DRAFT - Miami Beach Senior High School

MATHEMATICSRStandard LevelDPaper 111 pages

–2–SPEC/5/MATME/SP1/ENG/TZ0/XX/MInstructions to ExaminersAbbreviationsMarks awarded for attempting to use a correct Method.AMarks awarded for an Answer or for Accuracy; often dependent on preceding M marks.RMarks awarded for clear Reasoning.NMarks awarded for correct answers if no working shown.AGAnswer given in the question and so no marks are awarded.AFTMUsing the markscheme1GeneralAward marks using the annotations as noted in the markscheme eg M1, A2.2Method and Answer/Accuracy marksR Do not automatically award full marks for a correct answer; all working must be checked,and marks awarded according to the markscheme. It is generally not possible to award M0 followed by A1, as A mark(s) depend on the precedingM mark(s), if any. An exception to this rule is when work for M1 is missing, as opposed toincorrect (see point 4). Where M and A marks are noted on the same line, e.g. M1A1, this usually means M1 for anattempt to use an appropriate method (e.g. substitution into a formula) and A1 for using thecorrect values. Where there are two or more A marks on the same line, they may be awarded independently; so ifthe first value is incorrect, but the next two are correct, award A0A1A1. Where the markscheme specifies M2, N3, etc., do not split the marks, unless there is a note. Once a correct answer to a question or part-question is seen, ignore further working.N marksD3If no working shown, award N marks for correct answers. In this case, ignore mark breakdown(M, A, R). Do not award a mixture of N and other marks. There may be fewer N marks available than the total of M, A and R marks; this is deliberate as itpenalizes candidates for not following the instruction to show their working. If a candidate has incorrect working, which somehow results in a correct answer, do not award theN marks for this correct answer. However, if the candidate has indicated (usually by crossing out)that the working is to be ignored, award the N marks for the correct answer.

–3–4SPEC/5/MATME/SP1/ENG/TZ0/XX/MFollow through marks (only applied after an error is made)Follow through (FT) marks are awarded where an incorrect answer from one part of a question isused correctly in subsequent part(s) or subpart(s). Usually, to award FT marks, there must beworking present and not just a final answer based on an incorrect answer to a previous part.However, if the only marks awarded in a subpart are for the answer (i.e. there is no workingexpected), then FT marks should be awarded if appropriate.5AFT Within a question part, once an error is made, no further A marks can be awarded for work whichuses the error, but M marks may be awarded if appropriate. If the question becomes much simpler because of an error then use discretion to award fewerFT marks. If the error leads to an inappropriate value (e.g. probability greater than 1, use of r 1 for the sumof an infinite GP, sin 1.5 , non integer value where integer required), do not award the mark(s)for the final answer(s). The markscheme may use the word “their” in a description, to indicate that candidates may beusing an incorrect value. Exceptions to this rule will be explicitly noted on the markscheme. If a candidate makes an error in one part, but gets the correct answer(s) to subsequent part(s),award marks as appropriate, unless the question says hence. It is often possible to use a differentapproach in subsequent parts that does not depend on the answer to previous parts.Mis-readIf a candidate incorrectly copies information from the question, this is a mis-read (MR). Apply a MRpenalty of 1 mark to that questionR If the question becomes much simpler because of the MR, then use discretion to awardfewer marks. If the MR leads to an inappropriate value (e.g. probability greater than 1, use of r 1 for the sumof an infinite GP, sin 1.5 , non integer value where integer required), do not award the mark(s)for the final answer(s). Miscopying of candidates’ own work does not constitute a misread, it is an error.Alternative methodsCandidates will sometimes use methods other than those in the markscheme. Unless the questionspecifies a method, other correct methods should be marked in line with the markschemeD6 Alternative methods for complete questions are indicated by METHOD 1, METHOD 2, etc. Alternative solutions for part-questions are indicated by EITHER . . . OR.

–4–7SPEC/5/MATME/SP1/ENG/TZ0/XX/MAlternative formsUnless the question specifies otherwise, accept equivalent forms. As this is an international examination, accept all alternative forms of notation. In the markscheme, equivalent numerical and algebraic forms will generally be written inbrackets immediately following the answer. In the markscheme, simplified answers, (which candidates often do not write in examinations),will generally appear in brackets. Marks should be awarded for either the form preceding thebracket or the form in brackets (if it is seen).Accuracy of AnswersAFT8If the level of accuracy is specified in the question, a mark will be allocated for giving the answer tothe required accuracy.There are 2 types of accuracy errors, and the final answer mark should not be awarded if these errorsoccur. Rounding errors: only applies to final answers not to intermediate steps. Level of accuracy: when this is not specified in the question the general rule applies to finalanswers: unless otherwise stated in the question all numerical answers must be given exactly orcorrect to three significant figures.9CalculatorsRNo calculator is allowed. The use of any calculator on paper 1 is malpractice, and will result in nograde awarded.StyleThe markscheme aims to present answers using good communication, e.g. if the question asks to findthe value of k, the markscheme will say k 3 , but the marks will be for the correct value 3 – there isusually no need for the “ k ”. In these cases, it is also usually acceptable to have another variable,as long as there is no ambiguity in the question, e.g. if the question asks to find the value of p and of q,then the student answer needs to be clear. Generally, the only situation where the full answer isrequired is in a question which asks for equations – in this case the markscheme will say “must be anequation”.D10The markscheme often uses words to describe what the marks are for, followed by examples, using thee.g. notation. These examples are not exhaustive, and examiners should check what candidates havewritten, to see if they satisfy the description. Where these marks are M marks, the examples mayinclude ones using poor notation, to indicate what is acceptable.

–5–SPEC/5/MATME/SP1/ENG/TZ0/XX/MSECTION A1.(a) 1 AE AD2A1 attempt to find AD M1e.g. AB BD , u v 11 1AE (u v ) u v 22 2A1N2(b)AFT[3 marks] 1ED AE (u v )2A1DC 3vA1 attempt to find EC 1e.g. ED DC, (u v ) 3v2 17 1 EC u v (u 7v ) 22 2 M1A1N2[4 marks]Total [7 marks](a)min value of r is 1 , max value of r is 1R2.C(c)linear, strong negativeD(b)A1A1N2[2 marks]A1N1[1 mark]A1A1N2[2 marks]Total [5 marks]

–6–3.SPEC/5/MATME/SP1/ENG/TZ0/XX/M(a)4 (ms 1 )A1(b)recognising that acceleration is the gradient4 0e.g. a (1.5) 2 0M1a 2 (ms 2 )recognizing area under curvee.g. trapezium, triangles, integrationM1correct substitution61e.g. (3 6) 4, v(t ) dt02A1distance 18 (m)A1N1[2 marks]AFT(c)A1N1[1 mark]N2[3 marks]Total [6 marks]4.(a)new mean is 20 10 30A1N1(ii)new sd is 6A1N1[2 marks](i)new mean is 20 10 200A1N1(ii)METHOD 1A1A1N2R(b)(i)variance is 36new variance is 36 100 3600METHOD 2Dnew sd is 60new variance is 602 3600A1A1N2[3 marks]Total [5 marks]

–7–5.(a)SPEC/5/MATME/SP1/ENG/TZ0/XX/Mattempt to use substitution or inspectiondu exe.g. u 1 e x sodxM1correct workingA1due.g. ln uuln (1 e x ) CN3[3 marks]METHOD 1AFT(b)A1attempt to use substitution or inspectione.g. let u sin 3xdu 3cos3 xdx11 u2uu Cd3 3 2sin 2 3 xxxx Csin3cos3d 6M1A1A1A1N2[4 marks]METHOD 2attempt to use substitution or inspectione.g. let u cos 3 xRdu 3sin 3xdx11 u2 u du C33 2cos 2 3 xxxx Csin3cos3d 6M1A1A1A1N2[4 marks]DMETHOD 3recognizing double anglecorrect working1e.g. sin 6 x2 cos 6 x C61cos 6 x 2 sin 6 xdx 12 C sin 6 xdx M1A1A1A1N2[4 marks]Total [7 marks]

–8–6.(a)recognizing double anglee.g. 3 2sin x cos x , 3sin 2 xa 3, b n 3sin 2 x 3212finding the angle e.g. , 2 x 66sin 2 x N3[3 marks]M1A1AFTA1x 5 12A1N2Note: Award A0 if other values are included.[4 marks]Total [7 marks]7.(a)f ( x) x 2 f ( x) 2 x 3 f ( x) 6 x 4or 1 x22 x36 x4 24 or 5xor Rf (4) ( x) 24 x 5or f ( n ) ( x) ( 1) n n !or ( 1)n n! x ( n 1) n 1xD(b)A1N1A1N1A1N1A1N1[4 marks]A1A1A1N3[3 marks]Total [7 marks]

–9–SPEC/5/MATME/SP1/ENG/TZ0/XX/MSECTION B8.(a)f ( x) 3( x 2 2 x 1) 12A1 3 x 6 x 3 12 3x 2 6 x 92(i)vertex is ( 1, 12)A1A1N2(ii)y 9, or (0, 9)A1N1(iii)evidence of solving f ( x) 0e.g. factorizing, formulaM1correct workingA1e.g. 3( x 3)( x 1) 0, x 6 36 1086x 3, x 1, or ( 3, 0), (1, 0)R(c)N0[2 marks]AFT(b)A1AGA1A1N2[7 marks]A1A1A1N3DNote: Award A1 for a parabola opening upward,A1 for vertex in approximately correct position,A1 for intercepts in approximately correct positions.Scale and labelling not required.(d) p 1 , t 3 12q[3 marks]A1A1A1N3[3 marks]Total [15 marks]

– 10 –9.(a)(i)(ii)number of ways of getting X 6 is 55P ( X 6) 36number of ways of getting X 6 is 21P ( X 6) (iii)21 7 36 12 P ( X 7 X 6) 6 2 21 7 T[6 marks]attempt to find P ( X5 21e.g. 136 366)1036fair game if E (W ) 0 (may be seen anywhere)attempt to substitute into E ( X ) formulaP(X6) M1A1R1M1 5 21 10 e.g. 3 1 k 36 36 36 correct substitution into E (W ) 0A1 5 21 10 e.g. 3 1 k 0 36 36 36 M136 10k36k ( 3.6)10A1DRwork towards solvinge.g. 15 21 10k 0A1N4[8 marks]Total [14 marks]

– 11 –10.(a)f ( x) sin x 3 cos x(i)A1A1at A, f ( x) 0R1correct workinge.g. sin x 3 cos xA1tan x 3 4 x ,3 3attempt to substitute their x into f ( x)A1N2[2 marks]A1AFT(b)SPEC/5/MATME/SP1/ENG/TZ0/XX/MM1 4 4 3 sin e.g. cos 33(ii)correct substitution 13 e.g. 3 22A1correct working that clearly leads to 21 3e.g. 2 2A1q 2AGcorrect calculations to find f ( x) either side of x 4 3N0A1A1Re.g. f ( f f ( x) changes sign from negative to positiveso A is a minimumAG 3R1max when x D(c)correctly substituting x e.g. into f ( x)3N0[10 marks]A1 3 1 3 22max value is 2(d)R1r 2, a 3A1N1[3 marks]A1A1N2[2 marks]Total [17 marks]

SPEC/5/MATME/SP1/ENG/TZ0/XX/M 11 pages MARKSCHEME SPECIMEN MATHEMATICS Standard Level Paper 1 DRAFT