Cambridge Assessment International Education Cambridge International Advanced Subsidiary and Advanced Level MATHEMATICS 9709/7 Paper 7 Paper 7 MARK SCHEME Maximum Mark: 50 Published This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes should be read in conjunction with the question paper and the Principal Examiner Report for Teachers. Cambridge International will not enter into discussions about these mark schemes. Cambridge International is publishing the mark schemes for the series for most Cambridge IGCSE, Cambridge International A and AS Level components and some Cambridge O Level components. IGCSE is a registered trademark. This document consists of 9 printed pages. UCLES [Turn over
Mark Scheme Notes Marks are of the following three types: M 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. A B 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 FT 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 0. B//0 means that the candidate can earn anything from 0 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 9.8 or 9.8 instead of 0. UCLES Page of 9
The following abbreviations may be used in a mark scheme or used on the scripts: AEF/OE Any Equivalent Form (of answer is equally acceptable) / Or Equivalent AG CAO CWO ISW SOI SR Answer Given on the question paper (so extra checking is needed to ensure that the detailed working leading to the result is valid) 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 Seen or implied 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 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. PA This is deducted from A or B marks in the case of premature approximation. The PA penalty is usually discussed at the meeting. UCLES Page of 9
(a)(i) Po(.54) M seen or implied Po(540 0.00) e.54 ( +.54) M any λ Allow end error = 0.7 ( sf) A (a)(ii) n large and p small (or np (=.54) < 5) B n > 50, p < 0. (b) µ = 5.6 B σ =.7 ( sf) B Accept 5.6 (i) 40 480± z 5 M Must be a z value z =.6 B Accept.6 -.9 454/455 to 55/56 or 450 to 50 ( sf) A Must be an interval UCLES Page 4 of 9
(ii) x= 4840 B or width = 80 or half width = 40 4840 +.96 40 = 4980 OE M or 40 =.96 n 40 OE n n = 95 A CAO must be an integer (i) m = 98. 00 = 0.98 B Accept either 00 04.5 s = 0.98 (= 0.858) 99 00 or var = 0.0869 H 0 : Pop mean mass =.0 H : Pop mean mass <.0 M B not just mean, but allow just µ ± 0.98.0 0.858 00 M 0.98.0 ± M 0.8487 00 = 0.980 ( sf) accept ± A = 0.985 ( sfs) accept ± A Comp with z =.645 (or areas 0.65 > 0.05) No evidence that (mean) mass is less than.0 M Valid comparison of z s or area s A FT Correct conclusion FT their z 7 UCLES Page 5 of 9
(ii) Distr of X normal (so distr of X normal) Must state or imply No B X/parent population 4(i) a d x 0 k x = (k[x 0.5 ] 0 a = ) M Attempt int f(x) and = ignore limits A OE; a correct eqn in k & a after sub limits ka 0.5 = or a = 4k a x d x 0 k x = e.g. ka.5 = or a = 8 4k M Attempt int xf(x) and = A OE; a correct eqn in k and a after sub limits e.g. a = 8 or e.g. k = a = 9 8 4 9 M Attempt eliminate one letter A Convincingly obtained e.g. k = 9 54 k = 6 AG A 7 UCLES Page 6 of 9
4(ii) m x 6 d = 0.5 OE x 0 M Attempt int f(x), unknown limit and = 0.5 m 0.5 = 0.5 A a correct equn in m after sub limits m =.5 A 5(i) E(X Y) = 56-4 (= ) B Var(X Y) = 6 + 5 (= 6) M 0 (=.664) M Ignore any attempted cc/no SD/var mixes. '6' var must be attempt at a combination ɸ('.664') = ɸ('.664') M For area consistent with their working = 0.95 ( sf) A Similar scheme for use of Y X 5 UCLES Page 7 of 9
5(ii) E(M) = 56 +.5(4) (= 0.5) B Var(M) = 6 +.5 5 (= 9.5) M 5 0.5 (=.50) M Ignore any attempted cc/no SD/var mixes. '9.5 var must be attempt at a combination ɸ('.50') M For area consistent with their working = 0.0655 or 0.0656 or 6.55% or 6.56% ( sf) As final answer A Allow 6.6% or 6.5% or 7% if correct working seen 5 6(i) H 0 : Pop mean no. defectives = 5.5 H : Pop mean no. defectives < 5.5 P(X ) B or =.0 (per day) not just mean, but allow just λ or µ M Attempted. Any one term error/end error/incorrect λ/expression = e 5.5 ( + 5.5 + 5.5 M Correct expression attempted ) = 0. A Comp with 0. No evidence to believe mean no. of defectives has decreased M Valid comparison A FT Correct conclusion (FT their value) No contradictions 6 UCLES Page 8 of 9
6(ii) BOTH P(X ) = e 5.5 ( + 5.5) (= 0.057) AND P(X ) = = B* (Could be seen in (i)) e 5.5 ( + 5.5 + )= (0.) 5.5 Comp either with 0. P(Type I error) = 0.057 ( sf) DB One comparison with 0.0 (could be seen in (i)) B 6(iii) Actually mean =.0 but conclude that mean <.0 B Mean no. of defectives not reduced, but conclude that it is reduced. UCLES Page 9 of 9