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1 Thoms Whithm ith Form Pure Mthemtis Uit lger Trigoometr Geometr lulus lger
2 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
3 U U 5 U 7 U Oviousl first four terms re 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
4 Emple 96 P hs seo term 96 fifth term. Fi the ommo ifferee the first term the sum of the first te terms 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
5 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 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 This is P with = = l = = Usig l
6 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)
7 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
8 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
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
10 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( ) etre t rius Lie of smmetr
11 (- ) 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
12 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 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
13 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).
14 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 P 6 7 (ii) 7 si PQR.9.. si 6 7si 6 si PQR PQR or PQR.9.. It t e sie R woul e woul e the lrgest gle i the trigle ut R fes the smllest sie so is the smllest gle. Hee PQR.9
15 re si rule give re of trigle = si irulr mesures o. Rememer 6 r r r 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
16 Emple Grphs of trig futios (ll perioi). Grph of si 5.8. os.57 Perio si( ) si si -. Grph of os Perio os( ) os os -
17 . 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
18 os 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
19 (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
20 solve first for (v) si 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 (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..
21 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) (ii) Iefiite itegrtio is the reverse of ifferetitio. Ever iefiite itegrl must hve ritrr ostt e. peil ses worth rememerig
22 (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 re o grph s efiite itegrl (i)
23 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 -
24 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 = sketh P Q
25 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 ; h usig 6 orites
26 5 Notes
27 6 Notes Notes
28 7
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