Friday 20 May 2016 Morning

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1 Oxford Cambridge ad RSA Friday 0 May 06 Morig AS GCE MATHEMATICS (MEI) 4755/0 Further Cocepts for Advaced Mathematics (FP) QUESTION PAPER * * Cadidates aswer o the Prited Aswer Boo. OCR supplied materials: Prited Aswer Boo 4755/0 MEI Examiatio Formulae ad Tables (MF) Other materials required: Scietific or graphical calculator Duratio: hour 0 miutes INSTRUCTIONS TO CANDIDATES These istructios are the same o the Prited Aswer Boo ad the Questio Paper. The Questio Paper will be foud iside the Prited Aswer Boo. Write your ame, cetre umber ad cadidate umber i the spaces provided o the Prited Aswer Boo. Please write clearly ad i capital letters. Write your aswer to each questio i the space provided i the Prited Aswer Boo. Additioal paper may be used if ecessary but you must clearly show your cadidate umber, cetre umber ad questio umber(s). Use blac i. HB pecil may be used for graphs ad diagrams oly. Read each questio carefully. Mae sure you ow what you have to do before startig your aswer. Aswer all the questios. Do ot write i the bar codes. You are permitted to use a scietific or graphical calculator i this paper. Fial aswers should be give to a degree of accuracy appropriate to the cotext. INFORMATION FOR CANDIDATES This iformatio is the same o the Prited Aswer Boo ad the Questio Paper. The umber of mars is give i bracets [ ] at the ed of each questio or part questio o the Questio Paper. You are advised that a aswer may receive o mars uless you show sufficiet detail of the worig to idicate that a correct method is beig used. The total umber of mars for this paper is 7. The Prited Aswer Boo cosists of 6 pages. The Questio Paper cosists of 4 pages. Ay bla pages are idicated. INSTRUCTION TO EXAMS OFFICER / INVIGILATOR Do ot sed this Questio Paper for marig; it should be retaied i the cetre or recycled. Please cotact OCR Copyright should you wish to re-use this documet. OCR 06 [D/0/66] DC (LK/FD) 7059/ OCR is a exempt Charity Tur over

2 Sectio A (6 mars) 8 - The matrix M is give by M = c m, where p!- 4. p (i) Fid the iverse of M i terms of p. [] (ii) y x Fig. The triagle show i Fig. udergoes the trasformatio represeted by the matrix 8 - c m. Fid the area of the image of the triagle followig this trasformatio. [] The complex umber z is - 5 j ad the complex umber z is ( a - ) + ( - b)j, where a ad b are real. z * (i) Express i the form x + y j, givig x ad y i exact form. You must show clearly how you obtai your z aswer. * z (ii) Give that = z, fid the exact values of a ad b. [] z OCR /0 Ju6

3 m You are give that A = f matrix p, B = f p ad AB I =, where I is the # idetity (i) Fid the values of m ad. (ii) Hece fid B -. [] 4 (i) Use stadard series to show that / r ( r - p) = ( + )( + ( - p) - p), 6 r = where p is a costat. (ii) Give that the coefficiets of ad 4 i the expressio for / r ( r - p) are equal, fid the value of p. r = [] 5 The loci C ad C are give by z j = 5 ad arg ( z j) = r respectively. (i) Setch, o a sigle Argad diagram, the loci C ad C. [5] (ii) Write dow the complex umber represeted by the poit of itersectio of C ad C. [] (iii) Idicate, by shadig o your setch, the regio satisfyig z j H 5 ad r G arg( z j) G r. [] 4 6 A sequece is defied by u 8 = ad u + = u Prove by iductio that u = 4( ) - -. [6] OCR /0 Ju6 Tur over

4 4 4 Sectio B (6 mars) 7 The fuctio f( z) = z - 9z + Az + Bz - 6 has real coefficiets. The equatio f( z) = 0 has two real roots, a ad b, where a b, ad two complex roots, c ad d, where c = + j. (i) Show that a + b =- ad fid the value of ab. [5] (ii) Hece fid the two real roots a ad b. [] (iii) Fid the values of A ad B. w (iv) Write dow the roots of the equatio f c m = 0. [] j x A curve has equatio y =. x + x - 4 (i) Fid the equatios of the two vertical asymptotes ad the oe horizotal asymptote of this curve. [] (ii) State, with justificatio, how the curve approaches the horizotal asymptote for large positive ad large egative values of x. [] (iii) Setch the curve. x - 9 (iv) Solve the iequality x H 0. [] + x - 4 r 5 9 You are give that 4( r - ) - r + + 4( r + ) = +. ( r - )( r + )( r + ) (i) Use the method of differeces to show that / r + 5 = - ( r - )( r + )( r + ) 4( + ) + = 4( + ). [6] r (ii) Write dow the limit to which / r + 5 coverges as teds to ifiity. [] r = ( r - )( r + )( r + ) (iii) Fid the sum of the fiite series f +, 9 # 4 # 4 4 # 4 # 45 4 # 45 # # 0 # 0 givig your aswer to sigificat figures. [] [] END OF QUESTION PAPER Oxford Cambridge ad RSA Copyright Iformatio OCR is committed to seeig permissio to reproduce all third-party cotet that it uses i its assessmet materials. OCR has attempted to idetify ad cotact all copyright holders whose wor is used i this paper. To avoid the issue of disclosure of aswer-related iformatio to cadidates, all copyright acowledgemets are reproduced i the OCR Copyright Acowledgemets Boolet. This is produced for each series of examiatios ad is freely available to dowload from our public website ( after the live examiatio series. If OCR has uwittigly failed to correctly acowledge or clear ay third-party cotet i this assessmet material, OCR will be happy to correct its mistae at the earliest possible opportuity. For queries or further iformatio please cotact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB GE. 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 Ju6

5 4755 Mar Scheme Jue 06 Questio Aswer Mars Guidace Correct determiat correctly used (i) = 8+ p p 8 [] Correct re-arragemets of elemets (ii) Area of image = triagle area (8 + p) = ( 8 + p) Multiplyig a area by their determiat with p = (accept 8 + oly) = 68 (square uits) Or ew coords ad valid method to area cao [] (i) * z = + 5j * z (+ 5j) ( + 5j) 0 Correct use of cojugate = = + j 9 i deomiator z ( 5j )( + 5j) 9 9 All correct (ii) 0 a = ad b = 9 9 Equatig real ad imagiary parts 8 8 a =, b = 9 9 Both, ft their (i) (i) Either µ = ()(4) + (5)( 5) + ( )(8) = 5 Or µ = ( )( 4) + (4)(5) + ()( ) 9 λ + (6)(5) + ( 4)( ) = 5 Or 4 λ 0 = 0 Or -4 λ + 4 = 0 [] Multiplyig a row of A by a colum of B to fid λ or µ Multiplyig aother row of A by a colum of B to fid the other uow µ = 5 cao λ = cao 8

6 4755 Mar Scheme Jue 06 Questio Aswer Mars Guidace (ii) B = A with their λ ad their µ µ 6 4 cao B = 5 5 NB µ = /5 i (i) the 5A ears B0 4 whereas /5 A ears M0 [] 4 (i) r r p = r p r ( ) * Splittig ito sums a r ± b r r= r= r= 6 ( ) ( )( ) = + p + + o.e. Use of stadard results i terms of = ( ) ( ( ) p ( ) ) dep Attempt to factorise with + ( ) If from quartic i all steps justified, otherwise M0 = ( )( ( p ) p) AG 4 (ii) = 6 p p = [] Equatig their coefficiets of ad 4 9

7 4755 Mar Scheme Jue 06 Questio Aswer Mars Guidace 5 (i) Accept u-umbered evely spaced mars o axes to show scale Circle Cetre + 4j Radius = 5 (chec that the circle passes through O or ay valid poit or explicitly show) allow for cetre Half lie π from 6j ± Fully correct [5] 5 (ii) + 9j [] Not (-, 9j) or (-, 9) 0

8 4755 Mar Scheme Jue 06 Questio Aswer Mars Guidace 5 iii Regio outside circle, must have a border with the circumferece Correct regio show, boudary at π 4 reasoably well draw [] 6 Whe =, ( ) u = 4 = 8, (so true for = ) Showig use of u ( ) = 4 Assume u ( ) = 4 E Assumig true for = Allow let = ad (result) or If = ad (result) Do ot allow = or let = without the result quoted

9 4755 Mar Scheme Jue 06 Questio Aswer Mars Guidace u + = u + + 5= 4( ) u +, usig u ad attemptig to simplify + ( ) = 4 ( + ) ( ) But this is the give result with + replacig. Therefore if it is true for = the it is also true for = +. Sice it is true for =, it is true for all positive itegers. E E [6] Correct simplificatio or idetificatio with a target expressio usig = + The target shows this Depedet o ad previous E Depedet o ad previous E 7 (i) δ = j ad used 9 sum of roots = (correct sig i RHS) 9 α + β = ( + j) ( j) = αβ ( + j)( j) = (correct sig i RHS) Or through obtaiig two quadratic factors (AG) Or through obtaiig two quadratic factors αβ = [5] 7 (ii) α α = α + α = ( α )( α + ) = 0 α =, α =, β = 0 Attempt at solutio for α or β (from their αβ) Or from quadratic foud earlier Or z - (sum of roots)z + product of roots = 0 Oe root correct Both roots correct (codoe vice-versa) []

10 4755 Mar Scheme Jue 06 Questio Aswer Mars Guidace 7 (iii) f ( z) = (z )( z+ )( z ( + j))( z ( j)) = ( z + z )( z 6z+ ) 4 = z 9z + 6z + 5z 6 A valid method see to fid A or B (by factors or root relatios or substitutio) May be see earlier A = 6 [] B = 5 both cao 7 (iv) w f = 0 w = j, j, + j, +j j 8 (i) x = x = 4 y = [] [] Their roots FT j SC x =, -4 is B0 (ii) Evidece of method eeded e.g. evaluatig usig large values Large positive x, y Large egative x, y + [] SC for correct approaches without valid method see

11 4755 Mar Scheme Jue 06 Questio Aswer Mars Guidace (iii) [] braches correct relative to their asymptotes ad carefully draw Asymptotes correct ad labelled Itercepts correct ad labelled (allow.7 ad -.7) (iv) x< 4, x<, x B [] Oe mar for each. Correct iequality sigs. Allow.7 for (B the if more tha iequalities) 4

12 4755 Mar Scheme Jue 06 Questio Aswer Mars Guidace 9 (i) r + 5 = ( )( )( ) + Use of the give result (may be implied) r= r r+ r+ r= 4 ( r ) r+ 4 ( r+ ) = ( ) 4( + ) 4( ) + 4( + ) = = ( ) ( ) ( + ) ( + ) as required [6] Terms i full (first ad at least oe other) First term correct ad oe other correct Ay cosecutive terms fully correct Or cosecutive algebraic terms fully correct (eed ot be simplified) Valid attempt to cacel, icludig algebraic terms. Need to see the three algebraic fractios that remai after cacellig Covicigly show (AG) 9 (ii) = r= [] 9 (iii) r+ 5 = 45 r = 0 May be implied (eg by usig sum to 9 terms) r+ 5 = 05 r = 50 May be implied 50 9 Differece of their sum from r = to 50 = + + r= r= ad their sum from r = to 9 = Cao Ivalid method A0 Usee method SC 5