Markscheme May 2015 Calculus Higher level Paper 3

Size: px

Start display at page:

Download "Markscheme May 2015 Calculus Higher level Paper 3"

Cynthia Austin
5 years ago
Views:

1 M5/5/MATHL/HP3/ENG/TZ0/SE/M Markscheme May 05 Calculus Higher level Paper 3 pages

2 M5/5/MATHL/HP3/ENG/TZ0/SE/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 M5/5/MATHL/HP3/ENG/TZ0/SE/M Istructios to Examiers Abbreviatios M (M) A (A) R N AG Marks awarded for attemptig to use a valid 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 May 05. 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 correct workig. However, if further workig idicates a lack of mathematical uderstadig do ot award the fial. A exceptio to this may be i umerical aswers, where a correct exact value is followed by a icorrect decimal. However, if the icorrect decimal is carried through to a subsequet part, ad correct FT workig show, award FT marks as appropriate but do ot award the fial i that part.

4 4 M5/5/MATHL/HP3/ENG/TZ0/SE/M Examples Correct aswer see Further workig see Actio Award the fial 8 (icorrect decimal value) (igore the further workig). si 4 x si x Do ot award the fial 4 3. log a log b log ( a b) Do ot award the fial 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 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).

5 5 M5/5/MATHL/HP3/ENG/TZ0/SE/M 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. 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: f( x) cos(5x3) 5 0cos(5x 3) Award for cos(5x 3) 5, eve if 0cos(5x 3) is ot see. 0 Accuracy of Aswers 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.

6 6 M5/5/MATHL/HP3/ENG/TZ0/SE/M Calculators A GDC is required for paper 3, but calculators with symbolic maipulatio features (for example, TI-89) are ot allowed. Calculator otatio The Mathematics HL guide says: Studets must always use correct mathematical otatio, ot calculator otatio. Do ot accept fial aswers writte usig calculator otatio. However, do ot pealize the use of calculator otatio i the workig. 3 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.

7 7 M5/5/MATHL/HP3/ENG/TZ0/SE/M. f (0) 0 x x f ( x) e cosxe six M f (0) 0 (M) x f ( x) e six f (0) 0 (3) ( ) e x x f x si xe cos x f (3) (0) the first o-zero term is 3 x 3! 3 x 3 Note: Award o marks for usig kow series. [7 marks]. (a) METHOD dy f ( x)d x f ( x) dx x x MM dy x y f( x), x 0 dx AG Note: M for use of product rule, M for use of the fudametal theorem of calculus, for all correct. METHOD dy x y dx f( x) d( xy) dx f ( x) (M) xy f( x)dx M y f( x)dx x AG [3 marks] cotiued

8 8 M5/5/MATHL/HP3/ENG/TZ0/SE/M Questio cotiued (b) y x c x Note: for correct expressio apart from the costat, for icludig the costat i the correct positio. attempt to use the boudary coditio M c 4 y x 4 x [5 marks] Note: Codoe use of itegratig factor. Total [8 marks] 3. (a) METHOD (0 ), (for 3) l ( ) coverges by the compariso test ( coverges implies) coverges R (l ) Note: Metio of usig the compariso test may have come earlier. Oly award R if previous s have bee awarded. METHOD l lim lim 0 l coverges by the limit compariso test (if the limit is 0 ad the series represeted by the deomiator coverges, the so does the series represeted by the cotiued

9 9 M5/5/MATHL/HP3/ENG/TZ0/SE/M Questio 3 cotiued umerator, hece) coverges R (l ) Note: Metio of usig the limit compariso test may come earlier. Do ot award the R if icorrect justificatios are give, for example the series coverge or diverge together. Oly award R if previous s have bee awarded. [3 marks] (b) (i) EITHER l ( ) l l OR l ( ) l l ( ) l l( ) l( ) l ( ) THEN l ( ) AG (ii) l ( ) attempt to use the ratio test ( ) l( ) M as () l( ) l( ) M l ( ) l ( ) l (as ) () l ( ) (as ) hece ratio test is icoclusive ( ) l( ) R Note: A lik with the limit equallig ad the result beig icoclusive eeds to be give for R. [6 marks] (c) (i) cosider f ( x) xl x (for x ) f ( x ) is cotiuous ad positive ad is (mootoically) decreasig Note: If a cadidate uses rather tha x, award as follows is positive ad decreasig l is cotiuous for, > (oly award this mark if the l domai has bee explicitly chaged). cotiued

10 0 M5/5/MATHL/HP3/ENG/TZ0/SE/M Questio 3 cotiued (ii) R cosider d x M x l x l (l x) R (M) as R R hece series diverges Note: Codoe the use of i place of R. Note: If the lower limit is ot equal to, but the expressio is itegrated correctly award M0MR0A0. [8 marks] Total [7 marks] 4. (a) x x lim lim x x x e x e lim 0 x e x M M Note: Award M for a attempt at differetiatig for a secod time. [4 marks] (b) attempt to itegrate by parts M x x x x e dx x e xe dx () x x x x e xe e dx () x x x x e xe e ( c) R x e x dx R e R Re R e R M 0 x lim R x R 0 e dx Note: Award M for cosideratio of the limit ad for correct limitig value. M hece the improper itegral coverges AG Note: Do ot award the fial four marks to cadidates who do ot cosider R. [8 marks] Total [ marks]

11 M5/5/MATHL/HP3/ENG/TZ0/SE/M 5. (a) (i) f ( x) 3x 6x gradiet of chord 3c 6c 3 3 c (.5, 0.55) 3 Note: Accept ay aswers that roud to the correct sf aswers (., 0.5). (ii) award for correct shape ad clear idicatio of correct domai, for chord (from x = 3 to x = ) ad for two tagets draw at their values of c [7 marks] (b) (i) METHOD (if a theorem is true for the iterval [a, b], it is also true for ay iterval [ x, x ] which belogs to [a, b]) suppose x, x [ a, b] M f( x) f( x) by the MVT, there exists c such that f() c 0 x x M hece f ( x) f( x) R x, x are arbitrarily chose, f x as is costat o a, b Note: If the above is expressed i terms of a ad b award M0MA0R0. METHOD (if a theorem is true for the iterval [a, b], it is also true for ay iterval [ x, x ] which belogs to [a, b]) suppose x [ a, b] M cotiued

12 M5/5/MATHL/HP3/ENG/TZ0/SE/M Questio 5 cotiued by the MVT, there exists c such that hece f (x) f (a) costat f( x) f( a) f() c 0 x a M R M (ii) attempt to differetiate ( x) arccos x arccos x 4x x x 4x 0 4 x 4x 4x Note: Oly award for 0 if a correct attempt to simplify the deomiator is also see. f ( x) f (0) 0 AG Note: This is ot depedet o previous marks. Note: Allow ay value of x [0, ]. [9 marks] Total [6 marks]

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

N14/5/MATHL/HP1/ENG/TZ0/XX/M MARKSCHEME. November 2014 MATHEMATICS. Higher Level. Paper pages N4/5/MATHL/HP/ENG/TZ0/XX/M MARKSCHEME November 04 MATHEMATICS Higher Level Paper 0 pages N4/5/MATHL/HP/ENG/TZ0/XX/M This markscheme is the property of the Iteratioal Baccalaureate ad must ot be reproduced