Mark Scheme (Results) Summer GCE Further Pure Mathematics 3 (6669/01)

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1 Mark (Results) Summer GCE Further Pure Mathematics (6669/)

2 Edexcel ad BTEC Qualificatios Edexcel ad BTEC qualificatios come from Pearso, the world s leadig learig compay. We provide a wide rage of qualificatios icludig academic, vocatioal, occupatioal ad specific programmes for employers. For further iformatio, please visit our website at Our website subject pages hold useful resources, support material ad live feeds from our subject advisors givig you access to a portal of iformatio. If you have ay subject specific questios about this specificatio that require the help of a subject specialist, you may fid our Ask The Expert service helpful. Pearso: helpig people progress, everywhere Our aim is to help everyoe progress i their lives through educatio. We believe i every kid of learig, for all kids of people, wherever they are i the world. We ve bee ivolved i educatio for over 5 years, ad by workig across 7 coutries, i laguages, we have built a iteratioal reputatio for our commitmet to high stadards ad raisig achievemet through iovatio i educatio. Fid out more about how we ca help you ad your studets at: Summer Publicatios Code UA5977 All the material i this publicatio is copyright Pearso Educatio Ltd

3 Geeral Markig Guidace All cadidates must receive the same treatmet. Examiers must mark the first cadidate i exactly the same way as they mark the last. Mark schemes should be applied positively. Cadidates must be rewarded for what they have show they ca do rather tha pealised for omissios. Examiers should mark accordig to the mark scheme ot accordig to their perceptio of where the grade boudaries may lie. There is o ceilig o achievemet. All marks o the mark scheme should be used appropriately. All the marks o the mark scheme are desiged to be awarded. Examiers should always award full marks if deserved, i.e. if the aswer matches the mark scheme. Examiers should also be prepared to award zero marks if the cadidate s respose is ot worthy of credit accordig to the mark scheme. Where some judgemet is required, mark schemes will provide the priciples by which marks will be awarded ad exemplificatio may be limited. Crossed out work should be marked UNLESS the cadidate has replaced it with a alterative respose.

4 EDEXCEL GCE MATHEMATICS Geeral Istructios for Markig. The total umber of marks for the paper is 75.. The Edexcel Mathematics mark schemes use the followig types of marks: M marks: method marks are awarded for kowig a method ad attemptig to apply it, uless otherwise idicated. A marks: accuracy marks ca oly be awarded if the relevat method (M) marks have bee eared. B marks are ucoditioal accuracy marks (idepedet of M marks) should ot be subdivided.. Abbreviatios These are some of the traditioal markig abbreviatios that will appear i the mark schemes: bod beefit of doubt ft follow through the symbol will be used for correct ft cao correct aswer oly cso - correct solutio oly. There must be o errors i this part of the questio to obtai this mark isw igore subsequet workig awrt aswers which roud to SC: special case oe or equivalet (ad appropriate) dep depedet idep idepedet dp decimal places sf sigificat figures The aswer is prited o the paper The secod mark is depedet o gaiig the first mark 4. All A marks are correct aswer oly (cao.), uless show, for example, as A ft to idicate that previous wrog workig is to be followed through. After a misread however, the subsequet A marks affected are treated as A ft, but maifestly absurd aswers should ever be awarded A marks. 5. For misreadig which does ot alter the character of a questio or materially simplify it, deduct two from ay A or B marks gaied, i that part of the questio affected. 6. If a cadidate makes more tha oe attempt at ay questio: If all but oe attempt is crossed out, mark the attempt which is NOT crossed out. If either all attempts are crossed out or oe are crossed out, mark all the attempts ad score the highest sigle attempt. 7. Igore wrog workig or icorrect statemets followig a correct aswer. 8. I some istaces, the mark distributios (e.g., B ad A) prited o the cadidate s respose may differ from the fial mark scheme

5 Geeral Priciples for Pure Mathematics Markig (But ote that specific mark schemes may sometimes override these geeral priciples). Method mark for solvig term quadratic:. Factorisatio ( x bx c) ( x p)( x q), where pq c, leadig to x = ( ax bx c) ( mx p)( x q), where pq c ad m a, leadig to x. Formula Attempt to use correct formula (with values for a, b ad c).. Completig the square b Solvig x bx c : x q c, q, leadig to x =... Method marks for differetiatio ad itegratio:. Differetiatio Power of at least oe term decreased by. ( x x ). Itegratio Power of at least oe term icreased by. ( x x ) Use of a formula Where a method ivolves usig a formula that has bee leart, the advice give i recet examiers reports is that the formula should be quoted first. Normal markig procedure is as follows: Method mark for quotig a correct formula ad attemptig to use it, eve if there are mistakes i the substitutio of values. Where the formula is ot quoted, the method mark ca be gaied by implicatio from correct workig with values, but may be lost if there is ay mistake i the workig. Exact aswers Examiers reports have emphasised that where, for example, a exact aswer is asked for, or workig with surds is clearly required, marks will ormally be lost if the cadidate resorts to usig rouded decimals. Aswers without workig The rubric says that these may ot gai full credit. Idividual mark schemes will give details of what happes i particular cases. Geeral policy is that if it could be doe i your head, detailed workig would ot be required.

6 . (a) & (b) Mark (a) ad (b) together Sight of both of these (ca be ae ad implied by their work) a ( e ) 5 (allow ± ae = ± or ± ae = or ae = ±) Solves to obtai Elimiates e to reach a... or a... a...or a... B a Cao (ot ) uless - is rejected A e = / Uses their a to fid e or fids e by elimiatig a (Igore ± here) (Ca be implied by a correct aswer) : ( x )( ) a e ot eeded for this mark or is x ad eve allow y = ( ) a e here just x a ( ), e 44 look for use of a e ad e. A: x 44 equatio (Do ot allow with umerical a oe but must be a x ), A If they use the eccetricity equatio for the ellipse (b =a (-e )) allow the M s Total 6

7 . (a) x 9 karsih ( c ) or k l[ px ( p x p )] ( c) 4 (b) x arsih ( c ) or So: l 6 45 l 6 45 = 9 l[ px ( p x p )] ( c) A 4 = 6 45 l 6 45 Uses correct limits ad combies logs l (6 45) l 9 Correct method to ratioalise deomiator (may be implied) Method must be clear if aswer does ot follow their fractio = l[ 5] ( or l[9 4 5] ) Acso Note that the last marks ca be scored without the eed to ratioalise e.g l[ x (4x 9)] l(6 45) l l( ) : Uses the limits ad ad doubles : Combies Logs A: l[ 5] oe Alterative for x (a) x sih u. cosh u du k arsih ( c) 9sih u 9 x arsih ( c ) Alterative for (b) x arsih arsih arsih 5 l( 5) l( 5 ) l( ) 5 Uses correct limits ad combies logs l(. ) l( ) Correct method to ratioalise deomiator (may be implied) Method must be clear if aswer does ot follow their fractio l[9 4 5] A () () Total 5 Acso

8 . dx dy ( ) sih ad ( ) 4cosh d d Or equivalet correct derivatives A ( ) 4sih "sih " "4cosh " d A ( ) 4sih ( or "4 cosh " ( ).sih "sih " d Use of correct formula icludig replacig dx with "sih "d if chai rule used. Allow the omissio of the π here. A sih cosh d A (sih sih ) d Completely correct expressio for A with the square root removed This mark may be recovered later if the π is itroduced later : Valid attempt to itegrate a correct expressio or a multiple A = cosh of a correct expressio depedet o the first A: Correct expressio : Uses the limits ad correctly. Depedet o both = cosh previous M s A: Cao ad cso (o errors see) B B da dda (7) Example Alterative Itegratio for last 4 marks sih cosh d sih ( sih ) d (sih sih ) d A (sih sih sih ) d (sih sih ) d cosh cosh 4 d: sih cosh d pcosh qcosh 8 cosh cosh A: cosh cosh 4 8 (cosh cosh cosh cosh )... cosh : Uses the limits ad correctly. Depedet o both previous M s A: Cao da dda

9 . Alterative Cartesia Approach x y 8 Ay correct Cartesia equatio B dx y dy or dy 4 dx ( x ) Correct Derivative B y A. y ( )dy or 4 A. 8( x ) ( )dx x A Use of a correct formula y or A x d A : Covicig attempt to itegrate a relevat expressio depedet o the first but allow the omissio of π A: Completely correct expressio for A A 8 sih 8 or 8 cosh dd Correct use of limits ( 4sih for y or cosh for x) Use sih cosh to give cosh Use to give cosh cosh cosh A

10 4. dy 4 d x ( x ) - 9 Put d y dx = ad obtai x (Allow sig errors oly) y = x l{( 9) ( 9) } "4" dy p : q d x ( x ) A A: Cao 68 e.g. 8 : Square root 4 A: x or exact equivalet 9 (ot ± 4 9 ) 4 Substitutes x " " ito the curve 9 ad uses the logarithmic form of arcosh d A So y = 8l - 4 Cao A Total 7

11 5. (a) (i)&(ii) a a b c b c, ad so a = -, =, A, A - : Multiplies out matrix with first eigevector ad puts equal to times eigevector. A : Deduces a = -. A: Deduces = a a b c c, ad so c =,, A, A - : Multiplies out matrix with secod eigevector ad puts equal to times eigevector. A: Deduces c =. A: Deduces = (b)(i) b c so b = - : Uses b c with their to fid a value for b (They must have a equatio i b ad c from the first eigevector to score this mark) A A: b = - (a = -, b = -, c =, =, = ) (8) detp = -d - (ii) T P or miors cofactors d d d - d d d - - d -d - Allow d or ( + d) A correct (possibly u-simplified) determiat d+ d d - a correct first step or : Idetifiable full attempt at iverse icludig reciprocal of determiat. Could be idicated by at least 6 correct elemets. A: Two rows or two colums correct (igorig determiat) BUT MAA or MAA is ot possible A: Fully correct iverse B B A A (5) Total

12 6(a) 4 I (6 ) x x x dx 6( ) i.e. I I I : Obtais x(6 x ) prior to itegratio A: Correct uderlied A expressio (ca be implied by their itegratio) 4 4 I x (6 x ) x (6 x ) dx d d: Parts i the correct directio (Igore limits) 4 I (6 )(6 ) x x x dx Maipulates to obtai at least oe itegral i terms of I or I - o the rhs. 6( ) Collects terms i I I( ) I from both sides ( ) I 6( ) I * Prited aswer with o errors A*cso Way (6 x ) 6x x x (6 x ) dx x dx dx dx (6 x ) (6 x ) (6 x ) Way : Obtais x x(6 x ) d x x x(6 x ) dx A x(6 x ) prior to itegratio A: Correct expressios x (6 x ) 6( ) x (6 x ) dx 4 4 ( x (6 x ) ( ) x (6 x ) d x) d d: Parts i the correct directio o both (Igore limits) Maipulates to obtai at least oe I 6( ) I ( ) I itegral i terms of I or I - o the rhs. I ( ) 6( ) I Collects terms i I from both sides ( ) I 6( ) I * Prited aswer with o errors A* 4 4 (6 x ) x (6 x ) dx x x dx (6 x ) : Obtais x(6 x ) prior to itegratio A: Correct expressio 4 4 A x (6 x )(6 x ) (6( ) x ( ) x )(6 x ) d x d d: Parts i the correct directio (Igore limits) Maipulates to obtai at least oe I 6( ) I ( ) I itegral i terms of I or I - o the rhs. I ( ) 6( ) I Collects terms i I from both sides ( ) I 6( ) I * Prited aswer with o errors A* (6)

13 (b) 4 I x (6 x )dx = (6 ) 4 x = must come from correct work Usig x 4si : : Correct itegratio to fid I A: 64 or equivalet (May be implied by a later work they are ot asked explicitly for I ) A I 4si (6 6si ) 4cos d 64si cos d 64 cos : A complete substitutio ad attempt to substitute chaged limits A: 64 or equivalet I 64 5, 7 I I I 5 I Applies to apply reductio formula twice. First for I 5 i terms of I, secod for I i terms of I (Ca be implied) Ay exact equivalet (Depeds o all previous marks havig bee scored), A (5) Total

14 7(a) dx dy dy bcos ( asi ad bcos ) so d d dx asi : Differetiates both x ad y ad divides correctly A: Fully correct derivative Alterative: : x y cos x yy y b x b a a b a b a y a bsi Differetiates implicitly ad substitutes for x ad y bcos A: asi asi a y Correct perpedicular gradiet Normal has gradiet or bcos b x rule ( y bsi ) Correct straight lie method asi ( x acos ) usig a chaged gradiet which bcos is a fuctio of If y = mx + c is used eed to fid c for (b) ( a b )cos x a ( a b )si y b ( a b ) cos si ab A axsi by cos ( a b )si cos * A Fully correct completio to prited aswer Allow u-simplified Allow u-simplified = 4 ( a ) si : Area of triagle is " OA" " " OB ad uses double agle formula correctly A: Correct expressio for the area (must be positive) (c) Maximum area whe si so or 45 4 a b So the poit P is at, oe acos, b si scores BA 4 4 b ab Mark part (c) idepedetly Correct value for (may be implied by correct coordiates) : Substitutes their value of where or 9 ito their parametric coordiates A: Correct exact coordiates B B A B A (5) (4) () Total

15 8(a) (6i +j+k).(i 4j k ) 4 Attempt scalar product (6i + j+k).(i 4j k) 5 4 Use of correct formula 9 (ot 9) Correct distace (Allow 9 9 ) A (a) Way r (6i + j+k) (i 4j k) () (a) Way (b) For a cross product, if the method is uclear, out of compoets should be correct for Substitutes the parametric coordiates of the lie through (6,, ) perpedicular to the plae ito the cartesia equatio., 6, or i 4j k Solves for to obtai the required poit or vector. 9 Correct distace A Parallel plae cotaiig (6,, ) is r.(i 4j k) 4 4 Origi to this plae is r.(i 4j k ) r.(i 4j k) 5 5 Origi to plae is Correct distace A i j k : Attempts 5 9 ( i + j+ 5 k) ( i j k ) (cos ) 4 A: Ay multiple of i + j k (i - 4j+ k).(i + j- k) 9 A Attempts scalar product of ormal vectors icludig magitudes Obtais agle usig arccos 5 d A (depedet o previous ) Do ot isw ad mark the fial aswer e.g. 9 5 = 8 loses the A (5) (c) i j k : Attempt cross product of ormal vectors 5 A 4 A: Correct vector x :(,, ), y :(,, ), z :(,, ) A : Valid attempt at a poit o both plaes. A: Correct coordiates May use way to fid a poit o the lie : r dir = pos.vector dir (This r (-i +5j+k ) = 5i -5j+5k way roud) A A: Correct equatio (6)

16 (c) Way (c) Way x + y z = ad x 4y + z = 5 uses their cartesia form of ad elimiate x, or y or z ad substitutes back to obtai two of the variables i terms of the third x 5z 5 5 z ( x 5 y ad z 5 y) or ( y ad x ) or 5 5x 5 x ( y ad z ) Cartesia Equatios: 5 5 y z x z 5 or y or x y 5 Poits ad Directios: Directio ca be ay multiple (,, ), i j k or (,,), i + j+ k or (,,), i + j+ k 5 5 :Uses their Cartesia equatios correctly to obtai a poit ad directio A: Correct poit ad directio it may ot be clear which is which i.e. look for the correct umbers either as poits or vectors Equatio of lie i required form: e.g. r (-i +5j+k ) = 5i -5j+5k Or Equivalet , gives(,, ) , gives(,, ) 6 Directio 5 : Substitutes parametric form of ito the vector equatio of A: Correct equatio : Fids poits ad directio A: Correct coordiates ad directio z A A A (6) Total 4 A A Equatio of lie i required form: e.g. r (-i +5j+k ) = 5i -5j+5k Or Equivalet Do ot allow mixed methods mark the best sigle attempt NB for checkig, a geeral poit o the lie will be of the form: ( -, 5, + ) A

18 Further copies of this publicatio are available from Edexcel Publicatios, Adamsway, Masfield, Notts, NG8 4FN Telephoe Fax publicatio.orders@edexcel.com Order Code UA5977 Summer For more iformatio o Edexcel qualificatios, please visit our website Pearso Educatio Limited. Registered compay umber 8788 with its registered office at Ediburgh Gate, Harlow, Essex CM JE

Mark Scheme (Results) January International GCSE Further Pure Mathematics (4PM0/01)

Mark Scheme (Results) January International GCSE Further Pure Mathematics (4PM0/01) Mark Scheme (Results) Jauary 013 Iteratioal GCSE Further Pure Mathematics (4PM0/01) Edexcel ad BTEC Qualificatios Edexcel ad BTEC qualificatios come from Pearso, the world s leadig learig compay. We provide