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1 Mark Scheme (Results) Summer 01 GCE Core Mathematics C (6664) Paper 1

2 Edecel ad BTEC Qualificatios Edecel 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 Epert 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 150 years, ad by workig across 70 coutries, i 100 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 01 Publicatios Code UA All the material i this publicatio is copyright Pearso Educatio Ltd 01

3 Summer Core Mathematics C Mark Scheme Geeral Markig Guidace All cadidates must receive the same treatmet. Eamiers must mark the first cadidate i eactly 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. Eamiers 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. Eamiers should always award full marks if deserved, i.e. if the aswer matches the mark scheme. Eamiers 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 eemplificatio may be limited. Whe eamiers are i doubt regardig the applicatio of the mark scheme to a cadidate s respose, the team leader must be cosulted. Crossed out work should be marked UNLESS the cadidate has replaced it with a alterative respose.

4 EDEXCEL GCE MATHEMATICS Geeral Istructios for Markig 1. The total umber of marks for the paper is 75.. The Edecel 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) Marks should ot be subdivided. 3. Abbreviatios These are some of the traditioal markig abbreviatios that will appear i the mark schemes ad ca be used if you are usig the aotatio facility o epen. 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 eample, as 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 Geeral Priciples for Core Mathematics Markig (But ote that specific mark schemes may sometimes override these geeral priciples). Method mark for solvig 3 term quadratic: 1. Factorisatio ( + b + c) = ( + p)( + q), where pq = c, leadig to =... ( a + b + c) = ( m + p)( + q), where pq = c ad m = a, leadig to =. Formula Attempt to use correct formula (with values for a, b ad c), leadig to = 3. Completig the square Solvig + b + c = 0 b : ( ) ± ± q ± c, q 0, leadig to = Method marks for differetiatio ad itegratio: 1. Differetiatio Power of at least oe term decreased by 1. ( 1 ). Itegratio Power of at least oe term icreased by 1. ( +1 ) Use of a formula Where a method ivolves usig a formula that has bee leart, the advice give i recet eamiers 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. Eact aswers Eamiers reports have emphasised that where, for eample, a eact 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 Summer Core Mathematics Mark Scheme Questio umber Scheme Marks ( 3 ) =... + ( 3 ) + ( 3 ) +.., 3, 40, 70 = + B1,, Notes Total 4 : The method mark is awarded for a attempt at Biomial to get the secod ad/or third term eed correct biomial coefficiet combied with correct power of. Igore errors (or omissios) i powers of or 3 or sig or bracket errors. Accept ay otatio for 5 C 1 ad 5 C, 5 5 e.g. ad (usimplified) or 5 ad 10 from Pascal s triagle This mark may be give 1 if o workig is show, but either or both of the terms icludig is correct. Special Case Alterative Method Method 1: 5 B1: must be simplified to 3 ( writig just is B0 ). 3 must be the oly costat term i the fial aswer- so is B0 but may be eligible for A0A0. : is cao ad is for 40. (ot +-40) The is required for this mark : is c.a.o ad is for 70 (ca follow omissio of egative sig i workig) A list of correct terms may be give credit i.e. series appearig o differet lies 3 4 Igore etra terms i ad/or (isw) Special Case: Descedig powers of would be ( 3 ) + 5 ( 3 ) + ( 3 ) +.. i.e This is a 3 misread but award as s.c. B1A0A0 if completely correct or B0A0A0 for correct biomial coefficiet i ay form with the correct power of ( 3 ) = (1 + ( ) + ( ) +.. ) is B0A0A0 { The is 1 for the epressio i the bracket ad as i first method eed correct biomial coefficiet combied with correct power of. Igore bracket errors or errors (or omissios) i powers of or 3 or sig or bracket errors} aswers must be simplified to = + for full marks (awarded as before) 3, 40, ( 3 ) = (1 + ( ) + ( ) +.. ) would also be awarded B0A0A0 1 Method : Multiplyig out : B1 for 3 ad for other terms with awarded if or ^ term is correct. Completely correct is 4/4

7 Questio umber Scheme Marks log = log B1 log log ( ) = log = 9 o.e. Solves = 0 to give =... = 3, = 6 Notes B1 for this correct use of power rule (may be implied) : for correct use of subtractio rule (or additio rule) for logs N.B. log3 log 3( ) = log3 is M0. for correct equatio without logs (Allow ay correct equivalet icludig Total 5 3 istead of 9.) for attemptig to solve = 0 to give = (see otes o markig quadratics) for these two correct aswers Alterative Method log = + log ( ) is B1, Commo Slips log 3 ( ) = 3 eeds to be followed by so ( ) = 9( ) for Here is for complete method i.e.correct use of powers after logs are used correctly log log + log = may obtai B1 if log appears but the statemet is M0 ad so leads to o further marks log3 log 3( ) = so log3 log 3( ) = 1ad log3 = 1 ca ear for correct subtractio rule followig error, but o other marks Special Case log = leadig to = 9 ad the to =3, =6, usually ears B1M0A0, but may log( ) the ear (special case) so 3/5 [ This recovery after ucorrected error is very commo] Trial ad error, Use of a table or just statig aswer with both =3 ad =6 should be awarded B0M0A0 the fial i.e. /5

8 Questio umber Scheme Marks 3 Obtai ( ± 10) ad ( y ± 8) (a) Obtai ( 10) ad ( y 8) (b) Cetre is (10, 8). N.B. This may be idicated o diagram oly as (10, 8) See ( ± 10) + ( y ± 8) = 5 (= r ) or ( r = )"100" + "64" 139 (3) (c) (d) r = 5 * (this is a prited aswer so eed oe of the above two reasos) Use = 13 i either form of equatio of circle ad solve resultig quadratic to give y = ( ) ( y ) ( y ) e.g = = 5 8 = 16 or y y y y = = 0 so y= so y = y = 4 or 1 ( o EPEN mark oe correct value as A0 ad both correct as ) (), (3) Use of rθ with r = 5 ad θ = (may be implied by 9.75) Alteratives (a) OR (b) OR (c) Notes (a) (b) Perimeter PTQ = r + their arc PQ (Fidig perimeter of triagle is M0 here) = or 19.8 or 19.3 Method : From + y + g + fy + c = 0 Cetre is ( g, f ), ad so cetre is (10, 8). cetre is ( ± g, ± f ) Method 3: Use ay value of y to give two poits (L ad M) o circle. co-ordiate of mid poit of LM is 10 ad Use ay value of to give two poits (P ad Q) o circle. y co-ordiate of mid poit of PQ is 8 (Cetre chord theorem). (10,8) is Method : Usig r = 5 * g + f c or ( r = )"100" + "64" 139 Method 3: Use poit o circle with cetre to fid radius. Eg r = 5 * (13 10) + (1 8) Divide triagle PTQ ad use Pythagoras with r (13 "10") = h, the evaluate "8 ± h" - (N.B. Could use 3,4,5 Triagle ad 8 ± 4 ). Accuracy as before Mark (a) ad (b) together as i scheme ad ca be implied by ( ± 10, ± 8). Correct cetre (10, 8) implies for a correct method leadig to r =, or or for usig equatio of circle i (3) 11 marks, cao r = "100" + "64" 139 (ot ) ( 10) ( y 8) k ± + ± = form to idetify r= k = or r = + ) 3 rd r = 5 (NB This is a give aswer so should follow Special case: if cetre is give as (-10, -8) or (10, -8) or (-10, 8) allow for r = 5 worked correctly as r = (d) Full marks available for calculatio usig major sector so Use of rθ with r = 5 ad θ = 4.48 leadig to perimeter of 3.14 for major sector (3) ()

9 Questio umber Scheme Marks 3 4 (a) ( ) ( ) ( ) ( ) f = = 0 so (+) is a factor () (b) f ( ) = ( + )( ) f ( ) = ( + )( 3)( 4) d (4) 6 marks Notes (a) : Attempts f( ± ) (Log divisio is M0) : is for =0 ad coclusio Note: Statig hece factor or it is a factor or a (tick) or QED is fie for the coclusio. Note also that a coclusio ca be implied from a preamble, eg: If f ( ) = 0, ( + ) is a factor. (Not just f(-)=0) (b) 1 st : Attempts log divisio by correct factor or other method leadig to obtaiig ( ± a ± b), a 0, b 0, eve with a remaider. Workig eed ot be see as could be doe by ispectio. Or Alterative Method : 1 st 3 : Use ( + )( a + b + c) = with epasio ad compariso of coefficiets to obtai a = ad to obtai values for b ad c 1 st : For seeig ( ). [Ca be see here i (b) after work doe i (a)] d : Factorises quadratic. (see rule for factorisig a quadratic). This is depedet o the previous method mark beig awarded ad eeds factors d : is cao ad eeds all three factors together. Igore subsequet work (such as a solutio to a quadratic equatio.) 3 Note: Some cadidates will go from { + } + to { } ( ) ( 11 1) list all three factors. Award these resposes M0A0. =, =, 4, ad ot Fids = 4 ad = 1.5 by factor theorem, formula or calculator ad produces factors f ( ) = ( + )( 3)( 4) or f ( ) = ( + )( 1.5)( 4) o.e. is full marks f ( ) = ( + )( 1.5)( 4) loses last

10 Questio umber Method 1 5 (a) Puts 10 = 10 8 ad rearrages to give three term quadratic Solves their " = 0" usig acceptable method as i geeral priciples to give = Obtais =, = 9 (may be o diagram or i part (b) i limits) Substitutes their ito a give equatio to give y = (may be o diagram) Scheme Or puts y = y y 10(10 ) (10 ) 8 ad rearrages to give three term quadratic Solves their " y 9y + 8 = 0" usig acceptable method as i geeral priciples to give y = Obtais y = 8, y = 1 (may be o diagram) Substitutes their y ito a give equatio to give = (may be o diagram or i part (b)) Marks (b) y = 8, y = 1 =, = 9 (5) (10 8) d = 8 { + c } = = 90 = 88 or ( ) ( ) Area of trapezium = (8 + 1)(9 ) = 31.5 d B1 1 So area of R is = 57 or cao (7) 1 marks Notes (a) First : See scheme Secod : See otes relatig to solvig quadratics Third : This may be awarded if oe substitutio is made Two correct Aswers followig tables of values, or from Graphical calculator are 5/5 Just oe pair of correct coordiates o workig or from table is M0M0A0A0 1 (b) : + for ay oe term. 1 st : at least two out of three terms correct d : All three correct d: Substitutes 9 ad (or limits from part(a)) ito a itegrated fuctio ad subtracts, either way roud 3 10 (NB: If cadidate chages all sigs to get ( ) d = { + c } This is 3 The uses limits d ad trapezium is B1 Needs to chage sig of value obtaied from itegratio for fial so is M0A0 ) B1: Obtais 31.5 for area uder lie usig ay correct method (could be itegratio) or triagle mius triagle or rectagle plus triagle [may be implied by correct 57 1/6 ] : Their Area uder curve Their Area uder lie (if itegrate both eed same limits) : Accept 57.16recurrig but ot PTO for Alterative method 3

11 Method for (b) Area of R 9 = (10 8) (10 ) d 3 rd (i (b) ): Uses differece betwee two fuctios i itegral. 9 1 M: d + for ay oe term. at least two out of these three 3 11 simplified terms = + 18 { + c} 3 Correct itegratio. (Igore + c) = ( ) ( ) Substitutes 9 ad (or limits from part(a)) ito a itegrated fuctio ad subtracts, either way roud. d This mark is implied by fial aswer which rouds to 57. B1 See above workig(allow bracketig errors) to decide to award 3 rd mark for (b) here: ( 16 ) = 57 cao 3 6 (7) Special case of above method d 11 = + 18 { + c} = (...) (...) 3 This mark is implied by fial aswer which rouds to 57. (ot -57.) D B1 Special Case Notes Differece of fuctios implied (see above epressio) Itegrates epressio i y e.g ( 16 ) = 57 cao y y + = : This ca have first " 9 8 0" i part (b) ad o other marks. (It is ot a method for fidig this area) Take away trapezium agai havig used Method loses last two marks Commo Error: Itegrates Itegrates is likely to be A0dB0A0 11 is likely to e A0A0dB0A0 (7) Writig 9 (10 8) (10 ) d oly ears fial M mark

12 Questio umber Scheme Marks 6(a) States or uses si ta = cos si 5si si 5si cos 0 si (1 5cos ) 0 cos = = = () (b) si = 0 gives = 0, 180, 360 so = 0, 90, 180 B1 for two correct aswers, secod B1 for all three correct. Ecess i rage lose last B1 cos = gives = (or 78.5 or 78.4) or = (or 81.6) 1 5 = 39. (or 39.3), (or 141) B1, B1, (5) Notes 7 marks siθ (a) : Statemet that taθ = or Replacemet of ta (wherever it appears). Must be a correct cosθ statemet but may ivolve θ istead of. : the aswer is give so all steps should be give. 1 N.B. si 5si cos = 0 or 5si cos + si = 0 or si ( 5) 0 cos = o.e. must be see ad be followed by prited aswer for mark si = 5si cos is ot sufficiet. (b) Statemet of 0 ad 180 with o workig gets B1 B0 (bod) as it is two solutios : This mark for oe of the two statemets give (must relate to ot just to ), : first for 39., secod for Special case solvig cos = 1/ 5 givig = or 58.5 is awarded A0A omitted would give A0 Allow aswers which roud to 39. or 39.3 ad which roud to ad allow 141 Aswers i radias lose last awarded (These are 0, 0.68, 1.57,.46 ad 3.14) Ecess aswers i rage lose last Igore ecess aswers outside rage. All 5 correct aswers with o etras ad o workig gets full marks i part (b). The aswers imply the method here

13 Questio umber Scheme Marks 7 (a) y B1, B1 () (b) 1 0.5, { (1 ) ( ) } o.e. B1,, ft = Notes (a) first B1 for ad secod B1 for ( is B0 ) Wrog accuracy e.g. 1.49, 1.74 is B1B0 6 marks (b) B1: Need ½ of 0.5 or 0.15 o.e. : requires first bracket to cotai first plus last values ad secod bracket to iclude o additioal values from the three i the table. If the oly mistake is to omit oe value from secod bracket this may be regarded as a slip ad M mark ca be allowed ( A etra repeated term forfeits the M mark however) values: M0 if values used i brackets are values istead of y values ft follows their aswers to part (a) ad is for {correct epressio} Fial : Accept , or 1.50 oly after correct work. (No follow through ecept oe special case below followig i table) Separate trapezia may be used : B1 for 0.15, for 1 h( a + b) used 3 or 4 times (ad ft if it is all correct ) e.g ( ) ( ) ( ) is A0 equivalet to missig oe term i { } i mai scheme Special Case: Bracketig mistake: i.e. 0.15(1+) +( ) scores B1 A0 A0 for If the fial aswer implies that the calculatio has bee doe correctly i.e (the full marks ca be give). Need to see trapezium rule aswer oly (with o workig) is 0/4 ay doubts sed to review Special Case; Uses to give or or or 1.50 gets, B1 B0 B1ft the (lose 1 mark) NB Bracket is (4)

14 Questio umber Scheme Marks 8 60 B1 (a) ( h = ) or equivalet eact (ot decimal) epressio e.g. ( h = )60 π π (1) (b) ( A = )π + π h or ( A = )π r + π rh or ( A = )π r + π dh may ot be simplified ad may appear o separate lies (c) Either ( A) = π + π 60 π or As π h = the ( A = )π + 10 A = π + * cso da 10 ( ) = 4π or = 4π 10 d π = 0 implies B1 3 = (Use of > 0 or < 0 is M0 the M0A0) 10 = or aswers which roud to.1 ( -.1 is A0) d 4π (d) 10, A = π (.1) +, = 85 (oly ft = or.1 both give 85).1 (e) d A 40 Or (method ) cosiders gradiet to left ad right Either = 4π + ad sig 3 d of their.1 (e.g at ad.5) Notes cosidered ( May appear i (c) ) which is > 0 ad therefore miimum (most substitute.1 but it is ot essetial to see a substitutio ) (may appear i (c)) Or (method 3) cosiders value of A either side Fids umerical values for gradiets ad observes gradiets go from egative to zero to positive so cocludes miimum (a) B1: This epressio must be correct ad i part (a) OR fids umerical values of A, observig greater tha miimum value ad draws coclusio 60 π r is B0 (b) B1: Accept ay equivalet correct form may be o two or more lies. : substitute their epressio for h i terms of ito Area formula of the form : There should have bee o errors i part (b) i obtaiig this prited aswer (c) : At least oe power of decreased by 1 accept ay equivalet correct aswer k + ch : Settig d A 3 3 dy =0 ad fidig a value for ( = may be implied by aswer). Allow = 0 d d d: Usig cube root to fid : For ay equivalet correct aswer (eed 3sf or more) Correct aswer implies previous M mark (d) : Substitute the (+ve) value foud i (c) ito equatio for A ad evaluate. is for 85 oly 13 marks (e) : Complete method, usually oe of the three listed i the scheme. For first method A ( ) must be attempted ad sig cosidered : Clear statemets ad coclusio. (umerical substitutio of is ot ecessary i first method show, ad must be correct. Must ot see 85 substituted) or calculatio could be wrog but A ( ) (3) (5) () ()

15 Questio Scheme Marks 9 (a) 1 ( S = ) a + ar + ( ar ) ar 3 ad rs = ar + ar + ( ar )... + ar S rs = a ar S (1 r) = a(1 r ) d (b) (c) (d) Ad so result S a(1 r ) = (1 r) Divides oe term by other (either way) to give r =... the square roots to give r = * Or: ( Method ) Fids geometric mea i.e 3.4 ad divides oe term by 3.4 or 3.4 by oe term r =, r = 0.6 (igore 0.6) r = 0.6 (igore 0.6) 5.4 Uses Uses 5.4 r or r a = 15 4, to give a =, ft () 15 S =, to obtai , 0.6 (3) 11 marks Notes (a) : Lists both of these sums ( S =) may be omitted, r S (or rs) must be stated 1 st 1 two terms must be correct i each series. Last term must be ar or ar i first series ad the 1 correspodig ar or ar + i secod series. Must be ad ot a umber. Referece made to other terms e.g. space or dots to idicate missig terms : Subtracts series for rs from series for S (or other way roud) to give RHS = ± ( a ar ). This may have bee obtaied by followig a patter. If wrog power stated o lie 1 M0 here. (Igore LHS)M0M0M0A0 d: Factorises both sides correctly must follow from a previous (It is possible to obtai M0A0 or M0A0) : completes the proof with o errors see Special No errors see: First lie absolutely correct, omissio of secod lie, third ad fourth lies correct: Case M0 See et sheet of commo errors. Refer ay attempts ivolvig sigma otatio, or ay proofs by iductio to team leader. Also attempts which begi with the aswer ad work backwards. (b) : Deduces r by dividig either term by other ad attempts square root : ay correct equivalet for r e.g. 3/5 Aswer oly is / (Method ) Those who fid fourth term must use ab ad ot 1 ( a b) + the must use it i a divisio with give term to obtai r = (c) : May be doe i two steps or more e.g. 5.4 r the divided by r agai ft: follow through their value of r. Just a = 15 with o wrog workig implies (4) () Commo errors (d) : States sum to ifiity formula with values of a ad r foud earlier, provided r <1 : uses 15 ad 0.6 (or 3/5) (This is ot a ft mark) 5.4 (i) Fractio iverted i (b) r = ad (ii) Uses r = 0.36: (b)m0a0 (c)ft (d) A0A0 i.e. 3/7 (iii) Uses ar 3 5 : 37.5 or eact equivalet r = 1 3, the correct ft gives A0 ft M0A0A0 i.e. 3/7 = 5.4, ar = Likely to have (b) (c)m0a0 (d) A0A0 i.e.3/7

16 Further copies of this publicatio are available from Edecel Publicatios, Adamsway, Masfield, Notts, NG18 4FN Telephoe Fa Order Code UA Summer 01 For more iformatio o Edecel qualificatios, please visit our website Pearso Educatio Limited. Registered compay umber 8788 with its registered office at Ediburgh Gate, Harlow, Esse CM0 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