Cambridge International Examinations Cambridge International Advanced Level FURTHER MATHEMATICS 9/ Paper October/November 6 MARK SCHEME Maximum Mark: 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 October/November 6 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 5 printed pages. UCLES 6 [Turn over
Page Mark Scheme Syllabus Paper Cambridge International A Level October/November 6 9 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 9.8 or 9.8 instead of. UCLES 6
Page Mark Scheme Syllabus Paper Cambridge International A Level October/November 6 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 6
Page 4 Mark Scheme Syllabus Paper Cambridge International A Level October/November 6 9 = r r+ r r+ ( )( ) n = OE + 5 + + = + + ( r) r= n n n ( ) [4] n = = n+ n+ r= r ( ) [] αβ = 9 αβ = 4 Use of, e.g.: α αβγ = α ( α αβ ) α = α + α αβ αβγ or ( ) Correct substitution in formula αβγ = 7 Required cubic equation is x x x + 4 + 7= must see final equation ALT METHOD: S S +4S + r = r = + 4 r=7 [6] UCLES 6
Page 5 Mark Scheme Syllabus Paper Cambridge International A Level October/November 6 9 P =, D = soi P AP= D A= PDP soi P = PD = or DP = A =. ALT METHOD: Av = v, Av = v, Av = v, Multiply out, solve each equation,,, A = [7] UCLES 6
Page 6 Mark Scheme Syllabus Paper Cambridge International A Level October/November 6 9 4 n n + r r n! n! n! = + = + r! n r+! r! n r! r! n r! n r+ r ( ) ( ) ( ) ( ) ( ) n! r+ n r+ ( n! ) = = = ( r! ) ( n r)! r( n r ) r! ( n r! ) + n + + + r [] ( a+ x) = a+ x= a+ x H is true. Assume H k is true,i.e. k k k k k k k r r k k ( a+ x) = a + a x+ + a x + + x r k k r+ r k k k+ Multiplying by ( a+ x), the coefficient of a x is: + = r r r H k+ is true. Hence H n is true for all positive integers. [4] UCLES 6
Page 7 Mark Scheme Syllabus Paper Cambridge International A Level October/November 6 9 5 (i) 5 7 8 7 9 9 6 4 7 r(a) = 4 = 5 7 5. [] (ii) Basis for range space is 8, 6 4 oe [] (iii) x+ y+ 5z+ 7t = y z 5t = Solving 9 9 5 A basis for null space is, oe [4] UCLES 6
Page 8 Mark Scheme Syllabus Paper Cambridge International A Level October/November 6 9 m + 7m+ = m+ m+ 5 = m= or 5 6 ( )( ) CF: Ae + Be t 5t PI: x = psin t+ qcos t x = pcos t qsin t x= 4 psin t 4qcos t Substituting: 4 p+ 6q= 6p 4q= 6 Solving: p =, q = 7 x= sint 7cost GS: 5 e t t x A = + Be + sint 7cost [8] For large positive values of t x sint 7cost. [] UCLES 6
Page 9 Mark Scheme Syllabus Paper Cambridge International A Level October/November 6 9 x e 4 e 7 (i) M.V. = = ( =.49) [] (ii) Coordinates are x = xe e x x dx dx and e y = 4x dx x e d x x x 4 ( ) e dx = e = e (=.498 ) (N.B. As in (i)) x x x x 4 x x xe e e e 5e xe dx= + dx= = + =.7.. 4 4 4 4x 8 4x e e ( ) e d (.495 ) x = = = (not straight from calculator) 4 4 4 x =.46, y =.55 (Final answers in algebraic form SR) [9] UCLES 6
Page Mark Scheme Syllabus Paper Cambridge International A Level October/November 6 9 8 x+ 4 ( xy' + y) yy' = (*) y' = x+ 4y= x= y (AG) At stationary points 4y 8y y + = 5y = y=± Coordinates of stationary points are (4, ) and ( 4,) From (*) : x+ ( xy' + y) yy' = Differentiating: + ( xy + y + y ) yy [ y ] ( ) ' + = (or by quotient rule) At (4, ) with y = : + 8y + y = y = maximum. At ( 4, ) with y = : 8y y = y = minimum. [] [8] UCLES 6
Page Mark Scheme Syllabus Paper Cambridge International A Level October/November 6 9 9 x x x [ x x] x x [ x] sin d = cos + cos d = + sin = [] n n In = x cos x + nx cos xdx = n + n ( ) n x cos x nx sin x n n x sin xdx n ( ) = n n n I n [4] du x = cos u u = cos x = sin x ; u=, x= /, dx ( ) / cos u du = x sin xdx= x sin xdx= I / I = I = 6 4 (OE) [5] UCLES 6
Page Mark Scheme Syllabus Paper Cambridge International A Level October/November 6 9 z = cos nθ + isin nθ and z = cos nθ + isin( nθ) = cos nθ isin nθ n n ( ) z n n + = cos n θ and z = isin n θ. (AG) n n z z 4 z z+ = z z z z z z 4 = z + z + 4 z z 6 4 = z + z z 4 6 + 4 + + z z z 4 64sin θcos θ = cos 6θ 4cos 4θ cos θ + 4 sin θcos θ ( cos6θ cos 4θ cos θ ) 4 (Or st bracket expanded) (and nd bracket expanded) (Expanded and grouped) = + (AG) (Uses initial result) 4 4 sin cos d = sin 6 sin 4 sin + θ θ θ θ θ θ θ 6 4 4 = + = or.8 (sf) 6 9 ALT METHOD: Expand and group each bracket separately and sub trig:,, Multiply together two brackets and manipulate trig expressions, LHS = RHS [] [7] [] UCLES 6
Page Mark Scheme Syllabus Paper Cambridge International A Level October/November 6 9 E i j k = 4 5 Obtains three equations. E.g. from OP + PQ = OQ t s+ k = 4 t+ 4s k = t+ s+ 5k = 4 Solves: s =, t =, k = p = i + j + k and q= i+ j k (Both required) ALT METHOD: Find PQ in terms of s,t Calculate scalar products direction vectors for l and l,, Solve simultaneous equations p and q i j k 4 = 9 5 5 r = + λ 9 + µ 5 5 [7] [] UCLES 6
Page 4 Mark Scheme Syllabus Paper Cambridge International A Level October/November 6 9 Direction of l is i j k 4 4 4 = ~ Normal to Π is i j k 9 9 5 = = 4 5 6 Cartesian equation of Π is x 4y z = (Since P lies in Π ) 4 Vector equation of l is r = + λ (OE) (Since z =.) [4] UCLES 6
Page 5 Mark Scheme Syllabus Paper Cambridge International A Level October/November 6 9 4 4 O x + y = 6t + 8t + 8t = + 8t + 8t = ( + 9 t ) ( AG) (i) ( ) s = + t t = t+ t = 9 d [] [] 5 (ii) = ( )( + ) = ( + ) S t t 9t dt t 6t 7t dt 9 t t t 4 6 = + = y Substitute t = in one of the parametric equations to obtain e.g. x = y. x x Uses x= rcosθ and y rsinθ rcosθ = tan θ r = secθ tan θ. States domain for θ : θ, 6 = ( ) * D [] [4] 6 sec 6tan 9tan 4 d ( ) A = θ θ + θ θ 9 = tan tan tan + 5 5 θ θ θ 6 4 = 45 [] UCLES 6