数学历年真题

1、UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONSGCE Advanced Subsidiary Level and GCE Advanced LevelMARK SCHEME for the October/November 2008 question paperThis mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on whi

2、ch Examiners were instructed to award marks. It does notindicate thes of the discussions that took place at an Examiners meeting before marking began.All Examiners are instructed that alternative correct answers and unexpected approaches in candidates scripts must be given marks that fairly reflect

3、the relevant knowledge and skills demonstrated.Mark schemes must be examination.in conjunction with the question papers and the report on theCIE will not enter into discussions or correspondence in connection with these mark schemes.CIE is publishing the mark schemes for the October/November 2008 qu

4、estion papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.9709 MATHEMATICS9709/01Paper 1,um raw mark 75Mark Scheme NotesMarks are of the following three types:MMethod mark, awarded for a valid method applied to the problem. Method ma

5、rks are not lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting th

6、e relevantquties into the formula. Correct application of a formula without theformula being quoted obviously earns the M mark and in some cases an M mark can be implied from a correct answer.AAccuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot

7、be given unless the associated method mark is earned (or implied).BMark for a correct result or statement independent of method marks.When a part of a question has two or more "method" steps, the M marks are generally independent unless the scheme specifically says otherwise; and similarly

8、 when there are several B marks allocated. The notation DM or DB (or dep*) is used to indicate that a particular M or B mark is dependent on an earlier M or B (asterisked) mark in the scheme. When two or more steps are run together by the candidate, the earlier marks are implied and full credit is g

9、iven.The symbol implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A or B marks are given for correct work only. A and B marks are not given for fortuitously "correct" answers or results obtained from incorrect wo

10、rking.Note:B2 or A2 means that the candidate can earn 2 or 0.B2/1/0 means that the candidate can earn anything from 0 to 2.The marks indicated in the scheme may not be subdivided. If there is genuine doubt whether a candidate has earned a mark, allow the candidate the benefit of the doubt. Unless ot

11、herwise indicated, marks once gained cannot subsequently be lost,e.g. wrong working following a correct form of answer is ignored.Wrong or missing units in an answer should not lead to the loss of a mark unless the scheme specifically indicates otherwise.For a numerical answer, allow the A or B mark

12、 if a value is obtained which is correct to 3 s.f., or which would be correct to 3 s.f. if rounded (1 d.p. in the case of an angle). As stated above, an A or B mark is not given if a correct numerical answer arises fortuitously from incorrect working. For Mechanics questions, allow A or B marks for

13、correct answers which arise from taking g equal to 9.8 or 9.81 instead of 10.© UCLES 2008Page 2Mark SchemeSyllabusPaperGCE A/AS LEVEL October/November 2008970901The following abbreviations may be used in a mark scheme or used on the scripts:AEFAny Equivalent Form (of answer is equally acceptabl

14、e)AGAnswer Given on the question paper (so extra checking is needed toensure that theed working leading to the result is valid)BODBenefit of Doubt (allowed when the validity of a solution may not be absolutely clear)CAOCorrect Answer Only (emphasising that no "follow through" from a previo

15、us error is allowed)CWOCorrect Working Only - often written by a fortuitous' answerISWIgnore Subsequent WorkingMRMisPAPremature Approximation (resulting in basically correct work that is insufficiently accurate)SOSSee Other Solution (the candidate makes a better attempt at the same question)SRSp

16、ecial Ruling (ing the mark to be given for a specific wrongsolution, or a case where some standard marking practice is to be varied in the light of a particular circumstance)PenaltiesMR -1A penalty of MR -1 is deducted from A or B marks when the data of aquestion or part question are genuinely misan

17、d the object anddifficulty of the question remain unaltered. In this case all A and B marks then become "follow through " marks. MR is not applied when thecandidate miss his own figures - this is regarded as an error inaccuracy. An MR-2 penalty may be applied in particular cases if agreed

18、at the coordination meeting.PA -1This is deducted from A or B marks in the case of premature approximation. The PA -1 penalty is usually discussed at the meeting.© UCLES 2008Page 3Mark SchemeSyllabusPaperGCE A/AS LEVEL October/November 2008970901© UCLES 20086 x2 1 2 + x 422 x 2 Term in x &

19、#215;15 2 x 15Coeff = 4 or 3.75M1 A1A13Correct term needs powers 4 and 2 For × 152Ignore inclusion of x1 + sin xcos x22+cos x1 + sin xcos x(1 + sin x)2 + cos2 xLHScos x(1 + sin x)=2 + 2sin x cos x(1 + sin x)2=cos xM1M1 A1A14Reasonable algebra. Correct denominator and one term correct in numerat

20、or22Use of sin x + cos x = 1For 2 + 2sinxCo answer was given check preceding linest3 1 term = a = 6th5 term = a + 4d = 12 d = 1.5Sn = n (12 + (n 1)1.5) = 902 n2 + 7n 120 = 0 n = 8B1 M1DM1 A14Correct value of dUse of correct formula with his dCorrect method for soln of quadratic Co (ignore inclusion

21、of n = 15)4 (i)PA = 6i 8j 6kPN = 6i + 2j 6k(ii) PA . PN = 36 16 + 36 = 16cos APN16136 76 APN = 99°B1 B2, 13M1M1 M1A14Co column vectors ok One off for each error(all incorrect sign just one error) Use of x1x2 + y1y2 + z1z2Modulus worked correctly for either one Division of "16" by &quo

22、t;product of moduli"Allow more accuracyPage 4Mark SchemeSyllabusPaperGCE A/AS LEVEL October/November 2008970901© UCLES 20085 x a a bcosx(i) a + b = 10 and a b = 2 a = 4 and b = 6(ii) 4 6cosx = 0 cosx = 2/3 x = 48.2° or 311.8°(iii) (iii)M1 A1Al3M1A1 A13B2,12M1 for either correct.

23、A1 both correct Co(if a b = 10, and a + b = 2, treat as MR 1,(i) a = 4, b = 6, (ii) 131.8, 228.2,(iii) Sketch is mirror image in y = 4)Makes cosx subject and uses inv cos.stFor 1 angle. for 360° "his angle"Must be just one cycle Starts at 2 and ends at 2 Max at 10."V shapes "

24、; lose a mark. Parabolas lose 1 mark.6 (i)Using s = r, 9 = 5 = 1.8 rad.(ii) Uses POT. Halves the angle Uses tangent in POTPT = 5 tan0.9 = 6.30 cm (not 6.31)2(iii) area of sector = ½ × 5 × 1.8 (22.5)Area of POT = ½ × 5 × 6.30 (15.75)Shaded area = 2 triangles sector 9.00

25、(allow 8.95 to 9.05)Ml Al2Ml Ml Al3MlMl Al3Use of formula. coRealises the need to halveUse of tangent even ile not halved co2Use of A = ½r with 1.8 or 0.9.Use of ½bh and (2 triangles sector) co7 (i) 4x + 2r = 8022A = x + r( + 4)x 2160x + 1600 AdA2( + 4)x 160(ii)dx160= 0 when x =or 11.22( +

26、 4)B1 B1M1 A14Ml Al DM1Al4Connection of lengths Connection of areasEliminates r. co but answer given.Attempt at differentiation. co Ignore omission of .Sets to 0 and solves. coPage 5Mark SchemeSyllabusPaperGCE A/AS LEVEL October/November 2008970901© UCLES 200888 y = 5 x , P(2, l)dy8(i) dxx 2m o

27、f tan = 2m of normal = ½1Eqn of normal y 1= 2 (x 2) 2y + x = 48(ii) Sim eqns 2y + x = 4, y = 5 x x2 + 6x 16 = 0 or y2 7y + 6 = 0 (8, 6)(iii) Length =102 + 52125 11.2 (accept125 or 5 5 etc)B1 M1 M1A14M1 DM1 A13M1Al2Correct differentiation Use of m1m2 = 1 Correct method for line Answer givenCompl

28、ete elimination of x or ySoln of quadratic. coCorrect use of Pythagoras For his points.9 y =3x + 122 y1(i) A = x dy = 13dy y 3y 4= 93 9 (allow 0.44 to 0.45)3(3x + 1) 24or 2 3x + 1 dx = 2 3 × =239B1 B1 M1A12(ii) V y d 3x 2= + x from 0 to 1 22Vol of cylinder = × 2 × 1 = 4 Subtraction 1.

29、5 (4.71)M1AlDM1 A14M1A1B1 A14Uses integration wrt yIntegration correct Use of limits 0 to 1. coB1 for everything but ÷3. B1 for ÷3. M1 for "2" and use of limits 0 to 1.2M1 for correct formula used with y andintegration wrt x. (does not need ) A1 integration correct, including .2Or by integration of y = 4coPage 6Mark SchemeSyllabusPaperGCE A/AS LEVEL October/November 2008970901© UCLES 2008dy1 (3x + 1)× 31(iii)22dx33If x = 0, m = 2 . If x = 1, m = 4At x = 0, angle = 56.3° At x = 1, angle = 36