Cambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level MATHEMATICS 909/1 Paper 1 May/June 01 MARK SCHEME Maximum Mark: 5 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 will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the May/June 01 series for most Cambridge IGCSE, Cambridge International A and AS Level components and some Cambridge O Level components. IGCSE is the registered trademark of Cambridge International Examinations. This document consists of printed pages. UCLES 01 [Turn over
Page Mark Scheme Syllabus Paper Cambridge International AS/A Level May/June 01 909 1 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 0. B/1/0 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 (1 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 9.81 instead of 10. Cambridge International Examinations 01
Page Mark Scheme Syllabus Paper Cambridge International AS/A Level May/June 01 909 1 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 extra 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 Approximation (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 1 PA 1 A penalty of MR 1 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 approximation. The PA 1 penalty is usually discussed at the meeting. Cambridge International Examinations 01
Page 4 Mark Scheme Syllabus Paper Cambridge International AS/A Level May/June 01 909 1 1 1 5C ( x ) x 10( 1) 90 (x) ( ) ( x 1) π + dx 4 x ( π ) + x 4 π or 18.8 [] D Can be seen in expansion Identified as leading to answer Attempt to resolve to integrate y and attempt Applying limits 0 and. (Limits reversed: Allow M mark and allow A mark if final answer is π) (i) + k = k = 4 [1] (ii) x 4 ( y) = x (+c) 9 = + + c c must be present ( y) = x + x + 5 k ft on their k. Accept + x Sub (1,9) with numerical k. Dep on attempt Equation needs to be seen Sub (, ) c = 1½ scores A0 4 1 1 r = + d or + d or r = + d + d 1 correct equation in r and d only is sufficient ( d) ( 1d) + = + oe OR sub d = r Eliminate r or d using valid method ( 4) d( d ) = 0 OR r = 18r 15 r 1 r 5 ( )( ) D Attempt to simplify and solve quadratic d = r = 5 [5] Ignore d = 0 or r = 1 Do not allow 5 or ±5 Cambridge International Examinations 01
Page 5 Mark Scheme Syllabus Paper Cambridge International AS/A Level May/June 01 909 1 5 dy dx = [8] + [ ] [ ( x ) ( x ) 1 ] = 0 4 1 = 1 oe eg 1x 1x+ = 0 1 x = and 4 4 d y = 8 ( x 1) dx 1 When x = 4, d y ( = 4 ) and/or < 0 MAX dx When x = 4, d y ( = 4) and/or > 0 MIN dx B,1,0 * D D [] Set to zero, simplify and attempt to solve soi Needs both x values. Ignore y values ft to ( ) k x 1 where k > 0 Alt. methods for last marks (values either side of 1/4 & /4) must indicate which x-values and cannot use x = 1/. () 1 1 1 BAC = sin ( / 5) or cos (4 / 5) or tan ( / 4) 1 1 1 ABC = sin (4 / 5) or cos ( / 5) or tan (4 / ) Accept.8()º Accept 5.1()º ACB = π / (Allow 90º) Shaded area = ABC sectors (AEF + BEG + CFG) 1 ABC = 4 oe Sum sectors = 1 0.45 ) + 0.9 + 1 1.508] π OR ( ) ( ).8 0 + 5.1 + 1 90 5.5 = 0.44 [] dy 1/ x 5x 5 dx = + dy dx = 1/ x 5x + 5= ( ) 1/ x + = or equivalent -term x 5 0 quadratic Attempt to solve for ( x 1/ )( x 1/ ) 1 = 0 1/ x = / and 1 x = 9 / 4 and 1 1/ x e.g. D [] Equate their dy/dx to their or ½. Dep. on -term quadratic ALT ½ ( ) 5x = x+ 5x= x+ ( ) 4x 1x+ 9 = 0 x = 9 / 4 and 1 Cambridge International Examinations 01
Page Mark Scheme Syllabus Paper Cambridge International AS/A Level May/June 01 909 1 8 (i) sin x cos x+ cos x= 0 Use s = 1 c and simplify to -term quad cos x = / 4 and 1 Multiply by cos x Expect 4c c = 0 x =.4 (allow 0.π ) or 0 (extra in range max 1) [5] SC1 for 0. (or 0.π), π following 4c + c = 0 (ii) x = π their.4 or 0 18. x = 1.1 (0.85π), 1.9 (0.14/5π), 0, π (.14) (extra max 1) [] Expect x =.8 Any correct. Remaining correct. SCfor all 9., 110., 0, 180 (degrees) SC for.1, π/,.8 after 4c + c = 0 9 (i) 1 AB = OB OA = p + 4 4 CB = OB OC = 5 p ( p ) ( p ) 1+ 4+ + 4 = 1+ 5+ p = Ignore labels. Allow BA or BC (ii) AB.CB = 4+10 5 = 9 AB = 1+ 4+ 5 = 0, CB = 1 + 5 + 1 = 4 9 9 cos ABC = or 0 4 5 ABC = 5. or 1.1rads (ignore reflex angle 85 ) 10 (i) ( ) ax + b + = x 1 a =, b= 1 [] Use of xx 1 + yy 1 + zz 1 Product of moduli Allow one of AB, CB reversed - but award A0 (ii) x 1 0 or x 1 x i.e. (max) q = [] Allow = or or > or <. Ft from their a, b Must be in terms of q (eg q ) (iii) y ( y) ( ) 1 range is [1] Do not allow y >.Accept all other notations e.g. [, ) or [, ] Cambridge International Examinations 01
Page Mark Scheme Syllabus Paper Cambridge International AS/A Level May/June 01 909 1 y + 1 (iv) y = x 1 x= ( ± ) 1 x 1 = ( fg) ( x) + Domain is x [] Allow y =.... Must be a function of x ft from their part (iii) but x essential 11 (i) AB = + = 85, BC = + 9 = 85 ( isosceles) Or AB = BC = 85 etc AC = 8 + = 8 M = (, ) or BM = ( 85) (½ 8) Where M is mid-point of AC BM = + 8 = 8 or 85 1 = 8 Area 1 ABC = 8 8 = 4 [] (ii) Gradient of AB = / Equation of AB is y+ 1= ( x+ ) Gradient of CD = / 14 = x+ x Equation of CD is y+ = ( x ) Sim Eqns 4 x = = oe 85 5 [] Or y = ( x 4) Cambridge International Examinations 01