GCE Mathematics. Mark Scheme for June Unit 4723: Core Mathematics 3. Advanced GCE. Oxford Cambridge and RSA Examinations

Transcription

1 GCE Mathematics Unit 47: Core Mathematics Advanced GCE Mark Scheme for June 017 Oford Cambridge and RSA Eaminations

2 OCR (Oford Cambridge and RSA) is a leading UK awarding body, providing a wide range of qualifications to meet the needs of candidates of all ages and abilities. OCR qualifications include AS/A Levels, Diplomas, GCSEs, Cambridge Nationals, Cambridge Technicals, Functional Skills, Key Skills, Entry Level qualifications, NVQs and vocational qualifications in areas such as IT, business, languages, teaching/training, administration and secretarial skills. It is also responsible for developing new specifications to meet national requirements and the needs of students and teachers. OCR is a not-for-profit organisation; any surplus made is invested back into the establishment to help towards the development of qualifications and support, which keep pace with the changing needs of today s society. This mark scheme is published as an aid to teachers and students, to indicate the requirements of the eamination. It shows the basis on which marks were awarded by eaminers. It does not indicate the details of the discussions which took place at an eaminers meeting before marking commenced. All eaminers are instructed that alternative correct answers and unepected approaches in candidates scripts must be given marks that fairly reflect the relevant knowledge and skills demonstrated. Mark schemes should be read in conjunction with the published question papers and the report on the eamination. OCR will not enter into any discussion or correspondence in connection with this mark scheme. OCR 017

3 Annotations and abbreviations Annotation in scoris Meaning and BOD Benefit of doubt FT Follow through ISW Ignore subsequent working M0, Method mark awarded 0, 1 A0, Accuracy mark awarded 0, 1 B0, Independent mark awarded 0, 1 SC Special case ^ Omission sign MR Misread Highlighting Other abbreviations Meaning in mark scheme E1 Mark for eplaining U1 Mark for correct units G1 Mark for a correct feature on a graph dep* Method mark dependent on a previous mark, indicated by * cao Correct answer only oe Or equivalent rot Rounded or truncated soi Seen or implied www Without wrong working

4 Subject-specific Marking Instructions for GCE Mathematics Pure strand a Annotations should be used whenever appropriate during your marking. The A, M and B annotations must be used on your standardisation scripts for responses that are not awarded either 0 or full marks. It is vital that you annotate standardisation scripts fully to show how the marks have been awarded. For subsequent marking you must make it clear how you have arrived at the mark you have awarded b An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking incorrect solutions. Correct solutions leading to correct answers are awarded full marks but work must not be judged on the answer alone, and answers that are given in the question, especially, must be validly obtained; key steps in the working must always be looked at and anything unfamiliar must be investigated thoroughly. Correct but unfamiliar or unepected methods are often signalled by a correct result following an apparently incorrect method. Such work must be carefully assessed. When a candidate adopts a method which does not correspond to the mark scheme, award marks according to the spirit of the basic scheme; if you are in any doubt whatsoever (especially if several marks or candidates are involved) you should contact your Team Leader. c The following types of marks are available. M A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not usually 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, eg by substituting the relevant quantities into the formula. In some cases the nature of the errors allowed for the award of an M mark may be specified. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated Method mark is earned (or implied). Therefore M0 cannot ever be awarded. B Mark for a correct result or statement independent of Method marks. 4

5 E A given result is to be established or a result has to be eplained. This usually requires more working or eplanation than the establishment of an unknown result. Unless otherwise indicated, marks once gained cannot subsequently be lost, eg wrong working following a correct form of answer is ignored. Sometimes this is reinforced in the mark scheme by the abbreviation isw. However, this would not apply to a case where a candidate passes through the correct answer as part of a wrong argument. d e When a part of a question has two or more method steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. (The notation dep * is used to indicate that a particular mark is dependent on an earlier, asterisked, mark in the scheme.) Of course, in practice it may happen that when a candidate has once gone wrong in a part of a question, the work from there on is worthless so that no more marks can sensibly be given. On the other hand, when two or more steps are successfully run together by the candidate, the earlier marks are implied and full credit must be given. The abbreviation ft implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A and B marks are given for correct work only differences in notation are of course permitted. A (accuracy) marks are not given for answers obtained from incorrect working. When A or B marks are awarded for work at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such cases, eactly what is acceptable will be detailed in the mark scheme rationale. If this is not the case please consult your Team Leader. Sometimes the answer to one part of a question is used in a later part of the same question. In this case, A marks will often be follow through. In such cases you must ensure that you refer back to the answer of the previous part question even if this is not shown within the image zone. You may find it easier to mark follow through questions candidate-by-candidate rather than question-by-question. f g Wrong or missing units in an answer should not lead to the loss of a mark unless the scheme specifically indicates otherwise. Candidates are epected to give numerical answers to an appropriate degree of accuracy, with significant figures often being the norm. Small variations in the degree of accuracy to which an answer is given (e.g. or 4 significant figures where is epected) should not normally be penalised, while answers which are grossly over- or under-specified should normally result in the loss of a mark. The situation regarding any particular cases where the accuracy of the answer may be a marking issue should be detailed in the mark scheme rationale. If in doubt, contact your Team Leader. Rules for replaced work If a candidate attempts a question more than once, and indicates which attempt he/she wishes to be marked, then eaminers should do as the candidate requests. 5

6 If there are two or more attempts at a question which have not been crossed out, eaminers should mark what appears to be the last (complete) attempt and ignore the others. NB Follow these maths-specific instructions rather than those in the assessor handbook. h For a genuine misreading (of numbers or symbols) which is such that the object and the difficulty of the question remain unaltered, mark according to the scheme but following through from the candidate s data. A penalty is then applied; 1 mark is generally appropriate, though this may differ for some units. This is achieved by withholding one A mark in the question. Note that a miscopy of the candidate s own working is not a misread but an accuracy error. 6

7 Question Answer Marks Guidance 1 1 State volume is (4 4e e ) d Condone absence of d; no need for limits here; may be implied here by its appearance later in solution; integrand must be epanded Obtain integral of form 1 p qe re * With non-zero constants p, q, r; with or without here 1 Obtain correct 4 8e e or (4 8e e ) Apply limits 0 and 4 correctly to their integral Obtain 4 (e 8e 7) 1 [5] Or unsimplified equiv; condone presence of + c Dep *M; with at least one non-zero term obtained from use of limit 0; limits used the wrong way round is M0 Or simplified equiv; + c now is A0; ignore subsequent working if necessary i Attempt calculation of form k( y 4y y 4 y y ) Obtain k(ln1ln5 4lnln 7 ln5ln9 4ln 7ln11 ln9ln1) Use k Obtain 6.6 [4] Any non-zero constant k with attempts at y values (in terms of ln or decimals); M0 if attempt does not involve eactly four strips; M0 if each y value initially amended, to ln( 4) for eample Or equiv involving decimals indicating use of correct values Allow greater accuracy ; any value rounding to 6.6 with no errors seen ii State or imply that integrand now involves ln( 4) or both Obtain 5. or 5.4 as final answer ln or ft [] Following their Simpson rule answer from (i), ie times their answer; allow greater accuracy; correct answer with no working earns B; second use of Simpson s rule leading to correct answer earns B, but B0 if incorrect; concluding with 5. or 5.4 (perhaps with some reference to area below ais) is A0 7

8 Question Answer Marks Guidance i Draw V-shaped graph with verte on positive -ais 7 State ( a, 0) and (0, 7 a ) [] And graph etending at least a little into second quadrant; condone minimal smoothing at the verte; allow graph which is asymmetrical about vertical line through verte unless it is an etreme case Can be earned if first not awarded; allow for 7 a and 7a marked on aes of graph or cases where zero coordinates are not given but are clearly implied ii Attempt to find two critical values By squaring both sides (giving terms on left) and solving quadratic equation or by solving two linear equations (one with signs of and 4a the same and one with the signs different) or using graph with horizontal line representing y 4a Obtain a and 11 a iii Conclude with a 11 a Relate ln N to their upper limit of (ii) with a 1.5 or proceed directly from inequality in (iii) to ln N 16.5 State the single value 87 [] [] 11 Allow the logically correct a and a but not conclusions such as 11, a a ; giving a a particular value means only is available; use of signs is final A0 A0 for N 87 ; A0 for N i Use identity sec 1 tan Attempt solution of -term quadratic equation in tan Obtain at least tan 4 from the correct equation [] Identity must be used not merely quoted If using factorisation, earned if their factors correct; if using formula, earned if substitution of their values into correct formula correct; for incorrect equation and two values produced with no working, check that values are correct given their equation so that can be awarded Ignore second value given provided no error at this stage is involved; so and 4 is, 4 only is, only is A0, and 4 is A0 ; allow solution such as y 4 when clear that y is tan ; ignore subsequent work with angles 8

9 Question Answer Marks Guidance tan ii a Attempt substitution into 1 tan Using any value from (i) Use 4 to obtain 8 15 and no other value [] Or eact equiv; full details to be shown; indication of use of calculator is M0; finding tan for both angles is A0; answer 8 with no working is M0A0; 8 final answer is A b State or imply cot( 15 ) is 1 tan( 15 ) Attempt substitution of their value from (a) into 1 tan tan15 tan tan15 or into tan tan15 1 tan tan15 Obtain and no other value 7 [] Either at beginning of solution or towards the end Allow with tan15 still present Or eact equiv; full details to be shown; allow 7 5 Differentiate to obtain k(4 ) 1 For any non-zero constant k 1 Obtain correct (4 ) Use negative reciprocal of gradient to find intersection of normal with -ais 5 Obtain for gradient of normal and hence 9 or equiv such as base of triangle is Or unsimplified equiv Using their attempt at first derivative; either using equation of normal 5 45 ( y ) or relevant right-angled triangle Integrate to obtain Obtain correct 1 6 p(4 ) (4 ) Use limits 15 and 7 to obtain for area under curve 4 6 Use triangle area to obtain 155 for shaded area 6 [8] For any non-zero constant p Or unsimplified equiv Allow calculation apparently using only upper limit 9

10 Question Answer Marks Guidance 6 i Translation parallel to -ais by 1 Must use term translate or translation, not move, not shift, etc.; translate by 1 is acceptable; in direction is acceptable; translate in negative 0 direction by 1 is B0 Stretch parallel to y-ais, factor [] Must use term stretch ; in y direction is acceptable; condone in y ais ; the two transformations can be given in either order ii State either or 0 or both State correct 0 f ( ) or 0 y or 0 f [] Allow any letter; accept < or, > or here for method mark iii Obtain epression of form a b or a b Obtain correct 1 or Reflection in line y [] For non-zero constants a and b; or equiv in terms of y In terms of now Or clear equiv such as one is the mirror image of the other iv Either Attempt correct process to find ff ( ) Obtain 1 or 4 1 Solve to obtain 5 Or equiv Or Attempt 1 1 f f () Obtain 1 as first value Obtain 5 with their f 1 [] 10

11 Question Answer Marks Guidance ln y 7 i State ln a Ignore any subsequent manipulation of right-hand side Differentiate to obtain d 1 d must be used; quotient rule may be used but must be correct dy yln a dy Rearrange to confirm d y a d ln a [] AG at least one intermediate step needed ii a Obtain derivative 4 4 ln 4 Equate attempt at first derivative to 8 and rearrange to form... Confirm 1 4 ln 4 [] Or equiv Where epression under cube root involves two terms at least one of which involves ; allow if there is one sign slip AG necessary detail needed b Carry out iteration process Obtain 1.7 for -coordinate Showing at least values after 1 Condone correct value eventually obtained after error in iteration process; answer required to precisely dp; ( ); iterates must be present and showing at least dp; answer only and no iterates shown earns 0/; treat sequence starting at value other than 1 as mis-read Obtain.79 for y-coordinate [] Answer required to precisely dp; using 1.7 to obtain.77 is A0; A0 is possible where iterates shown are not to at least dp( but values are perhaps in calculator) 8 i Use sin sin cos Obtain 6sin 8cos Obtain R 10 Attempt appropriate trigonometry to find Obtain 5.1 [5] Must be used not merely stated May be implied From correct 6sin 8cos 6 Allow for tan or equiv 8 Or greater accuracy 5.1 ; with no errors seen 11

12 Question Answer Marks Guidance ii State or imply equation is 10sin( 6.1 ) Carry out correct process to find one value of Obtain 99.4 (or 14 ) ft Following their R and ; Not available for finding negative angle; must involve use of nd quadrant angle Or greater accuracy Carry out correct process to find second value of Obtain 14 (or 99.4 ) Must involve use of 5th quadrant angle Accept value rounding to 14 providing no error; and no others between 0 and 60 [5] [Note: Solving 10sin( 5.1 ) can earn if correct processes followed; if continue to find correct angles by subtracting 10, available; can be retrospectively given even if answers are wrong] 9 a Differentiate using quotient rule or equiv p( ) ( p q) Obtain or equiv ( ) Equate derivative to zero and attempt discriminant Obtain 4q 1 p and observe it is positive [4] With negative sign in numerator, with ( ) in denominator and at least one of the two terms in the numerator correct Provided equation is a -term quadratic with p and q present With at least one reference to squared value being positive b Differentiate to obtain form e ( p q) dy Obtain e ( a b) ae d d y 4 Obtain e (4a 10a 4b a b) d Equate coefficient of Confirm 5a b 0 e to zero [ Or equiv Or equiv Provided second derivative involves others AG necessary detail needed 4 e, e and e terms and no 1

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