Markscheme May 2016 Mathematics Standard level Paper 1

Transcription

1 M6/5/MATME/SP/ENG/TZ/XX/M Mrkscheme My 06 Mthemtics Stndrd level Pper 7 pges

2 M6/5/MATME/SP/ENG/TZ/XX/M This mrkscheme is the property of the Interntionl Bcclurete nd must not be reproduced or distributed to ny other person without the uthoriztion of the IB Assessment Centre.

3 M6/5/MATME/SP/ENG/TZ/XX/M Instructions to Exminers Abbrevitions M (M) A (A) R N AG Mrks wrded for ttempting to use vlid Method; working must be seen. Mrks wrded for vlid Method; my be implied by correct subsequent working. Mrks wrded for n Answer or for Accurcy; often dependent on preceding M mrks. Mrks wrded for n Answer or for Accurcy; my be implied by correct subsequent working. Mrks wrded for cler Resoning. Mrks wrded for correct nswers if no working shown. Answer given in the question nd so no mrks re wrded. Using the mrkscheme Generl Mrk ccording to RM ssessor instructions. Method nd Answer/Accurcy mrks Do not utomticlly wrd full mrks for correct nswer; ll working must be checked, nd mrks wrded ccording to the mrkscheme. It is generlly not possible to wrd M0 followed by A, s A mrk(s) depend on the preceding M mrk(s), if ny. An exception to this rule is when work for M is missing, s opposed to incorrect (see point 4). Where M nd A mrks re noted on the sme line, MA, this usully mens M for n ttempt to use n pproprite method ( substitution into formul) nd A for using the correct vlues. Where there re two or more A mrks on the sme line, they my be wrded independently; so if the first vlue is incorrect, but the next two re correct, wrd A0AA. Where the mrkscheme specifies (M), N, etc., do not split the mrks, unless there is note. Most M mrks re for vlid method, ie method which cn led to the nswer: it must indicte some form of progress towrds the nswer. Once correct nswer to question or prt-question is seen, ignore further. However, if further working indictes lck of mthemticl understnding do not wrd finl A. N mrks If no working shown, wrd N mrks for correct nswers this includes cceptble nswers (see ccurcy booklet). In this cse, ignore mrk brekdown (M, A, R). Where student only shows finl incorrect nswer with no working, even if tht nswer is correct intermedite nswer, wrd N0. Do not wrd mixture of N nd other mrks. There my be fewer N mrks vilble thn the totl of M, A nd R mrks; this is deliberte s it penlizes cndidtes for not following the instruction to show their working. There my not be direct reltionship between the N mrks nd the implied mrks. There re times when ll the mrks re implied, but the N mrks re not the full mrks: this indictes tht we wnt to see some of the working, without specifying wht.

4 4 M6/5/MATME/SP/ENG/TZ/XX/M For consistency within the mrkscheme, N mrks re noted for every prt, even when these mtch the mrk brekdown. If cndidte hs in, which somehow results in correct nswer, do not wrd the N mrks for this correct nswer. However, if the cndidte hs indicted (usully by crossing out) tht the working is to be ignored, wrd the N mrks for the correct nswer. 4 Implied nd must be seen mrks Implied mrks pper in brckets. Implied mrks cn only be wrded if the work is seen or if implied in subsequent working ( correct finl nswer does not necessrily men tht the implied mrks re ll wrded). There re questions where some working is required, but s it is ccepted tht not everyone will write the sme steps, ll the mrks re implied, but the N mrks re not the full mrks for the question. Normlly the correct work is seen in the next line. Where there is n followed by A for ech correct nswer, if no working shown, one correct nswer is sufficient evidence to wrd the. Must be seen mrks pper without brckets M. Must be seen mrks cn only be wrded if the work is seen. If must be seen mrk is not wrded becuse work is missing (s opposed to M0 or A0 for incorrect work) ll subsequent mrks my be wrded if pproprite. 5 Follow through mrks (only pplied fter n error is mde) Follow through (FT) mrks re wrded where n incorrect nswer (finl or intermedite) from one prt of question is used correctly in subsequent prt(s) or subprt(s). Usully, to wrd FT mrks, there must be working present nd not just finl nswer bsed on n incorrect nswer to previous prt. However, if the only mrks wrded in subprt re for the finl nswer, then FT mrks should be wrded if pproprite. Exminers re expected to check student work in order to wrd FT mrks where pproprite. Within question prt, once n error is mde, no further A mrks cn be wrded for work which uses the error, but M nd R mrks my be wrded if pproprite. (However, s noted bove, if n A mrk is not wrded becuse work is missing, ll subsequent mrks my be wrded if pproprite). Exceptions to this rule will be explicitly noted on the mrkscheme. If the question becomes much simpler becuse of n error then use discretion to wrd fewer FT mrks. If the error leds to n inpproprite vlue ( probbility greter thn, use of r > for the sum of n infinite GP, sin.5, non inter vlue where inter required), do not wrd the mrk(s) for the finl nswer(s). The mrkscheme my use the word their in description, to indicte tht cndidtes my be using n incorrect vlue. If cndidte mkes n error in one prt, but gets the correct nswer(s) to subsequent prt(s), wrd mrks s pproprite, unless the question sys hence. It is often possible to use different pproch in subsequent prts tht does not depend on the nswer to previous prts. In show tht question, if n error in previous subprt leds to not showing the required nswer, do not wrd the finl A. Note tht if the error occurs within the sme subprt, the FT rules my result in further loss of mrks.

5 5 M6/5/MATME/SP/ENG/TZ/XX/M 6 Mis-red If cndidte incorrectly copies informtion from the question, this is mis-red (MR). A cndidte should be penlized only once for prticulr mis-red. Use the MR stmp to indicte tht this is misred. Do not wrd the first mrk in the question, even if this is n M mrk, but wrd ll others (if pproprite) so tht the cndidte only loses one mrk for the misred. If the question becomes much simpler becuse of the MR, then use discretion to wrd fewer mrks. If the MR leds to n inpproprite vlue ( probbility greter thn, use of r > for the sum of n infinite GP, sin.5, non inter vlue where inter required), do not wrd the mrk(s) for the finl nswer(s). Miscopying of cndidtes own work does not constitute misred, it is n error. 7 Discretionry mrks (d) An exminer uses discretion to wrd mrk on the rre occsions when the mrkscheme does not cover the work seen. In such cses the nnottion DM should be used nd brief note written next to the mrk explining this decision. 8 Alterntive methods Cndidtes will sometimes use methods other thn those in the mrkscheme. Unless the question specifies method, other correct methods should be mrked in line with the mrkscheme. If in doubt, contct your tem leder for dvice. Alterntive methods for complete prts re indicted by METHOD, METHOD, etc. Alterntive solutions for prts of questions re indicted by EITHER... OR. Where possible, lignment will lso be used to ssist exminers in identifying where these lterntives strt nd finish. 9 Alterntive forms Unless the question specifies otherwise, ccept equivlent forms. As this is n interntionl exmintion, ccept ll lterntive forms of nottion. In the mrkscheme, equivlent numericl nd lgebric forms will generlly be written in brckets immeditely following the nswer. In the mrkscheme, simplified nswers, (which cndidtes often do not write in exmintions), will generlly pper in brckets. Mrks should be wrded for either the form preceding the brcket or the form in brckets (if it is seen). 0 Clcultors No clcultor is llowed. The use of ny clcultor on pper is mlprctice, nd will result in no grde wrded. If you see work tht suggests cndidte hs used ny clcultor, plese follow the procedures for mlprctice. Exmples: finding n ngle, given trig rtio of Style The mrkscheme ims to present nswers using good communiction, if the question sks to find the vlue of k, the mrkscheme will sy k =, but the mrks will be for the correct vlue there is usully no need for the k =. In these cses, it is lso usully cceptble to hve nother vrible, s long s there is no mbiguity in the question, if the question sks to find the vlue of p nd of q, then the student nswer needs to be cler. Generlly, the only sitution

6 6 M6/5/MATME/SP/ENG/TZ/XX/M where the full nswer is required is in question which sks for equtions in this cse the mrkscheme will sy must be n eqution. The mrkscheme often uses words to describe wht the mrks re for, followed by exmples, using the nottion. These exmples re not exhustive, nd exminers should check wht cndidtes hve written, to see if they stisfy the description. Where these mrks re M mrks, the exmples my include ones using poor nottion, to indicte wht is cceptble. A vlid method is one which will llow cndidte to proceed to the next step if qudrtic function is given in fctorised form, nd the question sks for the zeroes, then multiplying the fctors does not necessrily help to find the zeros, nd would not on its own count s vlid method. Cndidte work If cndidte hs drwn line through work on their exmintion script, or in some other wy crossed out their work, do not wrd ny mrks for tht work. Cndidtes re ment to write their nswers to Section A on the question pper (QP), nd Section B on nswer booklets. Sometimes, they need more room for Section A, nd use the booklet (nd often comment to this effect on the QP), or write outside the box. Tht is fine, nd this work should be mrked. The instructions tell cndidtes not to write on Section B of the QP. Thus they my well hve done some rough work here which they ssume will be ignored. If they hve solutions on the nswer booklets, there is no need to look t the QP. However, if there re whole questions or whole prt solutions missing on nswer booklets, plese check to mke sure tht they re not on the QP, nd if they re, mrk those whole questions or whole prt solutions tht hve not been written on nswer booklets.. Digrms The notes on how to llocte mrks for sketches usully refer to pssing through prticulr points or hving certin fetures. These mrks cn only be wrded if the sketch is pproximtely the correct shpe. All vlues given will be n pproximte guide to where these points/fetures occur. In some questions, the first A is for the shpe, in others, the mrks re only for the points nd/or fetures. In both cses, unless the shpe is pproximtely correct, no mrks cn be wrded (unless otherwise stted). However, if the grph is bsed on previous clcultions, FT mrks should be wrded if pproprite. 4. Accurcy of Answers If the level of ccurcy is specified in the question, mrk will be llocted for giving the finl nswer to the required ccurcy. When this is not specified in the question, ll numericl nswers should be given exctly or correct to three significnt figures. Do not ccept unfinished numericl finl nswers such s /0. (unless otherwise stted). As rule, numericl nswers with more thn one prt (such s frctions) should be given using inters ( 6/8). Clcultions which led to inters should be completed, with the exception of frctions which re not whole numbers. Intermedite vlues do not need to be given to the correct three significnt figures. But, if cndidtes work with rounded vlues, this could led to n incorrect nswer, in which cse wrd A0 for the finl nswer. Where numericl nswers re required s the finl nswer to prt of question in the mrkscheme, the mrkscheme will show truncted 6 sf vlue the exct vlue if pplicble, the correct sf nswer Units will pper in brckets t the end.

7 7 M6/5/MATME/SP/ENG/TZ/XX/M Section A. () h, k AA N [ mrks] (b), b 4 (or 4, b ) AA N [ mrks] (c) ttempt to substitute x 0 into their f (0 ), (0)(0 4) y 8 A N [ mrks] Totl [6 mrks]. () correct pproch 60 0 men 6 A N [ mrks] (b) (i) new men 4 A N (ii) vlid pproch vrince (4), 6, new stndrd devition 4 new vrince 48 A N [ mrks] Totl [5 mrks]. () correct pproch ln5 ln 5 ln y x A N [ mrks] (b) recognizing fctors of 45 (my be seen in log expnsion) ln(9 5), 5, log log5 correct ppliction of log( b) log log b ln9 ln5, lnln ln5, ln ln5 ln ln5, x x y ln 45 x y A N [4 mrks] Totl [6 mrks]

8 8 M6/5/MATME/SP/ENG/TZ/XX/M 4. METHOD vlid pproch 6 r x, ( x ) r 6, correct eqution in terms of x only 6 x, ( x)( x ) 6, x 6 x x 4, x x 4 ( x ) r x 6 x x 6 A vlid ttempt to solve their qudrtic eqution fctorizing, formul, completing the squre evidence of 69 ( x7)( x 6), x 7, x 6 A N4 METHOD (finding r first) vlid pproch 6 r x, 6 r x, ( x ) r x correct eqution in terms of r only 6 6r r, 6r 6r r, evidence of rr, r, 6r 5r 6 0 A r A substituting their vlues of r to find x x 6, x 6 x 7, x 6 A N4

9 9 M6/5/MATME/SP/ENG/TZ/XX/M 5. () METHOD correct substitution into formul for re of tringle (6) sin B, 6 sin B, (6) sin B 6 sin B, sin B (6) sin B (0 ) 6 ˆ 5 ABC (50 ) 6 A N METHOD (using height of tringle ABC by drwing perpendiculr sment from C to AD) correct substitution into formul for re of tringle ( h ), h h height of tringle is CBD ˆ (0 ) 6 ˆ 5 ABC (50 ) A N 6 (b) recognizing supplementry ngle CBD ˆ, sector ABC ˆ 6 correct substitution into formul for re of sector 0 6, A [5 mrks] re (cm ) A N [ mrks] Totl [8 mrks]

10 0 M6/5/MATME/SP/ENG/TZ/XX/M 6. () ttempt to form composite in ny order ( ) f g x, cos6x x 6cos x cos x correct ppliction of Pythgoren identity (do not ccept sin x cos x, 6cos xsin x, 6cos xsin x sin xcos x ) vlid pproch (do not ccept sin x cos x sin x) (cos xsin x ) h( x) sinx A N [5 mrks] (b) vlid pproch mplitude, sketch with mx nd min y-vlues lbelled, y correct rnge A N y, [, ], from to Note: Do not wrd A for y or for between nd. [ mrks] Totl [7 mrks]

11 M6/5/MATME/SP/ENG/TZ/XX/M 7. correct sclr product m n setting up their sclr product equl to 0 (seen nywhere) uv 0, (0) ( m) ( n) 0, m n correct interprettion of unit vector 0 m n, m n vlid ttempt to solve their equtions (must be in one vrible) ( n) n, n n 0, m ( m), m m 0 M n, m,, m n both correct pirs A N m nd n, m nd n, m 0.5. nd n 0.5., m nd n A Note: Awrd A0 for m, n indicte the correct pirs., or ny other nswer tht does not clerly [7 mrks]

12 M6/5/MATME/SP/ENG/TZ/XX/M Section B 8. () (i) p A N (ii) vlid pproch ( 0 ), 8 q 4 A N (iii) r 8, s 6 AA N [5 mrks] (b) (i) 4 7 A N (ii) vlid pproch 8 6, r s 4 A N [4 mrks] (c) (i) AAA N Note: Awrd A for ech correct bold nswer. (ii) 0 A N [4 mrks] Totl [ mrks]

13 M6/5/MATME/SP/ENG/TZ/XX/M 9. () correct substitution into the formul for volume A 6 y x x vlid pproch to eliminte y (my be seen in formul/substitution) 6 6 y, xy x x correct expression for surfce re xy xy xy x x, re xy x M A correct expression in terms of x only A 6 x x , x x, x x x x x x 08 A( x) x x AG N0 [4 mrks] 08 (b) A( x) 4x, x 4x 08 x AA N Note: Awrd A for ech term. [ mrks] (c) recognizing tht minimum is when A( x) 0 correct eqution x 0, 4x x x correct simplifiction 08 4x 0, x 7 4x 08 height (m) (ccept x ) A N [5 mrks] continued

14 4 M6/5/MATME/SP/ENG/TZ/XX/M Question 9 continued (d) ttempt to find re using their height 08 (), 9 9 minimum surfce re 54m (my be seen in prt (c)) A ttempt to find the number of tins 54 0, (tins) $0 A N [5 mrks] Totl [6 mrks]

15 5 M6/5/MATME/SP/ENG/TZ/XX/M 0. () (i) recognizing the need to find the grdient when x 0 (seen nywhere) R f (0) correct substitution 4(0) f (0) 0 f(0) correct eqution with grdient (do not ccept equtions of the form L x ) A N y x, y b ( x ), y x b (ii) METHOD ttempt to substitute x into their eqution of L y b A N METHOD equting grdients b b A N [6 mrks] continued

16 6 M6/5/MATME/SP/ENG/TZ/XX/M Question 0 continued (b) METHOD recognizing tht re f ( x )d x (seen nywhere) R 0 vlid pproch using substitution or inspection x udx, u x, du xdx x d d x x x u u, u udu f ( x)dx x c substituting limits nd subtrcting AR ( ) ( 0), ( ) AR METHOD AG N0 recognizing tht re f ( x )d x (seen nywhere) R 0 vlid pproch using substitution or inspection x udx, u x, du xdx x x x dx udu, A u udu new limits for u (even if intrtion is incorrect) 0 u u 0 nd u, u d u, 0 substituting limits nd subtrcting A AR 0, ( ) AR AG N0 [6 mrks] continued

17 7 M6/5/MATME/SP/ENG/TZ/XX/M Question 0 continued (c) METHOD vlid pproch to find re of tringle OQ PQ, b correct substitution into formul for A T, vlid ttempt to find k (must be in terms of ) k, k A T (seen nywhere) k A N METHOD vlid pproch to find re of tringle 0 x dx x, 0 vlid ttempt to find k (must be in terms of ) k, k k A N [4 mrks] Totl [6 mrks]