M13/5/MATHL/HP/ENG/TZ/XX/M MARKSCHEME May 013 MATHEMATICS Higher Level Paper 0 pages
M13/5/MATHL/HP/ENG/TZ/XX/M This markscheme is confidential and for the exclusive use of examiners in this examination session. It is the property of the International Baccalaureate and must not be reproduced or distributed to any other person without the authorization of the IB Assessment Centre.
3 M13/5/MATHL/HP/ENG/TZ/XX/M Instructions to Examiners Abbreviations M Marks awarded for attempting to use a correct Method; working must be seen. (M) Marks awarded for Method; may be implied by correct subsequent working. A (A) R N AG Marks awarded for an Answer or for Accuracy; often dependent on preceding M marks. Marks awarded for an Answer or for Accuracy; may be implied by correct subsequent working. Marks awarded for clear Reasoning. Marks awarded for correct answers if no working shown. Answer given in the question and so no marks are awarded. Using the markscheme 1 General Mark according to Scoris instructions and the document Mathematics HL: Guidance for e-marking May 013. It is essential that you read this document before you start marking. In particular, please note the following. Marks must be recorded using the annotation stamps. Please check that you are entering marks for the right question. If a part is completely correct, (and gains all the must be seen marks), use the ticks with numbers to stamp full marks. If a part is completely wrong, stamp A0 by the final answer. If a part gains anything else, it must be recorded using all the annotations. All the marks will be added and recorded by Scoris. Method and Answer/Accuracy marks Do not automatically award full marks for a correct answer; all working must be checked, and marks awarded according to the markscheme. It is not possible to award M0 followed by, as A mark(s) depend on the preceding M mark(s), if any. Where M and A marks are noted on the same line, eg M1, this usually means M1 for an attempt to use an appropriate method (eg substitution into a formula) and for using the correct values. Where the markscheme specifies (M), N3, etc., do not split the marks. Once a correct answer to a question or part-question is seen, ignore further working.
4 M13/5/MATHL/HP/ENG/TZ/XX/M 3 N marks Award N marks for correct answers where there is no working. 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 it penalizes candidates for not following the instruction to show their working. 4 Implied marks Implied marks appear in brackets eg, and can only be awarded if correct work is seen or if implied in subsequent working. Normally the correct work is seen or implied in the next line. Marks without brackets can only be awarded for work that is seen. 5 Follow through marks Follow through (FT) marks are awarded where an incorrect answer from one part of a question is used correctly in subsequent part(s). To award FT marks, there must be working present and not just a final answer based on an incorrect answer to a previous part. If the question becomes much simpler because of an error then use discretion to award fewer FT marks. If the error leads to an inappropriate value (eg sin 1.5 ), do not award the mark(s) for the final answer(s). Within a question part, once an error is made, no further dependent A marks can be awarded, but M marks may be awarded if appropriate. Exceptions to this rule will be explicitly noted on the markscheme. 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 has been a misread. Then deduct the first of the marks to be awarded, even if this is an M mark, but award all others so that the candidate only loses one mark. If the question becomes much simpler because of the MR, then use discretion to award fewer marks. If the MR leads to an inappropriate value (eg sin 1.5 ), do not award the mark(s) for the final answer(s). 7 Discretionary marks (d) An examiner uses discretion to award a mark on the rare occasions when the markscheme does not cover the work seen. In such cases the annotation DM should be used and a brief note written next to the mark explaining this decision.
5 M13/5/MATHL/HP/ENG/TZ/XX/M 8 Alternative methods Candidates will sometimes use methods other than those in the markscheme. Unless the question specifies a method, other correct methods should be marked in line with the markscheme. If in doubt, contact your team leader for advice. Alternative methods for complete questions are indicated by METHOD 1, METHOD, etc. Alternative solutions for part-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 markscheme, equivalent numerical and algebraic forms will generally be written in brackets 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 the bracket or the form in brackets (if it is seen). Example: for differentiating f ( x) sin (5x 3), the markscheme gives: ( ) cos(5 3) 5 10cos(5 x 3) f x x Award for cos(5x 3) 5, even if 10cos(5x 3) is not seen. 10 Accuracy of Answers Candidates should NO LONGER be penalized for an accuracy error (AP). If the level of accuracy is specified in the question, a mark will be allocated for giving the 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. Please check work carefully for FT. 11 Crossed out work If a candidate has drawn a line through work on their examination script, or in some other way crossed out their work, do not award any marks for that work.
6 M13/5/MATHL/HP/ENG/TZ/XX/M 1 Calculators A GDC is required for paper, but calculators with symbolic manipulation features (eg TI-89) are not allowed. Calculator notation The Mathematics HL guide says: Students must always use correct mathematical notation, not calculator notation. Do not accept final answers written using calculator notation. However, do not penalize the use of calculator notation in the working. 13 More than one solution Where a candidate offers two or more different answers to the same question, an examiner should only mark the first response unless the candidate indicates otherwise.
7 M13/5/MATHL/HP/ENG/TZ/XX/M SECTION A 1. (a) EITHER ˆ 3 AOB arcsin 4 or equivalent (eg ˆ 3 AOB arctan 7 ˆ 7 AOB arccos ) 4 OR ˆ 4 4 6 cos AOB 44 THEN 1 8 1.696 (correct to 4sf) (b) use of area of segment area of sector area of triangle 1 1 4 1.696 4 sin1.696 () 5.63 (cm ) [ marks] [3 marks] Total [5 marks]. (a) attempting to express the system in matrix form M1 0.1 1.7 0.9 x 4.4.4 0.3 3. y 1..5 0.6 3.7 z 0.8 Note: Award M1 for a correct augmented matrix. [ marks] (b) either direct GDC use, attempting elimination or using an inverse matrix. 93.4.40 389 x 68 y 1.6 (correct to sf) or 1.61 (correct to 3sf) or (exact) 389 z 1.6 1.57 61 389 A [3 marks] Total [5 marks]
8 M13/5/MATHL/HP/ENG/TZ/XX/M 3. (a) 13.5 9.5 X 13.5 9.5 10.4 X 16.6 Note: Accept 6.16. [ marks] (b) P( X 10) 0.1807 () estimate is 181 (correct to the nearest whole number). Note: Accept 180. [3 marks] Total [5 marks] 4. (a) x sec x d x x tan x 1 tan x d x M1 xtan x ln cos x ( c) ( x tan x ln sec x ( c) ) M1 [4 marks] (b) attempting to solve an appropriate equation eg m tan m ln(cos m) = 0.5 m 0.8 Note: Award if m 0.8 is specified with other positive solutions. [ marks] Total [6 marks]
9 M13/5/MATHL/HP/ENG/TZ/XX/M 5. (a) u v n 1 1.6 ( 1) 1.5 3 1. n 1 n n ( 1.5n 0.1 3 1. n ) [ marks] (b) attempting to solve un vn numerically or graphically. n.61,9.695 () So 3n 9 [3 marks] (c) The greatest value of un vn is 1.64. Note: Do not accept 1.64. [1 mark] Total [6 marks] 6. (a) attempting to solve for cosx or for u where u cos x or for x graphically. EITHER cos x (and ) 3 () OR x 48.1897 () THEN x 48 Note: Award ()A0 for x 48, 13. Note: Award ()A0 for 0.841 radians. [3 marks] (b) attempting to solve for sec x or for v where v sec x. sec x and 3 () sec x [3 marks] Total [6 marks]
10 M13/5/MATHL/HP/ENG/TZ/XX/M 7. (a) 0.5 1 ax d 0.5 (1 x )d 1 0 0.5 M1 5a (or equivalent) or a0.104... 1 48 Note: Award M1 for considering two definite integrals. Award for equating to 1. Award for a correct equation. The can be awarded in any order. a 9.6 AG [3 marks] (b) correct shape for 0 x 0.5 and f (0.5).4 correct shape for 0.5 x 1 and f (1) 0 (c) attempting to find P( X 0.6) direct GDC use or eg P(0 X 0.5) P(0.5 X 0.6) or 1 P(0.6 X 1) 77 P( X 0.6) 0.616 15 [ marks] [ marks] Total [7 marks]
11 M13/5/MATHL/HP/ENG/TZ/XX/M 8. n f n n is divisible by 576 for n n P( ): ( ) 5 4 1 for n 1, f (1) 5 4 1 0 Zero is divisible by 576, (as every non-zero number divides zero), and so P(1) is true. R1 Note: Award R0 for P(1) 0 shown and zero is divisible by 576 not specified. Note: Ignore P() 576 if P(1) 0 is shown and zero is divisible by 576 is specified. Assume P( k) is true for some k( f( k) N 576). M1 Note: Do not award M1 for statements such as let n k. consider EITHER OR THEN ( k 1) P ( k 1) : f ( k 1) 5 4 ( k 1) 1 M1 55 k 4k 5 5 (4k1N576) 4k 5 576k5 576N which is a multiple of 576 55 k 600k 5 600k 4k k 5 5 4k 1 576k (or equivalent) which is a multiple of 576 P(1) is true and P( k ) true P( k 1) true, so P( n ) is true for all n R1 Note: Award R1 only if at least four prior marks have been awarded. Total [7 marks]
1 M13/5/MATHL/HP/ENG/TZ/XX/M 9. (a) X Po(1.) X X P 3 P 0 0.08670.3011 0.061 (b) Three requests over two days can occur as (3, 0), (0, 3), (,1) or (1, ). R1 using conditional probability, for example P(3,0) P(,1) 0.15 or 0.375 P(3 requests, m =.4) P(3 requests, m =.4) M1 [ marks] expected income is 0.15US$10 0.375US$180 M1 Note: Award M1 for attempting to find the expected income including both (3, 0) and (,1) cases. US$30 US$135 US$165 [5 marks] Total [7 marks]
13 M13/5/MATHL/HP/ENG/TZ/XX/M 10. METHOD 1 dv 1 (60 v ) dt 40 dv dt attempting to separate variables 60 v 40 M1 t ln (60 v) c 40 c ln60 (or equivalent) attempting to solve for v when t 30 3 4 e v60 60 1 v 31.7 (ms ) METHOD dv 1 (60 v ) dt 40 dt 40 (or equivalent) dv 60 v M1 v f 40 dv 30 where v 0 f is the velocity of the car after 30 seconds. 60 v v f 40 attempting to solve dv 30 for v 0 f 60 v 1 v 31.7 (ms ) Total [6 marks]
14 M13/5/MATHL/HP/ENG/TZ/XX/M SECTION B 11. (a) (i) n (k 1) (or equivalent) k1 Note: n Award A0 for (n 1) or equivalent. n1 (ii) EITHER nn ( 1) n M1 OR n n (using n 1 1 1 n S u n d ) M1 OR n n (using S u u 1 1 n n 1 n ) M1 (iii) THEN n AG 47 14 013 [4 marks] (b) (i) EITHER a pentagon and five diagonals OR five diagonals (circle optional) (ii) Each point joins to n 3 other points. a correct argument for nn ( 3) R1 a correct argument for nn ( 3) R1 (iii) attempting to solve 1 nn ( 3) 1000000 for n. n 1415.7 () n 1416 [7 marks] continued
15 M13/5/MATHL/HP/ENG/TZ/XX/M Question 11 continued (c) (i) np 4 and npq 3 () attempting to solve for n and p 1 n 16 and p 4 (ii) X B(16,0.5) () P( X 1) 0.0534538 16 15 ( (0.5)(0.75) ) 1 () P( X 3) 0.07876 16 3 13 ( (0.5) (0.75) ) 3 () P( X 1) P( X 3) 0.61 [8 marks] Total [19 marks]
16 M13/5/MATHL/HP/ENG/TZ/XX/M 1. (a) (i) METHOD 1 dy sin x cos x dx dy y (cos x sin x)( sin x cos x) dx M1 cos x sin x cosx AG METHOD sin cos y x x dy y (cos x sin x)(cos x sin x) dx M1 dy y cos x sin x dx cosx AG (ii) attempting to separate variables y dy cosx dx 1 1 sin M1 y x C Note: Award for a correct LHS and for a correct RHS. 1 y (sin x A) (iii) sin x A (cos x sin x) (cos x sin x) cos x sin xcos x sin x use of sinx sin xcos x. A 1 [10 marks] continued
17 M13/5/MATHL/HP/ENG/TZ/XX/M Question 1 continued (b) (i) substituting so π x and y into 4 1 1 y (sin x A) M1 g( x) (sinx 3). range g is, Note: Accept 1.41,. Award for each correct endpoint and for the correct closed interval. (ii) 1 (sin 3) d 0 ().99 (iii) π (or equivalent) ()() π (sin x 3) dx π1 0 Note: Award ()() for (sin x )d x 0 π π 17.946 4.935 ( 3π π ) 13.0 Note: Award for π(π 1). [1 marks] Total [ marks]
18 M13/5/MATHL/HP/ENG/TZ/XX/M 13. (a) EITHER π arctan 8 arctan 13 θ x 0 x (or equivalent) M1 Note: 8 13 Accept 180 arctan θ x arctan (or equivalent). 0 x OR x 0 x θ arctan arctan 8 13 (or equivalent) M1 [ marks] 0 (b) (i) θ 0.994 ( arctan ) 13 5 (ii) θ 1.19 ( arctan ) (c) correct shape. correct domain indicated. [ marks] [ marks] continued
19 M13/5/MATHL/HP/ENG/TZ/XX/M Question 13 continued (d) attempting to differentiate one arctan f( x ) term M1 EITHER π arctan 8 arctan 13 θ x 0 x d 8 1 13 1 d x x 8 (0 x) 13 1 1 x 0 x OR x 0 x θ arctan arctan 8 13 1 1 d 8 13 d x 0 1 x 1 x 8 13 THEN 8 13 x 64 569 40x x 8(569 40 x x ) 13( x 64) ( x 64)( x 40x 569) M1 5(744 64 xx ) ( x 64)( x 40x 569) AG [6 marks] (e) Maximum light intensity at P occurs when d 0. dx either attempting to solve d d 0 for x or using the graph of either or dx dx x 10.05 (m) [3 marks] continued
0 M13/5/MATHL/HP/ENG/TZ/XX/M Question 13 continued (f) dx 0.5 dt () At x 10, d 5 0.000453 ( ). dx 11 09 () use of d d d x dt dx dt M1 d 0.0007 5 ) (rad s -1 ) dt 058 Note: Award () for d x 0.5 dt and for d 5 0.0007 ( ). dt 058 Note: Implicit differentiation can be used to find d. Award as above. dt [4 marks] Total [19 marks]