Cambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level. Published

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1 Cmbridge Interntionl Exmintions Cmbridge Interntionl Advnced Subsidiry nd Advnced Level MATHEMATICS 9709/ Pper October/November 06 MARK SCHEME Mximum Mrk: 75 Published This mrk scheme is published s n id to techers nd cndidtes, to indicte the requirements of the exmintion. It shows the bsis on which Exminers were instructed to wrd mrks. It does not indicte the detils of the discussions tht took plce t n Exminers meeting before mrking begn, which would hve considered the cceptbility of lterntive nswers. Mrk schemes should be red in conjunction with the question pper nd the Principl Exminer Report for Techers. Cmbridge will not enter into discussions bout these mrk schemes. Cmbridge is publishing the mrk schemes for the October/November 06 series for most Cmbridge IGCSE, Cmbridge Interntionl A nd AS Level components nd some Cmbridge O Level components. IGCSE is the registered trdemrk of Cmbridge Interntionl Exmintions. This document consists of 7 printed pges. UCLES 06 [Turn over

2 Pge Mrk Scheme Syllbus Pper Cmbridge Interntionl AS/A Level October/November Mrk Scheme Notes Mrks re of the following three types: M A B Method mrk, wrded for vlid method pplied to the problem. Method mrks re not lost for numericl errors, lgebric slips or errors in units. However, it is not usully sufficient for cndidte just to indicte n intention of using some method or just to quote formul; the formul or ide must be pplied to the specific problem in hnd, e.g. by substituting the relevnt quntities into the formul. Correct ppliction of formul without the formul being quoted obviously erns the M mrk nd in some cses n M mrk cn be implied from correct nswer. Accurcy mrk, wrded for correct nswer or intermedite step correctly obtined. Accurcy mrks cnnot be given unless the ssocited method mrk is erned (or implied). Mrk for correct result or sttement independent of method mrks. When prt of question hs two or more method steps, the M mrks re generlly independent unless the scheme specificlly sys otherwise; nd similrly when there re severl B mrks llocted. The nottion DM or DB (or dep*) is used to indicte tht prticulr M or B mrk is dependent on n erlier M or B (sterisked) mrk in the scheme. When two or more steps re run together by the cndidte, the erlier mrks re implied nd full credit is given. The symbol implies tht the A or B mrk indicted is llowed for work correctly following on from previously incorrect results. Otherwise, A or B mrks re given for correct work only. A nd B mrks re not given for fortuitously correct nswers or results obtined from incorrect working. Note: B or A mens tht the cndidte cn ern or 0. B//0 mens tht the cndidte cn ern nything from 0 to. The mrks indicted in the scheme my not be subdivided. If there is genuine doubt whether cndidte hs erned mrk, llow the cndidte the benefit of the doubt. Unless otherwise indicted, mrks once gined cnnot subsequently be lost, e.g. wrong working following correct form of nswer is ignored. Wrong or missing units in n nswer should not led to the loss of mrk unless the scheme specificlly indictes otherwise. For numericl nswer, llow the A or B mrk if vlue is obtined which is correct to s.f., or which would be correct to s.f. if rounded ( d.p. in the cse of n ngle). As stted bove, n A or B mrk is not given if correct numericl nswer rises fortuitously from incorrect working. For Mechnics questions, llow A or B mrks for correct nswers which rise from tking g equl to 9.8 or 9.8 insted of 0. UCLES 06

3 Pge Mrk Scheme Syllbus Pper Cmbridge Interntionl AS/A Level October/November The following bbrevitions my be used in mrk scheme or used on the scripts: AEF/OE Any Equivlent Form (of nswer is eqully cceptble) / Or Equivlent AG CAO CWO ISW SOI SR Answer Given on the question pper (so extr checking is needed to ensure tht the detiled working leding to the result is vlid) Correct Answer Only (emphsising tht no follow through from previous error is llowed) Correct Working Only often written by fortuitous nswer Ignore Subsequent Working Seen or implied Specil Ruling (detiling the mrk to be given for specific wrong solution, or cse where some stndrd mrking prctice is to be vried in the light of prticulr circumstnce) Penlties MR A penlty of MR is deducted from A or B mrks when the dt of question or prt question re genuinely misred nd the object nd difficulty of the question remin unltered. In this cse ll A nd B mrks then become follow through mrks. MR is not pplied when the cndidte misreds his own figures this is regrded s n error in ccurcy. An MR penlty my be pplied in prticulr cses if greed t the coordintion meeting. PA This is deducted from A or B mrks in the cse of premture pproximtion. The PA penlty is usully discussed t the meeting. UCLES 06

4 Pge 4 Mrk Scheme Syllbus Pper Cmbridge Interntionl AS/A Level October/November kx x = x k kx 4 x + k ( = 0) ( 4) 4 ( k)( k) soi Eliminte y nd rerrnge into - term qud b 4c. k >, k < co Allow (, ) etc. Allow <k< k [] ( / ) 0 ( x ), 0 ( x ) + soi Ech term cn include x = 00 oe = 4 Must hve terms nd include nd sin x 6cos x tn x = = or 4sin x = 6( sin x) 4 [tn x = (±).5 or sin x = (±) or cos x = (±)0.65] x= 50.8 (Allow (rd)) Another ngle correct x = 50.8, 9., 0.8, 09. [ 0.886,.5/6, 4.0, 5.40 (rd) ] Or ( x) 4 cos = 6cos x Or ny other ngle correct Ft from st ngle (Allow rdins) All 4 ngles correct in degrees f x = x 6x 9 soi 4 f = 0 or f 0 or f 0 soi ' ' Attempt to solve ( x) ( x) > ( x) ( x )( x+ ) or, seen or only seen Lest possible vlue of n is. Accept n =. Accept n With or without equlity/inequlity signs Must be in terms of n π 5 (i) cos0.9 = OE / 6 or = sin 0.9 oe OE = 6cos0.9 =.7 oe AG [] Other methods possible (ii) Use of (π.8 ) or equivlent method Are of lrge sector ½ 6 ( π.8) = oe Are of smll sector ½.7.8 Totl re = 80.7(0) +.5() = 9. Expect 4.48 Or π 6 ½6.8. Expect Expect.5 Other methods possible 6 (i) + x = n x = n m + y = 6 y = m [] No MR for (½(+n), ½(m 6)) n, Expect ( m) UCLES 06

5 Pge 5 Mrk Scheme Syllbus Pper Cmbridge Interntionl AS/A Level October/November (ii) Sub their x, y into y = x+ m= n + m + 6 = oe Not nested in n eqution n Eliminte vrible m= 9, n= * D Expect m+ n= Expect m n= 8 Note: other methods possible 7 (i) AB.AC = = 0 hence perpendiculr or 90 AB.AD = + 4 7= 0 hence perpendiculr or 90 AC.AD = 8+ 7= 0 hence perpendiculr or 90 AG [] or sum of prods etc must be seen Or single sttement: mutully perpendiculr or 90 seen t lest once. (ii) Are ABC = ( ½) ( ) + ( ) = ½ 6 Expect ½ 66 Vol. = ⅓ their ABC = = 6 Not.0 8 (i) ( x + ) + Cnnot score retrospectively in (iii) (ii) ( x) g = x+ co [] [] For =, b=, c= In (ii),(iii) Allow if from 4 x + + (iii) y = x+ + x+ = ± y or ft from (i) Or with x/y trnsposed. y x= ± or ft from (i) ( fg) ( x) = x co Note lt. method Domin is ( x ) > 0 ALT. method for first mrks: Trying to obtin g f ( x) ( x ) g = ½, f = x for x g f * D Or with x/y trnsposed Allow sign errors. Must be function of x. Allow y =... Allow (0, ), 0 < x < etc. but not with y or f or g involved. Not 0 Both required UCLES 06

6 Pge 6 Mrk Scheme Syllbus Pper Cmbridge Interntionl AS/A Level October/November () 6 = r + r r = S = 9 [] (b) cos sin 5 θ + θ = cosθ + ( cos² θ) = 8 6cos θ cosθ = 0 cosθ = / or / soi θ = 0.84,.09 Dep on previous * D Use of correct formul for sum of AP Use s = c & simplify to - term qud Accept 0.68π, π/. SR for 48., 0 Extr solutions in rnge 0 (i) t dy x = = + + = dx, or or y = x or y= x+ c = + c y = x or x co [] + or + seen nywhere in (i) Through (,) & with their grd s f() (ii) ( y) ½ ½ x x = + (+ c) ½ ½ sub x=, y= into d y /dx ½ 4x ½ c = ( y = x + ) c must be present. Expect = 4 + c (iii) 4 sub x= 6, y= 8 8 = = 0 = A = (4, ), B = (6, 8) AB = + 5 AB = * D Sub into their y Allow 6 in ddition UCLES 06

7 Pge 7 Mrk Scheme Syllbus Pper Cmbridge Interntionl AS/A Level October/November d y = + k = 0 dx (i) Attempt diffn. nd equte to 0 k( kx ) ( kx ) = or k x 6k x+ 8 k( = 0) 4 x = or k k d y = k kx dx d y When x ( k ) =, 0 k dx = < MAX All previous 4 d y When x ( k ) =, 0 k dx = > MIN working correct * D ** D D Must contin ( kx ) term(s) Simplify to qudrtic Legitimtely obtined + other Ft must contin Ak ( kx ) where A>0 Convincing lt. methods (vlues either side) must show which vlues used & cnnot use x = / k [7] (ii) V = ( π ) ( x ) + ( x) ( π ) ( x ) ( x ) dx = [ + + ]dx ( x ) ( x ) = ( π ) ( x ) + + = ( π ) = 40 π / oe or 4.9 Condone missing x * D Attempt to expnd y² nd then integrte Or x ( x ) + x + 9x+ x Apply limits 0 missing 8 π / scores A0A0 UCLES 06

4037 ADDITIONAL MATHEMATICS

4037 ADDITIONAL MATHEMATICS CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinry Level MARK SCHEME for the October/November 0 series 07 ADDITIONAL MATHEMATICS 07/ Pper, mximum rw mrk 80 This mrk scheme is published s n id to techers