Further Concepts for Advanced Mathematics (FP1) MONDAY 2 JUNE 2008

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1 ADVANCED SUBSIDIARY GCE 4755/0 MATHEMATICS (MEI) Further Cocepts for Advaced Mathematics (FP) MONDAY JUNE 008 Additioal materials: Aswer Booklet (8 pages) Graph paper MEI Examiatio Formulae ad Tables (MF) Morig Time: hour 30 miutes INSTRUCTIONS TO CANDIDATES Write your ame i capital letters, your Cetre Number ad Cadidate Number i the spaces provided o the Aswer Booklet. Read each questio carefully ad make sure you kow what you have to do before startig your aswer. Aswer all the questios. You are permitted to use a graphical calculator i this paper. Fial aswers should be give to a degree of accuracy appropriate to the cotext. INFORMATION FOR CANDIDATES The umber of marks is give i brackets [ ] at the ed of each questio or part questio. The total umber of marks for this paper is 7. You are advised that a aswer may receive o marks uless you show sufficiet detail of the workig to idicate that a correct method is beig used. This documet cosists of 4 prited pages. OCR 008 [D/0/663] OCR is a exempt Charity [Tur over

2 Sectio A (36 marks) (i) Write dow the matrix for reflectio i the y-axis. [] (ii) Write dow the matrix for elargemet, scale factor 3, cetred o the origi. [] (iii) Fidthematrixforreflectioithey-axis, followed by elargemet, scale factor 3, cetred o the origi. [] Idicate o a sigle Argad diagram (i) the set of poits for which ( 3 + j) =, [3] (ii) the set of poits for which arg( j) =π, [3] (iii) the two poits for which ( 3 + j) = adarg( j) =π. [] 3 Fid the equatio of the lie of ivariat poits uder the trasformatio give by the matrix M = ( ). [3] 4 Fid the values of A, B, C ad D i the idetity 3x 3 x + A(x ) 3 +(x 3 + Bx + Cx + D). [5] YouaregivethatA = ( 3 5) ad B = ( ) (i) Calculate AB. [3] (ii) Write dow A. [] 6 The roots of the cubic equatio x 3 + x 3x + = 0areα, β ad γ. Fid the cubic equatio whose roots are α, β ad γ, expressig your aswer i a form with iteger coefficiets. [5] 7 (i) Show that 3r 3r + 3 for all itegers r. [] (3r )(3r + ) (ii) Hece use the method of differeces to fid r= (3r )(3r + ). [5] OCR /0 Ju08

3 8 A curve has equatio y = x (x 3)(x + ). 3 Sectio B (36 marks) (i) Write dow the equatios of the three asymptotes. [3] (ii) Determie whether the curve approaches the horizotal asymptote from above or below for (A) large positive values of x, (B) large egative values of x. [3] (iii) Sketch the curve. [3] (iv) Solve the iequality x < 0. [3] (x 3)(x + ) 9 Two complex umbers, α ad β, aregivebyα = j ad β = + j. α ad β are both roots of a quartic equatio x 4 + Ax 3 + Bx + Cx + D = 0, where A, B, C ad D are real umbers. (i) Write dow the other two roots. [] (ii) Represet these four roots o a Argad diagram. [] (iii) Fid the values of A, B, C ad D. [7] 0 (i) Usig the stadard formulae for r= r= r ad r= r 3,provethat r (r + ) = ( + )( + )(3 + ). [5] (ii) Prove the same result by mathematical iductio. [8] OCR /0 Ju08

4 4 Permissio to reproduce items where third-party owed material protected by copyright is icluded has bee sought ad cleared where possible. Every reasoable effort has bee made by the publisher (OCR) to trace copyright holders, but if ay items requirig clearace have uwittigly bee icluded, the publisher will be pleased to make ameds at the earliest possible opportuity. OCR is part of the Cambridge Assessmet Group. Cambridge Assessmet is the brad ame of Uiversity of Cambridge Local Examiatios Sydicate (UCLES), which is itself a departmet of the Uiversity of Cambridge. OCR /0 Ju08

5 4755 Mark Scheme Jue (FP) Further Cocepts for Advaced Mathematics Qu Aswer Sectio A 0 (i) 0 Mark Commet (ii) (iii) = A [4] Multiplicatio, or other valid method (may be implied) c.a.o. B3 B3 Circle, ; cetre 3+ j, ; radius =, Lie parallel to real axis, ; through (0, ), ; correct half lie, 3-3 x x = y y x y= x, x+ y = y y = x [7] [3] Poits + j ad 5+ j idicated c.a.o. x x For = y y 4 3 3x 3 x + A( x ) + ( x 3 + Bx + Cx+ D) Ax Ax Ax A x Bx Cx D 3 ( A ) x ( B 3A) x ( 3A C) x ( D A) A=, B= 5, C = 6, D= 4 B4 [5] Attempt to compare coefficiets Oe for each correct value 9

6 4755 Mark Scheme Jue 008 5(i) AB = B3 [3] Mius each error to miimum of 0 5(ii) A = A [] Use of B c.a.o. 6 w w= x x = 3 w w w = 0 A Substitutio. For substitutio x = w give B0 but the follow through for a maximum of 3 marks Substitute ito cubic Correct substitutio 6 OR w 3 + w w+ = α + β + γ = αβ + αγ + βγ = αβγ = 3 A [5] Mius for each error (icludig = 0 missig), to a miimum of 0 Give full credit for iteger multiple of equatio All three Let ew roots be k, l, m the B k + l+ m = ( α + β + γ) = = A ( αβ αγ βγ ) kl + km + lm = = 6= D klm = 8αβγ = 4= A 3 ω + ω 6ω+ 4= 0 C A A A [5] Attempt to use sums ad products of roots of origial equatio to fid sums ad products of roots i related equatio Sums ad products all correct ft their coefficiets; mius oe for each error (icludig = 0 missig), to miimum of 0 Give full credit for iteger multiple of equatio 0

7 4755 Mark Scheme Jue 008 7(i) ( 3r )( 3r+ ) 3r+ ( 3r ) ( )( ) 3r 3r+ 3r 3r+ 3 A [] Attempt at correct method Correct, without fudgig 7(ii) = 3r 3r+ 3 3r 3r+ r ( )( ) = r= = = A A [5] Attempt to use idetity Terms i full (at least two) Attempt at cacellig A if factor of 3 missig, A max if aswer ot i terms of Sectio A Total: 36

8 4755 Mark Scheme Jue 008 Sectio B 8(i) x = 3, x =, y = [3] 8(ii) 8(iii) + Large positive x, y (e.g. cosider x = 00 ) Large egative x, y (e.g. cosider x = 00 ) [3] Evidece of method required Curve Cetral ad RH braches correct Asymptotes correct ad labelled LH brach correct, with clear miimum [3] y = 8(iv) < x < 3 x 0 B [3] B max if ay iclusive iequalities appear B3 for < x < 0 ad 0< x < 3,

9 4755 Mark Scheme Jue 008 9(i) 9(ii) + j ad j B [] mark for each B [] mark for each correct pair 9(iii) ( x j)( x + j)( x+ + j)( x+ j) ( x 4x 8)( x x ) = B A Attempt to use factor theorem Correct factors, mius each error if oly errors are sig errors Oe correct quadratic with real coefficiets (may be implied) = x + x + x 4x 8x 8x+ 8x + 6x = x x + x + 8x+ 6 Expadig A=, B=, C = 8, D= 6 OR A [7] Mius each error, A if oly errors are sig errors α = αβγδ = 6 * * * * * * αβ = αα + αβ + αβ + ββ + βα + β α * * * * * * αβγ = αα β + αα β + αββ + α ββ αβ =, αβγ = 8 A=, B =, C = 8, D= 6 OR A A [7] Both correct Mius each error, A if oly errors are sig errors Attempt to substitute i oe root Attempt to substitute i a secod root Equatig real ad imagiary parts to 0 Attempt to solve simultaeous equatios A=, B =, C = 8, D= 6 A A [7] Both correct Mius each error, A if oly errors are sig errors 3

10 4755 Mark Scheme Jue 008 Qu Aswer Sectio B (cotiued) 0(i) r r+ = r + r ( ) 3 r= r= r= ( ) ( )( ) ( ) 3 ( ) ( ) = = ( )( 3 7 ) = ( )( )( 3 ) = Mark A E [5] Commet Separatio of sums (may be implied) Oe mark for both parts Attempt to factorise (at least two liear algebraic factors) Correct Complete, covicig argumet 0(ii) r = ( + ) = ( + )( + )( 3 + ) r r =, LHS = RHS = Assume true for = k k r r k k r = k k k + r = r ( + ) = ( + )( + )( 3 + ) ( r+ ) k( k )( k )( 3k ) ( k ) ( k ) ( k )( k ) [ k( 3k ) ( k ) ] = = ( k )( k )( 3k 3k ) = ( k )( k )( k 3)( 3k 4) ( k ) (( k ) ) (( k ) ) ( 3( k ) ) = = But this is the give result with k + replacig k. Therefore if it is true for k it is true for k +. Sice it is true for k =, it is true for k =,, 3 ad so true for all positive itegers. E A A E E [8] must be see Assumig true for k (k + )th term Attempt to factorise Correct Complete covicig argumet Depedet o previous A ad previous E Depedet o first ad previous E Sectio B Total: 36 Total: 7 4

11 Report o the Uits take i Jue Further Cocepts for Advaced Mathematics (FP) Geeral commets Cadidates geerally performed well, demostratig a good kowledge of the syllabus. The paper eabled the cadidates to demostrate their kowledge, but also differetiated well betwee them. Some cadidates dropped marks through careless algebraic maipulatio, ad a smaller umber by failig to label diagrams ad graphs clearly. Commets o Idividual Questios ) There were may good aswers, but some surprisig errors, icludig vectors give as aswers. I (iii) cadidates must show the multiplicatio of the two matrices to avoid loss of a method mark if either of their earlier aswers was icorrect. ) (i) A circle of radius was usually clearly show. A icorrect cetre was the most commo error. (ii) A full lie was ofte show, istead of a half lie, ad may lies wet through - j istead of j. Most cadidates correctly drew lies parallel to the real axis. (iii) If (i) ad (ii) were correct, (iii) was usually correct too, though ot all cadidates showed the poits clearly o their Argad diagrams. 3) This was very well doe by about half the cadidates. Others either omitted it or made errors such as failig to use a ivariat vector, or settig the trasformed poit (-x y, x + y) to (0, 0). 4) There were may good aswers, but also may careless mistakes. Ofte ( x) 3 was expaded without multiplyig all terms by A (especially the -). There was also evidece of careless solvig of the equatio leadig to B. 5) (i) This was well doe by almost all cadidates. (ii) This was ofte badly doe, idicatig that may studets did ot properly uderstad iverse matrices. May tried to use a algorithm to ivert the matrix; some did this correctly, but wasted a lot of time i doig so; others tried to apply rules for ivertig a matrix to the 3 3case. 6) The most popular method ivolved usig sums ad products of roots, but there were w quite a few careless errors ivolvig sigs. Those who substituted x = were usually successful. Several cadidates omitted the = 0 from their equatio ad so lost the fial mark.

12 Report o the Uits take i Jue 008 7) (i) This was well doe by almost all cadidates. (ii) The factor of 3 was very ofte lost followig successful expasio ad cacellatio of terms. A few cadidates multiplied by 3 istead of dividig. A few left aswers i terms of r ot. 8) (i) This was well aswered, but y = ad y = 0 were quite frequet errors for the asymptote parallel to the x-axis. (ii) (iii) Most cadidates showed their method, usually by substitutig large positive ad egative values of x. The results were ot always put to correct use i (iii). The left-had brach was ofte icorrect. May cadidates seemed to hold the miscoceptio that the graph caot cross a asymptote. Eve where graphs were show crossig the asymptote, they ofte did ot show a clear miimum. (iv) May correct aswers, but x 0 was ofte omitted. A fairly commo error was x<, x> 3, presumably from a misiterpretatio of the < 0 i the questio. Several cadidates used algebraic methods whe the aswer could be most easily foud directly from the graph. 9) (i) ad (ii) were both well doe by almost all cadidates. (iii) (iv) Use of the sums ad products of roots was the most commo method, but cadidates foud difficulty i correctly fidig αβ ad αβγ. They also had problems i assigig the correct sigs to four coefficiets, ad got muddled by A B C D ad a b c d ad e. Multiplyig factors with cojugate pairs was a less popular but more successful method, apart from occasioal sig errors. A few cadidates tried to substitute roots, equatig the real ad imagiary parts to 0 ad the solvig simultaeous equatios. This was the least successful method.

13 Report o the Uits take i Jue 008 0) (i) This was usually started well, but a surprisig umber did ot take out the commo factors to ease the workig. As a result there were algebraic errors ad sometimes fudgig of the fial factorisatio of the quartic. (ii) (iii) (iv) This was well doe i may cases, but there were sigs that some cadidates were uder time pressure. The basic structure was well doe i may cases, but the presetatio ad otatio were ofte poor, with r, k ad ofte used i the wrog places. Summatio sigs were ofte omitted so that statemets effectively meaig last term = sum of k terms + last term frequetly appeared. Cadidates ofte failed to take out commo factors i the algebraic maipulatio, ad their proofs faltered as a result. The fial words of explaatio were ot always covicig. However, it was ecouragig that may perfect solutios were see, showig that cadidates uderstood the proof ad were able to commuicate their argumets clearly.

Further Concepts for Advanced Mathematics (FP1) MONDAY 2 JUNE 2008

Further Concepts for Advanced Mathematics (FP1) MONDAY 2 JUNE 2008 ADVANCED SUBSIDIARY GCE 4755/0 MATHEMATICS (MEI) Further Cocepts for Advaced Mathematics (FP) MONDAY JUNE 008 Additioal materials: Aswer Booklet (8 pages) Graph paper MEI Examiatio Formulae ad Tables (MF)