Further Applications of Advanced Mathematics (FP3) THURSDAY 14 JUNE 2007

Transcription

1 ADVANCED GCE UNIT 77/ MATHEMATICS (MEI) Further Applictios of Avce Mthemtics (FP) THURSDAY JUNE 7 Aitiol mterils: Aswer booklet (8 pges) Grph pper MEI Emitio Formule Tbles (MF) Afteroo Time: hour miutes INSTRUCTIONS TO CANDIDATES Write your me, cetre umber cite umber i the spces provie o the swer booklet. Aswer y three questios. You re permitte to use grphicl clcultor i this pper. Fil swers shoul be give to egree of ccurcy pproprite to the cotet. INFORMATION FOR CANDIDATES The umber of mrks is give i brckets [ ] t the e of ech questio or prt questio. The totl umber of mrks for this pper is 7. ADVICE TO CANDIDATES Re ech questio crefully mke sure you kow wht you hve to o before strtig your swer. You re vise tht swer my receive o mrks uless you show sufficiet etil of the workig to iicte tht correct metho is beig use. This ocumet cosists of 6 prite pges blk pges. HN/ OCR 7 [K//66] OCR is eempt Chrity [Tur over

2 Optio : Vectors Three ples P, Q R hve the followig equtios. Ple P: 8 - y - z Ple Q: 6 y - z 6 Ple R: y - z The lie of itersectio of the ples P Q is K. The lie of itersectio of the ples P R is L. (i) Show tht K L re prllel lies, fi the shortest istce betwee them. [9] (ii) Show tht the shortest istce betwee the lie K the ple R is 6. [] The lie M hs equtio r (i j) l (i j k). (iii) Show tht the lies K M itersect, fi the coorites of the poit of itersectio. [7] (iv) Fi the shortest istce betwee the lies L M. [] Optio : Multi-vrible clculus A surfce hs equtio z y y 7 6. z z (i) Fi [] y. (ii) Fi the coorites of the four sttiory poits o the surfce, showig tht oe of them is (,, 8). [8] (iii) Sketch, o seprte igrms, the sectios of the surfce efie by by y. Iicte the poit (,, 8) o these sectios, euce tht it is either mimum or miimum. [6] (iv) Show tht there re just two poits o the surfce where the orml lie is prllel to the vector 6i k, fi the coorites of these poits. [7] OCR 7 77/ Jue 7

3 Optio : Differetil geometry The curve C hs equtio y l, is costt with. (i) Show tht the legth of the rc of C for which is l. [] (ii) Fi the re of the surfce geerte whe the rc of C for which is rotte through p ris bout the y-is. [] (iii) Show tht the rius of curvture of C t the poit where is Ê Ë ˆ []. (iv) Fi the cetre of curvture correspoig to the poit (, ) o C. [] C is oe member of the fmily of curves efie by y p p l, where p is prmeter. (v) Fi the evelope of this fmily of curves. [] [Questios re prite overlef.] OCR 7 77/ Jue 7 [Tur over

4 Optio : Groups (i) Prove tht, for group of orer, every proper subgroup must be cyclic. [] The set M,,,,, 6, 7, 8, 9, moulo. is group uer the biry opertio of multiplictio (ii) Show tht M is cyclic. [] (iii) List ll the proper subgroups of M. [] The group P of symmetries of regulr petgo cosists of trsformtios A, B, C, D, E, F, G, H, I, J the biry opertio is compositio of trsformtios. The compositio tble for P is give below. A B C D E F G H I J A C J G H A B I F E D B F E H G B A D C J I C G D I F C J E B A H D J C B E D G F I H A E A B C D E F G H I J F H I D C F E J A B G G I H E B G D A J C F H D G J A H I B E F C I E F A J I H C D G B J B A F I J C H G D E Oe of these trsformtios is the ietity trsformtio, some re rottios the rest re reflectios. (iv) Ietify which trsformtio is the ietity, which re rottios which re reflectios. [] (v) Stte, givig reso, whether P is isomorphic to M. [] (vi) Fi the orer of ech elemet of P. [] (vii) List ll the proper subgroups of P. [] OCR 7 77/ Jue 7

5 Optio : Mrkov chis A computer is progrmme to geerte sequece of letters. The process is represete by Mrkov chi with four sttes, s follows. The first letter is A, B, C or D, with probbilities.,.,.. respectively. After A, the et letter is either C or D, with probbilities.8. respectively. After B, the et letter is either C or D, with probbilities..9 respectively. After C, the et letter is either A or B, with probbilities..6 respectively. After D, the et letter is either A or B, with probbilities..7 respectively. (i) Write ow the trsitio mtri P. [] (ii) Use your clcultor to fi P P 7. (Give elemets correct to eciml plces.) [] (iii) Fi the probbility tht the 8th letter is C. [] (iv) Fi the probbility tht the th letter is the sme s the 8th letter. [] (v) By ivestigtig the behviour of letter is A whe (A) is lrge eve umber, P whe is lrge, fi the probbility tht the ( )th (B) is lrge o umber. [] The progrm is ow chge. The iitil probbilities the trsitio probbilities re the sme s before, ecept for the followig. After D, the et letter is A, B or D, with probbilities.,.6. respectively. (vi) Write ow the ew trsitio mtri Q. [] (vii) Verify tht Q pproches limit s becomes lrge, hece write ow the equilibrium probbilities for A, B, C D. [] (viii) Whe is lrge, fi the probbility tht the ( )th, ( )th ( )th letters re DDD. [] OCR 7 77/ Jue 7

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7 7 BLANK PAGE OCR 7 77/ Jue 7

8 8 Permissio to reprouce items where thir-prty owe mteril protecte by copyright is iclue hs bee sought clere where possible. Every resoble effort hs bee me by the publisher (OCR) to trce copyright holers, but if y items requirig clerce hve uwittigly bee iclue, the publisher will be plese to mke mes t the erliest possible opportuity. OCR is prt of the Cmbrige Assessmet Group. Cmbrige Assessmet is the br me of Uiversity of Cmbrige Locl Emitios Syicte (UCLES), which is itself eprtmet of the Uiversity of Cmbrige. OCR 7 77/ Jue 7

9 Mrk Scheme 77 Jue 7

10 77 Mrk Scheme Jue 7 (i) 8 6 K [ ] 8 L [ ] Hece K L re prllel For poit o K, z,, y i.e. (,, ) For poit o L, z, 6, y 8 i.e. (6, 8, ) M* A* A M*A* A* Fiig irectio of K or L Oe irectio correct * These mrks c be ere ywhere i the questio Correctly show Fiig oe poit o K or L or ( 6,, ) or (, 8, ) etc Or ( 7,, ) or (, 6, ) etc Distce is 8 9 OR 6 λ 8 λ M λ 87 9λ, λ M Distce is 6 8 A (ii) Distce from (,, ) to R is (iii) K, M itersect if λ μ () λ μ () λ μ () Solvig () (): λ, μ 6 Check i (): LHS, RHS 8 Hece K, M itersect, t (,, ) OR M meets P whe M 8 ( λ ) ( λ) (λ) A M meets Q whe 6 ( λ ) ( λ) (λ) 6 A Both equtios hve solutio λ A Poit is o P, Q M; hece o K M M Poit of itersectio is (,, ) A 6 M M A MA ft A g M A ft MM MA A 9 7 For ( b ) Correct metho for fiig istce For ( b λ ). Fiig λ, the mgitue At lest eqs, ifferet prmeters Two equtios correct Itersectio correctly show C be wre fter MAMMM Itersectio of M with both P Q

11 77 Mrk Scheme Jue 7 (iv) Distce is 9 8 MA ft M A ft A For evlutig M L For ) (. ) ( M L c b Numericl epressio for istce 6

12 77 Mrk Scheme Jue 7 (i) z y 8y 6 z y y 6 B B Give B for terms correct (ii) z z At sttiory poits, y Whe, y 6 M M y ±6 ; poits (, 6, ) (, 6, ) Whe y, , Poits (,, ) (,, 8) AA M MA A 8 If A, give A for y ± 6 or y, A if y etr poits give (iii) Whe, z y 6y B Upright prbol B (,, 8) ietifie s miimum (i the first qurt) Whe y, z z whe B Negtive cubic curve B (,, 8) ietifie s sttiory poit The poit is miimum o oe sectio mimum o the other; so it is either mimum or miimum B B 6 Fully correct (umbiguous miimum mimum) (iv) z z Require 6 y Whe, y 6 6 y ; poit (,, ) Whe y, M M A M z 6 c er ll M mrks 8, gives (,, ) sme s bove gives (, 6, 7) M A A Solvig to obti (or y) or sttig o roots if pproprite (e.g. whe 6 hs bee use) 7 7

13 77 Mrk Scheme Jue 7 (i) 6 6 y Arc legth is [ ] l l M A M M A g For y (ii) Curve surfce re is s π 7 ) ( 87 π π π M A ft M A A Ay correct itegrl form (icluig limits) for (iii) B y y ρ B M A A g y form, i terms of or y form, i terms of or Formul for κ ρ or κ ρ or correct, i y form, i terms of or (iv) At 6 ) ( ),, ( ρ ˆ so, y 6 c M A M AA Fiig griet Correct orml vector (ot ecessrily uit vector); my be i terms of Cetre of curvture is 7, 6 OR MA for obtiig equtio of orml lie t geerl poit ifferetitig prtilly 8

14 77 Mrk Scheme Jue 7 (v) Differetitig prtilly w.r.t. p p l M A p l y l l y l M A 9

15 77 Mrk Scheme Jue 7 (i) By Lgrge s theorem, proper subgroup hs orer or A group of prime orer is cyclic Hece every proper subgroup is cyclic (ii) e.g., 8,,, 6 9, 7 7, 8, hs orer, hece M is cyclic (iii) {, } {,,,, (iv) (v) (vi) (vii) 9 } E is the ietity A, C, G, I re rottios B, D, F, H, J re reflectios P M re ot isomorphic M is beli, P is o-beli 9 6, A B C D E F G H I J M A M A M A A A B B B M A A B B Orer B { E, B }, { E, D }, { E, F }, { E, H }, { E, J } { E, A, C, G, I } M A ft B co Usig Lgrge (ee ot be metioe eplicitly) or equivlet For completio Cosierig orer of elemet Ietifyig elemet of orer (, 6, 7 or 8) Fully justifie For coclusio (c be wre fter MAA) Igore {} M Deuct mrk (from BB) for ech (proper) subgroup give i ecess of Cosierig elemets of orer (or equivlet) Implie by four of B, D, F, H, J i the sme set Give A if oe elemet is i the wrog set; or if two elemets re iterchge Vli reso e.g. M hs oe elemet of orer P hs more th oe Give B for 7 correct B for correct Igore { E } P Subgroups of orer Usig elemets of orer (llow two errors/omissios) Correct or ft. A if y others give Subgroups of orer greter th Deuct mrk (from B) for ech etr subgroup give

16 77 Mrk Scheme Jue 7 Pre-multiplictio by trsitio mtri (i) (ii) (iii) (iv) P P B B P.. B M 7.. P P(8th letter is C). A M M A ft A Give B for two colums correct Give B for two o-zero elemets correct to t lest p Give B for two o-zero elemets correct to t lest p 7 8 Usig P (or P ) iitil probs Usig probbilities for 8th letter Usig igol elemets from P (v)(a) P P( ( ) th letter is A). M A Approimtig eve P whe is lrge (B) P P( ( ) th letter is A). M A Approimtig o P whe is lrge (vi) Q B

17 77 Mrk Scheme Jue 7 (vii) Q Probbilities re.7,.,.69,.9 M M.87.. MM (viii). A A Cosierig Q for lrge OR t lest two eqs for equilib probs Probbilities from equl colums OR solvig to obti equilib probs Give A for two correct Usig.87.

18 77 Mrk Scheme Jue 7 Post-multiplictio by trsitio mtri (i) (ii) P 7 P P (iii) 7 (....) P ( ) (iv) P(8th letter is C) B B B M A M MA ft A Give B for two rows correct Give B for two o-zero elemets correct to t lest p Give B for two o-zero elemets correct to t lest p 7 8 Usig P (or P ) iitil probs Usig probbilities for 8th letter Usig igol elemets from P (v)(a) u P (....).. (..667 ) P( ( ) th letter is A). M A Approimtig eve P whe is lrge (B) u P (....).667 (... ) P( ( ) th letter is A). M A Approimtig o P whe is lrge (vi) Q..6 B..6.

19 77 Mrk Scheme Jue 7 (vii) Q Probbilities re.7,.,.69,.9 M M.87.. MM (viii). A A Cosierig Q for lrge OR t lest two eqs for equilib probs Probbilities from equl rows OR solvig to obti equilib probs Give A for two correct Usig.87.

20 Report o the Uits tke i Jue 7 77: Further Applictios of Avce Mthemtics (FP) Geerl Commets There were some ecellet scripts, with bout % of cites scorig more th 6 mrks (out of 7). However, lot of ble cites clerly fou swerig three log theme questios to be ifficult tsk, overll the mrks were somewht isppoitig. Whe thigs go stry prt-wy through questio, it is importt to crry o with the lter prts, but ot ll cites hve the cofiece to o this. Some cites iicte tht they were short of time; iee oly very few swere more th the three questios require. The five questios seeme to offer roughly comprble chlleges to the cites; the verge mrks (out of ) rge from bout for Q to bout 7 for Q Q. The most populr questio ws Q (ttempte by bout 8% of the cites) the lest populr ws Q (ttempte by bout % of the cites). The most commo combitios of questios seeme to be Q Q Q or Q Q Q or Q Q Q. Commets o Iiviul Questios Vectors (i) (ii) (iii) (iv) Most cites relise tht they shoul strt by fiig the irectios of, poits lyig o, the lies K L. Showig tht the lies re prllel ws usully oe correctly, but fiig the istce betwee them cuse problems, ws sometimes ot eve ttempte. Whe it ws recogise s the istce from poit to lie it ws ofte fou efficietly ccurtely. Quite umber of cites took L to be the lie of itersectio of Q R (iste of P R); fortutely this misre i ot sigifictly lter the work to be oe throughout the questio. Surprisigly, this prt ws quite ofte omitte, presumbly becuse it ws ot recogise s the simple problem of fiig the istce from poit to ple. The correct poit of itersectio ws very ofte fou, but my cites i ot check properly tht the lies o itersect. The metho for fiig the shortest istce betwee skew lies ws well uerstoo, usully pplie correctly. Multi-vrible clculus (i) (ii) Almost every cite fou the prtil erivtives correctly. The metho for fiig sttiory poits ws well kow, ws very ofte crrie out completely correctly. The cse ws sometimes overlooke; sometimes, hvig obtie or y, it ws ssume tht y whe. (iii) The sectio ws usully sketche correctly; but o the sectio y most cites showe (,, 8) s poit of iflectio iste of mimum.

21 Report o the Uits tke i Jue 7 (iv) This ws resobly well oe; lthough quite lrge proportio put z/ equl to 6 iste of 6. As this lso les to ectly two poits, cites were ot lerte to their error. Differetil geometry (i) The rc legth ws ofte fou correctly. However, my cites were uble to simplify s / this prevete success i this prt i prt (ii). (ii) Most cites beg correctly with π s for the surfce re, but my coul ot procee beyo this, eve whe prt (i) h bee swere correctly. (iii) (iv) (v) Most cites obtie correct epressio for the rius of curvture, but lrge umber file to er the fil mrk for simplifyig it to the require form. This ws quite well swere, with my cites fiig the cetre of curvture correctly. Most of the cites who ttempte this prt kew wht ws require, ofte fou the evelope correctly. Groups (i) (ii) (iii) (iv) (v) (vi) (vii) This ws quite well swere, lthough some cites simply stte tht ll groups of orer or re cyclic, without givig reso (for emple, tht re prime umbers). This ws lso well oe, with most cites selectig geertig elemet clcultig ll its powers. The correct subgroups were ofte fou, it ws quite rre for etr oes to be give; the subgroup of orer ws sometimes omitte. There ws sometimes uecessrily lrge mout of workig, such s fiig the subgroup geerte by ech of the elemets. E ws lmost lwys stte to be the ietity; the reflectios were very ofte correctly fou by cosierig the elemets of orer. Most cites stte tht the groups were ot isomorphic; whe reso ws give it ws usully bse o the orers of the elemets (for emple, P hs more elemets of orer, or P hs o elemet of orer ). Strgely, very few cites referre to the commuttivity of M the o-commuttivity of P. The orers of the elemets were usully fou correctly. Most cites use their swer to prt (vi) ppropritely to write ow the require subgroups. Mrkov chis (i) (ii) The trsitio mtri ws lmost lwys give correctly. Clcultors were ccurtely use, most cites obeye the istructio to give the elemets to eciml plces. 6

22 Report o the Uits tke i Jue 7 (iii) 8 The probbility ws usully fou correctly. Just few cites use P iste 7 of P. (iv) (v) (vi) (vii) Almost ll cites wrogly ssume tht the 8th th letters were iepeet whe fiig the probbility tht they were the sme. Oly hful of cites use the igol elemets of P s the require coitiol probbilities. This prt ws very well swere. The gret mjority wrote ow the ew trsitio mtri correctly. The worig of the questio ws itee to ecourge fiig the limitig mtri Q whe is lrge, hece writig ow the equilibrium probbilities from tht mtri, whe it ws use this metho ws usully successful. Nevertheless, lrge umber of cites preferre to fi the equilibrium probbilities by solvig simulteous equtios, this metho ws much more proe to error. (viii) Most cites clculte this probbility s p D iste of p... D 7