N13/5/MATHL/HP2/ENG/TZ0/XX/M MARKSCHEME. November 2013 MATHEMATICS. Higher Level. Paper pages

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1 N/5/MATHL/HP/ENG/TZ0/XX/M MARKSCHEME November 0 MATHEMATICS Higher Level Paper 0 pages

2 N/5/MATHL/HP/ENG/TZ0/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 N/5/MATHL/HP/ENG/TZ0/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 General Mark according to Scoris instructions and the document Mathematics HL: Guidance for e-marking November 0. 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, for example,, this usually means for an attempt to use an appropriate method (for example, substitution into a formula) and for using the correct values. Where the markscheme specifies (M), N, etc, do not split the marks. Once a correct answer to a question or part-question is seen, ignore further working.

4 N/5/MATHL/HP/ENG/TZ0/XX/M 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. Implied marks Implied marks appear in brackets, for example,, 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 (for example, sin.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 (for example, sin.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 5 N/5/MATHL/HP/ENG/TZ0/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, 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 ), the markscheme gives: ( ) cos(5 ) 5 0cos(5x ) f x x Award for cos(5x ) 5, even if 0cos(5x ) is not seen. 0 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. 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 6 N/5/MATHL/HP/ENG/TZ0/XX/M Calculators A GDC is required for paper, but calculators with symbolic manipulation features (for example, 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. 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.. Candidate work Candidates are meant to write their answers to Section A on the question paper (QP), and Section B on answer booklets. Sometimes, they need more room for Section A, and use the booklet (and often comment to this effect on the QP), or write outside the box. This work should be marked. The instructions tell candidates not to write on Section B of the QP. Thus they may well have done some rough work here which they assume will be ignored. If they have solutions on the answer booklets, there is no need to look at the QP. However, if there are whole questions or whole part solutions missing on answer booklets, please check to make sure that they are not on the QP, and if they are, mark those whole questions or whole part solutions that have not been written on answer booklets.

7 7 N/5/MATHL/HP/ENG/TZ0/XX/M SECTION A. AX B EITHER X A B OR attempting row reduction: eg THEN X 7 5 Total [ marks]. (a) METHOD a d and 76 a 9d d 7 a METHOD 76 6d d 7 a a [ marks] n () n () (b) n Note: Award for using either an equation, a graphical approach or a numerical approach. n 7 N [ marks] Total [7 marks]

8 8 N/5/MATHL/HP/ENG/TZ0/XX/M. (a) A correct graph shape for 0 x 0. maxima (.78, 0.88) and 9.70,.89 minimum (6., 0.885) x-axis intercepts (.97,0), (5.,0) and (7.,0) A Note: Award if two x-axis intercepts are correct. (b) 0 x x 7. [5 marks] [ marks] Total [7 marks]

9 9 N/5/MATHL/HP/ENG/TZ0/XX/M PZ 0.9 and PZ 0. use of inverse normal and () solving simultaneously Note: Award for attempting to solve an incorrect pair of equations eg, inverse normal not used hrs 6 mins ( 766 mins) 0 mins Total [6 marks] 5. (a) 7 6 P( F) () Note: Award for the sum of two products. ( 0.90 ) 6 [ marks] (b) Use of P S F P( S F) to obtain P( F) 7 P ( S F) Note: Award only if the numerator results from the product of two probabilities. 7 ( 0.58 ) [ marks] Total [5 marks]

10 0 N/5/MATHL/HP/ENG/TZ0/XX/M 6. (a) ai ai ai ai a ai a a i a a a (i) z is real when a 0 (ii) z is purely imaginary when a Note: Award A0A0 for a ai a i a a leading to a 0 in (i). [ marks] (b) METHOD attempting to find either z or eg z z a a a a a a a a z by expanding and simplifying z METHOD z a i a i z a z a [ marks] Total [6 marks]

11 N/5/MATHL/HP/ENG/TZ0/XX/M 7. (a) attempting to form (cos 6)(cos ) 7( sin ) 0 cos 7sin 7 0 sin 7sin 5 0 sin 7sin 0 AG [ marks] (b) attempting to solve algebraically (including substitution) or graphically for sin sin () 0.0 ( 9.5 ) [ marks] Total [6 marks] 8. (a) A 0 0 sin Note: Award for use of area of segment area of sector area of triangle sin AG [ marks] (b) METHOD 0 unshaded area 50( sin ) (or equivalent eg sin) 50 50sin ( sin ) () sin (rad) METHOD sin () sin (rad) [ marks] Total [5 marks]

12 N/5/MATHL/HP/ENG/TZ0/XX/M 9. (a) METHOD 5 for P on L, OP 5 require (or equivalent) OP L : r Note: Do not award the final if METHOD Calculating either OP or OP eg r is not seen. [5 marks] OP ( 5 ) ( ) ( ) d d Solving either OP 0 d or OP 0for. d OP L : r Note: Do not award the final if r is not seen. [5 marks]

13 N/5/MATHL/HP/ENG/TZ0/XX/M Question 9 continued continued (b) METHOD OP METHOD 5 shortest distance 0i 8 j k [ marks] Total [7 marks]

14 N/5/MATHL/HP/ENG/TZ0/XX/M 0. EITHER dx d sec u u sec u du tan u tan sec u du tan usec u du ( u sin u tan u or OR x u arctan du dx x sec u du ) tan u sec u tan u du tan u secudu tan u THEN sec u du tan u cos u cosec u cot u d u d u sin u cosec u ( C) ( C) sin u use of either tan u x or an appropriate trigonometric identity either sin u x x x or cosec u (or equivalent) x x ( C ) AG x Total [7 marks]

15 5 N/5/MATHL/HP/ENG/TZ0/XX/M. (a) (i) X ~ Po(0.6) P( 0) 0.59 X e 0.6 SECTION B (ii) P( X ) P( X ) () e e 0.6 e 0.0 (iii) Y ~ Po(.) P( Y 5) 0.96 (iv) Z ~ B(, ) () Note: Award for recognising binomial and for using correct parameters. P( Z ) 0.69 [9 marks] (b) (i) (ii) k ln xdx ( k ) k E( X ) kx ln xdx () attempting to evaluate their integral.7 (iii) x (iv) m k ln x d x 0.5 k[ xln x x] m 0.5 attempting to solve for m m. [9 marks] Total [8 marks]

16 6 N/5/MATHL/HP/ENG/TZ0/XX/M. (a) (i) METHOD t v cos dt t sin c t t 0, v v sin METHOD t t v cos dt 0 t v sin (ii) Note: Award for shape and domain 0t 8π. Award for (0, ) and (6, 0) (8.8, 0). Award for (, ) (6.8, ). (iii) METHOD 6 t sin dt (m) (m) METHOD t s sin dt t 8cos t c When t 0, s 0 and so c 8. When 6π s 7 (m) 7π 8 (m). t, [8 marks] continued

17 7 N/5/MATHL/HP/ENG/TZ0/XX/M Question continued (b) (i) METHOD dv d dv dt v v arctan t c v t () EITHER t 0, v c 8 v π arctan t OR v tan c t t 0, v tan c and so c 8 THEN π v tan t 8t v tan AG METHOD dv π8t sec dt π8t dv Substituting v tan v dt : into dv π8t tan dt π8t tan () π8t sec Verifying that v when t 0. continued

18 8 N/5/MATHL/HP/ENG/TZ0/XX/M (ii) METHOD dv v d v s dv v AG ds v METHOD dv dv dt ds dt ds dv v AG ds v (iii) METHOD π When v 0, t ( t ). 8 π 8 π8t s tan dt 0 s 0.7 (m) s ln (m) A METHOD v d v d s v EITHER ln v s c (or equivalent) v, s 0 c ln8 8 s ln v ln 8 s ln v () v 0 s ln (m) s 0.7 (m) OR v v 0 dvs (or equivalent) Note: Award for setting up a definite integral and award for stating correct limits. s 0.7 (m) s ln (m) A [ marks] Total [0 marks]

19 9 N/5/MATHL/HP/ENG/TZ0/XX/M. (a) (i) either counterexample or sketch or recognising that y k ( k ) intersects the graph of y f ( x) twice function is not (does not obey horizontal line test) R so f does not exist AG () f (ln ). () m 5 x x (ii) f ( x) e e f (ln ).67 EITHER 5 y x ln 0 y x ln OR 5 ln c 5 c ln 5 y x ln y 9x 0 9ln THEN 9xy 9ln 0 0 AG (iii) The tangent at a, f ( a ) has equation y f ( a) f ( a)( x a). f( a) f( a) (or equivalent) a () a a a a e e e e (or equivalent) a attempting to solve for a a.0 [ marks] continued

20 0 N/5/MATHL/HP/ENG/TZ0/XX/M Question continued x (b) (i) y e e x x x e ye 0 Note: Award for either attempting to rearrange or interchanging x and y. x y y e x e y y x y y ln f ( x) ln x x Note: Award for correct notation and for stating the positive branch. 5 V ln y y dy () (ii) Note: Award for attempting to use V x dy. d c 7. units [8 marks] Total [ marks]

M13/5/MATHL/HP2/ENG/TZ2/XX/M MARKSCHEME. May 2013 MATHEMATICS. Higher Level. Paper pages

M13/5/MATHL/HP2/ENG/TZ2/XX/M MARKSCHEME. May 2013 MATHEMATICS. Higher Level. Paper pages 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