GCE. Mathematics. Mark Scheme for January Advanced GCE Unit 4726: Further Pure Mathematics 2

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1 GCE Mathematics Advanced GCE Unit 76: Further Pure Mathematics Mark Scheme for January

2 OCR (Oford Cambridge and RSA) is a leading UK awarding body, providing a wide range of qualifications to meet the needs of pupils of all ages and abilities. OCR qualifications include AS/A Levels, Diplomas, GCSEs, OCR Nationals, 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 Any enquiries about publications should be addressed to: OCR Publications PO Bo Annesley NOTTINGHAM NG DL Telephone: Facsimile: 6 publications@ocr.org.uk

3 76 Mark Scheme January t tan dt sec d ( t )d d. dt sincos t t t t t d t ln t ( c) t B For correct result AEF (may be implied) M For substituting throughout for A For correct unsimplified t integral M For integrating (even incorrectly) to aln f( t ). Allow or ln k tan ( c) A For correct epression k may be numerical, c is not required (i) f( ) tanh, f( ), f ( ) ( ) (ii) f ( ) ( ).( ).. OR ( ) ( ) ( ) ( ) 8 ( ) 8 ( ) OR ( ) ( ) ( ) ( ) ( ) f(), f(), f () f () tanh M A M A For quoting f( ) differentiate f( ) For f ( ) correct WWW and attempting to For using quotient OR product rule on f ( ) For correct unsimplified f ( ) A For simplified f ( ) WWW AG B M A 8 For all values correct (may be implied) f.t. from (i) For evaluating f () and using Maclaurin epansion For correct series (i)(a) Asymptote y B For correct equation (allow -ais) (b) METHOD a y a y a a y b ac a a y y METHOD a dy a a y d a a M M A A For epressing as a quadratic in For using b ac For a a y For correct range seen or implied M* For differentiating y by quotient OR product rule dy A For correct values of a y d M For finding y values and giving argument for range Asymptote, sketch etc y A For correct range (*dep) (ii)(a) y B For correct equation (allow -ais) (b) B For correct maimum f.t. from (i)(b) Maimum, minimum B For correct minimum f.t. from (i)(b) Allow decimals (c) B For correct set of values (allow in words) 9

4 76 Mark Scheme January (i) 8 6 e e 6 e e e e e e 8sinh e e B seen or implied sinh e e 8 M For attempt to epand 6 by binomial theorem OR otherwise 6 M For grouping terms for cosh or cosh OR using e or e epressions from RHS cosh cosh A For correct epression AG SR may be done wholly from RHS to LHS M M Evidence of factorising required for nd M B A (ii) METHOD cosh cosh (8sinh cosh ) cosh M For using (i) and cosh sinh 8sinh sinh A For correct equation sinh sinh sinh M For solving their quartic for sinh A For correct sinh (ignore other roots) A For correct answers (and no more) (i) ln ln f.t. from their value(s) for sinh SR Similar scheme for 8cosh cosh cosh ln METHOD cosh cosh (cosh ) cosh M For using cosh cosh cosh cosh A For correct equation cosh ln M For solving for cosh A For correct cosh (ignore others) ln A For correct answers (and no more) f.t. from value(s) for cosh METHOD Put all into eponentials M For changing to e e e e e A For correct equation e e e M For solving for e A For correct e (ignore others) e ln A For correct answers (and no more) f.t. from value(s) for e n n n n n (ii) F( ) n n 6 9 M A A M M For attempt at N-R formula For correct N-R epression For correct answer (necessary details needed) AG Allow omission of suffies For using quotient OR product rule to find F( ) For factorising numerator to show k F( ) since A For correct eplanation of AG (iii).87,.879,.8,.8 B First iterate correct to at least d.p. OR 7 (α = ).8 B B SR For starting value leading to another root allow up to B B B 9 For equal iterates to at least d.p. For correct α to d.p. Allow answer rounding to.8 SR If not N-R, B B B

5 76 Mark Scheme January 6 (i) dy y ln y ln ln y d M For differentiating ln yorln w.r.t. dy A For ln ln d seen or implied ln e For correct -value from fully correct A working AG (ii)..7.9 A M For areas of lower rectangles A.88(88).88 A For lower bound rounding to AG (iii).7.9 A M For areas of upper rectangles A For upper bound rounding to.8 A.77(77).8 (iv) y O e. y M A B Consider rectangle of height f(e ) Use at least lower rectangle, height f(e ) Use at least upper rectangle, height f() SR If more than one rectangle is used for either bound, they must be shown correctly O e. 7 (i) cos cos( ) OR cos cos( ) for all θ M For a correct procedure for symmetry related to the equation OR to cos equation is unchanged, so symmetrical about A For correct eplanation relating to equation AG (ii) r cos M For obtaining equation for tangents, A A for any values A A for all, no etras (ignore outside range) (iii) cos (d ) B For correct integral with limits soi (limits may be, at any stage) cos cos d M* For multiplying out, giving at least terms For integration to cos cos6d M A Bsin Csin 6 AEF M For completing integration and substituting sin sin 6 (*dep) their limits into terms in cos sin n A For correct area WWW

6 76 Mark Scheme January 8 (i) METHOD M For appropriate use of sinh cosh sinh (cosh ) sinh cosh METHOD, cosh ln sinh ln A For correct verification to AG M For attempted use of ln forms of sinh and cosh sinh (cosh ) A For both ln epressions seen (ii) (iii) METHOD cosh ln M For use of ln form of cosh and definition of sinh ln ln sinh cosh e e A For correct verification to AG SR Other similar methods may be used Note that ln ln cosh n In d M* For attempt to integrate cosh.cosh n by parts n n sinh.cosh For correct first stage of integration (ignore sinh.( n)cosh d A limits) n n M sinh. cosh ( n) cosh cosh d For using sinh cosh (*dep) For n A ( n) In In correct ( n) In In B For n correct at any stage n nin ( n ) In A 6 For correct result AG I cosh d sinh B For correct value I M 8 I A For using (ii) with n OR n For I I OR I I A For correct value

7 OCR (Oford Cambridge and RSA Eaminations) Hills Road Cambridge CB EU OCR Customer Contact Centre 9 Qualifications (General) Telephone: 998 Facsimile: 67 general.qualifications@ocr.org.uk For staff training purposes and as part of our quality assurance programme your call may be recorded or monitored Oford Cambridge and RSA Eaminations is a Company Limited by Guarantee Registered in England Registered Office; Hills Road, Cambridge, CB EU Registered Company Number: 866 OCR is an eempt Charity OCR (Oford Cambridge and RSA Eaminations) Head office Telephone: Facsimile: OCR

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