A Level Futhe Mthemtics A (H45) Fomule Booklet Specime OCR 07 H45 Tu ove QN 603/35/0
Pue Mthemtics Aithmetic seies S ( l) { ( ) d} Geometic seies S S ( ) fo Biomil seies ( b) C b C b C b b ( ),! whee C!( )! ( ) ( ) ( ) ( )!!, Seies ( )( ), 6 Mclui seies ( ) 4 3 ( ) f (0) f (0) f( ) f(0) f (0)......!! e ep( )... fo ll!! 3 l( )... ( )... ( ) 3 3 5 si... ( )... fo ll 3! 5! ( )! Specime 4 cos ( ) fo ll! 4! ( )! ( ) ( ) ( ) ( ),!! Mti tsfomtios Reflectio i the lie y : 0 0 Aticlockwise ottio though bout O: cos si si cos OCR 07 H45
3 Rottios though θ bout the coodite es. The diectio of positive ottio is tke to be ticlockwise whe lookig towds the oigi fom the positive side of the is of ottio. 0 0 R 0 cos si 0 si cos R y cos 0 si 0 0 si 0 cos R z cos si 0 si cos 0 0 0 Diffeetitio f ( ) f ( ) t k k sec k sec sec t cot cosec cosec cosec cot csi o si ccoso cos cto t du dv v u u dy Quotiet ule y, d d v d v Diffeetitio fom fist piciples f ( h) f ( ) f ( ) lim h 0 h Itegtio f ( ) d l f ( ) c f ( ) f ( ) f ( ) d f ( ) c Itegtio by pts dv du u d uv v d d d Specime OCR 07 H45 Tu ove
4 The me vlue of f ( ) o the itevl [ b, ] is Ae of secto eclosed by pol cuve is d f( ) f( )d si ( ) t sih o l( ) cosh o l( ) ( ) b b f ( )d Numeicl methods b Tpezium ule: y d h {( y 0 y ) ( y y y ) b }, whee h f( ) The Newto-Rphso itetio fo solvig f( ) 0 : f ( ) Comple umbes Cicles: z k Hlf lies: g z Lies: z z b De Moive s theoem: { cos isi } cos isi Roots of uity: The oots of z e give by k z ep i fo 0,,,..., k Vectos d 3-D coodite geomety Ctesi equtio of the lie though the poit A with positio vecto i j 3k i diectio y z 3 u ui uj u3k is u u u 3 Ctesi equtio of ple is y 3z d 0 Specime Vecto poduct: b i b b3 3b b b j b 3b b 3 3 b 3 3 b 3 b b k OCR 07 H45
5 The distce betwee skew lies is D b lie d is mutul pepedicul to both lies, whee d b e positio vectos of poits o ech by c The distce betwee poit d lie is D b (, y) d the equtio of the lie is give by by c, whee the coodites of the poit e The distce betwee poit d ple is p D b, whee b is the positio vecto of the poit d the equtio of the ple is give by p Smll gle ppoimtios si, cos,t whee θ is smll d mesued i dis Tigoometic idetities si( A B) si Acos B cos Asi B cos( AB) cos Acos B si Asi B t A t B t( A B) A B k t At B Hypebolic fuctios cosh sih sih l[ ( )] cosh l[ ( )], th l, Simple hmoic motio si Acos t B t Rsit Sttistics Specime Pobbility P( A B) P( A) P( B) P( A B) P( A B) P( A)P( B A) P( B)P( A B) o P( A B) P( A B) P( B ) OCR 07 H45 Tu ove
6 Stdd devitio o f f f f Smplig distibutios Fo y vible X, E( X ), V( X ) lge eough (ppoimtely 5 ) X X the X ~ N, d ~ N(0, ) / If ~ N, Ubised estimtes of the popultio me d vice e give by d X is ppoimtely omlly distibuted whe is Epecttio lgeb Use the followig esults, icludig the cses whee b d/o c 0 :. E E E X by c X b Y c,. if X d Y e idepedet the V X by c V X b V Y. d Discete distibutios X is dom vible tkig vlues i i discete distibutio with P( X i ) p i Epecttio: E( X ) ipi V( ) ( ) i i i i Vice: X p p P( X ) E( X ) V( X ) Biomil B(, p ) p ( p ) p p( p) Uifom distibutio ove,,,, U ( ) Geometic distibutio Geo( ) p p p p p Poisso Po( ) e! Specime Cotiuous distibutios X is cotiuous dom vible with pobbility desity fuctio (p.d.f.) f( ) Epecttio: E( X ) f( )d Vice: V( X ) ( ) f( )d f( )d Cumultive distibutio fuctio F( ) P( X ) f ( t)dt OCR 07 H45
7 Cotiuous uifom distibutio ove b, p.d.f. E( X ) V( X ) b b b Epoetil e N Noml, e Pecetge poits of the oml distibutio If Z hs oml distibutio with me 0 d vice the, fo ech vlue of p, the tble gives the vlue of z such tht P( Z z) p. p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.9995 z 0.674.8.645.960.36.576.807 3.090 3.9 No-pmetic tests Goodess-of-fit test d cotigecy tbles: Appoimte distibutios fo lge smples ( Oi Ei) E i v ~ Wilcoo Siged Rk test: T ~ N ( ), ( )( ) 4 4 Wilcoo Rk Sum test (smples of sizes m d, with m ) : W ~ N m( m ), m( m ) Coeltio d egessio Fo smple of pis of obsevtios (, y ) i yi ( ), ( ), S i i Syy yi y yi i y i i i i S ( )( y y) y Poduct momet coeltio coefficiet: i i y i S y S S Sy ( i )( yi y) The egessio coefficiet of y o is b S ( ) yy Lest sques egessio lie of y o is y b whee y b Specime i i yi y i i i yi i y i Spem s k coeltio coefficiet: i 6 d s OCR 07 H45 Tu ove
8 Citicl vlues fo the poduct momet coeltio coefficiet, Citicl vlues fo Spem s k coeltio coefficiet, s 5% ½% % ½% -Til -Til 5% ½% % ½% 5% ½% % ½% Test Test 5% ½% % ½% 0% 5% % % -Til -Til 0% 5% % % 0% 5% % % Test Test 0% 5% % % - - - - 3 0.3009 0.3550 0.458 0.4556 - - - - 3 0.30 0.3560 0.485 0.4593 - - - - 3 0.960 0.3494 0.4093 0.4487 - - - - 3 0.96 0.3504 0.47 0.453 3 0.9877 0.9969 0.9995 0.9999 33 0.93 0.3440 0.403 0.44 3 - - - - 33 0.94 0.3449 0.4054 0.4455 4 0.9000 0.9500 0.9800 0.9900 34 0.869 0.3388 0.397 0.4357 4.0000 - - - 34 0.87 0.3396 0.3995 0.4390 5 0.8054 0.8783 0.9343 0.9587 35 0.86 0.3338 0.396 0.496 5 0.9000.0000.0000-35 0.89 0.3347 0.3936 0.438 6 0.793 0.84 0.88 0.97 36 0.785 0.39 0.386 0.438 6 0.886 0.8857 0.949.0000 36 0.788 0.3300 0.388 0.468 7 0.6694 0.7545 0.839 0.8745 37 0.746 0.346 0.380 0.48 7 0.743 0.7857 0.899 0.986 37 0.748 0.353 0.389 0.4 8 0.65 0.7067 0.7887 0.8343 38 0.709 0.30 0.3760 0.48 8 0.649 0.738 0.8333 0.880 38 0.70 0.309 0.3778 0.455 9 0.58 0.6664 0.7498 0.7977 39 0.673 0.360 0.37 0.4076 9 0.6000 0.7000 0.7833 0.8333 39 0.674 0.368 0.379 0.403 0 0.5494 0.639 0.755 0.7646 40 0.638 0.30 0.3665 0.406 0 0.5636 0.6485 0.7455 0.7939 40 0.640 0.38 0.368 0.405 0.54 0.60 0.685 0.7348 4 0.605 0.308 0.36 0.3978 0.5364 0.68 0.709 0.7545 4 0.606 0.3087 0.3636 0.400 0.4973 0.5760 0.658 0.7079 4 0.573 0.3044 0.3578 0.393 0.5035 0.5874 0.6783 0.773 4 0.574 0.305 0.3594 0.3955 3 0.476 0.559 0.6339 0.6835 43 0.54 0.3008 0.3536 0.3887 3 0.4835 0.5604 0.6484 0.7033 43 0.543 0.304 0.3550 0.3908 4 0.4575 0.534 0.60 0.664 44 0.5 0.973 0.3496 0.3843 4 0.4637 0.5385 0.664 0.679 44 0.53 0.978 0.35 0.3865 5 0.4409 0.540 0.593 0.64 45 0.483 0.940 0.3457 0.380 5 0.4464 0.54 0.6036 0.6536 45 0.484 0.974 0.3470 0.38 6 0.459 0.4973 0.574 0.66 46 0.455 0.907 0.340 0.376 6 0.494 0.509 0.584 0.6353 46 0.456 0.93 0.3433 0.378 7 0.44 0.48 0.5577 0.6055 47 0.49 0.876 0.3384 0.37 7 0.44 0.4877 0.566 0.676 47 0.49 0.880 0.3396 0.374 8 0.4000 0.4683 0.545 0.5897 48 0.403 0.845 0.3348 0.3683 8 0.404 0.476 0.550 0.5996 48 0.403 0.850 0.336 0.370 9 0.3887 0.4555 0.585 0.575 49 0.377 0.86 0.334 0.3646 9 0.39 0.4596 0.535 0.584 49 0.378 0.80 0.336 0.3664 0 0.3783 0.4438 0.555 0.564 50 0.353 0.787 0.38 0.360 0 0.3805 0.4466 0.58 0.5699 50 0.353 0.79 0.393 0.368 0.3687 0.439 0.5034 0.5487 5 0.39 0.759 0.349 0.3575 0.370 0.4364 0.509 0.5558 5 0.39 0.764 0.360 0.359 0.3598 0.47 0.49 0.5368 5 0.306 0.73 0.38 0.354 0.3608 0.45 0.4975 0.5438 5 0.307 0.736 0.38 0.3558 3 0.355 0.43 0.485 0.556 53 0.84 0.706 0.388 0.3509 3 0.358 0.460 0.486 0.536 53 0.84 0.70 0.398 0.354 4 0.3438 0.4044 0.476 0.55 54 0.6 0.68 0.358 0.3477 4 0.3443 0.4070 0.4757 0.509 54 0.6 0.685 0.368 0.349 5 0.3365 0.396 0.46 0.505 55 0.4 0.656 0.39 0.3445 5 0.3369 0.3977 0.466 0.508 55 0.4 0.659 0.339 0.3460 6 0.397 0.388 0.4534 0.4958 56 0. 0.63 0.30 0.345 6 0.3306 0.390 0.457 0.5009 56 0. 0.636 0.3 0.349 7 0.333 0.3809 0.445 0.4869 57 0.0 0.609 0.3074 0.3385 7 0.34 0.388 0.4487 0.495 57 0.0 0.6 0.3083 0.3400 8 0.37 0.3739 0.437 0.4785 58 0.8 0.586 0.3048 0.3357 8 0.380 0.3755 0.440 0.488 58 0.8 0.589 0.3057 0.3370 9 0.35 0.3673 0.497 0.4705 59 0.6 0.564 0.30 0.338 9 0.38 0.3685 0.435 0.4749 59 0.6 0.567 0.3030 0.334 30 0.306 0.360 0.46 0.469 60 0.44 0.54 0.997 0.330 30 0.3063 0.364 0.45 0.4670 60 0.44 0.545 0.3005 0.334 Specime OCR 07 H45
9 Citicl vlues fo the distibutio If X hs distibutio with v degees of feedom the, fo ech pi of vlues of p d v, the tble gives the vlue of such tht P X p. O p p 0.0 0.05 0.05 0.90 0.95 0.975 0.99 0.995 0.999 v 0.0 3 57 0.0 3 98 0.0 393.706 3.84 5.04 6.635 7.879 0.83 0.000 0.05064 0.06 4.605 5.99 7.378 9.0 0.60 3.8 3 0.48 0.58 0.358 6.5 7.85 9.348.34.84 6.7 4 0.97 0.4844 0.707 7.779 9.488.4 3.8 4.86 8.47 5 0.5543 0.83.45 9.36.07.83 5.09 6.75 0.5 6 0.87.37.635 0.64.59 4.45 6.8 8.55.46 7.39.690.67.0 4.07 6.0 8.48 0.8 4.3 8.647.80.733 3.36 5.5 7.53 0.09.95 6. 9.088.700 3.35 4.68 6.9 9.0.67 3.59 7.88 0.558 3.47 3.940 5.99 8.3 0.48 3. 5.9 9.59 3.053 3.86 4.575 7.8 9.68.9 4.73 6.76 3.6 3.57 4.404 5.6 8.55.03 3.34 6. 8.30 3.9 3 4.07 5.009 5.89 9.8.36 4.74 7.69 9.8 34.53 4 4.660 5.69 6.57.06 3.68 6. 9.4 3.3 36. 5 5.9 6.6 7.6.3 5.00 7.49 30.58 3.80 37.70 6 5.8 6.908 7.96 3.54 6.30 8.85 3.00 34.7 39.5 7 6.408 7.564 8.67 4.77 7.59 30.9 33.4 35.7 40.79 8 7.05 8.3 9.390 5.99 8.87 3.53 34.8 37.6 4.3 9 7.633 8.907 0. 7.0 30.4 3.85 36.9 38.58 43.8 Specime 0 8.60 9.59 0.85 8.4 3.4 34.7 37.57 40.00 45.3 8.897 0.8.59 9.6 3.67 35.48 38.93 4.40 46.80 9.54 0.98.34 30.8 33.9 36.78 40.9 4.80 48.7 3 0.0.69 3.09 3.0 35.7 38.08 4.64 44.8 49.73 4 0.86.40 3.85 33.0 36.4 39.36 4.98 45.56 5.8 5.5 3. 4.6 34.38 37.65 40.65 44.3 46.93 5.6 30 4.95 6.79 8.49 40.6 43.77 46.98 50.89 53.67 59.70 40.6 4.43 6.5 5.8 55.76 59.34 63.69 66.77 73.40 50 9.7 3.36 34.76 63.7 67.50 7.4 76.5 79.49 86.66 60 37.48 40.48 43.9 74.40 79.08 83.30 88.38 9.95 99.6 70 45.44 48.76 5.74 85.53 90.53 95.0 00.4 04..3 80 53.54 57.5 60.39 96.58 0.9 06.6.3 6.3 4.8 90 6.75 65.65 69.3 07.6 3. 8. 4. 8.3 37. 00 70.06 74. 77.93 8.5 4.3 9.6 35.8 40. 49.4 OCR 07 H45
0 Wilcoo siged k test W is the sum of the ks coespodig to the positive diffeeces, W is the sum of the ks coespodig to the egtive diffeeces, T is the smlle of W d W. Fo ech vlue of the tble gives the lgest vlue of T which will led to ejectio of the ull hypothesis t the level of sigificce idicted. Citicl vlues of T Level of sigificce Oe Til 0.05 0.05 0.0 0.005 Two Til 0.0 0.05 0.0 0.0 6 0 7 3 0 8 5 3 0 9 8 5 3 0 0 8 5 3 3 0 7 5 7 3 9 7 3 7 9 4 5 5 5 30 5 9 5 6 35 9 3 9 7 4 34 7 3 8 47 40 3 7 9 53 46 37 3 0 60 5 43 37 Fo lge vlues of, ech of W d W c be ppoimted by the oml distibutio with me d vice 4. 4 Specime OCR 07 H45
Wilcoo k sum test The two smples hve sizes m d, whee m. R m is the sum of the ks of the items i the smple of size m. W is the smlle of R d m m m R. m Fo ech pi of vlues of m d, the tble gives the lgest vlue of W which will led to ejectio of the ull hypothesis t the level of sigificce idicted. Citicl vlues of W Level of sigificce Oe Til 0.05 0.05 0.0 0.05 0.05 0.0 0.05 0.05 0.0 0.05 0.05 0.0 Two Til 0. 0.05 0.0 0. 0.05 0.0 0. 0.05 0.0 0. 0.05 0.0 m 3 m 4 m 5 m 6 3 6 - - 4 6 - - 0-5 7 6-0 9 7 6 6 8 7-3 0 8 7 8 6 4 7 8 7 6 4 3 0 8 9 7 5 8 9 8 6 5 4 3 9 3 9 7 9 0 8 7 6 4 3 4 0 33 3 8 0 0 9 7 7 5 3 6 3 35 3 9 Level of sigificce Oe Til 0.05 0.05 0.0 0.05 0.05 0.0 0.05 0.05 0.0 0.05 0.05 0.0 Two Til 0. 0.05 0.0 0. 0.05 0.0 0. 0.05 0.0 0. 0.05 0.0 m 7 m 8 m 9 m 0 7 39 36 34 8 4 38 35 5 49 45 9 43 40 37 54 5 47 66 6 59 0 45 4 39 56 53 49 69 65 6 8 78 74 Fo lge vlues of m d, the oml distibutio with me mm m m Specime should be used s ppoimtio to the distibutio of R m. d vice OCR 07 H45
Mechics Kiemtics Motio i stight lie Motio i two dimesios v u t v u t s ut t s u v t s ut t s u v t v u s v v uu s s vt t s vt t Newto s epeimetl lw Betwee two smooth sphees v v eu u Betwee smooth sphee with fied ple sufce v eu Motio i cicle Tgetil velocity is v Rdil cceletio is v Tgetil cceletio is v Cetes of mss Tigul lmi: 3 o towds the cete log medi fom vete Solid hemisphee, dius : 3 fom cete 8 Hemispheicl shell, dius : fom cete Cicul c, dius, gle t cete α: Specime si fom cete Secto of cicle, dius, gle t cete α: si fom cete 3 Solid coe o pymid of height h: h bove the bse o the lie fom cete of bse to vete 4 Coicl shell of height h: h bove the bse o the lie fom cete of bse to vete 3 OCR 07 H45
3 Discete Iclusio-eclusio piciple Fo sets A, B d C: A BC A B C A B AC B C A B C The hiechy of odes 3 O O! O O log O O log O O... Sotig lgoithms Bubble sot: Shuttle sot: Quick sot: Stt t the left hd ed of the list uless specified othewise. Compe the fist d secod vlues d swp if ecessy. The compe the (ew) secod vlue with the thid vlue d swp if ecessy. Cotiue i this wy util ll vlues hve bee cosideed. Fi the lst vlue the epet with the educed list util eithe thee is pss i which o swps occu o the list is educed to legth, the stop. Stt t the left hd ed of the list uless specified othewise. Compe the secod vlue with the fist d swp if ecessy, this completes the fist pss. Net compe the thid vlue with the secod d swp if ecessy, if swp hppeed shuttle bck to compe the (ew) secod with the fist s i the fist pss, this completes the secod pss. Net compe the fouth vlue with the thid d swp if ecessy, if swp hppeed shuttle bck to compe the (ew) thid vlue with the secod s i the secod pss (so if swp hppes shuttle bck gi). Cotiue i this wy fo psses, whee is the legth of the list. Specime The fist vlue i y sublist will be the pivot, uless specified othewise. Wokig fom left to ight, wite dow ech vlue tht is smlle th the pivot, the the pivot, the wok log the list d wite dow ech vlue tht is ot smlle th the pivot. This poduces two sublists (oe of which my be empty) with the pivot betwee them d completes the pss. Net pply this pocedue to ech of the sublists fom the pevious pss, uless they cosist of sigle ety, to poduce futhe sublists. Cotiue i this wy util o sublist hs moe th oe ety. OCR 07 H45
4 Netwok lgoithms Dijkst s lgoithm START with gph G. At ech vete dw bo, the lowe e fo tempoy lbels, the uppe left hd e fo the ode of becomig pemet d the uppe ight hd e fo the pemet lbel. STEP Mke the give stt vete pemet by givig it pemet lbel 0 d ode lbel. STEP Fo ech vete tht is ot pemet d is coected by c to the vete tht hs just bee mde pemet (with pemet lbel P ), dd the c weight to P. If this is smlle th the best tempoy lbel t the vete, wite this vlue s the ew best tempoy lbel. STEP 3 STEP 4 Pim s lgoithm (gphicl vesio) Choose the vete tht is ot yet pemet which hs the smllest best tempoy lbel. If thee is moe th oe such vete, choose y oe of them. Mke this vete pemet d ssig it the et ode lbel. If evey vete is ow pemet, o if the tget vete is pemet, use tce bck to fid the outes o oute, the STOP; othewise etu to STEP. START with bity vete of G. STEP Add edge of miimum weight joiig vete ledy icluded to vete ot ledy icluded. STEP If spig tee is obtied STOP; othewise etu to STEP. Pim s lgoithm (tbul vesio) START with tble (o mti) of weights fo coected weighted gph. STEP STEP Coss though the eties i bity ow, d mk the coespodig colum. Choose miimum ety fom the ucicled eties i the mked colum(s). STEP 3 If o such ety eists STOP; othewise go to STEP 4. STEP 4 Cicle the weight w ij foud i STEP ; mk colum j ; coss though ow i. STEP 5 Retu to STEP. Kuskl s lgoithm Specime START with ll the vetices of G, but o edges; list the edges i icesig ode of weight. STEP Add edge of G of miimum weight i such wy tht o cycles e ceted. STEP If spig tee is obtied STOP; othewise etu to STEP. OCR 07 H45
5 Neest eighbou method START t give vete of G. STEP Fid the lest weight c fom this vete to vete tht hs ot ledy bee icluded (o bck to the stt vete if evey vete hs bee icluded). STEP If o such c eists the the method hs stlled STOP; othewise dd this c to the pth. STEP 3 If cycle hs bee foud STOP; othewise etu to STEP. Lowe boud fo tvellig slespeso poblem START with ll vetices d cs of G. STEP Remove give vete d ll cs tht e diectly coected to tht vete, fid miimum spig tee fo the esultig educed etwok. STEP 3 Add the weight of this miimum coecto to the sum of the two lest weight cs tht hd bee deleted. This gives lowe boud. Route ispectio poblem START with list of the odd degee vetices. STEP Fo ech pi of odd odes, fid the coectig pth of lest weight. STEP Goup the odd odes so tht the sum of weights of the coectig pths is miimised. STEP 3 Add this sum to the totl weight of the gph STOP. The simple lgoithm START with tbleu i stdd fomt. Specime STEP Choose colum with egtive ety i the objective ow (o zeo i degeete cses). STEP The pivot ow is the oe fo which o-egtive vlue of the ety i the fil colum divided by the positive vlue of the ety i the pivot colum is miimised. The pivot elemet is the ety of the pivot ow i the chose colum. STEP 3 Divide ll eties i the pivot ow by the vlue of the pivot elemet. STEP 4 Add to, o subtct fom, ll othe old ows multiple of the ew pivot ow, so tht the pivot colum eds up cosistig of zeoes d sigle oe, d coespods to the ew bsic vible. STEP 5 If the objective ow hs o egtive eties STOP; othewise etu to STEP. OCR 07 H45
6 Additiol Pue Vecto poduct b = b si ˆ, whee, b, ˆ, i tht ode fom ight-hded tiple. Sufces Fo 3-D sufces give i the fom z f, y, the Hessi Mti is give by f fy H. f f y yy At sttioy poit of the sufce:. if H 0 d f 0, thee is (locl) miimum;. if H 0 d f 0, thee is (locl) mimum; 3. if H 0 thee is sddle-poit; 4. if H 0 the the tue of the sttioy poit cot be detemied by this test. The equtio of tget ple to the cuve t give poit, y, z, b, f, b Clculus Ac legth Sufce e of evolutio b dy s d d b s y dt z f (, b) ( )f (, b) y b f, b. b y is Specime dy d S y d S y dy d dy b d d dy d dy S y( t) dt S y ( t) dt dt dt dt dt c c d OCR 07 H45