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


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1 GCE Mathematics Unit 473: Core Mathematics 3 Advanced GCE Mark Scheme for June 06 Oxford Cambridge and RSA Examinations
2 OCR (Oxford 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 notforprofit 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 examination. It shows the basis on which marks were awarded by examiners. It does not indicate the details of the discussions which took place at an examiners meeting before marking commenced. All examiners are instructed that alternative correct answers and unexpected 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 examination. OCR will not enter into any discussion or correspondence in connection with this mark scheme. OCR 06
3 473 Mark Scheme June 06. Annotations and abbreviations Annotation in scoris Meaning and BOD Benefit of doubt FT Follow through ISW Ignore subsequent working M0, M Method mark awarded 0, A0, A Accuracy mark awarded 0, B0, Independent mark awarded 0, SC Special case ^ Omission sign MR Misread Highlighting Other abbreviations Meaning in mark scheme E Mark for explaining U Mark for correct units G Mark for a correct feature on a graph M 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 3
4 473 Mark Scheme June 06. Subjectspecific 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 unexpected 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 A cannot ever be awarded. B Mark for a correct result or statement independent of Method marks. 4
5 473 Mark Scheme June 06 E A given result is to be established or a result has to be explained. This usually requires more working or explanation 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, exactly 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 candidatebycandidate rather than questionbyquestion. f Wrong or missing units in an answer should not lead to the loss of a mark unless the scheme specifically indicates otherwise. Candidates are expected to give numerical answers to an appropriate degree of accuracy, with 3 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 3 is expected) should not normally be penalised, while answers which are grossly over or underspecified 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. 5
6 473 Mark Scheme June 06 g Rules for replaced work If a candidate attempts a question more than once, and indicates which attempt he/she wishes to be marked, then examiners should do as the candidate requests. If there are two or more attempts at a question which have not been crossed out, examiners should mark what appears to be the last (complete) attempt and ignore the others. NB Follow these mathsspecific 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; 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
7 473 Mark Scheme June 06 Question Answer Marks Guidance Differentiate to produce form m n kx( x ) kx ( x ) *M For positive integers k, k, m, n ; allow M if slip to, for example, ( x 3) in both brackets 6 5 Obtain 6 x( x ) 8 x ( x ) A Or unsimplified equiv Substitute x to obtain value A From correct work only Attempt equation of tangent (not normal) through point (, 3) M Dep *M; using nonzero numerical value of gradient; condone slip in use of coordinates Obtain y x 5 A [5] Answer required in y mx c form i k k3 Expand to produce form k x x M For nonzero constants k, k, k 3 ; allow if middle term appears as two, so far, unsimplified terms Obtain 4x 4ln x or 4x 4ln x x A Condoning absence of modulus signs but A0 x if expression involves ln x or 4ln x ii Integrate to obtain form 4 k(4x ) 3 M Any nonzero constant k Obtain (4x ) A With coefficient simplified Include + c or + k at least once anywhere in answer to question [5] Even if associated with incorrect integral 7
8 473 Mark Scheme June 06 Question Answer Marks Guidance 3 i Obtain 8 for value corresponding to 0 Allow any value rounding to 8 Obtain 65.5 for value corresponding to 5 Allow any value rounding to 65 or 66; whether obtained using powers of 0.8 or by use of formula [] ii Attempt to find formula for m of form 00e kt or 00 r t M Whether attempted in part (i) or (ii) If formula attempted in part (i), marks earned must be recorded in part (ii) (0.ln 0.8) t Obtain 00e t or 00e 0. or t or t A Or equiv Show correct process for solving equation of form 00e t 50 or 00r 50 M Obtain 3 A Or greater accuracy rounding to 3; ignore any units given; second M is implied by correct answer Special case: no formula anywhere and answer 3 (or greater accuracy) given, award B (i.e. /4 for part (ii)) [4] 4 Use identity sec A tan A Attempt solution of threeterm quadratic equation to obtain two values of tan A M Implied if correct values obtained; allow M for incorrect factorisation provided expansion would give correct first and third terms; allow M for incorrect use of formula if only one error present Obtain tan A 3 and tan A 4 A And no others; implied by A tan 3 and tan 4 ; A 3, 4 is A0 here unless subsequent work shows values used correctly Use correct identities to produce equation in tanb only M Equation might be t 3 7 or t 5 t 3 7t 7 0 State tan B 3 A And no others 8
9 473 Mark Scheme June 06 Question Answer Marks Guidance Substitute at least one pair of nonzero numerical values into tan A tan B tan Atan B M Must be the correct identity Obtain one of or exact equiv 3 4 A Obtain the other exact value or equiv A And no others [8] 5 i e x x 3 x Either State 8e and so e 8 Obtain e x and hence x ln AG; necessary detail needed Verifying by substitution of ln in each equation earns B0B0 Or State e x x 8e and so 3 e x 8 State 3x ln8, x ln8 3 and hence x ln AG; necessary detail needed Going immediately from x ln8 to 3 x ln does not earn the second Or State e x x 8e and x ln8 x State 3x ln8, x ln8 3 and hence x ln [] AG; necessary detail needed Going immediately from x ln8 to 3 x ln does not earn the second ii Integrate to obtain Obtain correct separately, 8e x and x x ke and ke x x e e x 8e or, if done M A Any nonzero constants k and k Apply limits 0 and ln correctly to their integral(s) Obtain at least 4 8 (or equivs) Obtain 5 or equiv M *A A [5] Condone one sign slip; earned by sight of ln ln 8e e 8 (or equivs if integrals treated separately) Final A dependent on *A M also implied by sight only of 4 8 (or equivs ) 9
10 473 Mark Scheme June 06 Question Answer Marks Guidance 6 State, at some stage, a(4 b) 8 Obtain derivative Obtain derivative Obtain correct 4 4x 7 for C b kx( x ) for C b ax( x ) M A Any nonzero constant k Equate derivatives with x Attempt values of a and b from two equations involving a and (4 b) M M Using correct process Correct equations are a(4 b) 8 and Obtain a 6 Obtain b 5 A A b a(4 ) 4 [8] 7 i Draw more or less correct sketch of y cos x existing in first and second quadrants Draw Ushaped parabola passing through origin and showing minimum point * * Ignore any curve outside 0 y ; condone no or wrong intercepts on axes Curve must exist in first and third quadrants Indicate one intersection in first quadrant by blob or reference in words or Dep *B *B [3] 0
11 473 Mark Scheme June 06 Question Answer Marks Guidance ii Obtain correct first iterate showing at least 4 s.f. rounded or truncated Show iterative process to produce at least three iterates in all showing at least 3 s.f. M Obtain at least four correct iterates in all showing at least 4 s.f. Conclude with value 0.4 A A [4] Implied by incorrect values apparently converging Allowing recovery after error Answer to be clearly indicated by underlining final value in sequence or by separate statement; answer required to precisely 3 s.f.; allow final A even if iterates have been shown to only 3 s.f.; answer only earns 0/ iii State y cos ( x) or y cos x State y x( x 5) or equiv State 0.4 for xcoordinate State.33 for ycoordinate FT Allow y x(( x) 5) or similar; condone missing y in each case Following their answer to (ii); allow greater accuracy here Allow value rounding to.33 [4] 8 i State range of f is f ( x) 3a or y 3a State range of g is all real numbers or equiv such as y R (real numbers) [] Allow f 3a or equiv expression in words but 3a to be included
12 473 Mark Scheme June 06 Question Answer Marks Guidance ii State function is not or different x values give same yvalue or equiv no credit for no inverse due to modulus nor for cannot be reflected across y x Obtain form k( y 4 a) or k( x 4 a) M for nonzero constant k Obtain ( 4 x 4 a A Must finally be in terms of x [3] iii Either Attempt composition of functions the right way round Obtain 5 x a a 3a or equiv M A Earned for 5(what they think f(x) is) 4a Or Apply their g to 3a Obtain x a 3a 7a or equiv M A Either Solve x a 4a and obtain 3 a Solve linear equation in which signs of (their) x and (their) 4a are different 5 Obtain a FT M A Following their x a ka Condone other sign slips 5 And no others; obtaining and then concluding 5 a is A0 a Or Square both sides and obtain 4x 4ax 5a 0 Solve 3term quadratic equation to obtain two values 5 3 Obtain a, a FT M A [5] Following their x a ka Allow M if factorisation wrong but expansion gives correct first and third terms; allow M if incorrect use of formula involves only one error And no others; continuing from two correct answers to conclude 5 a, 3 a is A0
13 473 Mark Scheme June 06 Question Answer Marks Guidance 9 i Use sin sin cos State sin cos or tan cos sin tan Perhaps as part of expression Simplify using correct identities Note that going directly from M sin cos to is M0 but Obtain correctly A AG; necessary detail needed (sin cos ) to is MA [4] ii a Obtain expression involving at least one of sin and sin 6 Obtain 4 sin sin 6 4 Obtain 4 or exact equiv M A A Or equiv involving cosecant Answer only is 0/3 [3] b Use sin 4 sin cos 5 Obtain cos or cos or sin 8 Obtain or 0.66 [3] Or greater accuracy; and no others between 0 and ; allow 0. but not ; answer only earns 0/3 c 4 k Express in form k sin 3 sin 4 8 Obtain 4sin and hence 3sin 3 sin M A [] A0 if ( sin ) involved in simplification 3
14 OCR (Oxford Cambridge and RSA Examinations) Hills Road Cambridge C EU OCR Customer Contact Centre Education and Learning Telephone: Facsimile: For staff training purposes and as part of our quality assurance programme your call may be recorded or monitored Oxford Cambridge and RSA Examinations is a Company Limited by Guarantee Registered in England Registered Office; Hills Road, Cambridge, C EU Registered Company Number: OCR is an exempt Charity OCR (Oxford Cambridge and RSA Examinations) Head office Telephone: Facsimile: OCR 06
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