth posto vector k j = d the drecto vector s gve k j =, the the strght le through A wth drecto vector hs crtes equto ) ( 3 3 = λ = = z The ple through A wth orml vector k j = hs crtes equto =. = d d z where 0 3 The ple through o-coller pots A, B d C hs vector equto c c r μ λ μ λ μ λ + + = + + = ) ( ) ( ) ( The ple through the pot wth posto vector d prllel to d c hs equto c r t + s + = The perpedculr dstce of ),, ( γ β α from 0 3 = d z s 3 3 d γ β α.

24 Formule Booklet Hperolc fuctos cosh sh = sh = sh cosh cosh = cosh + sh rcosh = l{ + } ( ) rsh = l{ + + } + rth = l ( < ) Cocs Ellpse Prol Hperol Rectgulr Hperol Stdrd Form + = = 4 = = c Prmetrc Form ( cosθ, sθ ) ( t, t) ( sec θ, t θ ) (± cosh θ, sh θ) ct, c t Eccetrct e < = ( e ) e = e > = e ( ) e = Foc ( ± e, 0) (, 0) ( ± e, 0) (± c, ± c) Drectrces = ± = e = ± + = ± c e Asmptotes oe oe = ± = 0, = 0 UA08598 Edecel AS/A level Mthemtcs Formule Lst: Further Pure Mthemtcs FP3 Issue Septemer Edecel GCE Mthemtcs Gettg Strted

25 Formule Booklet Dfferetto f() f () rcs rccos rct + sh cosh cosh sh th rsh rcosh sech + rth Itegrto (+ costt; > 0 where relevt) f() f( ) d sh cosh th + + cosh sh l cosh rcs rct ( < ) rcosh = l{ + } rsh = l + l = l + { + + } rth ( > ) ( < ) UA08598 Edecel AS/A level Mthemtcs Formule Lst Issue Septemer Edecel GCE Mthemtcs Gettg Strted

26 Edecel GCE Mthemtcs Gettg Strted Edecel GCE Mthemtcs Formule Booklet 4 UA08598 Edecel AS/A level Mthemtcs Formule Lst: Further Pure Mthemtcs FP3 Issue Septemer 007 Arc legth s d d d + = (crtes coordtes) t t t s d d d d d + = (prmetrc form) Surfce re of revoluto d d d d d d S s t t t π π = = +

27 Formule Booklet BLANK PAGE TURN OVER FOR MECHANICS & STATISTICS FORMULAE UA08598 Edecel AS/A level Mthemtcs Formule Lst Issue Septemer Edecel GCE Mthemtcs Gettg Strted 3

28 Formule Booklet Mechcs M There re o formule gve for M ddto to those cddtes re epected to kow. Cddtes sttg M m lso requre those formule lsted uder Core Mthemtcs C. Mechcs M Cddtes sttg M m lso requre those formule lsted uder Core Mthemtcs C, C d C3. Cetres of mss For uform odes: Trgulr lm: 3 log med from verte r sα Crculr rc, rdus r, gle t cetre α : from cetre α r sα Sector of crcle, rdus r, gle t cetre α : from cetre 3α Mechcs M3 Cddtes sttg M3 m lso requre those formule lsted uder Mechcs M, d lso those formule lsted uder Core Mthemtcs C C4. Moto crcle Trsverse veloct: v = r θ Trsverse ccelerto: v = r θ v Rdl ccelerto: r θ = r Cetres of mss For uform odes: Sold hemsphere, rdus r: Hemsphercl shell, rdus r: Sold coe or prmd of heght h: Cocl shell of heght h: Uversl lw of grvtto Gm m Force = d 3 r from cetre 8 r from cetre h ove the se o the le from cetre of se to verte 4 h ove the se o the le from cetre of se to verte 3 6 UA08598 Edecel AS/A level Mthemtcs Formule Lst: Mechcs M M3 Issue Septemer Edecel GCE Mthemtcs Gettg Strted

29 Formule Booklet Mechcs M4 There re o formule gve for M4 ddto to those cddtes re epected to kow. Cddtes sttg M4 m lso requre those formule lsted uder Mechcs M d M3, d lso those formule lsted uder Core Mthemtcs C C4 d Further Pure Mthemtcs FP. Mechcs M5 Cddtes sttg M5 m lso requre those formule lsted uder Mechcs M d M3, d lso those formule lsted uder Core Mthemtcs C C4 d Further Pure Mthemtcs FP. Momets of ert For uform odes of mss m: Th rod, legth l, out perpedculr s through cetre: ml Rectgulr lm out s ple sectg edges of legth l: ml Th rod, legth l, out perpedculr s through ed: ml Rectgulr lm out edge perpedculr to edges of legth l: Rectgulr lm, sdes d, out perpedculr s through cetre: m ( + ) 3 Hoop or cldrcl shell of rdus r out s through cetre: mr Hoop of rdus r out dmeter: mr Dsc or sold clder of rdus r out s through cetre: Dsc of rdus r out dmeter: mr Sold sphere, rdus r, out dmeter: 4 mr Sphercl shell of rdus r out dmeter: 5 mr 3 mr ml Prllel es theorem: I A = I G + m(ag) Perpedculr es theorem: I = I + I (for lm the - ple) Momets s vectors The momet out O of F ctg t r s z r F UA08598 Edecel AS/A level Mthemtcs Formule Lst: Mechcs M4 M5 Issue Septemer Edecel GCE Mthemtcs Gettg Strted 5

30 Formule Booklet Sttstcs S Prolt 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 ) Dscrete dstrutos For dscrete rdom vrle X tkg vlues wth proltes P(X = ) Epectto (me): E(X) = μ = Σ P(X = ) Vrce: Vr(X) = σ = Σ( μ ) P(X = ) = Σ P(X = ) μ For fucto g(x ) : E(g(X)) = Σg( ) P(X = ) Cotuous dstrutos Stdrd cotuous dstruto: Dstruto of X P.D.F. Me Vrce Norml N( μ, σ ) μ σ e σ π μ σ 8 UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S Issue Septemer Edecel GCE Mthemtcs Gettg Strted

31 Edecel GCE Mthemtcs Gettg Strted 7 Edecel GCE Mthemtcs Formule Booklet UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S Issue Septemer Correlto d regresso For set of prs of vlues ), ( S ) ( ) ( Σ = Σ = Σ S ) ( ) ( Σ = Σ = Σ S ) )( ( ) )( ( Σ Σ = Σ = Σ The product momet correlto coeffcet s Σ Σ Σ Σ Σ Σ Σ = Σ Σ Σ = = S S S r ) ( ) ( ) )( ( ) ( ) ( ) )( ( } }{ { The regresso coeffcet of o s ) ( ) )( ( S S Σ Σ = = Lest squres regresso le of o s + = where =

32 Formule Booklet THE NORMAL DISTRIBUTION FUNCTION The fucto tulted elow s Φ(z), defed s Φ(z) = z t π e dt. z Φ(z) z Φ(z) z Φ(z) z Φ(z) z Φ(z) UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S Issue Septemer Edecel GCE Mthemtcs Gettg Strted

33 Formule Booklet PERCENTAGE POINTS OF THE NORMAL DISTRIBUTION The vlues z the tle re those whch rdom vrle Z N(0, ) eceeds wth prolt p; tht s, P(Z > z) = Φ(z) = p. p z p z UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S Issue Septemer 007 Edecel GCE Mthemtcs Gettg Strted 9

34 Formule Booklet Sttstcs S Cddtes sttg S m lso requre those formule lsted uder Sttstcs S, d lso those lsted uder Core Mthemtcs C d C. Dscrete dstrutos Stdrd dscrete dstrutos: Dstruto of X P( X = ) Me Vrce Boml B(, p) p ( p) p p( p) Posso Po(λ) λ λ e λ λ! Cotuous dstrutos For cotuous rdom vrle X hvg prolt dest fucto f Epectto (me): E( X ) = μ = f( ) d σ Vrce: Vr( X ) = = ( ) f( )d = f( )d For fucto g(x ) : μ E(g( X )) = g( ) f( ) d Cumultve dstruto fucto: F( ) = P( X ) = 0 Stdrd cotuous dstruto: μ 0 f ( t) d 0 Dstruto of X P.D.F. Me Vrce t Uform (Rectgulr) o [, ] ( + ) ( ) UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S Issue Septemer Edecel GCE Mthemtcs Gettg Strted

35 Formule Booklet BINOMIAL CUMULATIVE DISTRIBUTION FUNCTION The tulted vlue s P(X ), where X hs oml dstruto wth de d prmeter p. p = = 5, = = 6, = = 7, = = 8, = = 9, = = 0, = UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S Issue Septemer Edecel GCE Mthemtcs Gettg Strted 3

36 Formule Booklet p = =, = = 5, = = 0, = UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S Issue Septemer Edecel GCE Mthemtcs Gettg Strted

37 Formule Booklet p = = 5, = = 30, = UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S Issue Septemer Edecel GCE Mthemtcs Gettg Strted 33

38 Formule Booklet p = = 40, = UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S Issue Septemer Edecel GCE Mthemtcs Gettg Strted

39 Formule Booklet p = = 50, = UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S Issue Septemer Edecel GCE Mthemtcs Gettg Strted 35

40 Formule Booklet POISSON CUMULATIVE DISTRIBUTION FUNCTION The tulted vlue s P(X ), where X hs Posso dstruto wth prmeter λ. λ = = λ = = UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S Issue Septemer Edecel GCE Mthemtcs Gettg Strted

41 Formule Booklet Sttstcs S3 Cddtes sttg S3 m lso requre those formule lsted uder Sttstcs S d S. Epectto lger For depedet rdom vrles X d Y E( XY ) = E( X ) E( Y ), Vr( X ± Y ) = Vr( X ) + Vr( Y ) Smplg dstrutos For rdom smple X,, X, X of depedet oservtos from dstruto hvg me d vrce σ σ X s used estmtor of, wth Vr( X ) = S s used estmtor of σ, where For rdom smple of oservtos from N( μ, σ ) X μ ~ N(0,) σ / S Σ( X X ) = For rdom smple of oservtos from N( μ, ) d, depedetl, rdom smple of oservtos from N( μ, σ ) ( X Y ) ( μ μ ) ~ N(0,) σ σ + σ Correlto d regresso Sperm s rk correlto coeffcet s r s = 6Σd ( ) No-prmetrc tests Goodess-of-ft test d cotgec tles: ( O E ) E ~ χ ν UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S3 Issue Septemer Edecel GCE Mthemtcs Gettg Strted 37

42 Formule Booklet PERCENTAGE POINTS OF THE χ DISTRIBUTION The vlues the tle re those whch rdom vrle wth the χ dstruto o ν degrees of freedom eceeds wth the prolt show. ν UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S3 Issue Septemer Edecel GCE Mthemtcs Gettg Strted

43 Formule Booklet CRITICAL VALUES FOR CORRELATION COEFFICIENTS These tles cocer tests of the hpothess tht populto correlto coeffcet ρ s 0. The vlues the tles re the mmum vlues whch eed to e reched smple correlto coeffcet order to e sgfct t the level show, o oe-tled test. Product Momet Coeffcet Sperm s Coeffcet Level Smple Level Level UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S3 Issue Septemer Edecel GCE Mthemtcs Gettg Strted 39

44 Formule Booklet RANDOM NUMBERS UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S3 Issue Septemer Edecel GCE Mthemtcs Gettg Strted

45 Formule Booklet Sttstcs S4 Cddtes sttg S4 m lso requre those formule lsted uder Sttstcs S, S d S3. Smplg dstrutos For rdom smple of oservtos from N( μ, σ ) ( ) S σ X μ ~ S / ~ χ t (lso vld mtched-prs stutos) For rdom smple of oservtos from N( μ, ) d, depedetl, rdom smple of oservtos from N( μ, σ ) σ S S / σ / σ ~ F, If σ = σ = σ (ukow) the ( X Y ) ( μ μ ) ~ t S + p + where ( S = p ) S + ( + ) S UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S4 Issue Septemer Edecel GCE Mthemtcs Gettg Strted 4

46 Formule Booklet PERCENTAGE POINTS OF STUDENT S t DISTRIBUTION The vlues the tle re those whch rdom vrle wth Studet s t dstruto o ν degrees of freedom eceeds wth the prolt show. ν UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S4 Issue Septemer Edecel GCE Mthemtcs Gettg Strted

47 Formule Booklet PERCENTAGE POINTS OF THE F DISTRIBUTION The vlues the tle re those whch rdom vrle wth the F dstruto o ν d ν degrees of freedom eceeds wth prolt 0.05 or 0.0. Prolt ν /ν If upper percetge pot of the F dstruto o ν d ν degrees of freedom s f, the the correspodg lower percetge pot of the F dstruto o ν d ν degrees of freedom s / f. UA08598 Edecel AS/A level Mthemtcs Formule Lst: Sttstcs S4 Issue Septemer Edecel GCE Mthemtcs Gettg Strted 43