General Certificate of Education

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1 Fomule ttstcl Tbles fo GCE Mthemtcs GCE ttstcs Fst Issue eptembe 004 Fo the ew specfctos fo fst techg fom eptembe 004 GCE Mthemtcs ADVANCED UBIDIARY MATHEMATIC (56) ADVANCED UBIDIARY PURE MATHEMATIC (566) ADVANCED UBIDIARY FURTHER MATHEMATIC (57) ADVANCED MATHEMATIC (66) ADVANCED PURE MATHEMATIC (666) ADVANCED FURTHER MATHEMATIC (67) GCE ttstcs ADVANCED UBIDIARY TATITIC (58) ADVANCED TATITIC (68) Geel Cetfcte of Eucto 9MPM

2 Futhe copes of ths booklet e vlble fom: AQA Logstcs Cete (Gulfo), Deco Fel Offce, tg Hll House, Gulfo, ue, GU 7XJ Telephoe: F: o owlo fom the AQA webste Copght 00 AQA ts lcesos. All ghts eseve. COPYRIGHT AQA ets the copght o ll ts publctos, clug the specme uts mk schemes/ teches gues. Howeve, egstee cetes of AQA e pemtte to cop mtel fom ths booklet fo the ow tel use, wth the followg mpott ecepto: AQA cot gve pemsso to cetes to photocop mtel tht s ckowlege to th pt eve fo tel use wth the cete. et publshe b the Assessmet Qulfctos Allce. The Assessmet Qulfctos Allce (AQA) s comp lmte b gutee egstee Egl Wles 6447 egstee cht umbe 074. Regstee ess AQA, Devs teet, Mcheste, M5 6EX.

3 Cotets Pge 4 Pue Mthemtcs 9 Mechcs 0 Pobblt ttstcs ttstcl Tbles 5 Tble Cumultve Boml Dstbuto Fucto Tble Cumultve Posso Dstbuto Fucto 4 Tble Noml Dstbuto Fucto 5 Tble 4 Pecetge Pots of the Noml Dstbuto 6 Tble 5 Pecetge Pots of the tuet s t-dstbuto 7 Tble 6 Pecetge Pots of the χ Dstbuto 8 Tble 7 Pecetge Pots of the F-Dstbuto 0 Tble 8 Ctcl Vlues of the Pouct Momet Coelto Coeffcet Tble 9 Ctcl Vlues of pem s Rk Coelto Coeffcet Tble 0 Ctcl Vlues of the Wlcoo ge Rk ttstc Tble Ctcl Vlues of the M-Whte ttstc 4 Tble Cotol Chts fo Vblt 5 Tble Rom Numbes klj

4 klm 4 PURE MATHEMATIC Mesuto ufce e of sphee π 4 Ae of cuve sufce of coe heght slt π Athmetc sees [ ] l u ) ( ) ( ) ( Geometc sees fo ) ( < u ummtos ( ) ) )( ( 6 4 ( ) Tgoomet the Cose ule A bc c b cos Boml ees b b b b b ( ) ( ) whee )!!(! C <, (. ) ( ) (. ) ( ) ( ) Logthms epoetls l e Comple umbes ) s (cos )} s (cos { θ θ θ θ θ θ θ s cos e The oots of z e gve b k z π e, fo,,,, 0 k N R

5 Mclu s sees f( ) f(0) f (0) ( ) f (0) f (0)!! e ep( )! l( )! fo ll ( ) ( < ) 5 s ( )! 5! ( )! fo ll 4 cos ( )! 4! ()! fo ll Hpebolc fuctos cosh sh sh sh cosh cosh cosh sh cosh l{ } ( sh l{ } th l ( < ) ) Cocs Ellpse Pbol Hpebol Rectgul hpebol t fom b 4 c b Asmptotes oe oe ± 0, 0 b Tgoometc ettes s( A ± B) s Acos B ± cos As B cos( A ± B) cos Acos B s As B ( A ± B ( )π) t A ± t B t( A ± B) t At B k 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 klj 5

6 Vectos The esolve pt of the ecto of b s.b b The posto vecto of the pot vg AB the to λ : μ s μ λb λ μ Vecto pouct: b b sθ ˆ j k b b b b b b b b b If A s the pot wth posto vecto j k the ecto vecto b s gve b b b b j bk, the the stght le though A wth ecto vecto b hs ctes equto z λ b b b The ple though A wth oml vecto z whee. j k hs ctes equto The ple though o-colle pots A, B C hs vecto equto λ ( b ) μ( c ) ( λ μ) λb μc The ple though the pot wth posto vecto pllel to b c hs equto s b tc Mt tsfomtos cosθ sθ Atclockwse otto though θ bout O: sθ cosθ cos θ s θ Reflecto the le (tθ ) : s θ cos θ The mtces fo ottos ( thee mesos) though gle θ bout oe of the es e 0 0 cosθ 0 sθ 0 sθ fo the -s cosθ cosθ 0 sθ 0 0 fo the -s sθ 0 cosθ cosθ sθ 0 sθ cosθ fo the z-s 6 klm

7 Dffeetto f( ) f ( ) s cos t t k k sec k cosec cosec cot sec sec t cot cosec sh cosh cosh sh th sech sh cosh th f ( ) g( ) f ( ) g( ) f( ) g ( ) (g( )) Itegto ( costt; > 0 whee elevt) f( ) f( ) t l sec cot l s cosec l cosec cot l t( ) sec sec t l t( π) sec k k k t sh cosh cosh sh th l cosh l 4 INTEGRATION FORMULAE CONTINUE OVER THE PAGE klj 7

8 klm 8 ) ( s < t ) ( l o cosh } { > } { l o sh ) ( th l < l u v uv v u Ae of secto θ A (pol cootes) Ac legth s (ctes cootes) t t s t (pmetc fom) ufce e of evoluto π (ctes cootes) t t t π (pmetc fom) Numecl tegto The tpezum ule: b h 0 )} ( ) {(, whee b h The m-ote ule: b h ) (, whee b h mpso s ule: ( ) ( ) ( ) { } b h whee b h s eve

9 Numecl soluto of ffeetl equtos Fo f ( ) smll h, ecuece eltos e: Eule s metho: hf ( ); Fo f(, ) : h Eule s metho: h f(, ) Impove Eule metho: k k ), whee k h f(, ), k h f( h, ) Numecl soluto of equtos ( k The Newto-Rphso teto fo solvg f( ) 0 : f( ) f ( ) MECHANIC Moto ccle Tsvese veloct: v θ Tsvese cceleto: v θ Rl cceleto: θ v Cetes of mss Fo ufom boes Tgul lm: log me fom vete ol hemsphee, us : 8 fom cete Hemsphecl shell, us : fom cete Ccul c, us, gle t cete α : ecto of ccle, us, gle t cete α : sα fom cete α sα fom cete α ol coe o pm of heght h: 4 h bove the bse o the le fom cete of bse to vete Cocl shell of heght h: h bove the bse o the le fom cete of bse to vete Momets of et Fo ufom boes of mss m Th o, legth l, bout pepecul s though cete: ml Rectgul lm bout s ple bsectg eges of legth l: Th o, legth l, bout pepecul s though e: 4 ml Rectgul lm bout ege pepecul to eges of legth l: Rectgul lm, ses b, bout pepecul s though cete: m ( b ) ml 4 ml MOMENT OF INERTIA FORMULAE CONTINUE OVER THE PAGE klj 9

10 Hoop o clcl shell of us bout s: Hoop of us bout mete: m Dsc o sol cle of us bout s: Dsc of us bout mete: 4 m ol sphee, us, bout mete: 5 m phecl shell of us bout mete: m m m Pllel es theoem: I A I G m(ag) Pepecul es theoem: I I I (fo lm the - ple) Geel moto two mesos Rl veloct Tsvese veloct θ Rl cceleto θ Tsvese cceleto θ θ ( θ ) t Momets s vectos z The momet bout O of F ctg though the pot wth posto vecto s Uvesl lw of gvtto Gmm Foce F PROBABILITY TATITIC Pobblt P( A B) P( A) P( B) P( A B) P( A B) P( A) P( B A) P ( A B) j P ( A ) P( B A ) P Epectto lgeb j ( A ) P( B A ) j Covce: Cov( X, Y ) E(( X μ )( Y μ )) E( XY) μ μ V( X ± by ) V( X ) b V( Y ) ± b Cov( X, Y ) X Y X Y Pouct momet coelto coeffcet: Cov( X, Y ) ρ σ σ X Y Fo epeet om vbles X Y E( XY ) E( X ) E( Y ) V( X ± by ) V( X ) b V( Y ) 0 klm

11 Dscete stbutos Fo scete om vble X tkg vlues wth pobbltes p Epectto (me): E( X ) μ p Vce: V( X ) σ ( μ) p Fo fucto g(x ) : E(g( X )) g( ) p t scete stbutos: p μ E( X ) μ Dstbuto of X P( X ) Me Vce Boml B(, p) Posso ) Po(λ Geometc Geo( p ) o,, p ( p) p p( p) λ λ e λ λ! p ( p) p p p Cotuous stbutos Fo cotuous om vble X hvg pobblt est fucto f() Epectto (me): E( X ) μ f( ) Vce: V( X ) σ ( μ) f( ) f( ) Fo fucto g(x ) : E(g( X )) g( ) f( ) μ E( X ) μ Cumultve stbuto fucto: F( ) P( X ) f( t) t t cotuous stbutos: Dstbuto of X Ufom (Rectgul) o [, b] Noml N(, σ ) Epoetl Pobblt est fucto b μ σ ( ) μ e σ π Me ( b ) λ e λ λ Vce ( b ) μ σ λ klj

12 mplg stbutos Fo om smple X, X,, X of epeet obsevtos fom stbuto hvg me μ vce σ X s ubse estmto of μ, wth V( X ) σ s ubse estmto of σ, whee ( X X ) Fo om smple of obsevtos fom N( μ, σ ) X μ ~ N(0,) σ ( ) σ ~ χ X μ ~ t (lso vl mtche-ps stutos) If X s the obseve umbe of successes epeet Beoull tls ech of whch the pobblt of success s p, Y X, the E( Y ) p V( Y ) p( p) Fo om smple of obsevtos fom N( μ, σ ), epeetl, om smple of obsevtos fom N( μ, σ ) ( X Y ) ( μ μ ) ~ N(0,) σ σ / σ / σ ~ F, If σ σ σ (ukow), ( X Y ) ( μ μ ) the ~ t p whee p ( ) ( ) klm

13 Coelto egesso Fo set of ps of vlues, ) ( ( ) ) ( ( )( ) ( ) ( ) ( )( ) The pouct momet coelto coeffcet s ( )( ) { ( ) }{ ( ) } ( )( ) ( ) ( ) pem s k coelto coeffcet s the pouct momet coelto coeffcet betwee ks 6 Whe thee e o te ks t m be clculte usg s ( ) The egesso coeffcet of o s b ( )( ( ) ) Lest sques egesso le of o s b, whee b Alss of vce Oe-fcto moel: j μ α ε, whee ~ N(0, σ ) j ε j T Totl sum of sques T j j Betwee goups sum of sques B T T Two-fcto moel (wth m ows colums): μ α β ε, whee ~ N(0, σ ) Totl sum of sques, Betwee ows sum of sques, T j Betwee colums sum of sques, R j T m R C j T m C j T j m m j j ε j klj

14 Dstbuto-fee (o-pmetc) tests ( O E Gooess-of-ft tests cotgec tbles: ) E s ppomtel stbute s χ Wlcoo sge k test T s the sum of the ks of obsevtos wth the sme sg M-Whte test ( ) U T whee T s the sum of the ks of the smple of sze Kuskl-Wlls test T H ( ) ( ) N N N whee T s the sum of the ks of smple of sze H s ppomtel stbute s χ wth k egees of feeom whee k s the umbe of smples N 4 klm

15 TABLE CUMULATIVE BINOMIAL DITRIBUTION FUNCTION The tbulte vlue s P(X ), whee X hs boml stbuto wth pmetes p. p p klj 5

16 p p klm

17 p p klj 7

18 p p klm

19 p p klj 9

20 p p klm

21 THERE I NO TEXT PRINTED ON THI PAGE klj

22 TABLE CUMULATIVE POION DITRIBUTION FUNCTION The tbulte vlue s P(X ), whee X hs Posso stbuto wth me λ. λ λ λ λ klm

23 λ λ klj

24 TABLE NORMAL DITRIBUTION FUNCTION The tble gves the pobblt, p, tht omll stbute om vble Z, wth me 0 vce, s less th o equl to z. z z klm

25 TABLE 4 PERCENTAGE POINT OF THE NORMAL DITRIBUTION The tble gves the vlues of z stsfg P(Z z) p, whee Z s the omll stbute om vble wth me 0 vce. p p p p klj 5

26 TABLE 5 PERCENTAGE POINT OF THE TUDENT' t-ditribution The tble gves the vlues of stsfg P(X ) p, whee X s om vble hvg the tuet's t-stbuto wth ν egees of feeom. p p ν ν klm

27 TABLE 6 PERCENTAGE POINT OF THE χ DITRIBUTION The tble gves the vlues of stsfg P(X ) p, whee X s om vble hvg the χ stbuto wth ν egees of feeom. p p ν ν klj 7

28 TABLE 7 PERCENTAGE POINT OF THE F-DITRIBUTION The tbles gve the vlues of stsfg P(X ) p, whee X s om vble hvg the F-stbuto wth ν egees of feeom the umeto ν egees of feeom the eomto. F-Dstbuto (p0.995) Use fo oe-tl tests t sgfcce level 0.5% o two-tl tests t sgfcce level %. ν ν ν ν F-Dstbuto (p0.99) Use fo oe-tl tests t sgfcce level % o two-tl tests t sgfcce level %. ν ν ν ν klm

29 F-Dstbuto (p0.975) Use fo oe-tl tests t sgfcce level.5% o two-tl tests t sgfcce level 5%. ν ν ν ν F-Dstbuto (p0.95) Use fo oe-tl tests t sgfcce level 5% o two-tl tests t sgfcce level 0%. ν ν ν ν klj 9

30 TABLE 8 CRITICAL VALUE OF THE PRODUCT MOMENT CORRELATION COEFFICIENT The tble gves the ctcl vlues, fo ffeet sgfcce levels, of the pouct momet coelto coeffcet,, fo vg smple szes,. Oe tl 0% 5%.5% % 0.5% Oe tl Two tl 0% 0% 5% % % Two tl klm