AS Mathematics. MFP1 Further Pure 1 Mark scheme June Version: 1.0 Final

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1 AS Mathematics MFP Futhe Pue Mak scheme 0 Jue 07 Vesio:.0 Fial

2 Mak schemes ae pepaed by the Lead Assessmet Wite ad cosideed, togethe with the elevat questios, by a pael of subject teaches. This mak scheme icludes ay amedmets made at the stadadisatio evets which all associates paticipate i ad is the scheme which was used by them i this examiatio. The stadadisatio pocess esues that the mak scheme coves the studets esposes to questios ad that evey associate udestads ad applies it i the same coect way. As pepaatio fo stadadisatio each associate aalyses a umbe of studets scipts. Alteative aswes ot aleady coveed by the mak scheme ae discussed ad legislated fo. If, afte the stadadisatio pocess, associates ecoute uusual aswes which have ot bee aised they ae equied to efe these to the Lead Assessmet Wite. It must be stessed that a mak scheme is a wokig documet, i may cases futhe developed ad expaded o the basis of studets eactios to a paticula pape. Assumptios about futue mak schemes o the basis of oe yea s documet should be avoided; whilst the guidig piciples of assessmet emai costat, details will chage, depedig o the cotet of a paticula examiatio pape. Futhe copies of this mak scheme ae available fom aqa.og.uk Copyight 07 AQA ad its licesos. All ights eseved. AQA etais the copyight o all its publicatios. Howeve, egisteed schools/colleges fo AQA ae pemitted to copy mateial fom this booklet fo thei ow iteal use, with the followig impotat exceptio: AQA caot give pemissio to schools/colleges to photocopy ay mateial that is ackowledged to a thid paty eve fo iteal use withi the cete.

3 MARK SCHEME AS MATHEMATICS MFP JUNE 07 Key to mak scheme abbeviatios M mak is fo method m o dm mak is depedet o oe o moe M maks ad is fo method A mak is depedet o M o m maks ad is fo accuacy B mak is idepedet of M o m maks ad is fo method ad accuacy E mak is fo explaatio o ft o F follow though fom pevious icoect esult CAO coect aswe oly CSO coect solutio oly AWFW aythig which falls withi AWRT aythig which ouds to ACF ay coect fom AG aswe give SC special case OE o equivalet A, o (o 0) accuacy maks x EE deduct x maks fo each eo NMS o method show PI possibly implied SCA substatially coect appoach c cadidate sf sigificat figue(s) dp decimal place(s) No Method Show Whee the questio specifically equies a paticula method to be used, we must usually see evidece of use of this method fo ay maks to be awaded. Whee the aswe ca be easoably obtaied without showig wokig ad it is vey ulikely that the coect aswe ca be obtaied by usig a icoect method, we must awad full maks. Howeve, the obvious pealty to cadidates showig o wokig is that icoect aswes, howeve close, ea o maks. Whee a questio asks the cadidate to state o wite dow a esult, o method eed be show fo full maks. Whee the pemitted calculato has fuctios which easoably allow the solutio of the questio diectly, the coect aswe without wokig eas full maks, uless it is give to less tha the degee of accuacy accepted i the mak scheme, whe it gais o maks. Othewise we equie evidece of a coect method fo ay maks to be awaded. of

4 MARK SCHEME AS MATHEMATICS MFP JUNE 07 Q Solutio Mak Total Commet DO NOT ALLOW ANY MISREADS IN THIS QUESTION M Attempt to fid h y h y { y. } A 8.0 OE { y. } y. 0. y dm Attempt to fid y. 0. y... ; must see evidece of umeical expessio if coect ft [0.08 +c s y(.)] value is ot obtaied AF PI ft o c s value fo y(.); dp (ouded o tucated) o bette y. (to dp) A CAO Must be 8.08 idetified as y(.) 8.08 Total o as c s fial aswe o as c s highlighted aswe. of

5 MARK SCHEME AS MATHEMATICS MFP JUNE 07 Q Solutio Mak Total Commet (a) p q B; B ; p ; q 0 p q M Elimiatig to fom a eq i p ad q oly, dep o at least B scoed above. M0 00 if > idep eo i pocess befoe the lie whee has bee elimiated Alt p 00( ) p 0q 00 q A AG Be coviced p p 0q ( x ) 0 (B) PI Equatig oe coect oot to ad the othe coect oot to (B) PI (M) Elimiatig p 0q to fom a eq i p ad q oly, codoe sig eo i oots of eq 0 p 0q 0 p 0q 00 (A) () AG Be coviced Alt p q 0 (b)(i) ad (B) Both equied if a B ot scoed fom p q 0 mai scheme. Subtact eqs to get 0.p (B) OE liea eq i ad p oly 0.p p 0.p q 0 (M) Elimiatig to fom a eq i p ad q oly, codoe sig eo i d B mak 0 0.0p q 0 so p 0q 00 (A) () AG Be coviced q 8 ] S[ 8 B A coect expessio fo the sum of the ew oots i tems of q oly P q [ ( ) ] B A coect expessio fo the poduct of the ew oots i tems of q oly q q x 8x 0 BF Ft c s S ad P to fom a quadatic eq i tems of q with o squae oots. Alt Subst y x gives y p y q 0 (B) p y y q (B) OE with o squae oot y 0q 00y q 0 (B) () ACF of quadatic eq i tems of q ad the vaiable oly with elevat tems gouped (ii) q q q 8 0 M Use of B AC 0 OE to obtai a liea eq i q. q 0 A q 0 NMS / (ii) Alt ( ) (M) q ( ) 0 (A) () q 0 NMS / Total 9 (b)(ii) Both maks ca be scoed without (b)(i) beig coect. of

6 MARK SCHEME AS MATHEMATICS MFP JUNE 07 Q Solutio Mak Total Commet z i i i = i i i i = i i i M Attempt to expad all backets z i ( ) ( i) M i used at least oce at ay stage i pat (a) z i A i obtaied covicigly SC i NMS (a) (b) z i i BF c s k + i. PI by ext lie i * i z BF c' s k i i i i m (#) Re: m ; Im: m m m M A Attemptig to equate, without mixig eal ad imagiay tems, both the Re pats ad the Im pats to fom two eqs each i m ad fo the c s eq (#). A coect eq i eithe m oly o i oly PI by coect values fo both m ad. m, A Both equied, be coviced. Total 8 Q Solutio Mak Total Commet (a) B. dx = x x x dx x x x see o used (igoe eos i dealig with the coefficiet ½ ) 0. = x (+ costat) B 0. x OE Itegatio coect d dx = c x x d c B OE (b) (i) as c fiite value ( ) c 0 so itegal has o E OE Ft o kc, 0 afte itegatio (ii) 0 as d M d OE Ft o kd, 0 afte itegatio so dx 9 A x x Total (b)(i)(ii) Do NOT allow examples whee c=0 eg o whee d eg 0 0 (b)(i)(ii) If 0/ SC if i (i) afte itegatio cad has kx, >0 the eg c 0, so o fiite value o eg c 0, so udefied of

7 MARK SCHEME AS MATHEMATICS MFP JUNE 07 Q Solutio Mak Total Commet (a) B ta ta OE stated o used. x M Ft c s ta. Codoe 80 i place of x AF Ft c s ta. No degees peset (b) x A OE fom with costat tems combied si x si B Must be fom coect GS si x si ad B OE exact values; eed both. Must be fom coect GS si x si x ad B OE exact foms; eed both SC if 0/ scoed awad mak if (i) cad gets ( possible values fo six ad oly oe possible value fo six) o (ii) cad obtais the two coect exact values by just cosideig specific values of i the coect GS NMS Mak as / max. Total 7 Alt Those usig, must be cosideig sepaately a agle i st quadat ad a agle i d quadat eg x ad x OE befoe M ca be awaded (a) eg ta allow B oly. 7 of

8 MARK SCHEME AS MATHEMATICS MFP JUNE 07 Q Solutio Mak Total Commet (a) Vetical tagets: x, x M Idetificatio of the tagets eithe stated Hoizotal tagets: y, y o show o a diagam. PI by coect aea. Aea of ectagle = 8 A NMS / (b) B, B else B fo, k 0, k 0 k (c) (i) (c) (ii) Taslatio maps (,0) to ( 7,*) ad (,0) to (,*) M Eithe pai; o statemet idicatig move to the ight. PI by coect value fo a. a A Coect value fo a. ( x a) ( y b) E : Elimiatig deomiatos to get ( x a) ( y b) M ( x a) ( y b) OE see o used. PI by p a ad eithe q 8b o a b x y ax 8by a b Compae with x y px qy B p a p Coect value fo p. Accept eithe fom compaig with ( x ) o with ( x a) Compaig coefficiets of y ad costat tems: q 8b ; a b M OE Both attempted with at least oe p q coect o OE b b q 8 A Coect values fo q. Total 0 (c)(ii) Alt a fo M (Taslate E oto E usig taslatio ): b x a y b p x a q y b see o used (M) PI by p a ad eithe q 8b o a b 8 of

9 MARK SCHEME AS MATHEMATICS MFP JUNE 07 9 of Q7 Solutio Mak Total Commet (a) M = see/used. = dm Substitutio of coect expessios fo ad = dm Takig out facto o othe poduct of factos i fom the coect expessio = = A covicigly obtaied (b) Seies = ] ) (... [ ) (... M PI by the ext lie i sol = 8 A PI by the ext lie i sol 8 = B o bette = A covicigly obtaied Alt (b) Seies = ] ) (... [ ) (... (M) PI by the ext lie i sol, but must see diffeece betwee two seies = = = (A) PI by the ext lie i sol (B) see o used = (A) () covicigly obtaied Total 8 (b) = ) )( / ( scoes M0 B0 as o diffeece betwee seies

10 MARK SCHEME AS MATHEMATICS MFP JUNE 07 Q8 Solutio Mak Total Commet (a). If ot B awad B fo eithe D = B, (i) elemets coect o. (ii) D = see o 7 (iii) D = see o (iv) D = 0. 0 see (b) Reflectio i the lie y x. E OE eg y x ta (c)(i) cos B see o used B = BF Ft oly o wog sig fo cos. Values must be exact (ii) BA = 0 o A = 0 M See o used. Codoe oe aithmetical slip i evaluatig the poduct of coect matices. 0 8 BA = o B 8 = 0 (P has coodiates) 8, A 8 A At least oe elemet of coect, ad coectly obtaied. 8, SC If 0/ awad mak fo eithe o 7 i matix o 7 coodiate fom Total 8 SC i (c)(ii), 7 fom wog sig fo cos ad,7 fom usig AB istead of BA. 0 of

11 MARK SCHEME AS MATHEMATICS MFP JUNE 07 Q9 Solutio Mak Total Commet (a) x ; x ; y B,,0 OE. Each must be a equatio. B fo two coect equatios ad o moe tha oe icoect equatio. (b) x x k k x x x x x x k x (k ) 0 M Elimiatio of y to fom a equatio i k ad x. Codoe oe sig eo if the deomiato has bee expaded. ( k ) x (*) A OE i fom ax bx c 0 y k itesects C so oots of (*) ae eal b ac ( k ) ( k )( k ) M b ac i tems of k; ft o c s quadatic povided a, b ad c ae all i tems of k. ( k ) ( k )( k ) 0 A A coect iequality obtaied coectly whee k is the oly ukow k k k k 0 k k 0 A CSO AG Be coviced (c) ( k )( k ) 0 (**) M Method to fid citical values fom pited iequality i (b). Codoe oe sig eo. PI by coect two citical values Citical values ae 0. ad A Sub k= 0. i (*), 9x x 0 OE Sub k= i (*) gives x x 0 OE dm Subst of eithe 0. o ito quadatic eq to each a quadatic i x with equal oots k= 0., x ;, is a statioay poit k=, x ; (, ) is a statioay poit A A Coect coespodig values fo k ad x o coect coodiates Coect coespodig values fo k ad x o coect coodiates PQ = dm OE A coect umeical expessio fo eithe PQ o PQ. Ft o c s wog x values PQ = A ACF povided aswe is exact value. ISW if is followed by a decimal. 7 NMS scoes 0/7; Usig diffeetiatio scoes 0/7 Total of

Version 1.0. General Certificate of Education (A-level) June Mathematics MM04. (Specification 6360) Mechanics 4. Final.

Version 1.0. General Certificate of Education (A-level) June Mathematics MM04. (Specification 6360) Mechanics 4. Final. Vesion 1.0 Geneal Cetificate of Education (A-level) June 011 Mathematics MM04 (Specification 660) Mechanics 4 Final Mak Scheme Mak schemes ae pepaed by the Pincipal Examine and consideed, togethe with