A Level Further Mathematics A

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1 Ofod Cmbdge d RSA Advced GCE (H45) * * A evel The fomto ths booklet s fo the use of cddtes followg the Advced GCE (H45) couse. The fomule booklet wll be pted fo dstbuto wth the emto ppes. Copes of ths booklet m be used fo techg. Ths documet cossts of 6 pges. OCR 08 [603350] DC (SCT) Fomule Booklet Istuctos to Ems OffceIvglto Do ot sed ths Fomule Booklet fo mkg; t should be eted the cete o destoed. Copght Ifomto OCR s commtted to seekg pemsso to epoduce ll thd-pt cotet tht t uses ts ssessmet mtels. OCR hs ttempted to detf d cotct ll copght holdes whose wok s used ths ppe. To vod the ssue of dsclosue of swe-elted fomto to cddtes, ll copght ckowledgemets e epoduced the OCR Copght Ackowledgemets Booklet. Ths s poduced fo ech sees of emtos d s feel vlble to dowlod fom ou publc webste ( fte the lve emto sees. If OCR hs uwttgl fled to coectl ckowledge o cle thd-pt cotet ths ssessmet mtel, OCR wll be hpp to coect ts mstke t the elest possble oppotut. Fo quees o futhe fomto plese cotct the Copght Tem, Fst Floo, 9 Hlls Rod, Cmbdge CB GE. OCR s pt of the Cmbdge Assessmet Goup; Cmbdge Assessmet s the bd me of Uvest of Cmbdge ocl Emtos Sdcte (UCES), whch s tself deptmet of the Uvest of Cmbdge. CST3

2 ue Mthemtcs Athmetc sees S = ( + l) = " + ( - ) d, Geometc sees ( - ) S = - S 3 = - fo Boml sees ( + b) = + C b+ C b + f+ C b + f + b (! ),! whee C C = = O =!( - )! ( - ) ( -) f( - + ) ( + ) = f + + f ^,! Rh!! Sees 3 = ( + )( + ), = ( + ) 6 4 = Mclu sees = () fm() 0 f () 0 f() = f() 0 + f l() f+ + f!! e = ep() = f+ + ffo ll!! 3 + l( l + ) = f+ (- ) + f( - # ) s = f+ (-) + ffo ll 3! 5! ( + )! 4 cos = f+ (- ) + ffo ll! 4! ( )! ( - ) ( -) f( - + ) ( + ) = f + + f ^,! Rh!! Mt tsfomtos 0! Reflecto the le =! :! 0 O cos Atclockwse otto though bout O : s OCR 09 -s cos O

3 3 Rottos though bout the coodte es. The decto of postve otto s tke to be tclockwse whe lookg towds the og fom the postve sde of the s of otto. R R R z 0 0 = 0 cos -s 0 s cos = = > H cos 0 -s 0 0 s 0 cos > H cos -s s cos > H Dffeetto f() f l() t k sec cot cosec cs o s ccos o cos ct o t du dv u d v - u Quotet ule =, = d d v d v Dffeetto fom fst pcples ( ) () f () lm f + h - f l = h " 0 h ksec k sect -cosec -coseccot Itegto f l() d = lf() + c f() + fl() ^f() h d = ^f() h + c + Itegto b pts dv du u d = uv- v d d d OCR 09 Tu ove

4 b The me vlue of f() o the tevl [, b] s f() d b- 4 Ae of secto eclosed b pol cuve s d f() f() d s ^ O h t - O sh - O o l( + + ) cosh - O o l( + - ) ( ) umecl methods b b Tpezum ule: d. h{( + ) + ( + + f + )} 0 -, whee h = - f( ) The ewto-rphso teto fo solvg f() = 0 : = - + f l( ) Comple umbes Ccles: z- = k Hlf les: g( z- ) = es: z- = z- b De Move s theoem: { ( cos+ s )} = ( cos+ s ) Roots of ut: The oots of z k = e gve b z = ep O fo k = 0,,, f, - Vectos d 3-D coodte geomet Ctes equto of the le though the pot A wth posto vecto = + j+ k decto z- u = u + u j+ u k s 3 = = 3 ^= mh u u u 3 Ctes equto of ple + + z+ d = 0 3 b b b-b O O 3 3 O Vecto poduct: # b = b j b b b O # O = = - 3 3O O b O k b b-bo OCR 09

5 5 ^b- h. The dstce betwee skew les s D =, whee d b e posto vectos of pots o ech le d s mutul pepedcul to both les The dstce betwee pot d le s D = the equto of the le s gve b + b = c The dstce betwee pot d ple s equto of the ple s gve b. = p + b - c, whee the coodtes of the pot e (, ) d + b D = b. - p, whee b s the posto vecto of the pot d the Smll gle ppomtos s., cos. -, t. whee s smll d mesued ds Tgoometc dettes s( A! B) = s AcosB! cos As B cos( A! B) = cos AcosB" s As B ta! t B t( A! B) = ( A! B! ( k + ) ) " tat B Hpebolc fuctos cosh - sh = - sh = l[ + ( + )] cosh th - - = l[ + ( - ) ], $ + = l, - - O Smple hmoc moto = Acos( ~ t) + Bs( ~ t) = Rs( ~ t + {) Sttstcs obblt ( A, B) = ( A) + ( B) - ( A+ B) ( A+ B) ( A+ B) = ( A) ( B A) = ( B) ( A B) o ( AB) = ( B) OCR 09 Tu ove

6 6 Stdd devto ^- h = - o f^- h f f = - f Smplg dstbutos v Fo vble X, E( X ) =, V( X ) = d X s ppomtel omll dstbuted whe s lge eough (ppomtel 5) If X + ^, vh the X +, v X - O d + ( 0,) v Ubsed estmtes of the populto me d vce e gve b d - - O O Epectto lgeb Use the followg esults, cludg the cses whee = b =! do c = 0 :. E( X + by + c) = E( X) + be( Y) + c,. f X d Y e depedet the V( X + by + c) = V( X) + b V( Y). Dscete dstbutos X s dom vble tkg vlues dscete dstbuto wth ^X = h = p Epectto: E( X) p = = Vce: v = V( X) = ( - ) p = p - Boml B(, p) Ufom dstbuto ove,,,, U() Geometc dstbuto Geo(p) osso o(m) X ( = ) E( X ) V( X) ( ) p - p - O p p( - p) ( - p) e + -m - m! p p m ^ - h - p p m Cotuous dstbutos X s cotuous dom vble wth pobblt dest fucto (p.d.f.) f() Epectto: E( X) f() d = = Vce: v = V( X) = ( - ) f() d = f() d - Cumultve dstbuto fucto F() = ( X # ) = f() t dt -3 OCR 09

7 7 p.d.f. E( X ) V( X) Cotuous ufom dstbuto ove [, b] b- ( + b) ( b- ) Epoetl me - m m m oml ^, vh e v - - v k v ecetge pots of the oml dstbuto If Z hs oml dstbuto wth me 0 d vce the, fo ech vlue of p, the tble gves the vlue of z such tht Z ( # z) = p. p z o-pmetc tests Goodess-of-ft test d cotgec tbles: Appomte dstbutos fo lge smples ( O - E ) E + Wlcoo Sged Rk test: T + ( + ), ( + )( + ) k Wlcoo Rk Sum test (smples of szes m d, wth m W + m( m+ + ), m( m+ + ) k Coelto d egesso Fo smple of ps of obsevtos (, ) S 4 4 v # ) : k k = ( - ) = -, S = ( ) - = -, S = ( -)( - ) = - S oduct momet coelto coeffcet: = = S S S ( - )( - ) The egesso coeffcet of o s b = = S ( - ) est sques egesso le of o s = + b whee = -b R S S S T - k O - - O V OW k W OW X Spem s k coelto coeffcet: 6 d = - ( - ) OCR 09 s Tu ove

8 OCR 09 Ctcl vlues fo the poduct momet coelto coeffcet, 5% ½% % ½% 0% 5% % % -Tl Test -Tl Test 5% ½% % ½% 5% ½% % ½% 0% 5% % % 0% 5% % % Ctcl vlues fo Spem s k coelto coeffcet, s -Tl Test -Tl Test 5% ½% % ½% 0% 5% % %

9 9 Ctcl vlues fo the dstbuto If X hs dstbuto wth v degees of feedom the, fo ech p of vlues of p d v, the tble gves the vlue of such tht ( X # ) = p. p O OCR 09 p v = Tu ove

10 0 Wlcoo sged k test W + s the sum of the ks coespodg to the postve dffeeces, W - s the sum of the ks coespodg to the egtve dffeeces, T s the smlle of W + d W -. Fo ech vlue of the tble gves the lgest vlue of T whch wll led to ejecto of the ull hpothess t the level of sgfcce dcted. Ctcl vlues of T evel of sgfcce Oe Tl Two Tl = Fo lge vlues of, ech of W + d W - c be ppomted b the oml dstbuto wth me 4 ( + ) d vce ( + )( + ). 4 OCR 09

11 Wlcoo k sum test The two smples hve szes m d, whee m #. R m s the sum of the ks of the tems the smple of sze m. W s the smlle of R m d mm ( + + )- R. m Fo ech p of vlues of m d, the tble gves the lgest vlue of W whch wll led to ejecto of the ull hpothess t the level of sgfcce dcted. Ctcl vlues of W evel of sgfcce Oe Tl Two Tl m = 3 m = 4 m = 5 m = evel of sgfcce Oe Tl Two Tl m = 7 m = 8 m = 9 m = Fo lge vlues of m d, the oml dstbuto wth me mm ( + + ) d vce m ( m + + ) should be used s ppomto to the dstbuto of R m. OCR 09 Tu ove

12 Mechcs emtcs Moto stght le Moto two dmesos v = u+ t v = u+ t s = ut+ t s = ut+ t s = ( u+ vt ) s = ( u+ v)t v = u + s v. v = u. u+. s s = vt- t s = vt- t ewto s epemetl lw Betwee two smooth sphees v - v =-eu ^ -u h Betwee smooth sphee wth fed ple sufce v Moto ccle Tgetl veloct s v = o Rdl cceleto s v o o towds the cete Tgetl cceleto s vo = p =-eu Cetes of mss Tgul lm: 3 log med fom vete 3 Sold hemsphee, dus : 8 fom cete Hemsphecl shell, dus : fom cete s Ccul c, dus, gle t cete : fom cete s Secto of ccle, dus, gle t cete : fom cete 3 Sold coe o pmd of heght h: 4 h bove the bse o the le fom cete of bse to vete Cocl shell of heght h: 3 h bove the bse o the le fom cete of bse to vete OCR 09

13 3 Dscete Mthemtcs Icluso-ecluso pcple Fo sets A, B d C: ^AjBjCh= ^Ah+ ^Bh+ ^Ch-^AkBh-^AkCh- B ^ kch+ ^AkBk Ch The hech of odes 3 O^hf O^loghf O^hf O^log hf O^ hf O^ hf f f O^ h f O^! h Sotg lgothms Bubble sot: Stt t the left hd ed of the lst uless specfed othewse. Compe the fst d secod vlues d swp f ecess. The compe the (ew) secod vlue wth the thd vlue d swp f ecess. Cotue ths w utl ll vlues hve bee cosdeed. F the lst vlue the epet wth the educed lst utl ethe thee s pss whch o swps occu o the lst s educed to legth, the stop. Shuttle sot: Stt t the left hd ed of the lst uless specfed othewse. Compe the secod vlue wth the fst d swp f ecess, ths completes the fst pss. et compe the thd vlue wth the secod d swp f ecess, f swp hppeed shuttle bck to compe the (ew) secod wth the fst s the fst pss, ths completes the secod pss. et compe the fouth vlue wth the thd d swp f ecess, f swp hppeed shuttle bck to compe the (ew) thd vlue wth the secod s the secod pss (so f swp hppes shuttle bck g). Cotue ths w fo - psses, whee s the legth of the lst. Quck sot: The fst vlue sublst wll be the pvot, uless specfed othewse. Wokg fom left to ght, wte dow ech vlue tht s smlle th the pvot, the the pvot, the wok log the lst d wte dow ech vlue tht s ot smlle th the pvot. Ths poduces two sublsts (oe of whch m be empt) wth the pvot betwee them d completes the pss. et ppl ths pocedue to ech of the sublsts fom the pevous pss, uless the cosst of sgle et, to poduce futhe sublsts. Cotue ths w utl o sublst hs moe th oe et. OCR 09 Tu ove

14 4 etwok lgothms Djkst s lgothm START wth gph G. At ech vete dw bo, the lowe e fo tempo lbels, the uppe left hd e fo the ode of becomg pemet d the uppe ght hd e fo the pemet lbel. STE Mke the gve stt vete pemet b gvg t pemet lbel 0 d ode lbel. STE STE 3 STE 4 Fo ech vete tht s ot pemet d s coected b c to the vete tht hs just bee mde pemet (wth pemet lbel = ), dd the c weght to. If ths s smlle th the best tempo lbel t the vete, wte ths vlue s the ew best tempo lbel. Choose the vete tht s ot et pemet whch hs the smllest best tempo lbel. If thee s moe th oe such vete, choose oe of them. Mke ths vete pemet d ssg t the et ode lbel. If eve vete s ow pemet, o f the tget vete s pemet, use tce bck to fd the outes o oute, the STO; othewse etu to STE. m s lgothm (gphcl veso) START wth bt vete of G. STE Add edge of mmum weght jog vete led cluded to vete ot led cluded. STE If spg tee s obted STO; othewse etu to STE. m s lgothm (tbul veso) START wth tble (o mt) of weghts fo coected weghted gph. STE STE Coss though the etes bt ow, d mk the coespodg colum. Choose mmum et fom the uccled etes the mked colum(s). STE 3 If o such et ests STO; othewse go to STE 4. STE 4 Ccle the weght w j foud STE ; mk colum ; coss though ow. STE 5 Retu to STE. uskl s lgothm START wth ll the vetces of G, but o edges; lst the edges cesg ode of weght. STE Add edge of G of mmum weght such w tht o ccles e ceted. STE If spg tee s obted STO; othewse etu to STE. OCR 09

15 5 eest eghbou method START t gve vete of G. STE STE Fd the lest weght c fom ths vete to vete tht hs ot led bee cluded (o bck to the stt vete f eve vete hs bee cluded). If o such c ests the the method hs stlled STO; othewse dd ths c to the pth. STE 3 If ccle hs bee foud STO; othewse etu to STE. owe boud fo tvellg slespeso poblem START wth ll vetces d cs of G. STE STE Remove gve vete d ll cs tht e dectl coected to tht vete, fd mmum spg tee fo the esultg educed etwok. Add the weght of ths mmum coecto to the sum of the two lest weght cs tht hd bee deleted. Ths gves lowe boud. Route specto poblem START wth lst of the odd degee vetces. STE STE STE 3 Fo ech p of odd odes, fd the coectg pth of lest weght. Goup the odd odes so tht the sum of weghts of the coectg pths s mmsed. Add ths sum to the totl weght of the gph STO. The smple lgothm START wth tbleu stdd fomt. STE STE STE 3 STE 4 Choose colum wth egtve et the objectve ow (o zeo degeete cses). The pvot ow s the oe fo whch o-egtve vlue of the et the fl colum dvded b the postve vlue of the et the pvot colum s mmsed. The pvot elemet s the et of the pvot ow the chose colum. Dvde ll etes the pvot ow b the vlue of the pvot elemet. Add to, o subtct fom, ll othe old ows multple of the ew pvot ow, so tht the pvot colum eds up cosstg of zeoes d sgle oe, d coespods to the ew bsc vble. STE 5 If the objectve ow hs o egtve etes STO; othewse etu to STE. OCR 09 Tu ove

16 6 Addtol ue Mthemtcs Vecto poduct # b = bs t, whee, b, t, tht ode, fom ght-hded tple. Sufces Fo 3-D sufces gve the fom z = f(, ), the Hess Mt s gve b H = f f At stto pot of the sufce: f O f O.. f H 0 d f 0, thee s (locl) mmum;. f H 0 d f 0, thee s (locl) mmum; 3. f H 0 thee s sddle-pot; 4. f H = 0 the the tue of the stto pot cot be detemed b ths test. The equto of tget ple to the cuve t gve pot (,, z) = ^, b, f(, b) h s Clculus Ac legth b d s = + O d d s = ^o + o hdt Sufce e of evoluto b z = f(, b) + ( - ) f (, b) + ( - b) f (, b). b d S = + d O d S d d S = + d d O b d d d = () t + dt dt O d d O S = () t + dt dt dt O O dt c c OCR 09