Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE Mathematics and Further Mathematics
|
|
|
- Amberlynn Thompson
- 5 years ago
- Views:
Transcription
1 Peso Edecel Level Advced Subsidiy d Advced GCE Mthemtics d Futhe Mthemtics Mthemticl fomule d sttisticl tbles Fo fist cetifictio fom Jue 08 fo: Advced Subsidiy GCE i Mthemtics (8MA0) Advced GCE i Mthemtics (9MA0) Advced Subsidiy GCE i Futhe Mthemtics (8FM0) Fo fist cetifictio fom Jue 09 fo: Advced GCE i Futhe Mthemtics (9FM0) This copy is the popety of Peso. It is ot to be emoved fom the emitio oom o mked i y wy.
2
3 Cotets Itoductio AS Level i Mthemtics Pue Mthemtics Sttistics Mechics A Level i Mthemtics 4 Pue Mthemtics 4 Sttistics 6 Mechics 7 4 AS Level i Futhe Mthemtics 8 Pue Mthemtics 8 Sttistics Mechics 4 5 A Level i Futhe Mthemtics 5 Pue Mthemtics 5 Sttistics Mechics 5 6 Sttisticl Tbles 6 Biomil Cumultive Distibutio Fuctio 6 Pecetge Poits Of The Noml Distibutio Poisso Cumultive Distibutio Fuctio Pecetge Poits of the Distibutio Citicl Vlues fo Coeltio Coefficiets 4 Rdom Numbes 5 Pecetge Poits of Studet s t Distibutio 6 Pecetge Poits of the F Distibutio 7
4
5 Itoductio The fomule i this booklet hve bee ged by qulifictio. Studets sittig AS o A Level Futhe Mthemtics ppes my be equied to use the fomule tht wee itoduced i AS o A Level Mthemtics ppes. It my lso be the cse tht studets sittig Mechics d Sttistics ppes will eed to use fomule itoduced i the ppopite Pue Mthemtics ppes fo the qulifictio they e sittig. Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
6 AS Level i Mthemtics Pue Mthemtics Mesutio Sufce e of sphee = 4 Ae of cuved sufce of coe = slt height Biomil seies ( b) b b b b ( N) whee! C! ( )! Logithms d epoetils logb log log l e b Diffeetitio Fist Piciples f ( h) f ( ) f ( ) =lim h0 h Sttistics Pobbility P(A) = P(A) Stdd devitio Stdd devitio = (Vice) Itequtile ge = IQR = Q Q Fo set of vlues,,...,... ( i ) S ( i ) i i Stdd devitio = S o Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
7 Sttisticl tbles The followig sttisticl tbles e equied fo A Level Mthemtics: Biomil Cumultive Distibutio Fuctio (see pge 5) Rdom Numbes (see pge 4) Mechics Kiemtics Fo motio i stight lie with costt cceletio: v = u + t s = ut + ½ t s = vt - ½ t v = u + s s = ½ (u + v)t Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
8 A Level i Mthemtics Pue Mthemtics Mesutio Sufce e of sphee = 4 Ae of cuved sufce of coe = slt height Aithmetic seies S = ( + l) = [ + ( )d] Biomil seies ( b ) b b b b ( N) whee! C! ( )! ( ) ( ) ( ) ( ) (, ) Logithms d epoetils logb log log l e b Geometic seies S = ( ) S = fo < Numeicl itegtio The tpezium ule: b y d h{(y 0 + y ) + (y + y y )}, whee b h 4 Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
9 Tigoometic idetities si ( A B) siacos B cosa sib cos( AB) cosa cosb sia sib ta tb t ( A B) ( A B ( k ) ) tatb A B A B siasib si cos A B A B siasib cos si A B A B cosacosb cos cos A B A B cosa cosb si si Diffeetitio Fist Piciples f ( h) f ( ) f ( ) =lim h0 h f() t k seck cotk cosec k f( ) g( ) f () k sec k kseck tk kcosec k kcosec k cot k f ( ) g( ) f( ) g ( ) (g( )) Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics 5 Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
10 Itegtio (+ costt) f() f( ) d sec k t k k tk cot k coseck seck l seck k l si k k l cosec k cot k, l t( k) k k l sec k t k, l t( k 4 ) k k dv du u d uv v d d d Numeicl solutio of equtios The Newto-Rphso itetio fo solvig f( ) 0 : f( ) f ( ) Sttistics Pobbility P(A) = P(A) P( A B) P( A) P( B) P( A B) P( AB) P( A)P( B A) P( B A)P( A) P( A B) P( B A)P( A) P( B A)P( A) Fo idepedet evets A d B, P(BA) = P(B), P(AB) = P(A), P(A B) = P(A) P(B) 6 Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
11 Stdd devitio Stdd devitio = (Vice) Itequtile ge = IQR = Q Q Fo set of vlues,,...,... i S ( ) i ( ) i i Stdd devitio = S o Discete distibutios Distibutio of X P(X = ) Me Vice Biomil B(, p) p ( p ) p p( p) Smplig distibutios Fo dom smple of obsevtios fom N(, ) X ~ N(0, ) / Sttisticl tbles The followig sttisticl tbles e equied fo A Level Mthemtics: Biomil Cumultive Distibutio Fuctio (see pge 5) Pecetge Poits Of The Noml Distibutio (0) Citicl Vlues fo Coeltio Coefficiets: Poduct Momet Coefficiet (see pge ) Rdom Numbes (see pge 4) Mechics Kiemtics Fo motio i stight lie with costt cceletio: v = u + t s = ut + ½ t s = vt - ½ t v = u + s s = ½ (u + v)t Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics 7 Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
12 4 AS Level i Futhe Mthemtics Studets sittig AS Level Futhe Mthemtics ppe my lso equie those fomule listed fo A Level Mthemtics i Sectio. Pue Mthemtics Summtios 6 ( )( ) 4 ( ) Mti tsfomtios Aticlockwise ottio though bout O: cos si si cos Reflectio i the lie cos si y (t ) : si cos Ae of secto (pol coodites) A = d Comple umbes { (cos isi )} (cos isi ) The oots of z e give by k i z e, fo k 0,,,, 8 Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
13 Mclui s d Tylo s Seies ( ) f( ) f(0) f (0) f (0) f (0)!! e ep( )!! fo ll l ( ) ( ) ( ) si! 5 5! ( ) ( )! fo ll cos! 4 4! ( ) ()! fo ll 5 ct ( ) ( ) 5 Vectos Vecto poduct: b b si ˆ i b b j k b b b b b b b. ( bc) b b b b. ( c) c. ( b) c c If A is the poit with positio vecto b bi b jbk b y b c i j k d the diectio vecto b is give by, the the stight lie though A with diectio vecto b hs ctesi equtio z b ( ) The ple though A with oml vecto y z d 0 whee d. i j k hs ctesi equtio The ple though o-collie poits A, B d C hs vecto equtio ( b ) ( c ) ( ) b c The ple though the poit with positio vecto d pllel to b d c hs equtio sb tc The pepedicul distce of,, ) ( fom y z d 0 is d Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics 9 Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
14 Hypebolic fuctios cosh sih sih sih cosh cosh cosh sih cosh l{ } ( ) sih l{ } th l ( ) Diffeetitio f() csi f() ccos ct sih cosh cosh sih th sech sih cosh th 0 Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
15 Itegtio (+ costt; 0 whee elevt) f() f( ) d sih cosh cosh sih th l cosh csi ( ) ct cosh, l{ } ( ) sih, l { } l th ( ) l Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
16 Sttistics Discete distibutios Fo discete dom vible X tkig vlues Epecttio (me): E(X) = = i P(X = i ) Vice: V(X) = = ( i ) P(X = i ) = i with pobbilities P(X = i) P(X = i ) i Discete distibutios Stdd discete distibutios: Distibutio of X P(X = ) Me Vice p Biomil B(, p) p p p( p) Poisso Po ( ) e! Cotiuous distibutios Fo cotiuous dom vible X hvig pobbility desity fuctio f Epecttio (me): E( X ) f( ) d X Vice: V( ) ( ) f( )d f( )d Fo fuctio ) g( X : E(g( X )) g( ) f( ) d Cumultive distibutio fuctio: F( ) P( X ) f ( t) dt Stdd cotiuous distibutio: Distibutio of X P.D.F. Me Vice Noml N(, ) e Uifom (Rectgul) o [, b] b ( b) ( b ) Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
17 Coeltio d egessio Fo set of pis of vlues, y ) ( i ) S ( i ) i ( yi ) S yy ( yi y) yi ( i i ( i)( yi) S y ( i )( yi y) i yi The poduct momet coeltio coefficiet is: ( i)( yi) S y ( )( ) i i y i yi y S S yy { ( i ) }{ ( yi y) } ( i) ( yi) i yi The egessio coefficiet of y o is S ( )( y y) b S ( ) y i i i Lest sques egessio lie of y o is y b whee y b Residul Sum of Sques (RSS) = Sy Syy Syy S Spem s k coeltio coefficiet is s 6d ( ) No-pmetic tests Goodess-of-fit test d cotigecy tbles: ( O i Ei E ) i ~ Sttisticl tbles The followig sttisticl tbles e equied fo AS Level Futhe Mthemtics: Biomil Cumultive Distibutio Fuctio (see pge 5) Poisso Cumultive Distibutio Fuctio (see pge ) Pecetge Poits of the Distibutio (see pge ) Citicl Vlues fo Coeltio Coefficiets: Poduct Momet Coefficiet d Spem s Coefficiet (see pge ) Rdom Numbes (see pge 4) Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
18 Mechics Cetes of mss Fo uifom bodies: Tigul lmi: log medi fom vete Cicul c, dius, gle t cete : si fom cete Secto of cicle, dius, gle t cete si : fom cete 4 Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
19 5 A Level i Futhe Mthemtics Studets sittig A Level Futhe Mthemtics ppe my lso equie those fomule listed fo A Level Mthemtics i Sectio. Pue Mthemtics Summtios 6 ( )( ) 4 ( ) Mti tsfomtios Aticlockwise ottio though bout O: cos si si cos Reflectio i the lie cos si y (t ) : si cos Ae of secto (pol coodites) A = d Comple umbes { (cos isi )} (cos isi ) The oots of z e give by k i z e, fo k 0,,,, Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics 5 Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
20 Mclui s d Tylo s Seies ( ) f( ) f(0) f (0) f (0) f (0)!! e ep( )!! fo ll l ( ) ( ) ( ) si! 5 5! ( ) ( )! fo ll cos! 4 4! ( ) ()! fo ll 5 ct ( ) ( ) 5 Vectos Vecto poduct: b b si ˆ i b b j k b b b b b b b. ( bc) b b b b. ( c) c. ( b) c c If A is the poit with positio vecto b bi b jbk y z b b b c i j k d the diectio vecto b is give by, the the stight lie though A with diectio vecto b hs ctesi equtio ( ) The ple though A with oml vecto y z d 0 whee d. i j k hs ctesi equtio The ple though o-collie poits A, B d C hs vecto equtio ( b ) ( c ) ( ) b c The ple though the poit with positio vecto d pllel to b d c hs equtio sb tc The pepedicul distce of (,, ) fom y z d 0 is d 6 Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
21 Hypebolic fuctios cosh sih sih sih cosh cosh cosh sih cosh l{ } ( ) sih l{ } th l ( ) Coics Ellipse Pbol Hypebol Rectgul Hypebol Stdd Fom b y y y 4 b y c Pmetic Fom ( cos, bsi ) ( t, t) ( sec, b t ) ( cosh, b sih ) ct, c t Ecceticity e b ( e ) e e b ( e ) e = Foci ( e, 0) (, 0) ( e, 0) ( c, c ) Diectices e + y = c e Asymptotes oe oe y 0, y 0 b Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics 7 Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
22 Diffeetitio f() csi f() ccos ct sih cosh cosh sih th sech sih cosh th 8 Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
23 Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics 9 Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07 Itegtio (+ costt; 0 whee elevt) f() d ) f( sih cosh cosh sih th cosh l ) ( csi ct ) ( l, cosh } { { } l, sih ) ( th l l Ac legth y s d d d (ctesi coodites) t t y t s d d d d d (pmetic fom) d d d s (pol fom)
24 Sufce e of evolutio S = dy y d d (ctesi coodites) S = d dy y dt dt dt (pmetic fom) S = d si d d (pol fom) 0 Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
25 Sttistics Discete distibutios Fo discete dom vible X tkig vlues Epecttio (me): E(X) = = i P(X = i) i with pobbilities P(X = i) Vice: V(X) = = ( i ) P(X = i) = P(X = i) Fo fuctio g( X ) : E(g(X)) = g( i) P(X = i) The pobbility geetig fuctio of X is G ( t ) E t X X i d E( X) G () d V( X) G () G () G () X X X X Fo Z = X + Y, whee X d Y e idepedet: G ( t) G ( t) G ( t) Z X Y Discete distibutios Stdd discete distibutios: Distibutio of X P(X = ) Me Vice P.G.F. p Biomil B(, p) p p p( p) p pt Poisso Po ( ) e e t! Geometic Geo(p) p o,, p p p p pt ( pt ) Negtive biomil o, +, p ( p) p ( p) p pt ( pt ) Cotiuous distibutios Fo cotiuous dom vible X hvig pobbility desity fuctio f Epecttio (me): E( X ) f( ) d X Vice: V( ) ( ) f( )d f( )d Fo fuctio ) g( X : E(g( X )) g( ) f( ) d Cumultive distibutio fuctio: F( ) P( X ) f ( t) dt Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
26 Stdd cotiuous distibutio: Distibutio of X P.D.F. Me Vice Noml N(, ) e Uifom (Rectgul) o [, b] b ( b) ( b ) Coeltio d egessio Fo set of pis of vlues, y ) ( i ) S ( i ) i ( yi ) S yy ( yi y) yi ( i i ( i)( yi) S y ( i )( yi y) i yi The poduct momet coeltio coefficiet is ( i)( yi) S y ( )( ) i i y i yi y S S yy { ( i ) }{ ( yi y) } ( i) ( yi) i yi The egessio coefficiet of y o is S ( )( y y) b S ( ) y i i i Lest sques egessio lie of y o is y b whee y b Residul Sum of Sques (RSS) = Sy Syy Syy S Spem s k coeltio coefficiet is s 6d ( ) Epecttio lgeb Fo idepedet dom vibles X d Y E( XY ) E( X ) E( Y), V( X by ) V( X ) b V( Y) Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
27 Smplig distibutios (i) Tests fo me whe is kow Fo dom smple me d vice X, X,, X of idepedet obsevtios fom distibutio hvig : X is ubised estimto of, with V( X ) S is ubised estimto of, whee S ( X X ) i Fo dom smple of obsevtios fom X N(, ), ~ N(0, ) / Fo dom smple of obsevtios fom N(, ) d, idepedetly, dom smple of obsevtios fom N(, ( X Y ) ( ) ) y y y, y ~ N(0,) y y (ii) Tests fo vice d me whe is ot kow Fo dom smple of obsevtios fom N(, ) : ( ) S ~ X ~ S / t (lso vlid i mtched-pis situtios) Fo dom smple of obsevtios fom N(, ) d, idepedetly, dom smple of obsevtios fom N(, ) y y y S S y / / y ~ F, y If y (ukow) the ( X Y) ( y) ~ t S p y y whee y y ( ) S ( ) S S p y Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
28 No-pmetic tests Goodess-of-fit test d cotigecy tbles: ( O i Ei E ) i ~ Sttisticl tbles The followig sttisticl tbles e equied fo A Level Futhe Mthemtics: Biomil Cumultive Distibutio Fuctio (see pge 5) Pecetge Poits Of The Noml Distibutio (see pge 0) Poisso Cumultive Distibutio Fuctio (see pge ) Pecetge Poits of the Distibutio (see pge ) Citicl Vlues fo Coeltio Coefficiets: Poduct Momet Coefficiet d Spem s Coefficiet (see pge ) Rdom Numbes (see pge 4) Pecetge Poits of Studet s t Distibutio (see pge 5) Pecetge Poits of the F Distibutio (see pge 6 4 Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
29 Mechics Cetes of mss Fo uifom bodies: Tigul lmi: log medi fom vete Cicul c, dius, gle t cete : si fom cete Secto of cicle, dius, gle t cete si : fom cete Solid hemisphee, dius : 8 fom cete Hemispheicl shell, dius : fom cete Solid coe o pymid of height h: h 4 bove the bse o the lie fom cete of bse to vete Coicl shell of height h: h bove the bse o the lie fom cete of bse to vete Motio i cicle Tsvese velocity: v Tsvese cceletio: v Rdil cceletio: v Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics 5 Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
30 6 Sttisticl Tbles Biomil Cumultive Distibutio Fuctio The tbulted vlue is P(X ), whee X hs biomil distibutio with ide d pmete p. p = = 5, = = 6, = = 7, = = 8, = = 9, = = 0, = Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
31 p = =, = = 5, = = 0, = Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics 7 Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
32 p = = 5, = = 0, = Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
33 p = = 40, = Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics 9 Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
34 p = = 50, = Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
35 Pecetge Poits Of The Noml Distibutio The vlues z i the tble e those which dom vible Z N(0, ) eceeds with pobbility p; tht is, P(Z > z) = (z) = p. p z p z Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
36 Poisso Cumultive Distibutio Fuctio The tbulted vlue is P(X ), whee X hs Poisso distibutio with pmete. = = = = Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
37 Pecetge Poits of the Distibutio The vlues i the tble e those which dom vible with the distibutio o degees of feedom eceeds with the pobbility show Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
38 Citicl Vlues fo Coeltio Coefficiets These tbles coce tests of the hypothesis tht popultio coeltio coefficiet is 0. The vlues i the tbles e the miimum vlues which eed to be eched by smple coeltio coefficiet i ode to be sigifict t the level show, o oe-tiled test. Poduct Momet Coefficiet Smple Level Spem s Coefficiet Level Level Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
39 Rdom Numbes Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics 5 Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
40 Pecetge Poits of Studet s t Distibutio The vlues i the tble e those which dom vible with Studet s t distibutio o degees of feedom eceeds with the pobbility show Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
41 Pecetge Poits of the F Distibutio The vlues i the tble e those which dom vible with the F distibutio o d degees of feedom eceeds with pobbility 0.05 o 0.0. Pobbility If uppe pecetge poit of the F distibutio o d degees of feedom is f, the the coespodig lowe pecetge poit of the F distibutio o d degees of feedom is / f. Peso Edecel Level Advced Subsidiy d Advced GCE i Mthemtics d Futhe Mthemtics 7 Mthemticl Fomule d Sttisticl Tbles Vesio.0 Accedited pe-publictio Mch 07 Peso Eductio Limited 07
42
43 Mch 07 Fo ifomtio bout Edecel, BTEC o LCCI qulifictios visit qulifictios.peso.com Edecel is egisteed tdemk of Peso Eductio Limited Peso Eductio Limited. Registeed i Egld d Wles No Registeed Office: 80 Std, Lodo WCR 0RL. VAT Reg No GB 78 57
44 Fo ifomtio bout Edecel, BTEC o LCCI qulifictios visit qulifictios.peso.com Edecel is egisteed tdemk of Peso Eductio Limited Peso Eductio Limited. Registeed i Egld d Wles No Registeed Office: 80 Std, Lodo WCR 0RL VAT Reg No GB 78 57
Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE Mathematics and Further Mathematics
Peso Edecel Level Advced Subsidiy d Advced GCE Mthemtics d Futhe Mthemtics Mthemticl fomule d sttisticl tbles Fo fist cetifictio fom Jue 08 fo: Advced Subsidiy GCE i Mthemtics (8MA0) Advced GCE i Mthemtics
Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE Mathematics and Further Mathematics
Peso Edecel Level 3 Advced Subsidiy d Advced GCE Mthemtics d Futhe Mthemtics Mthemticl fomule d sttisticl tbles Fo fist cetifictio fom Jue 08 fo: Advced Subsidiy GCE i Mthemtics (8MA0) Advced GCE i Mthemtics
DRAFT. Formulae and Statistical Tables for A-level Mathematics SPECIMEN MATERIAL. First Issued September 2017
Fist Issued Septembe 07 Fo the ew specifictios fo fist techig fom Septembe 07 SPECIMEN MATERIAL Fomule d Sttisticl Tbles fo A-level Mthemtics AS MATHEMATICS (7356) A-LEVEL MATHEMATICS (7357) AS FURTHER
PhysicsAndMathsTutor.com
PhysicsAdMthsTuto.com 5. () Show tht d y d PhysicsAdMthsTuto.com Jue 009 4 y = sec = 6sec 4sec. (b) Fid Tylo seies epsio of sec π i scedig powes of 4, up to d 3 π icludig the tem i 4. (6) (4) blk *M3544A08*
PhysicsAndMathsTutor.com
PhysicsAMthsTuto.com . M 6 0 7 0 Leve lk 6 () Show tht 7 is eigevlue of the mti M fi the othe two eigevlues of M. (5) () Fi eigevecto coespoig to the eigevlue 7. *M545A068* (4) Questio cotiue Leve lk *M545A078*
PhysicsAndMathsTutor.com
PhysicsAdMthsTuto.com PhysicsAdMthsTuto.com Jue 009 3. Fid the geel solutio of the diffeetil equtio blk d si y ycos si si, d givig you swe i the fom y = f(). (8) 6 *M3544A068* PhysicsAdMthsTuto.com Jue
PhysicsAndMathsTutor.com
PhysicsAdMthsTuto.com PhysicsAdMthsTuto.com Jue 009 7. () Sketch the gph of y, whee >, showig the coodites of the poits whee the gph meets the es. () Leve lk () Solve, >. (c) Fid the set of vlues of fo
PhysicsAndMathsTutor.com
PhsicsAMthsTuto.com 6. The hpeol H hs equtio, whee e costts. The lie L hs equtio m c, whee m c e costts. Leve lk () Give tht L H meet, show tht the -cooites of the poits of itesectio e the oots of the
A Level Further Mathematics A (H245) Formulae Booklet. Specimen. OCR 2017 H245 Turn over QN 603/1325/0
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!(
Advanced Higher Maths: Formulae
: Fomule Gee (G): Fomule you bsolutely must memoise i ode to pss Advced Highe mths. Remembe you get o fomul sheet t ll i the em! Ambe (A): You do t hve to memoise these fomule, s it is possible to deive
GCE AS/A Level MATHEMATICS GCE AS/A Level FURTHER MATHEMATICS
GCE AS/A Level MATHEMATICS GCE AS/A Level FURTHER MATHEMATICS FORMULA BOOKLET Fom Septembe 07 Issued 07 Mesuto Pue Mthemtcs Sufce e of sphee = 4 Ae of cuved sufce of coe = slt heght Athmetc Sees S l d
GCE AS and A Level MATHEMATICS FORMULA BOOKLET. From September Issued WJEC CBAC Ltd.
GCE AS d A Level MATHEMATICS FORMULA BOOKLET Fom Septeme 07 Issued 07 Pue Mthemtcs Mesuto Suce e o sphee = 4 Ae o cuved suce o coe = heght slt Athmetc Sees S = + l = [ + d] Geometc Sees S = S = o < Summtos
Advanced Higher Maths: Formulae
Advced Highe Mths: Fomule Advced Highe Mthemtics Gee (G): Fomule you solutely must memoise i ode to pss Advced Highe mths. Rememe you get o fomul sheet t ll i the em! Ame (A): You do t hve to memoise these
PhysicsAndMathsTutor.com
PhysicsAdMthsTutor.com PhysicsAdMthsTutor.com Jue 009 4. Give tht y rsih ( ), > 0, () fid d y d, givig your swer s simplified frctio. () Leve lk () Hece, or otherwise, fid 4 d, 4 [ ( )] givig your swer
The formulae in this booklet have been arranged according to the unit in which they are first
Fomule Booklet Fomule Booklet The fomule ths ooklet hve ee ge ccog to the ut whch the e fst touce. Thus cte sttg ut m e eque to use the fomule tht wee touce peceg ut e.g. ctes sttg C mght e epecte to use
[5 points] (c) Find the charge enclosed by the cylindrical surface of radius ρ 0 = 9 mm and length L = 1 m. [2
![[5 points] (c) Find the charge enclosed by the cylindrical surface of radius ρ 0 = 9 mm and length L = 1 m. [2](/thumbs/85/91744669.jpg "[5 points] (c) Find the charge enclosed by the cylindrical surface of radius ρ 0 = 9 mm and length L = 1 m. [2")
STUDENT NAME: STUDENT ID: ELEC ENG FH3: MIDTERM EXAMINATION QUESTION SHEET This emitio is TWO HOURS log. Oe double-sided cib sheet is llowed. You c use the McMste ppoved clculto Csio f99. You c tke y mteil
For use in Edexcel Advanced Subsidiary GCE and Advanced GCE examinations
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 Ju 008 UA08598 TABLE
The formulae in this booklet have been arranged according to the unit in which they are first
Fomule Booklet Fomule Booklet The fomule ths ooklet hve ee ge og to the ut whh the e fst toue. Thus te sttg ut m e eque to use the fomule tht wee toue peeg ut e.g. tes sttg C mght e epete to use fomule
BINOMIAL THEOREM SOLUTION. 1. (D) n. = (C 0 + C 1 x +C 2 x C n x n ) (1+ x+ x 2 +.)
BINOMIAL THEOREM SOLUTION. (D) ( + + +... + ) (+ + +.) The coefficiet of + + + +... + fo. Moeove coefficiet of is + + + +... + if >. So. (B)... e!!!! The equied coefficiet coefficiet of i e -.!...!. (A),
SULIT 3472/2. Rumus-rumus berikut boleh membantu anda menjawab soalan. Simbol-simbol yang diberi adalah yang biasa digunakan.
SULT 347/ Rumus-umus eikut oleh memtu d mejw sol. Simol-simol yg diei dlh yg is diguk. LGER. 4c x 5. log m log m log 9. T d. m m m 6. log = log m log 0. S d m m 3. 7. log m log m. S, m m logc 4. 8. log.
x a y n + b = 1 0<b a, n > 0 (1.1) x 1 - a y = b 0<b a, n > 0 (1.1') b n sin 2 + cos 2 = 1 x n = = cos 2 6 Superellipse (Lamé curve)
6 Supeellipse (Lmé cuve) 6. Equtios of supeellipse A supeellipse (hoizotlly log) is epessed s follows. Implicit Equtio y + b 0 0 (.) Eplicit Equtio y b - 0 0 (.') Whe 3, b, the supeellipses fo
physicsandmathstutor.com
physicsadmathstuto.com physicsadmathstuto.com Jue 005. A cuve has equatio blak x + xy 3y + 16 = 0. dy Fid the coodiates of the poits o the cuve whee 0. dx = (7) Q (Total 7 maks) *N03B034* 3 Tu ove physicsadmathstuto.com
Students must always use correct mathematical notation, not calculator notation. the set of positive integers and zero, {0,1, 2, 3,...
Appedices Of the vrious ottios i use, the IB hs chose to dopt system of ottio bsed o the recommedtios of the Itertiol Orgiztio for Stdrdiztio (ISO). This ottio is used i the emitio ppers for this course
2a a a 2a 4a. 3a/2 f(x) dx a/2 = 6i) Equation of plane OAB is r = λa + µb. Since C lies on the plane OAB, c can be expressed as c = λa +
-6-5 - - - - 5 6 - - - - - - / GCE A Level H Mths Nov Pper i) z + z 6 5 + z 9 From GC, poit of itersectio ( 8, 9 6, 5 ). z + z 6 5 9 From GC, there is o solutio. So p, q, r hve o commo poits of itersectio.
physicsandmathstutor.com
physicsadmathstuto.com physicsadmathstuto.com Jauay 2009 2 a 7. Give that X = 1 1, whee a is a costat, ad a 2, blak (a) fid X 1 i tems of a. Give that X + X 1 = I, whee I is the 2 2 idetity matix, (b)
UNIT V: Z-TRANSFORMS AND DIFFERENCE EQUATIONS. Dr. V. Valliammal Department of Applied Mathematics Sri Venkateswara College of Engineering
UNIT V: -TRANSFORMS AND DIFFERENCE EQUATIONS D. V. Vllimml Deptmet of Applied Mthemtics Si Vektesw College of Egieeig TOPICS:. -Tsfoms Elemet popeties.. Ivese -Tsfom usig ptil fctios d esidues. Covolutio
Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Statistics
Pearso Edecel Level 3 Advaced Subsidiary ad Advaced GCE i Statistics Statistical formulae ad tables For first certificatio from Jue 018 for: Advaced Subsidiary GCE i Statistics (8ST0) For first certificatio
Appendix A: Mathematical Formulae and Statistical Tables
Aedi A: Mathematical Formulae ad Statistical Tables Pure Mathematics Mesuratio Surface area of shere = r Area of curved surface of coe = r J slat height Trigoometry a * b & c ' bc cos A Arithmetic Series
Advanced Higher Formula List
Advaced Highe Fomula List Note: o fomulae give i eam emembe eveythig! Uit Biomial Theoem Factoial! ( ) ( ) Biomial Coefficiet C!! ( )! Symmety Idetity Khayyam-Pascal Idetity Biomial Theoem ( y) C y 0 0
SOLUTIONS ( ) ( )! ( ) ( ) ( ) ( )! ( ) ( ) ( ) ( ) n r. r ( Pascal s equation ). n 1. Stepanov Dalpiaz
STAT UIU Pctice Poblems # SOLUTIONS Stepov Dlpiz The followig e umbe of pctice poblems tht my be helpful fo completig the homewo, d will liely be vey useful fo studyig fo ems...-.-.- Pove (show) tht. (
2002 Quarter 1 Math 172 Final Exam. Review
00 Qute Mth 7 Fil Exm. Review Sectio.. Sets Repesettio of Sets:. Listig the elemets. Set-uilde Nottio Checkig fo Memeship (, ) Compiso of Sets: Equlity (=, ), Susets (, ) Uio ( ) d Itesectio ( ) of Sets
[Q. Booklet Number]
![[Q. Booklet Number]](/thumbs/82/85468362.jpg "[Q. Booklet Number]")
6 [Q. Booklet Numer] KOLKATA WB- B-J J E E - 9 MATHEMATICS QUESTIONS & ANSWERS. If C is the reflecto of A (, ) i -is d B is the reflectio of C i y-is, the AB is As : Hits : A (,); C (, ) ; B (, ) y A (,
Objective Mathematics
. o o o o {cos 4 cos 9 si cos 65 } si 7º () cos 6º si 8º. If x R oe of these, the mximum vlue of the expressio si x si x.cos x c cos x ( c) is : () c c c c c c. If ( cos )cos cos ; 0, the vlue of 4. The
ENGINEERING PROBABILITY AND STATISTICS
ENGINEERING PROBABILITY AND STATISTICS DISPERSION, MEAN, MEDIAN, AND MODE VALUES If X, X,, X represet the vlues of rdom smple of items or oservtios, the rithmetic me of these items or oservtios, deoted
Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE Mathematics and Further Mathematics
Pearson Edecel Level 3 Advanced Subsidiary and Advanced GCE Mathematics and Further Mathematics Mathematical formulae and statistical tables First certification from 08 Advanced Subsidiary GCE in Mathematics
physicsandmathstutor.com
physicsadmathstuto.com physicsadmathstuto.com Jue 005 5x 3 3. (a) Expess i patial factios. (x 3)( x ) (3) (b) Hece fid the exact value of logaithm. 6 5x 3 dx, givig you aswe as a sigle (x 3)( x ) (5) blak
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION. (2014 Admn. onwards) III Semester. B.Sc. Mathematics CORE COURSE CALCULUS AND ANALYTICAL GEOMETRY
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION (0 Adm. owrds) III Semester B.Sc. Mthemtics CORE COURSE CALCULUS AND ANALYTICAL GEOMETRY Questio Bk & Aswer Key. l l () =... 0.00 b) 0 c). l d =... c
We show that every analytic function can be expanded into a power series, called the Taylor series of the function.
10 Lectue 8 We show tht evey lytic fuctio c be expded ito powe seies, clled the Tylo seies of the fuctio. Tylo s Theoem: Let f be lytic i domi D & D. The, f(z) c be expessed s the powe seies f( z) b (
Summary: Binomial Expansion...! r. where
Summy: Biomil Epsio 009 M Teo www.techmejcmth-sg.wes.com ) Re-cp of Additiol Mthemtics Biomil Theoem... whee )!!(! () The fomul is ville i MF so studets do ot eed to memoise it. () The fomul pplies oly
PROGRESSION AND SERIES
INTRODUCTION PROGRESSION AND SERIES A gemet of umbes {,,,,, } ccodig to some well defied ule o set of ules is clled sequece Moe pecisely, we my defie sequece s fuctio whose domi is some subset of set of
Mark Scheme (Results) January 2008
Mk Scheme (Results) Jnuy 00 GCE GCE Mthemtics (6679/0) Edecel Limited. Registeed in Englnd nd Wles No. 4496750 Registeed Office: One90 High Holbon, London WCV 7BH Jnuy 00 6679 Mechnics M Mk Scheme Question
ANSWER KEY PHYSICS. Workdone X
ANSWER KEY PHYSICS 6 6 6 7 7 7 9 9 9 0 0 0 CHEMISTRY 6 6 6 7 7 7 9 9 9 0 0 60 MATHEMATICS 6 66 7 76 6 6 67 7 77 7 6 6 7 7 6 69 7 79 9 6 70 7 0 90 PHYSICS F L l. l A Y l A ;( A R L L A. W = (/ lod etesio
Mathematical Statistics
7 75 Ode Sttistics The ode sttistics e the items o the dom smple ed o odeed i mitude om the smllest to the lest Recetl the impotce o ode sttistics hs icesed owi to the moe equet use o opmetic ieeces d
Expansion by Laguerre Function for Wave Diffraction around an Infinite Cylinder
Joul of Applied Mthemtics d Physics, 5, 3, 75-8 Published Olie Juy 5 i SciRes. http://www.scip.og/joul/jmp http://dx.doi.og/.436/jmp.5.3 Expsio by Lguee Fuctio fo Wve Diffctio oud Ifiite Cylide Migdog
Important Facts You Need To Know/Review:
Importt Fcts You Need To Kow/Review: Clculus: If fuctio is cotiuous o itervl I, the its grph is coected o I If f is cotiuous, d lim g Emple: lim eists, the lim lim f g f g d lim cos cos lim 3 si lim, t
sin m a d F m d F m h F dy a dy a D h m h m, D a D a c1cosh c3cos 0
Q1. The free vibrtio of the plte is give by By ssumig h w w D t, si cos w W x y t B t Substitutig the deflectio ito the goverig equtio yields For the plte give, the mode shpe W hs the form h D W W W si
MATHEMATICS IV 2 MARKS. 5 2 = e 3, 4
MATHEMATICS IV MARKS. If + + 6 + c epesents cicle with dius 6, find the vlue of c. R 9 f c ; g, f 6 9 c 6 c c. Find the eccenticit of the hpeol Eqution of the hpeol Hee, nd + e + e 5 e 5 e. Find the distnce
Tuanaki, Kaupae 3, 2012
See ck cove fo Eglish tsltio of this cove LCALCMF 9908 Tuki, Kupe, 0.00 Rāhi 6 Whiig-ā-gi 0 TE PUKAITI O NGĀ TIKANGA TĀTAI ME NGĀ TŪTOHI mō 9065M, 9066M, 9068M me 9069M Tiohi tēei pukiti hei whkutu i gā
STD: XI MATHEMATICS Total Marks: 90. I Choose the correct answer: ( 20 x 1 = 20 ) a) x = 1 b) x =2 c) x = 3 d) x = 0
STD: XI MATHEMATICS Totl Mks: 90 Time: ½ Hs I Choose the coect nswe: ( 0 = 0 ). The solution of is ) = b) = c) = d) = 0. Given tht the vlue of thid ode deteminnt is then the vlue of the deteminnt fomed
Numerical integration
Numeicl itegtio Alyticl itegtio = ( ( t)) ( t) Dt : Result ( s) s [0, t] : ( t) st ode odiy diffeetil equtio Alyticl solutio ot lwys vilble d( ) q( ) = σ = ( d ) : t 0 t = Numeicl itegtio 0 t t 2. t. t
For this purpose, we need the following result:
9 Lectue Sigulities of omplex Fuctio A poit is clled sigulity of fuctio f ( z ) if f ( z ) is ot lytic t the poit. A sigulity is clled isolted sigulity of f ( z ), if f ( z ) is lytic i some puctued disk
National Quali cations AHEXEMPLAR PAPER ONLY
Ntiol Quli ctios AHEXEMPLAR PAPER ONLY EP/AH/0 Mthemtics Dte Not pplicble Durtio hours Totl mrks 00 Attempt ALL questios. You my use clcultor. Full credit will be give oly to solutios which coti pproprite
Week 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:
Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the
Add Maths Formulae List: Form 4 (Update 18/9/08)
Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Fuctios Asolute Vlue Fuctio f ( ) f( ), if f( ) 0 f( ), if f( ) < 0 Iverse Fuctio If y f( ), the Rememer: Oject the vlue of Imge the vlue of y or f() f()
Schrödinger Equation Via Laplace-Beltrami Operator
IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume 3, Issue 6 Ver. III (Nov. - Dec. 7), PP 9-95 www.iosrjourls.org Schrödiger Equtio Vi Lplce-Beltrmi Opertor Esi İ Eskitşçioğlu,
1. The 0.1 kg particle has a speed v = 10 m/s as it passes the 30 position shown. The coefficient of kinetic friction between the particle and the
1. The 0.1 kg pticle h peed v = 10 m/ it pe the 30 poitio how. The coefficiet of kietic fictio betwee the pticle d the veticl ple tck i m k = 0.0. Detemie the mgitude of the totl foce exeted by the tck
Rahul Chacko. IB Mathematics HL Revision Step One
IB Mthemtics HL Revisio Step Oe Rhul Chcko Chpte. Aithmetic sequeces d seies; sum of fiite ithmetic seies; geometic sequeces d seies; sum of fiite d ifiite geometic seies. Sigm ottio. Aithmetic Sequeces
BITSAT MATHEMATICS PAPER. If log 0.0( ) log 0.( ) the elogs to the itervl (, ] () (, ] [,+ ). The poit of itersectio of the lie joiig the poits i j k d i+ j+ k with the ple through the poits i+ j k, i
INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
Mthemtics Revisio Guides Itegrtig Trig, Log d Ep Fuctios Pge of MK HOME TUITION Mthemtics Revisio Guides Level: AS / A Level AQA : C Edecel: C OCR: C OCR MEI: C INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
1 Tangent Line Problem
October 9, 018 MAT18 Week Justi Ko 1 Tget Lie Problem Questio: Give the grph of fuctio f, wht is the slope of the curve t the poit, f? Our strteg is to pproimte the slope b limit of sect lies betwee poits,
Time: 2 hours IIT-JEE 2006-MA-1. Section A (Single Option Correct) + is (A) 0 (B) 1 (C) 1 (D) 2. lim (sin x) + x 0. = 1 (using L Hospital s rule).
IIT-JEE 6-MA- FIITJEE Solutios to IITJEE 6 Mthemtics Time: hours Note: Questio umber to crries (, -) mrks ech, to crries (5, -) mrks ech, to crries (5, -) mrks ech d to crries (6, ) mrks ech.. For >, lim
MATH Midterm Solutions
MATH 2113 - Midtem Solutios Febuay 18 1. A bag of mables cotais 4 which ae ed, 4 which ae blue ad 4 which ae gee. a How may mables must be chose fom the bag to guaatee that thee ae the same colou? We ca
AS and A Level Further Mathematics B (MEI)
fod Cmbdge d RSA *3369600* AS d A evel Futhe Mthemtcs B (MEI) The fomto ths booklet s fo the use of cddtes followg the Advced Subsd Futhe Mthemtcs B (MEI)(H635) o the Advced GCE Futhe Mthemtcs B (MEI)
Optimization. x = 22 corresponds to local maximum by second derivative test
Optimiztion Lectue 17 discussed the exteme vlues of functions. This lectue will pply the lesson fom Lectue 17 to wod poblems. In this section, it is impotnt to emembe we e in Clculus I nd e deling one-vible
National Quali cations SPECIMEN ONLY
AH Ntiol Quli ctios SPECIMEN ONLY SQ/AH/0 Mthemtics Dte Not pplicble Durtio hours Totl mrks 00 Attempt ALL questios. You my use clcultor. Full credit will be give oly to solutios which coti pproprite workig.
ANSWERS, HINTS & SOLUTIONS HALF COURSE TEST VII (Main)
AIITS-HT-VII-PM-JEE(Mai)-Sol./7 I JEE Advaced 06, FIITJEE Studets bag 6 i Top 00 AIR, 7 i Top 00 AIR, 8 i Top 00 AIR. Studets fom Log Tem lassoom/ Itegated School Pogam & Studets fom All Pogams have qualified
0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.
. Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric
13.5. Torsion of a curve Tangential and Normal Components of Acceleration
13.5 osion of cuve ngentil nd oml Components of Acceletion Recll: Length of cuve '( t) Ac length function s( t) b t u du '( t) Ac length pmetiztion ( s) with '( s) 1 '( t) Unit tngent vecto '( t) Cuvtue:
ENGINEERING MATHEMATICS I QUESTION BANK. Module Using the Leibnitz theorem find the nth derivative of the following : log. e x log d.
ENGINEERING MATHEMATICS I QUESTION BANK Modle Usig the Leibit theoem id the th deivtive o the ollowig : b si c e d e Show tht d d! Usig the Leibit theoem pove the ollowig : I si b the pove tht b I si show
Notation List. For Cambridge International Mathematics Qualifications. For use from 2020
Notatio List For Cambridge Iteratioal Mathematics Qualificatios For use from 2020 Notatio List for Cambridge Iteratioal Mathematics Qualificatios (For use from 2020) Mathematical otatio Eamiatios for CIE
physicsandmathstutor.com
physicsadmathstuto.com 2. Solve (a) 5 = 8, givig you aswe to 3 sigificat figues, (b) log 2 ( 1) log 2 = log 2 7. (3) (3) 4 *N23492B0428* 3. (i) Wite dow the value of log 6 36. (ii) Epess 2 log a 3 log
CITY UNIVERSITY LONDON
CITY UNIVERSITY LONDON Eg (Hos) Degree i Civil Egieerig Eg (Hos) Degree i Civil Egieerig with Surveyig Eg (Hos) Degree i Civil Egieerig with Architecture PART EXAMINATION SOLUTIONS ENGINEERING MATHEMATICS
CHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method
CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece 5.3.1 Divegece Test 5.3. Itegal Test 5.3.3
Topics for Review for Final Exam in Calculus 16A
Topics fo Review fo Finl Em in Clculus 16A Instucto: Zvezdelin Stnkov Contents 1. Definitions 1. Theoems nd Poblem Solving Techniques 1 3. Eecises to Review 5 4. Chet Sheet 5 1. Definitions Undestnd the
Inference on One Population Mean Hypothesis Testing
Iferece o Oe Popultio Me ypothesis Testig Scerio 1. Whe the popultio is orml, d the popultio vrice is kow i. i. d. Dt : X 1, X,, X ~ N(, ypothesis test, for istce: Exmple: : : : : : 5'7" (ull hypothesis:
: : 8.2. Test About a Population Mean. STT 351 Hypotheses Testing Case I: A Normal Population with Known. - null hypothesis states 0
8.2. Test About Popultio Me. Cse I: A Norml Popultio with Kow. H - ull hypothesis sttes. X1, X 2,..., X - rdom smple of size from the orml popultio. The the smple me X N, / X X Whe H is true. X 8.2.1.
New Advanced Higher Mathematics: Formulae
Advcd High Mthmtics Nw Advcd High Mthmtics: Fomul G (G): Fomul you must mmois i od to pss Advcd High mths s thy ot o th fomul sht. Am (A): Ths fomul giv o th fomul sht. ut it will still usful fo you to
M5. LTI Systems Described by Linear Constant Coefficient Difference Equations
5. LTI Systes Descied y Lie Costt Coefficiet Diffeece Equtios Redig teil: p.34-4, 245-253 3/22/2 I. Discete-Tie Sigls d Systes Up til ow we itoduced the Fouie d -tsfos d thei popeties with oly ief peview
IFYFM002 Further Maths Appendix C Formula Booklet
Ittol Foudto Y (IFY) IFYFM00 Futh Mths Appd C Fomul Booklt Rltd Documts: IFY Futh Mthmtcs Syllbus 07/8 Cotts Mthmtcs Fomul L Equtos d Mtcs... Qudtc Equtos d Rmd Thom... Boml Epsos, Squcs d Ss... Idcs,
2015/2016 SEMESTER 2 SEMESTRAL EXAMINATION
05/06 SEMESTER SEMESTRA EXAMINATION Course : Diplom i Electroics, Computer & Commuictios Egieerig Diplom i Aerospce Systems & Mgemet Diplom i Electricl Egieerig with Eco-Desig Diplom i Mechtroics Egieerig
Things I Should Know In Calculus Class
Thigs I Should Kow I Clculus Clss Qudrtic Formul = 4 ± c Pythgore Idetities si cos t sec cot csc + = + = + = Agle sum d differece formuls ( ) ( ) si ± y = si cos y± cos si y cos ± y = cos cos ym si si
Content: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B.
Review Sheet: Chpter Cotet: Essetil Clculus, Erly Trscedetls, Jmes Stewrt, 007 Chpter : Fuctios d Limits Cocepts, Defiitios, Lws, Theorems: A fuctio, f, is rule tht ssigs to ech elemet i set A ectly oe
FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),
FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To -periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the
Remarks: (a) The Dirac delta is the function zero on the domain R {0}.
Sectio Objective(s): The Dirc s Delt. Mi Properties. Applictios. The Impulse Respose Fuctio. 4.4.. The Dirc Delt. 4.4. Geerlized Sources Defiitio 4.4.. The Dirc delt geerlized fuctio is the limit δ(t)
Semiconductors materials
Semicoductos mteils Elemetl: Goup IV, Si, Ge Biy compouds: III-V (GAs,GSb, ISb, IP,...) IV-VI (PbS, PbSe, PbTe,...) II-VI (CdSe, CdTe,...) Tey d Qutey compouds: G x Al -x As, G x Al -x As y P -y III IV
Quality control. Final exam: 2012/1/12 (Thur), 9:00-12:00 Q1 Q2 Q3 Q4 Q5 YOUR NAME
Qulity contol Finl exm: // (Thu), 9:-: Q Q Q3 Q4 Q5 YOUR NAME NOTE: Plese wite down the deivtion of you nswe vey clely fo ll questions. The scoe will be educed when you only wite nswe. Also, the scoe will
Introduction to mathematical Statistics
More ttistics tutoril t wwwdumblittledoctorcom Itroductio to mthemticl ttistics Chpter 7 ypothesis Testig Exmple (ull hypothesis) : the verge height is 5 8 or more ( : μ 5'8" ) (ltertive hypothesis) :
Sharjah Institute of Technology
For commets, correctios, etc Plese cotct Ahf Abbs: hf@mthrds.com Shrh Istitute of Techolog echicl Egieerig Yer Thermofluids sheet ALGERA Lws of Idices:. m m + m m. ( ).. 4. m m 5. Defiitio of logrithm:
Multiplicative Versions of Infinitesimal Calculus
Multiplictive Versios o Iiitesiml Clculus Wht hppes whe you replce the summtio o stdrd itegrl clculus with multiplictio? Compre the revited deiitio o stdrd itegrl D å ( ) lim ( ) D i With ( ) lim ( ) D
1 Using Integration to Find Arc Lengths and Surface Areas
Novembe 9, 8 MAT86 Week Justin Ko Using Integtion to Find Ac Lengths nd Sufce Aes. Ac Length Fomul: If f () is continuous on [, b], then the c length of the cuve = f() on the intevl [, b] is given b s
MAHESH TUTORIALS SUBJECT : Maths(012) First Preliminary Exam Model Answer Paper
SET - GSE tch : 0th Std. Eg. Medium MHESH TUTILS SUJET : Mths(0) First Prelimiry Exm Model swer Pper PRT -.. () like does ot exist s biomil surd. () 4.. 6. 7. 8. 9. 0... 4 (c) touches () - d () -4 7 (c)
BRAIN TEASURES INDEFINITE INTEGRATION+DEFINITE INTEGRATION EXERCISE I
EXERCISE I t Q. d Q. 6 6 cos si Q. Q.6 d d Q. d Q. Itegrte cos t d by the substitutio z = + e d e Q.7 cos. l cos si d d Q. cos si si si si b cos Q.9 d Q. si b cos Q. si( ) si( ) d ( ) Q. d cot d d Q. (si
Macroscopic Fields in Accelerators
48 E hs simil effect s v. d q E v Fo eltivistic ticles T hs simil effect s dt E c 3 8 V/m such Electic field is beod techicl limits. Electic fields e ol used fo ve lo eegies o Fo setig to coute ottig bems
USE OF STATISTICAL TECHNIQUES FOR CRITICAL GAPS ESTIMATION
Sessio 1. Sttistic Methods d Thei Applictios Poceedigs of the 12 th Itetiol Cofeece Relibility d Sttistics i Tspottio d Commuictio (RelStt 12, 17 20 Octobe 2012, Rig, Ltvi, p. 20 26. ISBN 978-9984-818-49-8
WELSH JOINT EDUCATION COMMITTEE CYD-BWYLLGOR ADDYSG CYMRU MATHEMATICS. FORMULA BOOKLET (New Specification)
WELSH JOINT EDUCATION COMMITTEE CYD-BWYLLGOR ADDYSG CYMRU Gl Ciic o Eucio Avc Lvl/Avc Susii Tssgi Asg Giol So Uwch/Uwch Gol MATHEMATICS FORMULA BOOKLET Nw Spciicio Issu 004 Msuio Suc o sph 4π A o cuv suc
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION
School Of Distce Eductio Questio Bk UNIVERSITY OF ALIUT SHOOL OF DISTANE EDUATION B.Sc MATHEMATIS (ORE OURSE SIXTH SEMESTER ( Admissio OMPLEX ANALYSIS Module- I ( A lytic fuctio with costt modulus is :
2017/2018 SEMESTER 1 COMMON TEST
07/08 SEMESTER COMMON TEST Course : Diplom i Electroics, Computer & Commuictios Egieerig Diplom i Electroic Systems Diplom i Telemtics & Medi Techology Diplom i Electricl Egieerig with Eco-Desig Diplom
1.3 Continuous Functions and Riemann Sums
mth riem sums, prt 0 Cotiuous Fuctios d Riem Sums I Exmple we sw tht lim Lower() = lim Upper() for the fuctio!! f (x) = + x o [0, ] This is o ccidet It is exmple of the followig theorem THEOREM Let f be
Chapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
,... are the terms of the sequence. If the domain consists of the first n positive integers only, the sequence is a finite sequence.
Chpter 9 & 0 FITZGERALD MAT 50/5 SECTION 9. Sequece Defiitio A ifiite sequece is fuctio whose domi is the set of positive itegers. The fuctio vlues,,, 4,...,,... re the terms of the sequece. If the domi