STEP Solutions Mathematics STEP 9465/9470/9475

Transcription

1 STEP Solio Mhmi STEP 96/97/97 Novmr

2 Th Cmrig Am Gro i Ero' lrg m g l lig rol i rrhig, vloig livrig m ro h glo Or qlifiio r livr i ovr ori hrogh or hr mjor m or Cmrig Am i h r m of h Uivri of Cmrig Lol Emiio Si, rm of h Uivri of Cmrig Cmrig Am i o-for-rofi orgiio Thi mr hm i lih i o hr, o ii h rqirm of h miio I how h i o whih mr wr wr h Emir I o o ii h il of h iio whih oo l Emir mig for mrig omm All Emir r ir h lriv orr wr roh i i ri m giv mr h firl rfl h rlv owlg ill mor Mr hm hol r i ojio wih h lih qio r h Ror o h Emiio Cmrig Am will o r io iio or orro i oio wih hi mr hm UCLES Mor iformio o STEP fo : h://wwworg

3 Co STEP Mhmi 96, 97, 97 Ror Pg STEP Mhmi I STEP Mhmi II 8 STEP Mhmi III 8

4 STEP I Hi Solio Qio Th rliohi for oi iig h oori of h hr ow oi io h qio of h li, or ig h forml for llig h gri from ir of oi Th forml for h m of h i i h o fi iffriio wih r o will llow h miimm vl o fo Similrl, h i for h o r wri gi iffriio o fi h vl of for whih h miimm or Si h miimm vl of or h m vl of h miimm vl of, h iffriio imlifi j iffriig wih r o I i h iml mr o wri hi wr i form imilr o h of h fir r Qio To h h grh i i imor o ow whr h ior oi r Eihr oirig h grh of mimm or iffriig i h hr r wo miim o Th qio i h o r rrrg o how h h olio orro o h irio of h grh of righ li Th rqirig r i h wo olio hi m il h righ li whih oh h wo miim For h r of h qio, iffriio of h qio wi l qil o h wo oil vl of Boh h o oir, i hol lr h o grh i rflio of h ohr i h i, o h of vl for will h m for oh Th fil grh hol lrl hv miimm for om giv vl of will ill, o hr will wo oi of iflio Qio Th h of h grh, ilig hor g hol o mh iffil Aig h li hol how h h r r h grh li w wo rigl, oh wih of lgh wih high Igrig h fio w h wo limi h rrrgig will giv h orr rliohi For h o r of h qio iffr igrm i, hi im howig h r r rv oi wihi rzim wih rzim oi wihi i Th iffriio

5 igrio of will ro h rio rqir i h fil wr o h vril li o fi h rgio Qio Th qio of h g hol o fi h h olio of imlo qio will giv h rqir oori for Similrl, h qio for h orml hol o fi, i i mor iffil o fi imlifi rio for h olio o h imlo qio whih r fl for h fil r of h qio Th foriio of h iffr w wo, will fl for voiig lo of lgri milio If hi rio i imlifi orrl h h fil rl om righforwr Qio Th mo ovio roh o h fir igrio r o igr r or o h ii h m iio Th o igrio lo vl igrig r, h ii for i o fl For h hir igrio i i r o rwri h igrl i form from whih h rvio rl li Th fir oi o o i h h rio wihi h logrihm i o iml oi fio o h fir o mig h rio imilr o ho rviol i o rwri i i h form O hi i o, h iio of, wih om owlg of h rliohi w i oi grh hol r h igrl o omiio of h wo rvio o Qio 6 B wriig ow rio for h high of h ol ig h g of h of h gl of lvio, h rolm i qil r o wo imiol rolm o li wihi irl Th iml w o l hi i o orv h h rigl i h igrm r imilr o h ohr, roh worig wih vrio righ gl rigl lo l o h orr olio Th roof of h ii hol righ forwr for ho fmilir wih h ommol rigoomri rliohi h iqli i h il fo oirig h oq of h ori l o q Giv h o i rig fio i h rqir rg of vl, h fil rl follow il Qio 7 B iig for ll of h rm i h rrr rlio h rl for h fir r hol follow il For h o r, h wo vl i i oh orr io h rlio, givig wo qio i Solvig h wo qio h l o h orr of oiilii

6 Followig h m riil, h iio of, io h rlio o i hr iffr w, lig o hr imlo qio Solvig h qio giv qio i, whih olv o giv h wo iffr Th whr i o h, for h whr i hol oi h hi ol or if Qio 8 Th w iffril qio follow qi il o h iio giv h follow I i ml of iffril qio h olv rig h vril o vlig h wo igrl h r rh h rqir form oi Th m iio will lo r h o iffril qio o imlr form i i gi ml h olv rig h vril I hol lr h ril frio r rori mho for vlig h igrl h i rqir Qio 9 Th fir r of h qio i righforwr llio ivolvig h qio for moio r iform lrio I i imor o li h ro for hooig h oiiv qr roo howvr Th fil rl follow from orr of ii for I i lo o fi rio for h rg o h im h ll frhr liio of ii for will giv rio for hi i rm of Diffriio wih r o will h giv h rl h h mimm vl or wh Th fil r of h qio olv iig h rori vl of h rg io h forml for Qio Alhogh h qio loo omli, llio for h im h i h o o ro h im for h o o rr llow h fir rliohi o qi il Th o rliohi how imlifig h rio for howig h i i ql o Thi h rrrg o giv Th fil r of h qio i h iio of h vl giv io h forml o oi h im Qio Whil h igrm m loo lil mor omli h h r qio o hi oi, h fir io of hi qio rqir h l o lih ir of imlo qio Th iffr i h o r of h qio i h h lrio o m o o h ll i h mil of h igrm i l o mov h imor r rliohi h i i h rliohi w h lrio h hr oi h wo ril h ll

7 Qio Th roilii for filr i h r log rio o ll vlig h igrl from h i i oil o or r igrm from whih h roili ll Th fil r of h qio i iml h llio of oiiol roili A lw wih oiiol roili h imor i o whih wo roilii o ll Qio Thr r mr of w o roh hi rolm Th mo ovio i o wor o how m oiilii hr r for h mr of igi A lr mho for goriig h i o wor o h mr of oiilii i h For ml, if hr r iffr igi h hr r fiv hoi for h igi o for hoi for h igi o r Thr r h hoi for h oiio of h r igi! hoi for h orr of h rmiig igi Thi giv loghr

8 Hi & Solio for Pr 97 STEP II J Qio To ho, h iomil io of, i h =,, r o frql wihi AS A-lvl h h hol fmilir o ll i ig STEP Rlig i o gr frhr l 6 Th grl rm i i il o + 6 h 6 rm i om from , o h h offii of i =, riig from m of riglr mr Th, h offii of i if 6 {,,, } if 6 if 6 whih i mo il ri wiho ig irl, hr I ii, f = om from h rm givig h offii of = Howvr, hr r lo of w o go o oig hi For i No h, vr o-mlil-of- owr i r i r, h rm om from 6 oirig f = = Agi, vr o-mlil-of-6 owr i hi r i lo r, o o migh oir ol f = 7 9 from whih h offii of i iml ll = Thi rl, i om form or ohr, giv h w of worig o h offii of 6 for o-giv igr I i immil ovio h i i rr whih r o o h m r r 6 r Th roof of hi rl ol iio or ir milio of h r rl for r r Th off of i, h m for, i h r ol ri from rlig,,, i h fir r rm h of h worig h offii r ql i h I h wh =, h offii of 66 i = 6

9 Qio Firl, q h gr m i Dg[ = Dg[ = & Dg[ = Dg[LHS m, whil RHS i of gr Thrfor h LHS i o o o = i lir Sig = + = + + = + + = [ = Th [ + + [ + W hv, h, h = or I ihr, rirr vl h olio r h = + = + ii Dg[RHS = whil Dg[LHS m,,, o i follow h = i qri Sig = + +, w hv = = = = Th, LHS = whil h RHS = Eqig rm giv + = = or, + = = + = = wh = ; i = + OR = wh = ; i = No h hr r wo of oiio o, o h rl oi o h viil for oi: + + = h + = h lo Qio I hl grl o gi wih, o o h if, h Th h giv h rl, from whih w fi hgig h limi :, :,, o h f = f = f, rqir For h fir igrl, I = Th I = =, w r ig f = = i h rl lih iiill =

10 I h of h o igrl, h iio = = Alo h rqir hg of limi il,, W h hv I = i = [No h imor of hgig o = = W ow o, mhig hi wih h iiil rl, h w r ig f =, o h I = = 6 = = 7 Qio i Thi fir rl i il lih: For, >, + > + > o i ll rm r oiiv Th l rl whih i vli i < i h r rm i oiiv, ig A h rvio rl Eoiig h giv ii A i of rlimir log wor l o h l + rl o l = l l = = i ll rm fr h fir r oiiv Agi, o h w hol jif h h ri i vli for < i > i orr o jif h of h giv ri I h follow h l >, ig = + h ril fil giv l

11 iii Thi fil r ol rqir firl iforml rgm, h il ill rqir lil i of r i orr o voi ig oo vg A, io h m limi from oh ov low from ov i qz Qio Wih rv-hig qio of hi i, i i imor o gr ho fr h r imor igor ho h r For i, hrogho hi qio, h oiio of h -i i irl immril: i ol rw hrogh rh of h rv i qio or, i, r mo So h ll il of h -ir,, i r i, o o hl i wh h fio i o Th mo, rig oi lrl imor i r ii i h r ifill rq, mmri r imor Th ohr fr o i o r h hor-rm wh i mll h log-rm hvior I i, hr r vril mo = = + ; whil h -i i horizol mo Thr i mmr i h li = oq of whih i h mimm TP i h mil rh h log-rm hvior of h rv i h i liml rml h grh of = ii Diffriig h fio i ii giv g Foriig il ig h mror =, o h or I h fir, whr > + i + <, hr r fiv rh of h rv, wih vril mo: = = A h fio hg ig i ro h mo, h log-rm hvior i ill o rml =, h rh lr ov low h -i, wih mmr i I h o, whr = + i + =, h vr mil io h oll, lvig ol h for rh, h rv i ohrwi ill hg from h rvio Qio 6 A A qi igrm hl hr, lig o h imor orvio, from h GCSE gomr rl ooi gl of li q o 8 o, h BCD = 8 o Th, ig h Coi Rl wi oig h o8 o = o : D i BAD: BD = + o B i BCD: BD = + + o Eqig for BD r-rrgig giv o = C

12 N, h wll-ow forml for rigl r, C i, wi, giv Q = i i, i i = i Rrrgig h giv i = Q or Q U of i + o = 6 Q = hi h giv h ri rl, 6Q = Th, 6Q = h iffr-of-woqr foriio = [ [ [ [ = [ [ [ [ [ [ [ [ ig h iffr-of-wo-qr foriio i h lrg r = Sliig h 6 io for o r r ig = Q = Fill, for rigl gr li, lig Or Or l D A, w g h rl ow Hro Forml: = Qio 7 M of o will ow h hi oi G, hr, i h roi of h rigl, h oiio vor g = Th g GX o GY, whr > Alo GY OG OY [ [, h fir ri rl Th rll riil orvio hr i h h irl r O, ri h qio = or =, whr h v of oi o h irl Th, i OY OY, w hv 9 9 A >,, ig, Similrl,

13 GX Uig GY i i GX i =,,, GX GX GY GY GY i 9 = = GX i [Irigl, hi rl grli o oi o irl: GY i 9 i Qio 8 q > q rl, q q q h oig q q i h rl i r ll g qig for q Now hi giv whih i o i of Cllig hi o giv, rqir I orr o rmi, w ol o h f h = for ll, o w hoo h fir fw rm o wor wih = Alrivl, = = q q q = = = q q q q q q = q q q = q = q i q o-zro giv Si i oi for ho,, h roof follow iivl o wo oiv rm of h q Fill, o o h giv q If > + q, h iiil rl > > H, if > h o i Si >, roof follow iivl

14 If = + q h = o h h q i AP Alo, =, = + q, = + q, h ommo iffr i q w ill hv ril irig q, i q > giv Qio 9 I h r w, w h o-lrio forml o g = o = h i g g Wh =, Siig hi io h qio for h o A > h hi oi h ll, mig i o ril, i ov h, w g g h h, rqir g For h r, w = i = h i g i i 6gh h oiiv roo i rqir g Sig = o oig h <, g i gh i o o olv qri i o g i gh i g Thr r vrl w o ro from hr, hi i rh h mo righforwr g Sqrig i gh g i g Cllig i oh i & iviig g h h R-rrgig for g h h h Uig h fir rl,, i hr g g g R-rrgig for h h, whih l o h rqir fil wr Howvr, i i r i w migh ohrwi iviig qi h ol giv o li h > w r ow o h ohr i of h o h rojio oi l h irio of h iqli wol rvr

15 Qio A wih m i rolm, goo igrm i il o fl rogr Th hr r rlivl fw mhil riil o li rolvig wi, ig mom, h r Friio Lw I i, of or, lo imor o g h gl righ Tig mom o M : R F R i = R i + F o + F o Uig h Friio Lw : F = R F = R R O Diviig o r-rrgig r R = R + R + R B R R = R + R r A M F W or mg For h o r, i m lil h w will hv o rolv wi o hvig hi rilr of ool, hogh w ol mom o om ohr oi i l of o rolio Thr i lo h qio of whih irio o rolv i hr, i hol lr vr qil h horizoll vrill will ol il om vr m rl Mom o O : R R r = W r i i Rolvig // AB : R R o + R + R i = W i Giv o A hr if ll orr r from ig Rolvig r AB : R + R i R R o = W o No h ol wo of h r ll rqir, i m ir o wri hm ll ow fir h i whih wo r R R o R R i Diviig h l wo q R R i R R o R R o R R i Uig fir rl, R + R = R R R R i R R o o i Thr i o o o h R R for h h ro wol hv o i o oiio mmrill i h lir Mlilig hrogho o o i =, ig i o o o r o giv r r r r R R Fill,, from h fir rl, < < R R

16 Qio Agi, igrm i rll fl for hlig o hogh i orr; lo, w r goig o hv o oir wh i goig o grll o j r-o or w h li Bfor i h olliio Afr i h olliio i V i V i P i B i B i M i m i i m M i i m Uig h riil of Corvio of Lir Momm, CLM m + M V i = M + im V i NB V = l o V, V, V,, V Alrivl, CLM for ll ril giv m m m m mv, rrrgig for V = V il V Th l olliio or wh V, i N N N N N N hr r N olliio Now, h ol KE of ll h P i i N i N i m = m i i i N Th fil KE of h lo i N N mvn = N N m m N N N Thrfor, h lo i KE i h iffr: m N m i i N N N Si, h lo i KE i N N m N N = m i i

17 Qio Thi ro ow io mor for r, hr i o o: Pligh o = + = 8, h h oiiol roili PHll o = 8 8 = To m rogr wih hi r of h qio, i i imor o rogi h rlig iomil iriio, h h rr o h Broilli ril W r h lig wih B7,, whr = i h rviol giv wr For h mol vl o, w m hv P < P < P; h i, Uig = giv Qio Worig wih h iriio P o =, Po rmr = PY < = Diffriig wr o fi h f of Y f = π, giv Th EY = π Uig Igrio Pr wriig π π giv EY = = + I i fl o il o h imlifig iio = hi g o g = h giv rl, rlig o h r orml iriio f, h vr giig of h qio N, EY = π, ig Igrio Pr, i imilr w o rlir, wriig π π, EY = = + = ig rvio rl, or iio = VrY =, h giv wr, rqir

18 STEP Hi Solio Th m igr o giv for r i ig, h m giv whih olv for z ig h iiil oiio, h igr m i fir orr iffril qio for, whih wh olv, gi wih h iiil oiio, ro h rqir rl Pr ii follow h m r, wih i, whih h olio Th imlifiio i h oig i, oi r of h iffr of wo qr A iml lgri rrrgm, follow ig limi, h logrihm of oh rio, iffriio ro h ohr hr rl i r i Pr ii oi rlig i l l from r i, foriig h iffr h m of rig h r i rl for iffriig A lriv i o rli r i ig i h ro, h lil r r i rqir wih h rrrgm, oirio of h limi Th wo rol, wih vri ori i h irio of h oiiv, oh i h hir qr i, i h fir qr i h ohr hr I, hr r irio i h o hir qr, i i h hir forh, i, h rii, h r i h hir qr i h fir, w h ohig oi h vr of h rol o h i Th fir rl of r iii oi limiig w h wo qio, h o iffriig qig gri, h hir, limiig,, from h fir rl ig h o Th h ri r,,,,,,,, Wriig h llig h fir frio, giv oil ri!!! Similrl, wri, 8, h lr givig h rl Uig ril frio, wri, h fir rm givig GP, h ohr wo, log ri Th rl for r ii i h 6 l For o igr riol oi, i m o vl of o i o Phgor ril h,, or,,, Th hiq for i for ii, mrl hgig h vl of m, whr for ii, lighl mor ivolv rio i h o i, i o For ii, hr r wo lriv h wor il, oh ih, ih oh or,

19 A omll iffr roh for h l r of h qio i o wri 7 o hoo wih rl hoi of riol oil Th,, mr of h form r fil ovr h orio, h o of h orr form, h olvig for, h liwi hv o of h rqir form Som oil olio r i,,,,,,, ig,o ii,, ig, o,, ig,, o,, ig, oh, ih or, 6 Siig for z i h qri qio qig rl imgir r il h fir wo rl, h imgir giv wo iio, o rqir h ohr i io h rl giv h o Th fir of h wo rl i io h rl giv irl ri, r h origi, whil h o giv h rl i wiho h origi Th o qri qio m o h m roh givig h rl i wiho h origi gi, irl r, ri lo omiig h origi Th m roh i h hir il h rl i wih oirig h irimi, O h ohr h, ro 7 Th o orr iffril qio 7 6 m oi iffriig h fir qio, h iig for ig h ohr qio, h oig liwi for ig h fir gi Th olio for follow i h l mr Pr i ii il rivl Pr iii mrl rqir h m o wo gomri rogrio mrg o h 8,,, o oh rio i r i ql Coirig ihr or, lig h rrr rlio, oh fo o zro \o h giv rio i how o if i o if i v Th omo gl forml l h roof i r iii o oml o h iiil rrr rlio h rl from r ii hv li o h rio oi followig lgri imlifiio Rrrgig h rl iig io h rqir m giv, h mho of iffr,

20 9 Elimiig,,, w h fiv qio,,,, il h rqir rl for Th ol iffr i h o r i h h o qio om, o h m limiio il, whih o mllr h h i r i Th fir for qio i hi o r giv, o, h rqir rl follow Afr moio omm, h r oiio h h rig o h vril Corvig rg w h wo r oiio giv Corvig rg for h grl oiio rolvig rill, rig i mi h h gl of h ri o h vril i wi h gl of h rig o h vril, ig ol gl forml giv h rqir rl Th irimi ig giv or omlig h qr mor h h rio for i lw oiiv Vrio roh o fi h rg rm If oil rg zro lvl i o P, h h iiil oil rg i Wh h ril h fll i, h ii rg of h ril i, h oil rg of h ril i, h oil rg of h r of h ior i of rig of lgh i, h oil rg of h rmiig i of ol rig i, h ii rg of h horr movig i i Thi il h fir rl iffriio of i il h o A,, h omior i wi qr, h fil rl follow Th h rgio oi AB, h wo li gm oig A B o h r of h rigl A iml roh for h f i vi, fiig h o of roorioli h iffriig o giv h rqir rl For h o r, 9, o Th il h r i rl Similrl, h il h fir rl of r ii, h il h o Uig, o