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1 UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Level MARK SCHEME for the May/June question paper for the guidance of teachers FURTHER MATHEMATICS / Paper, maimum raw mark This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the eamination. It shows the basis on which Eaminers were instructed to award marks. It does not indicate the details of the discussions that took place at an Eaminers meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes must be read in conjunction with the question papers and the report on the eamination. CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.

2 Page Mark Scheme: Teachers version Syllabus Paper GCE A LEVEL May/June Mark Scheme Notes Marks are of the following three types: M A B Method mark, awarded for a valid method applied to the problem. Method marks are not 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, e.g. by substituting the relevant quantities into the formula. Correct application of a formula without the formula being quoted obviously earns the M mark and in some cases an M mark can be implied from a correct answer. 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). Mark for a correct result or statement independent of method marks. When a part of a question has two or more "method" steps, the M marks are generally independent unless the scheme specifically says otherwise; and similarly when there are several B marks allocated. The notation DM or DB (or dep*) is used to indicate that a particular M or B mark is dependent on an earlier M or B (asterisked) mark in the scheme. When two or more steps are run together by the candidate, the earlier marks are implied and full credit is given. The symbol implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A or B marks are given for correct work only. A and B marks are not given for fortuitously "correct" answers or results obtained from incorrect working. Note: B or A means that the candidate can earn or. B// means that the candidate can earn anything from to. The marks indicated in the scheme may not be subdivided. If there is genuine doubt whether a candidate has earned a mark, allow the candidate the benefit of the doubt. Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. Wrong or missing units in an answer should not lead to the loss of a mark unless the scheme specifically indicates otherwise. For a numerical answer, allow the A or B mark if a value is obtained which is correct to s.f., or which would be correct to s.f. if rounded ( d.p. in the case of an angle). As stated above, an A or B mark is not given if a correct numerical answer arises fortuitously from incorrect working. For Mechanics questions, allow A or B marks for correct answers which arise from taking g equal to.8 or.8 instead of. UCLES

3 Page Mark Scheme: Teachers version Syllabus Paper GCE A LEVEL May/June The following abbreviations may be used in a mark scheme or used on the scripts: AEF AG BOD CAO CWO ISW MR PA SOS SR Any Equivalent Form (of answer is equally acceptable) Answer Given on the question paper (so etra checking is needed to ensure that the detailed working leading to the result is valid) Benefit of Doubt (allowed when the validity of a solution may not be absolutely clear) Correct Answer Only (emphasising that no "follow through" from a previous error is allowed) Correct Working Only often written by a fortuitous' answer Ignore Subsequent Working Misread Premature Approimation (resulting in basically correct work that is insufficiently accurate) See Other Solution (the candidate makes a better attempt at the same question) Special Ruling (detailing the mark to be given for a specific wrong solution, or a case where some standard marking practice is to be varied in the light of a particular circumstance) Penalties MR PA A penalty of MR is deducted from A or B marks when the data of a question or part question are genuinely misread and the object and difficulty of the question remain unaltered. In this case all A and B marks then become "follow through " marks. MR is not applied when the candidate misreads his own figures this is regarded as an error in accuracy. An MR penalty may be applied in particular cases if agreed at the coordination meeting. This is deducted from A or B marks in the case of premature approimation. The PA penalty is usually discussed at the meeting. UCLES

4 Page Mark Scheme: Teachers version Syllabus Paper GCE A LEVEL May/June Relevant working from A 5I or the equivalent in equations to obtain an eigenvector of the form An eigenvalue of A A is 5 5 Corresponding eigenvector, as above M A [] B B [] No penalty for not hence methods. Accept λ with and λ with Accept linear scaling of eigenvectors. n N sinα sin( nα ) cosα cos[ ( N ) α ] displayed result (AG) MA MA [] cos ( N ) π / oscillates finitely as n sin( nπ / ) Require π α, oscillate or values of cos( ) does not converge (CWO) B n π N given as or [] H : > for some k B k k ( 8) /( ) k k k MA k 8 > k > H k H A k > H is true B CWO Completion of the induction argument A [] UCLES

5 Page 5 Mark Scheme: Teachers version Syllabus Paper GCE A LEVEL May/June Alternatively for lines and : k MA k ( ) k > H k k H A k OR MA k k k k ( ) > A k > k OR k k k MA ( k ) > > A k > k k Minimum conclusion is Hence true for n [. d / dt t cost, dy / dt t sin t (both) B ( d dt) ( dy / dt) t / B π / S π ( t sin t t cost)dt (AEF) MA t sin t dt t cost sin t B t cost dt t sin t t cost sin t B S ( π / )[ π ] Accept forms such as π π /, etc. A [] OR for lines and 5 π / π [( cost) cost dt {( t sin t) t sin t dt} ] t (LNR) M π / [ t cost sin t t sin t t cost sin t] π (LNR) A UCLES

6 Page Mark Scheme: Teachers version Syllabus Paper GCE A LEVEL May/June 5 ( ) ( ) ( ) 5 ( ) ( ) ( ) 5 5 c c is c s c is 5 c is 5 c is is M 5 θ 5c s c s s M 5 ( s ) s ( s ) 5s s M 5 s s 5s (AG) A OEW [] sinθ sin5θ / M Roots are sin qπ where q /, /, /, 5/, / A A for root: A for further roots: A for completion [] CWO CWO Alternative answers 5 5 q,,,, or 55 q,,,, (i) One asymptote is B y /( ) M Require y non-zero constant. Other asymptote is y A [] Alternatively for last two marks: OR k ( ) /( ) for large ( k ) k for large M k k other asymptote is y A OR ( ) /( ) m c ( m ) ( m c ) c M Put m, m c to obtain other asymptote is y A (ii) Obtains any correct result for dy/d M [( ) ] /( ) / ( ) K dy / d A No comment required. [] (iii) Aes and asymptotes correctly placed Right-hand branch Left-hand branch B B B Deduct overall for bad form(s) at infinity [] UCLES

7 Page Mark Scheme: Teachers version Syllabus Paper GCE A LEVEL May/June t (i) d / dt ( t t ) e, dy / dt ( t ) t e (both, AEF) B Obtains displayed result (AG) MA [] (ii) Any correct result for d ( dy d) / dt / in terms of t (*) MAA M Quotient Rule AA for terms in numerator d y / d () dt / d epressed in terms of t M ( t t ) Simplify to, e.g., t ( t ) e t Two terms in numerator must be combined. [5] A 8 Complementary function is Ae Be MA Particular integral of form P sin Q cos, so that 5 P 5Q and 5P 5Q M P / 5, Q / 5 A General solution is y Ae Be.sin.cos A [5] Ae Be as, M tan φ M R., φ π arctan ( / ). 8 BA Accept R, but must be positive. [] UCLES

8 Page 8 Mark Scheme: Teachers version Syllabus Paper GCE A LEVEL May/June n n D[ s c] s ( n ) s c, n where D d / d, s sin, c cos M ( ) n n s ( n ) s n K A Integrates w.r.t. over the range [, π / ] to obtain ( n ) I n ( n ) I n M Result (AG) π / n π / n OR In [ cs ] ( n ) c s d A [] M ( n ) I n ( n ) I n (n [) MA [( n ) / n] I n I n [( n ) /( n )] I n n I for n [ A OR Starts with I n and relates to I n directly: mark as above OR I n π / sin n θ cosec θ dθ π n π / / n [ sin θ θ ] cot ( n ) sin θ cosθ cotθdθ (LNR) M π / n ( n ) sin θ ( sin θ ) dθ (LR) M ( ) I n ( n ) I n n A n ( n ) I ( n ) n I A UCLES

9 Page Mark Scheme: Teachers version Syllabus Paper GCE A LEVEL May/June y π / m π / m let u m y π / π / sin sin sin sin 8 8 m d m d u du u du M (LR) M A I π π π π I I B 5 π 5 π 5π I I B I 8 5 y (or.5) B I [] OR for last marks I I (oe) MA 5 y A (i) / y y y cy y MA [] (ii) α c MA α c A (M is for use of α ( α ) αβ in either part.) [] (iii) ( ) S α α α 8 c α MA Aft is for adding answers to (ii) correctly and subtracting 8. [] (iv) c S so that if all roots are real then α ± and similarly for β, γ, δ MA CWO This is impossible since e.g.,, αβγδ or any other contradiction A CWO [] UCLES

10 Page Mark Scheme: Teachers version Syllabus Paper GCE A LEVEL May/June (i) ( ) dr / dθ a / θ M Since a >, then dr / dθ <, points of C A [] OR a ( > ) constant and as ( > ) θ increases, θ increases M a θ decreases A (ii) a sinθ /( θ ) y M ( θ ) cosθ sin dy / dθ K θ A tan θ θ (AG) A tanθ θ.5 when θ.: tanθ θ. when θ. B [] OR equivalent argument for B such as: (.). <., tan(.).. tan > B or f ( θ ) tanθ ( θ ) f (.). f (.). (iii) Sketch: Approimately correct shape and placement for Y θ Y π/ and passing through a, and (.a, π / ), approimately, indicated in some way BB ( ) Maimum in interval ( /, π / ) π / (iv) A ( a / ) ( θ ) dθ π B [] M π / [ θ ] ( / ) ( ) a A ( ) K πa / π A Do not accept double minus signs or fractions in the denominator for the final mark. [] UCLES

11 Page Mark Scheme: Teachers version Syllabus Paper GCE A LEVEL May/June EITHER (i) (i j) (j k) i j k PQ (i j k).(i j k)/ / MA MA [] (ii) ( λ)/ ( λ µ)/( ) ( µ)/ (AEF) OR ( λ) ( λ µ), ( λ µ ( µ), both MA K µ MA Position vector of Q is i j k A [5] Parts (i) and (ii) together: λ PQ λ µ µ B PQ. 5 λ µ MA PQ. λ 5µ A λ, µ MA PQ. PQ MA OQ A [] UCLES

12 Page Mark Scheme: Teachers version Syllabus Paper GCE A LEVEL May/June UCLES (iii) (i j) (i j k) i j k MA ( )( ) [ ] /. k i k j i p M K A [] for final marks : z y π perpendicular distance or any other method. M is for a complete strategy and A for. OR (i) (a) K OR Establishes equivalent result for M T OR Establishes linearly independent relations for rows or columns MA ( ) ( ) dim R r M A (b) Basis for R T is, MA [5] (ii) Let 5 q p Solves any of: p q, p q 5, p q, p q to obtain p, q MA Checks consistency with other equations A []

13 Page Mark Scheme: Teachers version Syllabus Paper GCE A LEVEL May/June UCLES (iii) 5 M OR Works with equations (M written out fully.) (AG) B OEW e and e basis vectors for, Me Me T M µ λ µ λ µ λ,, 5 5 Me Me M A [] (iv) Need λ µ M µ λ A [] Accept a parametric answer e.g. t Augmented Matri Method (i) (a) ~ ~ 5 MA ( ) ( ) dim r R M A (b) Basis for R is, MA [5]

14 Page Mark Scheme: Teachers version Syllabus Paper GCE A LEVEL May/June UCLES (ii) and (iii) 5 t z y t z y MA µ λ t z y A, t z y µ λ B 5 M Me Me M 5 e e M µ λ A (iv) λ µ t z y MA [8]

9709 MATHEMATICS. 9709/11 Paper 1, maximum raw mark 75

9709 MATHEMATICS. 9709/11 Paper 1, maximum raw mark 75 CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Subsidiary Level and GCE Advanced Level MARK SCHEME for the May/June 0 series 9709 MATHEMATICS 9709/ Paper, maimum raw mark 75 This mark scheme is published