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1 C3 Eponnials and logarihms - Eponnial quaions. Rabbis wr inroducd ono an island. Th numbr of rabbis, P, yars afr hy wr inroducd is modlld by h quaion P = 3 0, 0 (a) Wri down h numbr of rabbis ha wr inroducd o h island. () Find h numbr of yars i would ak for h numbr of rabbis o firs cd 000. dp (c) Find. d d p (d) Find P whn = 0. d (Toal marks) 2. Th funcion f is dfind by 2 f( ) = +,, 4, 2 ( + 4) ( 2)( + 4) (a) Show ha 3 f( ) = 2 () Edcl Inrnal Rviw

2 C3 Eponnials and logarihms - Eponnial quaions Th funcion g is dfind by 3 g ( ) =,, n 2 2 Diffrnia g() o show ha g' ( ) =, 2 ( 2) (c) Find h ac valus of for which g () = (4) (Toal 2 marks) 3. Th poin P lis on h curv wih quaion Th y -coordina of P is. y = (a) Find, in rms of ln 2, h -coordina of P. Find h quaion of h angn o h curv a h poin P in h form y = a + b, whr a and b ar ac consans o b found. (4) (Toal 6 marks) 4. Th amoun of a crain yp of drug in h bloodsram hours afr i has bn akn is givn by h formula = D, whr is h amoun of h drug in h bloodsram in milligrams and D is h dos givn in milligrams. A dos of 0 mg of h drug is givn. Edcl Inrnal Rviw 2

3 C3 Eponnials and logarihms - Eponnial quaions (a) Find h amoun of h drug in h bloodsram hours afr h dos is givn. Giv your answr in mg o 3 dcimal placs. A scond dos of 0 mg is givn afr hours. Show ha h amoun of h drug in h bloodsram hour afr h scond dos is 3.49 mg o 3 dcimal placs. No mor doss of h drug ar givn. A im T hours afr h scond dos is givn, h amoun of h drug in h bloodsram is 3 mg. (c) Find h valu of T. (Toal 7 marks). A had mal ball is droppd ino a liquid. As h ball cools, is mpraur, T C, minus afr i nrs h liquid, is givn by T = , 0. (a) Find h mpraur of h ball as i nrs h liquid. Find h valu of for which T = 300, giving your answr o 3 significan figurs. () (4) (c) Find h ra a which h mpraur of h ball is dcrasing a h insan whn = 0. Giv your answr in C pr minu o 3 significan figurs. (d) From h quaion for mpraur T in rms of, givn abov, plain why h mpraur of h ball can nvr fall o 20 C. () (Toal 9 marks) Edcl Inrnal Rviw 3

4 C3 Eponnials and logarihms - Eponnial quaions 6. A paricular spcis of orchid is bing sudid. Th populaion p a im yars afr h sudy sard is assumd o b p = 200a + a , whr a is a consan. Givn ha hr wr 300 orchids whn h sudy sard, (a) show ha a = 0.2, us h quaion wih a = 0.2 o prdic h numbr of yars bfor h populaion of orchids rachs 0. (4) 336 (c) Show ha p = () (d) Hnc show ha h populaion canno cd 200. (Toal 0 marks) 7. Find, giving your answr o 3 significan figurs whr appropria, h valu of for which (a) 3 =, log 2 (2 + ) log 2 = 2, (4) (c) ln sin = ln sc, in h inrval 0 < < 90. (Toal 0 marks) Edcl Inrnal Rviw 4

5 C3 Eponnials and logarihms - Eponnial quaions. Evry of mony invsd in a savings schm coninuously gains inrs a a ra of 4% pr yar. Hnc, afr yars, h oal valu of an iniial invsmn is y, whr y =.04. (a) Skch h graph of y =.04, 0. Calcula, o h nars, h oal valu of an iniial 00 invsmn afr 0 yars. (c) Us logarihms o find h numbr of yars i aks o doubl h oal valu of any iniial invsmn. (Toal 7 marks) 9. Find h ac soluions of (i) = 6, (ii) ln (3 + 2) = 4. (Toal 6 marks) Edcl Inrnal Rviw

6 C3 Eponnials and logarihms - Eponnial quaions. (a) P = 0 = 0 P = 0 o = 0() = 0 0 B P = = 0 0 rarrangs quaion o mak h subjc. M 000 = ln 0 = awr 2.6 or 3 yars A 2 No = 2 or = awr 2.6 =2 will scor A0 (c) dp = 6 d k and k 0. l 6 M A 2 (d) 0 = 6 0 = ln 6 { } Using 0 = d P and d an amp o solv o find h valu of or M P = 0 l 0 ln 6 or P = 0 ( ) Subsius hir valu of back ino h quaion for P. dm 0(0) P = = or awr 20 A 3 6 [] Edcl Inrnal Rviw 6

7 C3 Eponnials and logarihms - Eponnial quaions 2. (a) 2 f() = + ( + 4) ( 2)( + 4) R, 4, 2. ( 2)( + 4) 2( 2) + f() = ( 2) ( + 4) An amp o combin o on fracion Corrc rsul of combining all hr fracions M A = ( 2)( + 4) = Simplifis o giv h corrc [( + 4)( 2)] numraor. Ignor omission of dnominaor ( + 4)( 3) = [( + 4)( 2)] An amp o facoris h dm numraor. ( 3) = ( 2) Corrc rsul A cso AG A g() = 3 2 R, ln 2. Apply quoin rul: u = 3 du = d v = 2 dv = d g () = ( 2) ( 2 ( 2) 3) vu' uv' Applying 2 v Corrc diffrniaion M A = 2 2 ( 2) = Corrc rsul A AG 2 ( 2) cso 3 Edcl Inrnal Rviw 7

8 C3 Eponnials and logarihms - Eponnial quaions (c) g () = = = 2 ( 2) =( 2) 2 Pus hir diffrniad numraor M qual o hir dnominaor. = = A ( 4)( ) = 0 Amp o facoris M or solv quadraic in = 4 or = = ln 4 or = 0 boh = 0, ln 4 A 4 [2] 3. (a) 2+ = = ln2 = 2 (ln2 ) A 2 M dy d = 2+ B dy = (ln2 ) 2 d = 6 B y = 6 (ln 2 ) 2 M y = ln 2 A 4 [6] 4. (a) D = 0, =, = 0 M =.33 awr A 2 D = 0 + 0, =, = M = 3.49 (*) Acso 2 Edcl Inrnal Rviw

9 C3 Eponnials and logarihms - Eponnial quaions Al. = M = 3.49 (*) Acso (Main schm M is for (0 + 0 ), or {0 + hir(a)} N.B. Th answr is givn. Thr ar many corrc answrs sn which dsrv M0A0 or MA0 T (c) = 3 M 3 T = = T = ln M T = or 3. or 3 A 3 T s M is for ( ) = 3 o.. 2 nd T M is for convring = k (k > 0) o T k = ln indpndn of s M.. This is Trial and improvmn: M as schm, M corrc procss for hir quaion (wo qual o 3 s.f.) A as schm [7]. (a) 42 C B 300 = = 27 M sub. T = 300 and amp o rarrang o 0.0 = a, whr a Q = 400 A M corrc applicaion of logs M = 7.49 A 4 dt (c) = (M for k 0.0 ) M A d A = 0, ra of dcras = (±).64 C / min A 3 Edcl Inrnal Rviw 9

10 C3 Eponnials and logarihms - Eponnial quaions (d) T > 2, (sinc as ) B [9] 200a 6. (a) Sing p 300 a = = M + a (300 = 200a); a = 0.2 (c.s.o.) (*) dm A (0.2) 0 = ; 0.2 = 6.2 MA Corrcly aking logs o 0.2 = ln k M = 4 (3.9..) A 4 (c) Corrc drivaion: B (Showing division of num. and dn. by 0.2 ; using a) (d) Using, 0.2 0, M p 336 = 200 A [0] 7. (a) log3 = log M aking logs log = log 3 or log3 = log A =.46 cao A = log = = 4 = 2 or quivaln; 4 muliplying by o g a linar quaion M B M A 4 Edcl Inrnal Rviw 0

11 C3 Eponnials and logarihms - Eponnial quaions (c) sc = / cos B sin = cos an = = 4 M, A 3 us of an [0]. (a) y 0 Shap B domain, inrcp B M A 2 cao (c).04 = 2 M ln 2 = (yars) ln.04 M A 3 accp 7.7, 7 yars monhs [7] 9. (i) = = ln 6 M = (ln 6 3) 2 M A 3 (ii) ln (3 + 2) = = 4 M = ( 4 2) 3 M A 3 [6] Edcl Inrnal Rviw

12 C3 Eponnials and logarihms - Eponnial quaions. This qusion was wll answrd by h ovrwhlming majoriy of candidas who dmonsrad hir confidnc in working wih ponnials. Par (a) was almos univrsally answrd corrcly, alhough a fw candidas did ry o subsiu = ino h quaion for P in ordr o find h numbr of rabbis inroducd o h island. In par, mos candidas wr abl o us naural logarihms in ordr o find = 2.6 or = Alhough h pcd answr was 3 yars, any answr ha roundd o 2.6 yars was also accpd. Thos candidas who coninud o round hir answr down o 2 or sad in h 2h yar wr no awardd h final accuracy mark as h qusion rquird candidas o find h numbr of yars for h numbr of rabbis o cd 000. A fw candidas applid a rial and rror mhod in his par and wr usually succssful in gaining boh marks. In par (c), mos candidas corrcly sad dp as 6. Common rrors in his par wr d candidas giving answrs of h form k or6. A fw candidas rid o apply h produc rul for diffrniaion and usually sruggld o gain boh marks. dp In par (d), h vas majoriy of candidas quad hir found in par (c) o 0 and d procdd o solv for. A numbr of candidas faild a his poin o us hir valu for o find P as rquird in h qusion. I was plasing o s a significan minoriy of candidas dp who dducd ha P = 0 = 6 = = 20. d 4 2. Many candidas wr abl o obain h corrc answr in par (a) wih a significan numbr of candidas making mor han on amp o arriv a h answr givn in h qusion. Thos candidas who ampd o combin all hr rms a onc or hos who combind h firs wo rms and hn combind h rsul wih h hird rm wr mor succssful in his par. Ohr candidas who sard by rying o combin h scond and hird rms had problms 2 2 daling wih h ngaiv sign in fron and usually addd o + 4 ( + 4 ) ( 2)( + 4) bfor combining h rsul wih. I was plasing o s ha vry fw candidas usd ( + 4) 2 ( 2)as hir common dnominaor whn combining all hr rms. In par, mos candidas wr abl o apply h quoin rul corrcly bu a numbr of candidas faild o us bracks proprly in h numraor and hn found som difficully in arriving a h givn answr. In par (c), many candidas wr abl o qua h numraor o h dnominaor of h givn fracion and many of hs candidas wn ono form a quadraic in which hy usually solvd. A significan numbr of candidas ihr faild o spo h quadraic or pandd ( 2) 2 and hn ook h naural logarihm of ach rm on boh sids of hir rsuling quaion. In ihr or boh of pars and (c), som candidas wro 2 in hir working insad of 2 Such candidas usually los h final accuracy mark in par and h firs accuracy mark in par (c). Edcl Inrnal Rviw 2

13 C3 Eponnials and logarihms - Eponnial quaions 3. Par (a) was usually compld succssfully and h gra majoriy wr abl o ak logs dy corrcly o find. In par, mos could diffrnia corrcly and valua o find h d d y gradin of h angn. A fw faild o valua, giving a non-linar quaion for h d angn, and his los h las hr marks. Th majoriy dmonsrad a corrc mhod. Howvr h final mark was ofn los. Incorrc rmoval of h bracks, lading o y = 6 ln 2, was frqunly sn and if, as hr, h qusion asks for ac valus of a and b, giving b = 0.4 loss h final mark, unlss h ac soluion, ln 2, is also givn. 4. Par (a) was wll answrd, alhough candidas who gav h answr o 3 significan figurs los a mark. In par hos candidas who ralisd ha = usually gaind boh marks, bu a common misconcpion was o hink ha 0 should b addd o h answr o par (a). Par (c) provd a challnging final qusion, wih usually only h vry good candidas scoring all hr marks. From hos who rid o solv his in on sag i was mor common o s D = or 0 or 20 or 3.49, insad of han.326.., subsiud ino = D. Many candidas spli up h doss bu his, unforunaly, ofn ld o a compl prssion in T, T 3 = 0 + 0, which only h vry bs candidas wr abl o solv. On mark was a common scor for his par.. Calculaor work was gnrally accura in his qusion and i was ncouraging o s mos candidas giv hir answrs o h rquird dgr of accuracy. Th vas majoriy of candidas gav h corrc answr of 42 C in par (a). Many candidas wr abl o subsiu T = 300 in par and corrcly chang an quaion of h form a = b o a = ln b Wakr candidas showd a lack of undrsanding of logarihms by failing o simplify hir iniial quaion o h form a = b and using an incorrc samn of h form a = b + c ln a = ln b + ln c No all candidas undrsood h nd o diffrnia in par (c) and found h gradin dt of a chord insad of finding. Th mos common rror mad by candidas who did d diffrnia was o giv h diffrnial as Candidas ofn had difficuly giving prcis planaions in par (d). Alhough many rfrrd o h +2 rm in hir answrs, far fwr gav adqua rasons as o why his man ha h mpraur could nvr fall o 20 C, paricularly wih rgard o 0.0 > 0. Lack of undrsanding of h concp of limi ld som o wri (in words or symbols) T 2 rahr han T > 2. Edcl Inrnal Rviw 3

14 C3 Eponnials and logarihms - Eponnial quaions 6. (a) Th valu of a was calculad accuraly by mos candidas, bu a significan group did no subsiu = 0. Also a numbr of answrs wr producd whr a = 0.2 was subsiud o giv p=300, and his was no givn full crdi. Candidas could no always cop wih h algbra manipulaion and som found difficuly using logs corrcly;.g. wriing 0 = hn ln0 =ln4 ln 0.2. (c) Candidas frqunly did no show convincing work hr, wih h answr following from h qusion wih no working in bwn. I was ncssary o show h division of numraor and dnominaor by ^0.2 o jusify ging h givn rsul. (d) Candidas rarly usd h concp of limiing valus. Many candidas simply subsiud P = 200 in h formula givn in c) and showd ha -0.2 = 0, which was insufficin wihou furhr samns. Thr wr numbr of inqualiy rrors sn Thr wr howvr som clln soluions which clarly indicad ha 0.2 > 0 implid ha h dnominaor was >0.2 and ha h fracion was < 200. Som illusrad hir soluion wih a graph of an incrasing funcion nding o an asympo. 7. Th majoriy of candidas gaind h marks in par (a) alhough a fw did no givr hir answr o 3 significan figurs. Par was wll answrd by hos who undrsood logs. Mos did combin h logs corrcly, bu som did sill spli i up ino log log 2 or log 2 (2 + ). Many candidas found h combinaion of logs and rig funcions byond hm. log 2 sin = -/sc was a frqun indicaor of poor undrsanding, hough many did display hy knw sc = /cos. Quoin lins ofn slippd, ln /cos bcoming /ln cos.. Th purpos of his skch was o show h ovrall shap and orinaion of h graph. Many skchs sn wr indisinguishabl from a sraigh lin and som candidas had clarly bn using graphic calculaors wihou bing how augh o us appropria scal facors. Many who did produc curvs faild o no ha h domain was rsricd o = 0. Par was wll don alhough i was surprising o s a his lvl a subsanial numbr who carrid ou n spara muliplicaions by.04. This mhod did gain h wo marks bu was no good im 0 managmn. A fw calculad prssions lik 32 and faild o noic ha was an unrasonabl answr. In par (c) many could no handl h fac ha no iniial sum was spcifid and wr unabl o rduc h problm o an quaion in a singl variabl. Thos who did obain.04 = 2 wr usually abl o compl h qusion. Edcl Inrnal Rviw 4

15 C3 Eponnials and logarihms - Eponnial quaions 9. No Rpor availabl for his qusion. Edcl Inrnal Rviw