N14/5/MATHL/HP1/ENG/TZ0/XX/M MARKSCHEME. November 2014 MATHEMATICS. Higher Level. Paper pages

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1 N4/5/MATHL/HP/ENG/TZ0/XX/M MARKSCHEME November 04 MATHEMATICS Higher Level Paper 0 pages

2 N4/5/MATHL/HP/ENG/TZ0/XX/M This markscheme is the property of the Iteratioal Baccalaureate ad must ot be reproduced or distributed to ay other perso without the authorizatio of the IB Assessmet Cetre.

3 3 N4/5/MATHL/HP/ENG/TZ0/XX/M Istructios to Examiers Abbreviatios M (M) A (A) R N AG Marks awarded for attemptig to use a correct Method; workig must be see. Marks awarded for Method; may be implied by correct subsequet workig. Marks awarded for a Aswer or for Accuracy; ofte depedet o precedig M marks. Marks awarded for a Aswer or for Accuracy; may be implied by correct subsequet workig. Marks awarded for clear Reasoig. Marks awarded for correct aswers if o workig show. Aswer give i the questio ad so o marks are awarded. Usig the markscheme Geeral Mark accordig to RM Assessor istructios ad the documet Mathematics HL: Guidace for e- markig November 04. It is essetial that you read this documet before you start markig. I particular, please ote the followig: Marks must be recorded usig the aotatio stamps. Please check that you are eterig marks for the right questio. If a part is completely correct, (ad gais all the must be see marks), use the ticks with umbers to stamp full marks. If a part is completely wrog, stamp A0 by the fial aswer. If a part gais aythig else, it must be recorded usig all the aotatios. All the marks will be added ad recorded by RM Assessor. Method ad Aswer/Accuracy marks Do ot automatically award full marks for a correct aswer; all workig must be checked, ad marks awarded accordig to the markscheme. It is ot possible to award M0 followed by, as A mark(s) deped o the precedig M mark(s), if ay. Where M ad A marks are oted o the same lie, eg M, this usually meas M for a attempt to use a appropriate method (eg substitutio ito a formula) ad for usig the correct values. Where the markscheme specifies (M), N3, etc., do ot split the marks. Oce a correct aswer to a questio or part-questio is see, igore further workig. 3 N marks Award N marks for correct aswers where there is o workig. Do ot award a mixture of N ad other marks. There may be fewer N marks available tha the total of M, A ad R marks; this is deliberate as it pealizes cadidates for ot followig the istructio to show their workig.

4 4 N4/5/MATHL/HP/ENG/TZ0/XX/M 4 Implied marks Implied marks appear i brackets eg (M), ad ca oly be awarded if correct work is see or if implied i subsequet workig. Normally the correct work is see or implied i the ext lie. Marks without brackets ca oly be awarded for work that is see. 5 Follow through marks Follow through (FT) marks are awarded where a icorrect aswer from oe part of a questio is used correctly i subsequet part(s). To award FT marks, there must be workig preset ad ot just a fial aswer based o a icorrect aswer to a previous part. If the questio becomes much simpler because of a error the use discretio to award fewer FT marks. If the error leads to a iappropriate value (eg si.5), do ot award the mark(s) for the fial aswer(s). Withi a questio part, oce a error is made, o further depedet A marks ca be awarded, but M marks may be awarded if appropriate. Exceptios to this rule will be explicitly oted o the markscheme. 6 Mis-read If a cadidate icorrectly copies iformatio from the questio, this is a mis-read (MR). A cadidate should be pealized oly oce for a particular mis-read. Use the MR stamp to idicate that this has bee a misread. The deduct the first of the marks to be awarded, eve if this is a M mark, but award all others so that the cadidate oly loses oe mark. If the questio becomes much simpler because of the MR, the use discretio to award fewer marks. If the MR leads to a iappropriate value (eg si.5), do ot award the mark(s) for the fial aswer(s). 7 Discretioary marks (d) A examier uses discretio to award a mark o the rare occasios whe the markscheme does ot cover the work see. I such cases the aotatio DM should be used ad a brief ote writte ext to the mark explaiig this decisio. 8 Alterative methods Cadidates will sometimes use methods other tha those i the markscheme. Uless the questio specifies a method, other correct methods should be marked i lie with the markscheme. If i doubt, cotact your team leader for advice. Alterative methods for complete questios are idicated by METHOD, METHOD, etc. Alterative solutios for part-questios are idicated by EITHER... OR. Where possible, aligmet will also be used to assist examiers i idetifyig where these alteratives start ad fiish.

5 5 N4/5/MATHL/HP/ENG/TZ0/XX/M 9 Alterative forms Uless the questio specifies otherwise, accept equivalet forms. As this is a iteratioal examiatio, accept all alterative forms of otatio. I the markscheme, equivalet umerical ad algebraic forms will geerally be writte i brackets immediately followig the aswer. I the markscheme, simplified aswers, (which cadidates ofte do ot write i examiatios), will geerally appear i brackets. Marks should be awarded for either the form precedig the bracket or the form i brackets (if it is see). Example: for differetiatig f ( x) si(5x 3), the markscheme gives: 0 Accuracy of Aswers ( ) cos(5 3) 5 0cos(5x 3) f x x Award for cos(5x 3) 5, eve if 0cos(5x 3) is ot see. Cadidates should NO LONGER be pealized for a accuracy error (AP). If the level of accuracy is specified i the questio, a mark will be allocated for givig the aswer to the required accuracy. Whe this is ot specified i the questio, all umerical aswers should be give exactly or correct to three sigificat figures. Please check work carefully for FT. Crossed out work If a cadidate has draw a lie through work o their examiatio script, or i some other way crossed out their work, do ot award ay marks for that work. Calculators No calculator is allowed. The use of ay calculator o paper is malpractice, ad will result i o grade awarded. If you see work that suggests a cadidate has used ay calculator, please follow the procedures for malpractice. Examples: fidig a agle, give a trig ratio of More tha oe solutio Where a cadidate offers two or more differet aswers to the same questio, a examier should oly mark the first respose uless the cadidate idicates otherwise. 4. Cadidate work Cadidates are meat to write their aswers to Sectio A o the questio paper (QP), ad Sectio B o aswer booklets. Sometimes, they eed more room for Sectio A, ad use the booklet (ad ofte commet to this effect o the QP), or write outside the box. This work should be marked. The istructios tell cadidates ot to write o Sectio B of the QP. Thus they may well have doe some rough work here which they assume will be igored. If they have solutios o the aswer booklets, there is o eed to look at the QP. However, if there are whole questios or whole part solutios missig o aswer booklets, please check to make sure that they are ot o the QP, ad if they are, mark those whole questios or whole part solutios that have ot bee writte o aswer booklets.

6 6 N4/5/MATHL/HP/ENG/TZ0/XX/M. (a) g( x) x 3 SECTION A Note: Award for x 3 i the deomiator ad for the. [ marks] (b) x 3 y [ marks] Total [4 marks]. (a) usig the formulae for the sum ad product of roots: (i) 4 (ii) Note: Award A0A0 if the above results are obtaied by solvig the origial equatio (except for the purpose of checkig). [ marks] (b) METHOD required quadratic is of the form x x (M) 4 q q 8 p ( ) M 4 p 6 Note: Accept the use of exact roots cotiued

7 7 N4/5/MATHL/HP/ENG/TZ0/XX/M Questio cotiued METHOD replacig x with x M 8 0 x x () x x x 6x 8 0 p 6 ad q 8 Note: Award A0 for x 6x8 0 ie, if p 6 ad q 8 are ot explicitly stated. [4 marks] Total [6 marks]

8 8 N4/5/MATHL/HP/ENG/TZ0/XX/M 3. METHOD OP ( s) (3 s) ( s) ( 6s s ) Note: Award if the square of the distace is foud. EITHER attempt to differetiate: s attemptig to solve s OR d OP M ds d OP ds 0 for s (M) () attempt to differetiate: d 6s 6 OP ds 6s s M attemptig to solve d OP ds 0 for s (M) s () OR attempt at completig the square: miimum value occurs at s OP 6( s ) 5 M (M) () THEN the miimum legth of OP is 5 METHOD the legth of OP is a miimum whe OP is perpedicular to (R) s 3 s 0 s attemptig to solve s6 4ss 0 (6s6 0) for s (M) s () OP 5 Total [5 marks]

9 9 N4/5/MATHL/HP/ENG/TZ0/XX/M 4. (a) (i) use of P( A B) P( A) P( B) (M) P( AB) (ii) use of P( A B) P( A) P( B) P( A)P( B) (M) P( AB) (b) P( A B) P( A B) P( B) P( A B ) is a maximum whe P( A B) P( A) P( A B ) is a miimum whe P( A B) 0 0P( A B) 0.4 Note: for each edpoit ad for the correct iequalities. [4 marks] [3 marks] Total [7 marks] 5. use of the quotiet rule or the product rule M 3 t t t 6 t C () t 4t or 3 t 3 t 3 t 3 t Note: Award for a correct umerator ad for a correct deomiator i the quotiet rule, ad for each correct term i the product rule. attemptig to solve C() t 0for t (M) t 3 (miutes) 3 C 3 mg l or equivalet. 3 Total [6 marks]

10 0 N4/5/MATHL/HP/ENG/TZ0/XX/M 6. du dx x dx ( u )du Note: Award the for ay correct relatioship betwee dx ad du. x ( u ) dx du x u Note: Award the M for a attempt at substitutio resultig i a itegral oly ivolvig u. (M) u du u () u 4ul u( C) x x 3 l x ( C ) Note: Award the for a correct expressio i x, but ot ecessarily fully expaded/simplified. Total [6 marks] 7. (a) p(3) f(3) g(3) g(3) f(3) (M) Note: Award M if the derivative is i terms of x or [ marks] h( x) g f( x) f( x) (M)() (b) h() g() f() [4 marks] Total [6 marks]

11 N4/5/MATHL/HP/ENG/TZ0/XX/M 8. let P( ) be the propositio that ( )! (!), cosider P(): ad!! so P() is true R assume P( k ) is true ie ( k)! (!) k k, k M Note: Do ot award M for statemets such as let k. cosider P( k ) : k k k k ( )! ( )( )( )! M ( k )! (k )(k )( k!) k ( k )(k )( k!) k k ( k )( k )( k!) sice k k k k ( )! R P( k ) is true wheever P ( k ) is true ad P () is true, so P ( ) is true for R Note: To obtai the fial R, four of the previous marks must have bee awarded. Total [7 marks]

12 N4/5/MATHL/HP/ENG/TZ0/XX/M 9. (a) t correct for [, ] t correct for [, 3] [ marks] (b) EITHER let q be the lower quartile ad let q 3 be the upper quartile let d q q 3 ad so IQR d by symmetry use of area formulae to obtai d 4 (or equivalet) M d or the value of at least oe q. OR let q be the lower quartile q cosider ( t) dt M 4 obtai q THEN IQR Note: Oly accept this fial aswer for the. [4 marks] Total [6 marks]

13 3 N4/5/MATHL/HP/ENG/TZ0/XX/M 0. (a) use of the additio priciple with 3 terms (M) to obtai 4 C C3 C3 ( 40 0) umber of possible selectios is 34 [3 marks] (b) EITHER recogitio of three cases: ( odd ad eve or odd ad 3 eve or 0 odd ad 4 eve) (M) 5 C 4 C 5 C 4 C 5 C 4 C ( 60 0 ) (M) OR recogitio to subtract the sum of 4 odd ad 3 odd ad eve from the total (M) C C C C ( 65 40) (M) THEN umber of possible selectios is 8 [4 marks] Total [7 marks]

14 4 N4/5/MATHL/HP/ENG/TZ0/XX/M. (a) (i) x 3 e y SECTION B M Note: The M is for switchig variables ad ca be awarded at ay stage. Further marks do ot rely o this mark beig awarded. takig the atural logarithm of both sides ad attemptig to traspose f ( x) (lx ) 3 M (ii) x or equivalet, for example x 0. [4 marks] (b) l x (l x) l x l x (or equivalet) M l x (or equivalet) x e coordiates of P are e, [5 marks] (c) coordiates of Q are (, 0) see aywhere dy dx x M at Q, d y dx AG [3 marks] cotiued

15 5 N4/5/MATHL/HP/ENG/TZ0/XX/M Questio cotiued (d) let the required area be A e A xdx l xdx e M Note: The M is for a differece of itegrals. Codoe absece of limits here. attemptig to use itegratio by parts to fid lxdx (M) e x x [ xl xx] e Note: Award for x x ad for xl x x. Note: The secod M ad secod are idepedet of the first M ad the first. e e e e [5 marks] (e) (i) METHOD cosider for example h( x) x lx h() 0 ad h( x) x () as h( x) 0 for x, the h( x) 0 for x R as h( x) 0 for 0 x, the h( x) 0 for 0 x R so g( x) x, x METHOD g( x) x g( x) 0 (cocave dow) for x R the graph of y g( x) is below its taget ( y x at x ) R so g( x) x, x Note: The reasoig may be supported by draw graphical argumets. AG AG cotiued

16 6 N4/5/MATHL/HP/ENG/TZ0/XX/M Questio cotiued METHOD 3 clear correct graphs of y x ad l x for x 0 statemet to the effect that the graph of l x is below the graph of its taget at x RAG (ii) replacig x by x to obtai x l x x x x l x x x x l x x x x so g( x), x x M () AG [6 marks] Total [3 marks]

17 7 N4/5/MATHL/HP/ENG/TZ0/XX/M. (a) (i) AM AC (M) ( ca ) (ii) BM BAAM M ab ( ca) BM ab c AG [4 marks] (b) (i) (ii) RA BA 3 ( ab ) 3 RT RS 3 RA AS 3 (M) ( ) ( ) ( ab) ( ca) RT AG [5 marks] (c) BT BRRT BA RT 3 (M) 4 a b a b c BT 9 poit B is commo to BT ad BM 8 ad BT BM 9 RR so T lies o [BM] AG [5 marks] Total [4 marks]

18 8 N4/5/MATHL/HP/ENG/TZ0/XX/M 3. (a) (i) METHOD si si ( i ta ) ( i ta ) ( i ) ( i ) cos cos M cos isi cos isi cos cos by de Moivre s theorem cos i si cos isi = cos cos recogitio that cos i si is the complex cojugate of cos i si use of the fact that the operatio of complex cojugatio commutes with the operatio of raisig to a iteger power: cos i si cos isi = cos cos ( i ta ) ( i ta ) cos cos METHOD (M) (R) ( i ta ) ( i ta ) ( i ta ) ( i ta ( )) (M) (cos i si ) cos ( ) i si ( ) = cos cos Note: Award M for covertig to cosie ad sie terms. AG M use of de Moivre s theorem (M) cos isi cos ( ) isi ( ) cos cos as cos( ) cos ad si ( ) si RAG cos cotiued

19 9 N4/5/MATHL/HP/ENG/TZ0/XX/M Questio 3 cotiued (ii) 3π 4 4 cos 4 3π 3π 8 ita ita π cos 8 () 3π cos 4 3π cos 8 3π 0 as cos 0 R Note: The above workig could ivolve theta ad the solutio of cos (4 ) 0. so 3π ita 8 is a root of the equatio AG (iii) either 3π ita or 8 π ita or 8 π ita 8 Note: Accept 5π ita 8 or 7π ita 8. Accept i or i or i. [0 marks] (b) (i) π ta π ta 8 4 π ta 8 (M) π π ta ta π let t ta 8 attemptig to solve t t 0 for t M t π is a first quadrat agle ad ta is positive i this quadrat, so 8 π ta 0 R 8 π AG ta 8 cotiued

20 0 N4/5/MATHL/HP/ENG/TZ0/XX/M Questio 3 cotiued (ii) cos 4x cos x x x cos M x 4 4cos 4cos x x 4 8cos 8cos AG Note: Accept equivalet complex umber derivatio. (iii) cos4x 8cos x8cos x d dx cos x cos x π π x 0 0 use of π 8 8cos x 8 sec 0 d Note: The M is for a itegrad ivolvig o fractios. x x M cos x (cos x ) M π 8 4cosx 4 sec 0 d x x 4six 8x ta x π π (or equivalet) [3 marks] Total [3 marks]

Markscheme May 2015 Calculus Higher level Paper 3

Markscheme May 2015 Calculus Higher level Paper 3 M5/5/MATHL/HP3/ENG/TZ0/SE/M Markscheme May 05 Calculus Higher level Paper 3 pages M5/5/MATHL/HP3/ENG/TZ0/SE/M This markscheme is the property of the Iteratioal Baccalaureate ad must ot be reproduced or