Markscheme May 2017 Mathematics Standard level Paper 1

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1 M17/5/MATME/SP1/ENG/TZ/XX/M Marscheme May 017 Mathematics Standard level Paper 1 17 pages

2 M17/5/MATME/SP1/ENG/TZ/XX/M This marscheme is the property of the International Baccalaureate and must not be reproduced or distributed to any other person without the authorization of the IB Global Centre, Cardiff.

3 M17/5/MATME/SP1/ENG/TZ/XX/M Instructions to Examiners Abbreviations M (M) A (A) R N AG Mars awarded for attempting to use a valid Method; woring must be seen. Mars awarded for a valid Method; may be implied by correct subsequent woring. Mars awarded for an Answer or for Accuracy; often dependent on preceding M mars. Mars awarded for an Answer or for Accuracy; may be implied by correct subsequent woring. Mars awarded for clear Reasoning. Mars awarded for correct answers if no woring shown. Answer given in the question and so no mars are awarded. Using the marscheme 1 General Mar according to RM assessor instructions. Method and Answer/Accuracy mars Do not automatically award full mars for a correct answer; all woring must be checed, and mars awarded according to the marscheme. It is generally not possible to award M0 followed by, as A mar(s) depend on the preceding M mar(s), if any. An exception to this rule is when wor for M1 is missing, as opposed to incorrect (see point 4). Where M and A mars are noted on the same line, M1, this usually means M1 for an attempt to use an appropriate method ( substitution into a formula) and for using the correct values. Where there are two or more A mars on the same line, they may be awarded independently; so if the first value is incorrect, but the next two are correct, award A0. Where the marscheme specifies (M), N, etc., do not split the mars, unless there is a note. Most M mars are for a valid method, ie a method which can lead to the answer: it must indicate some form of progress towards the answer. Once a correct answer to a question or part-question is seen, ignore further correct woring. However, if further woring indicates a lac of mathematical understanding do not award final. N mars If no woring shown, award N mars for correct answers this includes acceptable answers (see accuracy boolet). In this case, ignore mar breadown (M, A, R). Where a student only shows a final incorrect answer with no woring, even if that answer is a correct intermediate answer, award N0. Do not award a mixture of N and other mars. There may be fewer N mars available than the total of M, A and R mars; this is deliberate as it penalizes candidates for not following the instruction to show their woring. There may not be a direct relationship between the N mars and the implied mars. There are times when all the mars are implied, but the N mars are not the full mars: this indicates that we want to see some of the woring, without specifying what.

4 4 M17/5/MATME/SP1/ENG/TZ/XX/M For consistency within the marscheme, N mars are noted for every part, even when these match the mar breadown. If a candidate has incorrect woring, which somehow results in a correct answer, do not award the N mars for this correct answer. However, if the candidate has indicated (usually by crossing out) that the woring is to be ignored, award the N mars for the correct answer. 4 Implied and must be seen mars Implied mars appear in bracets. Implied mars can only be awarded if the wor is seen or if implied in subsequent woring (a correct final answer does not necessarily mean that the implied mars are all awarded). There are questions where some woring is required, but as it is accepted that not everyone will write the same steps, all the mars are implied, but the N mars are not the full mars for the question. Normally the correct wor is seen in the next line. Where there is an followed by for each correct answer, if no woring shown, one correct answer is sufficient evidence to award the. Must be seen mars appear without bracets M1. Must be seen mars can only be awarded if the wor is seen. If a must be seen mar is not awarded because wor is missing (as opposed to M0 or A0 for incorrect wor) all subsequent mars may be awarded if appropriate. 5 Follow through mars (only applied after an error is made) Follow through (FT) mars are awarded where an incorrect answer (final or intermediate) from one part of a question is used correctly in subsequent part(s) or subpart(s). Usually, to award FT mars, there must be woring present and not just a final answer based on an incorrect answer to a previous part. However, if the only mars awarded in a subpart are for the final answer, then FT mars should be awarded if appropriate. Examiners are expected to chec student wor in order to award FT mars where appropriate. Within a question part, once an error is made, no further A mars can be awarded for wor which uses the error, but M and R mars may be awarded if appropriate. (However, as noted above, if an A mar is not awarded because wor is missing, all subsequent mars may be awarded if appropriate). Exceptions to this rule will be explicitly noted on the marscheme. If the question becomes much simpler because of an error then use discretion to award fewer FT mars. If the error leads to an inappropriate value ( probability greater than 1, use of r > 1 for the sum of an infinite GP, sinθ = 1.5, non-inter value where inter required), do not award the mar(s) for the final answer(s). The marscheme may use the word their in a description, to indicate that candidates may be using an incorrect value. If a candidate maes an error in one part, but gets the correct answer(s) to subsequent part(s), award mars as appropriate, unless the question says hence. It is often possible to use a different approach in subsequent parts that does not depend on the answer to previous parts. In a show that question, if an error in a previous subpart leads to not showing the required answer, do not award the final. Note that if the error occurs within the same subpart, the FT rules may result in further loss of mars.

5 5 M17/5/MATME/SP1/ENG/TZ/XX/M 6 Mis-read If a candidate incorrectly copies information from the question, this is a mis-read (MR). A candidate should be penalized only once for a particular mis-read. Use the MR stamp to indicate that this is a misread. Do not award the first mar in the question, even if this is an M mar, but award all others (if appropriate) so that the candidate only loses one mar for the misread. If the question becomes much simpler because of the MR, then use discretion to award fewer mars. If the MR leads to an inappropriate value ( probability greater than 1, use of r > 1 for the sum of an infinite GP, sinθ = 1.5, non- inter value where inter required), do not award the mar(s) for the final answer(s). Miscopying of candidates own wor does not constitute a misread, it is an error. 7 Discretionary mars (d) An examiner uses discretion to award a mar on the rare occasions when the marscheme does not cover the wor seen. In such cases the annotation DM should be used and a brief note written next to the mar explaining this decision. 8 Alternative methods Candidates will sometimes use methods other than those in the marscheme. Unless the question specifies a method, other correct methods should be mared in line with the marscheme. If in doubt, contact your team leader for advice. Alternative methods for complete parts are indicated by METHOD 1, METHOD, etc. Alternative solutions for parts of questions are indicated by EITHER... OR. Where possible, alignment will also be used to assist examiners in identifying where these alternatives start and finish. 9 Alternative forms Unless the question specifies otherwise, accept equivalent forms. As this is an international examination, accept all alternative forms of notation. In the marscheme, equivalent numerical and algebraic forms will generally be written in bracets immediately following the answer. In the marscheme, simplified answers, (which candidates often do not write in examinations), will generally appear in bracets. Mars should be awarded for either the form preceding the bracet or the form in bracets (if it is seen). 10 Calculators No calculator is allowed. The use of any calculator on paper 1 is malpractice, and will result in no grade awarded. If you see wor that suggests a candidate has used any calculator, please follow the procedures for malpractice. Examples: finding an angle, given a trig ratio of Style The marscheme aims to present answers using good communication, if the question ass to find the value of, the marscheme will say =, but the mars will be for the correct value there is usually no need for the =. In these cases, it is also usually acceptable to have another variable, as long as there is no ambiguity in the question, if the question ass to find the value of p and of q, then the student answer needs to be clear. Generally, the only situation

6 6 M17/5/MATME/SP1/ENG/TZ/XX/M where the full answer is required is in a question which ass for equations in this case the marscheme will say must be an equation. The marscheme often uses words to describe what the mars are for, followed by examples, using the notation. These examples are not exhaustive, and examiners should chec what candidates have written, to see if they satisfy the description. Where these mars are M mars, the examples may include ones using poor notation, to indicate what is acceptable. A valid method is one which will allow candidate to proceed to the next step if a quadratic function is given in factorised form, and the question ass for the zeroes, then multiplying the factors does not necessarily help to find the zeros, and would not on its own count as a valid method. 1 Candidate wor If a candidate has drawn a line through wor on their examination script, or in some other way crossed out their wor, do not award any mars for that wor. Candidates are meant to write their answers to Section A on the question paper (QP), and Section B on answer boolets. Sometimes, they need more room for Section A, and use the boolet (and often comment to this effect on the QP), or write outside the box. That is fine, and this wor should be mared. The instructions tell candidates not to write on Section B of the QP. Thus they may well have done some rough wor here which they assume will be ignored. If they have solutions on the answer boolets, there is no need to loo at the QP. However, if there are whole questions or whole part solutions missing on answer boolets, please chec to mae sure that they are not on the QP, and if they are, mar those whole questions or whole part solutions that have not been written on answer boolets. 1. Diagrams The notes on how to allocate mars for setches usually refer to passing through particular points or having certain features. These mars can only be awarded if the setch is approximately the correct shape. All values given will be an approximate guide to where these points/features occur. In some questions, the first is for the shape, in others, the mars are only for the points and/or features. In both cases, unless the shape is approximately correct, no mars can be awarded (unless otherwise stated). However, if the graph is based on previous calculations, FT mars should be awarded if appropriate. 14. Accuracy of Answers If the level of accuracy is specified in the question, a mar will be allocated for giving the final answer to the required accuracy. When this is not specified in the question, all numerical answers should be given exactly or correct to three significant figures. Do not accept unfinished numerical final answers such as /0.1 (unless otherwise stated). As a rule, numerical answers with more than one part (such as fractions) should be given using inters ( 6/8). Calculations which lead to inters should be completed, with the exception of fractions which are not whole numbers. Intermediate values do not need to be given to the correct three significant figures. But, if candidates wor with rounded values, this could lead to an incorrect answer, in which case award A0 for the final answer. Where numerical answers are required as the final answer to a part of a question in the marscheme, the marscheme will show a truncated 6 sf value the exact value if applicable, the correct sf answer Units will appear in bracets at the end.

7 7 M17/5/MATME/SP1/ENG/TZ/XX/M Section A 1. (a) attempt to subtract terms d = u u, 7 1 d = 4 N [ mars] (b) correct approach () u 10 = + 9(4) u 10 = 9 N [ mars] (c) correct substitution into sum () 10 S 10 = 5( + 9), S 10 = ( + 9 4) S 10 = 10 N [ mars] [Total 6 mars]. (a) evidence of scalar product M1 a b, 4( + ) + recognizing scalar product must be zero a b = 0, = 0 correct woring (must involve combining terms) 6 + 1, 6 = 1 = N [4 mars] (b) attempt to substitute their value of (seen anywhere) + b =, b = 4 correct woring 4 +, c = N [ mars] () () [Total 7 mars]

8 8 M17/5/MATME/SP1/ENG/TZ/XX/M. (a) P ( X > 107) = 0.4 = 6, 4% 5 N1 [1 mar] (b) P ( X > 100) = 0.5, P ( X > 100) P ( X > 107) correct woring , P (100 < X < 107) = 0.6 = 1, 6% 50 () N [ mars] (c) 0.6, 1 (0.4), P (9 < X < 100) = P (100 < X < 107) P (9 < X < 107) = 0.5 = 1, 5% 5 N [ mars] [Total 6 mars] 4. (a) (i) p = 6 N1 (ii) q = 5 N1 [ mars] (b) correct approach () p q, 5 6 = 0 N [ mars] (c) correct approach () rows = n + 1, columns = n An ( ) = nn ( + 1) ( = n + n) (cm ) N [ mars] [Total 6 mars]

9 9 M17/5/MATME/SP1/ENG/TZ/XX/M 5. f dx, x ( x + 1) 5 dx correct intration by substitution/inspection 1 f( x) ( x 1) 4 1 = + + c, 4 4 x + 1 ( ) 4 correct substitution into their intrated function (must include c) 1 1 = + c, c = ( ) 4 A M1 Note: Award M0 if candidates substitute into f or f. 5 c = 4 1 ( ) 4 5 f( x) = x ( x ) ( x ) = + 4, 4 4( x + 1) () N4 [6 mars] 6. (a) expressing h (1) as a product of f (1) and g (1) () f(1) g(1), (9) h (1) = 18 N [ mars] (b) attempt to use product rule (do not accept h = f g ) h = fg + gf, h (8) = f (8) g(8) + g (8) f(8) correct substitution of values into product rule h (8) = 4(5) + ( ), h (8) = 14 N [ mars] () [Total 5 mars]

10 10 M17/5/MATME/SP1/ENG/TZ/XX/M 7. correct application of log a + log b = log ab () log (sin xcos x ), log + log (sin x) + log (cos x) correct equation without logs sin xcos x=, sin xcos x=, sin x = 4 recognizing double-angle identity (seen anywhere) 1 log (sin x ), sin xcos x= sin x, sin x = evaluating sin π = (0 ) () correct woring π x = + π, 1 5π 9π x =,, 750, , π 5π x = and x =, one correct final answer 1 1 5π 9π x =, (do not accept additional values) 1 1 A N0 [7 mars]

11 11 M17/5/MATME/SP1/ENG/TZ/XX/M Section B 8. (a) (i) evidence of median position 80th employee 40 hours N (ii) 10 employees N1 [ mars] (b) (i) 0 N1 (ii) splitting into 40 and hours more, 10 correct woring () 50 N [4 mars] (c) 00 is less than 0 so 8 pounds/hour, 00 8, 5, 00 x = employees A N [ mars] (d) hours wored () correct woring 40(8) + 0(10), = 50 N [4 mars] () [Total 14 mars]

12 1 M17/5/MATME/SP1/ENG/TZ/XX/M 9. (a) recognizing t = 0 at A A is (4, 1, ) N [ mars] (b) (i) METHOD , ( 6,, 1) correct approach to find AB 6 4 AO + OB, B A, 1 1 AB = 4 4 () N METHOD recognizing AB is two times the direction vector correct woring 1 AB = AB = 4 4 () N (ii) correct substitution () AB = , , 6 AB = 6 N [5 mars] continued

13 1 M17/5/MATME/SP1/ENG/TZ/XX/M Question 9 continued (c) METHOD 1 (vector approach) involving AB and AC AB AC, BA AC AB AC finding scalar product and AC ()() scalar product () + 4(0) 4(4) ( = 10) AC = ( = 5) substitution of their scalar product and magnitudes into cosine formula ˆ cos BAC = ˆ 10 1 cos BAC = = 0 METHOD (triangle approach) involving cosine rule AB AC BC cos BAC ˆ + = AB AC finding lengths AC and BC AC = 5, BC = 9 substitution of their lengths into cosine formula cos BAC ˆ + = 5 6 ˆ 0 cos BAC = 60 1 = ()() N N [5 mars] continued

14 14 M17/5/MATME/SP1/ENG/TZ/XX/M Question 9 continued (d) Note: Award relevant mars for woring seen to find BC in part (c) (if cosine rule used in part (c)). METHOD 1 (using cosine rule) recognizing need to find BC choosing cosine rule c = a + b ab cos C correct substitution into RHS 1 BC = (6) + (5) (6) (5), distance is 9 N METHOD (finding magnitude of recognizing need to find BC BC ) attempt to find OB correct woring 1 BC = 4, 8 or OC, 1 CB = 4, 8 6 OB = or = 81 7 OC = 1, BA+ AC 7 distance is 9 N METHOD (finding coordinates and using distance formula) recognizing need to find BC attempt to find coordinates of B or C, B (6,, 1) or C(7, 1, 7) correct substitution into distance formula BC (6 7) ( 1) ( 1 7) = + +, ( ) = 81 distance is 9 N [4 mars] [Total 16 mars]

15 15 M17/5/MATME/SP1/ENG/TZ/XX/M 10. (a) (i) f ( x) = x N1 (ii) attempt to substitute x= into their derivative (b) METHOD 1 gradient of L is N [ mars] attempt to substitute coordinates of A and their gradient into equation of a line = ( ) + b correct equation of L in any form y = ( x+ ), y = x y = 0 correct substitution into L equation = x,0= x () correct woring x = x = AG N0 METHOD y gradient = x y, x 1 1 rise = run recognizing y = 0 at B () attempt to substitute coordinates of A and B into slope formula 0 x, x+ correct equation 0 =, x correct woring x = x+ =, = ( x+ ) x = AG N0 [5 mars] continued

16 16 M17/5/MATME/SP1/ENG/TZ/XX/M Question 10 continued (c) to find area of triangle 1 ( ) area of ABC = 4 N [ mars] (d) METHOD 1 ( f triangle ) to find area from to 0 0 x d x, f 0 correct intration (seen anywhere, even if M0 awarded) 0 x 1, x substituting their limits into their intrated function and subtracting ( ) 0, area from to 0 is Note: Award M0 for substituting into original or differentiated function. attempt to find area of R 0 f ( x )d x triangle correct woring for R, R = 4 1 () correct substitution into triangle = pr () p =, = p p = N continued

17 17 M17/5/MATME/SP1/ENG/TZ/XX/M Question 10 continued ) METHOD ( ( f L) to find area of R 0 + x ( x )dx x dx, ( f L) + f correct intration (seen anywhere, even if M0 awarded) x + x + x, x x + x + x A substituting their limits into their intrated function and subtracting ( ) ( ) + ( ) + (0) Note: Award M0 for substituting into original or differentiated function. correct woring for R +, , R = 1 () correct substitution into triangle = pr () p = +, = p p = N [7 mars] [Total 17 mars]

Markscheme May 2016 Mathematics Standard level Paper 2

Markscheme May 2016 Mathematics Standard level Paper 2 M16/5/MATME/SP/ENG/TZ/XX/M Markscheme May 016 Mathematics Standard level Paper 16 pages M16/5/MATME/SP/ENG/TZ/XX/M This markscheme is the property of the International Baccalaureate and must not be reproduced