Thoms Whithm ith Form Pure Mthemtis Uit lger Trigoometr Geometr lulus lger
equees The ifiite sequee of umers U U U... U... is si to e () overget if U L fiite limit s () iverget to if U s Emple The sequee... hs s Emple s 5 7 U U sequee is overget with limit. The sequee 8 6. hs U U sequee is iverget to. ome sequees osillte Emple Ivestigte sequees with th terms give (i) U (ii) U Best here to list few terms (i) sequee is - -5 is lerl ot overget ut iverget (ii) sequee is... is lerl overget with limit =. 9 6 ome sequees re perioi Emple U si equee is - -. This sequee hs perio (repetig itself ever terms) ome sequees might e efie reurree reltio Emple U r U r U. Fi the first four terms U U U U 5
U U 5 U 7 U Oviousl first four terms re 5 7 9.. eries rithmeti Progressios (Ps) The sequee of umers... is si to e i P = first term = ommo ifferee U th term = sum of terms = l where l is the lst term l = Proof of formul for LERN!! Write l l l... Rewrite l l l... l l l l l l... l l l Prolem solvig: Kowig sequee is... : Use of formule for U : rememer tht kowlege of two terms will give
Emple 96 P hs seo term 96 fifth term. Fi the ommo ifferee the first term the sum of the first te terms. 8 76 9 76 5 68 5 7 6 Geometri Progressio (GPs) The sequee of umers r r r... is si to e i GP. = first term r = ommo rtio U th term = r sum of terms = r Proof of formul for LERN!! r Write r r r r r r r r... r... r r r r utrt r r r r r r Prolem solvig: Kowig sequee is r r... : Use of formule for U : rememer tht kowlege of two terms will give r
Emple GP hs seo term 96 fifth term. Fi the ommo rtio the first term orret to the erest whole umer the sum of the first te terms. r 96 r r r 96 96 96 6.5 75 6.5 7 r 96 8 r ome GPs hve ifiite umer of terms et hve fiite sum. osier the formul for write s r Now if r s r r. This will e the se so log s r r ( r ) ; the ifiite GP is the si to e overget with fiite sum r Usig Do t e put off! Just ep it shoul revel either P or GP. Emple r r 5 7... This is P with = = l = = Usig l
r Emple... 5 This is ifiite GP with r Usig r Epoetil Log grphs log is horizotl smptote Logrithms is vertil smptote Defiitio The log of umer to give se is the power to whih the se must e rise i orer to oti the umer. o if log N the N ( ) NB log is useful (elimitig N from the ove) log N N (elimitig from the ove) Lws of logs Use proofs require log log M log N log MN (i) M M log N log (ii) N p log M plog M (iii)
Let log M log N M N 6 Proof of (i) MN log (lws of iies) MN log MN log M log N Proof of (ii) M N log log (lws of iies) M N M N log M log N p p Proof of (iii) p M (lws of iies) log M p p p log M plog M Emple implif log 6 log 5 log 6 log 6 log 5 log log 6 log 5 log log 5 log 6 log ommo Logs re logs to se. Where for emple log 5 is writte se is uerstoo. log N N
7 Nturl logs re logs to se e. Where for emple l 5 is writte se e is uerstoo. l N N e You shoul e le to uerst tht log l e logn l N e N N tht lws (i) (ii) (iii) ppl to these speil ses. pplitios to equtios of the form similr. Emple olve the equtios (i) (i) (ii) (ii) l l l l l. 58 l l l l l l l l l r Geometr The irle l 6 l l 6. l gles i semiirle is 9
8 Perpeiulr to hor from etre of irle isets the hor. etre rius form of equtio r etre ( ) rius = r Emple etre ( -) rius equtio 9 Emple etre ( ) touhig Equtio ( ) Geerl form of equtio f g To fi etre rius use the metho of T to hge ito etre/rius form. Emple 5
9 etre rius = 5 Tgets gle etwee tget rius rw to poit of ott is 9 Tgets rw from etee poit Emple Fi the equtio of the tget to the irle 5 t the poit P( ) 5 5 5 etre t rius Lie of smmetr
(- ) Griet P = griet of tget t P = P( ) Equtio 5 Touhig irles T is stright lie T irles touh eterll if the iste etwee etres is equl to the sum of the rii i.e. = r + r T is stright lie T irles touh iterll if the iste etwee etres is equl to the ifferee of the rii i.e. = r r
Emple how tht the irles 8 8 touh. etre/ rius of st irle = r 8 etre/ rius of irle = r 8 Differee i rii = irles touh Trigoometr Trig rtios for 6 5 6 5 si os6 si 5 os 5 si 6 t 6 os t 5 t Trig rtios for ll gles NB the T DIGRM For the sig of trig rtio ll positive i first qurt ie (ol) i seo qurt Et T
T Emple Without usig lultor fi (i) os 5 (ii) t (iii) si (i) (ii) (iii) os5 5 os T t t si si 6 T 6 - Trig of lee trigles ie rule B si si B si Give use it to fi seo sie Give use it to fi seo gle (ut tke re to hoose the gle size ppropritel it oul e ute or otuse).
osie rule B os os Both formule with two more sets. Give use it to fi the thir sie Give use it to fi gle (o possile miguit here). Emple Trigle PQR hs PR = m QR = 7m Q PR ˆ 6 Q Fi (i) QR usig the osie rule the (ii) sie rule. PQ ˆ R usig the R (i) QR 9 9 os 6... QR.9...9 P 6 7 (ii) 7 si PQR.9.. si 6 7si 6 si PQR.89....9.. PQR 57.86.. or PQR.9.. It t e 57.8.. sie R woul e 86.9.. woul e the lrgest gle i the trigle ut R fes the smllest sie so is the smllest gle. Hee PQR.9
re si rule give re of trigle = si irulr mesures o. Rememer 6 r r r 8 8 8 o B 9 o 6 et. r legth & re of setor o o 5 6 o r r r r r [ i ris] = re setor re r r r si Essetil to ler formule for r legth setor re tht is i RDIN! The formul for segmet might e lert! I trigles where gles re give i or re require i ris set our lultor ito RD moe
Emple 5.8 9.57. 5 5.8..5 Grphs of trig futios (ll perioi). Grph of si 5.8. os.57 Perio si( ) si si -. Grph of os Perio os( ) os os -
. Grph of t 6 Perio t( ) t Vertil smptotes t et Vertil smptotes Bour vlues of trig rtios Verif these from grphs T- = = T =T= = - =T= = Two importt trig ietities T si t si os os = - = T- 8 Emple Give is otuse si 7 fi the vlues of os t. si os os si
7 5 89 6 89 os 5 7 8 si 7 8 t t 5 5 os 7 T NB Ler how to rerrge the ietities si os t os si t os si si os omplemetr gles re those whih up to 9 si( 9 ) os os( 9 ) si t( 9 ) ot upplemetr gles re those whih up to 8 si( 8 ) si os( 8 ) os t( 8 ) t Trig equtios Rememer tht from our lultor t Emple give the priipl vlue (p.v.) olve the equtios (i) t. 5 for (ii) si. 5 for (iii) os (iv) si 6 os 8 8 si for 6 si si os for 6 for (v) si 8 8 8
(i) t.5 8 (ii) si. 5..first solve for for 5; 5 75 ; 5 65 (iii) (I this emple use os os si si si si si si or 9 si si si si 9 si si ) T 6 6 T T PV = -56. PV = PV = - (iv) Do t el out si si os si si os si. Brig to LH ftorise si si os si or si os 8 si os t 7 7 T PV = 6.56
8 7 7 9 solve first for (v) si 8 8 6 6 6 PV = 6 T I the et emple gles re i ris. The ri sig is sometimes omitte ut is implie whe the itervl otis. Emple form olve the followig equtios (i) os. for swers orret to.p. (ii) t for swers i et (i) os. put lultor ito RD moe..66....7 5..66... (ii) I et terms mes i terms of. The implitio is tht the gles will e et form i egrees. o work i egrees first the overt to ris. t 6 solve first for T T PV = 6 PV =.66..
lulus Itegrtio Iefiite itegrls w v u w v u f ) ( res the (iefiite) itegrl of ) ( f with respet to () f is lle the itegr. is the ifferetil of the itegrtio must ever e omitte. Emple Fi (i) (ii) (iii) (i) 6 9 6 9 (ii) Iefiite itegrtio is the reverse of ifferetitio. Ever iefiite itegrl must hve ritrr ostt e. peil ses worth rememerig
(iii) Not misprit!. Defiite itegrls If F f I ) ( ) ( the the efiite itegrl f ) ( is the ifferee i the vlue of I whe. i.e. ) ( ) ( ) ( F F f o ostt! The limits of the efiite itegrl re (lower limit) (upper limit). Note the use of squre rkets. ) ( ) ( ) ( F F F Emple Evlute 9 8 9 re o grph s efiite itegrl (i)
i.e. the vlue of the efiite itegrl will e egtive if is egtive for (iii) B B (iv) (v) NB for NB (i) most ertil will e teste (iv) oul e. (ii) (iii) most ulikel to e ilue. (v) most likel re etwee lie urve. Emple Fi the re elose etwee the grph of 9 is the orites t = the -
9 9 9 9 9 9 Emple The igrm shows the sketh of grph of. Fi the oorites of the poits of itersetio P Q of the grphs. lulte the she re. For P Q he re = 6 6 6 9 sketh P Q
pproimte Itegrtio h The re etwee orites t is ivie ito strips of equl with. The res of the strips re pproimte trpezi. Hee the trpezium rue s stte elow.... h.. Emple Fi the pproimte vlue of 6 rites 5 strips eh of with. 5 Hee there will e orites t..8..6.. ; h usig 6 orites..97 5.6.6..56
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