The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72.

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1 ADVANCED GCE UNIT / MATHEMATICS (MEI Further Methods for Advanced Mathematics (FP THURSDAY JUNE Additional materials: Answer booklet (8 pages Graph paper MEI Eamination Formulae and Tables (MF Morning Time: hour minutes INSTRUCTIONS TO CANDIDATES Write your name, centre number and candidate number in the spaces provided on the answer booklet. Answer all the questions in Section A and one question from Section B. You are permitted to use a graphical calculator in this paper. Final answers should be given to a degree of accuracy appropriate to the contet. INFORMATION FOR CANDIDATES The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is. ADVICE TO CANDIDATES Read each question carefully and make sure you know what you have to do before starting your answer. You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used. This document consists of printed pages. HN/ OCR [H//] OCR is an eempt Charity [Turn over

2 Section A ( marks Answer all the questions (a A curve has polar equation r a( cos q, where a is a positive constant. (i Sketch the curve. [] (ii Find the area of the region enclosed by the section of the curve for which and the line [] q p. Û (b Use a trigonometric substitution to show that d. [] Ù ı - (c In this part of the question, f( arccos (. (i Find f (. [] (ii Use a standard series to epand f (, and hence find the series for f( in ascending powers of, up to the term in. [] ( q p (a Use de Moivre s theorem to show that sin q sin q sin q sin q. [] (b (i Find the cube roots of j in the form re jq where r and p. [] p q These cube roots are represented by points A, B and C in the Argand diagram, with A in the first quadrant and ABC going anticlockwise. The midpoint of AB is M, and M represents the comple number w. (ii Draw an Argand diagram, showing the points A, B, C and M. [] (iii Find the modulus and argument of w. [] (iv Find w in the form a bj. [] OCR / June

3 Let Ê ˆ M -. Á Ë - - (i Show that the characteristic equation for M is l l 8l. [] Ê ˆ You are given that - is an eigenvector of M corresponding to the eigenvalue. Á Ë (ii Find the other two eigenvalues of M, and corresponding eigenvectors. [8] (iii Write down a matri P, and a diagonal matri D, such that P M P D. [] (iv Use the Cayley-Hamilton theorem to find integers a and b such that M am bm. [] Section B (8 marks Answer one question Option : Hyperbolic functions Û (a Find Ù d, giving your answer in an eact logarithmic form. [] ı 9 + (b (i Starting from the definitions of sinh and cosh in terms of eponentials, prove that sinh sinh cosh. [] (ii Show that one of the stationary points on the curve has coordinates (ln, 9, and find the coordinates of the other two stationary points. [] ln Û (iii Show that Ù ( cosh - cosh d. [] ı -ln y cosh cosh [Question is printed overleaf.] OCR / June [Turn over

4 Option : Investigation of curves This question requires the use of a graphical calculator. The curve with equation y k k is to be investigated for different values of k. k (i Use your graphical calculator to obtain rough sketches of the curve in the cases k, k. and k. [] (ii Show that the equation of the curve may be written as y k k (k. k Hence find the two values of k for which the curve is a straight line. [] (iii When the curve is not a straight line, it is a conic. (A Name the type of conic. [] (B Write down the equations of the asymptotes. [] (iv Draw a sketch to show the shape of the curve when k 8. This sketch should show where the curve crosses the aes and how it approaches its asymptotes. Indicate the points A and B on the curve where and k respectively. [] Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Eaminations Syndicate (UCLES, which is itself a department of the University of Cambridge. OCR / June

5 Mark Scheme June (a(i B Must include a sharp point at O and have infinite gradient at θ Give B for r increasing from zero for < θ <, or decreasing to zero for < θ < (ii Area is r dθ a ( cosθ d θ a a (b Put sinθ ( cos θ + ( + cos θ dθ a sin sin θ θ + θ ( Integral is tanθ ( sin θ cosθ dθ 8cos θ (cosθ dθ sec θ dθ M A B BB ft B M A M A ag For integral of ( cosθ For a correct integral epression including limits (may be implied by later work Using cos θ ( + cos θ Integrating Accept.8a or cosθ Limits not required For a + b cosθ and k cos θ sec θ dθ tanθ SR If tanhu is used M for sinh( ln A for ( (ma / 8 (c(i f ( B Give B for any non-zero real multiple of this (or for etc sin y (ii f ( ( ( + + f ( C f ( C +... f ( M A M A Binomial epansion ( terms, n Epansion of ( correct (accept unsimplified form Integrating series for f ( Must obtain a non-zero C not required term

6 Mark Scheme June OR by repeated differentiation ( Finding f ( Evaluating ( f ( ( 88 f ( +... M M A ft f ( +... A Must obtain a non-zero value ft from (c(i when B given 8

7 Mark Scheme June (a (cosθ + jsinθ c + jc s c s jc s + cs + js Equating imaginary parts sin θ c ( s s c s + s s ( s s + s M M A M s s sinθ sin + s s + s θ + sin θ + s A ag (b(i + j 8, arg( + j r θ θ, BB B ft B ft M A Accept.8;., (Implies B for 8 One correct (Implies B for Adding or subtracting Accept θ + k, k,, (ii B Give B for two of B, C, M in the correct quadrants Give B ft for all four points in the correct quadrants (iii w arg w ( + B ft B Accept. Accept.8 (iv w ( 8 arg( w w 8 j ( cos + jsin 8 M A ft A Obtaining either modulus or argument Both correct (ft Allow from arg w etc SR If B, C interchanged on diagram (ii B (iii B B for (iv MAA 9

8 Mark Scheme June (i det( M λ I ( λ[( λ( λ ] [( λ + ] + [ ( λ] ( λ ( + λ + λ ( λ + ( + λ 8 + 9λ + λ λ λ + λ 8λ + λ λ Characteristic equation is λ λ 8λ (ii λ ( λ 8( λ + (iii (iv Other eigenvalues are 8, When λ 8, + y + z 8 ( + y z 8y y z 8z y and z ; eigenvector is When λ, + y + z + y z y P M y, z ; eigenvector is D 8 M 8M M M + 8M M M (M M + 8M + 8M + 8M + 9M M A M A ag M A M M A M M A B ft M A M M A 8 Obtaining det( M λ I Any correct form Simplification Solving to obtain a non-zero value Two independent equations Obtaining a non-zero eigenvector ( + y+ z 8 etc can earn MM Two independent equations Obtaining a non-zero eigenvector B if P is clearly singular Order must be consistent with P when B has been earned

9 Mark Scheme June (a d arsinh 9 + arsinh 9 ln( + + ln M A A M A For arsinh or for any sinh substitution For or for sinh u For or for du OR M For ln( k + k +... [ Give M for ln( a + b +... ] ln( AA or ln( ln 8 ln ln A (b(i sinh cosh ( e ( e sinh e e ( e + e M A (e e (e For completion + e (e e (ii dy sinh sinh d For stationary points, sinh sinh cosh, y ( ± ln( sinh ( cosh sinh or cosh + 9 ln 9 y ln, y 9 BB M A A ag A ag B When eponential form used, give B for any terms correctly differentiated d y Solving to obtain a value of d sinh, cosh or e ( or stated Correctly obtained Correctly obtained The last AA ag can be replaced by BB ag for a full verification (iii sinh sinh 8 ln ln 9 9 BB M A ag When eponential form used, give B for any terms correctly integrated Eact evaluation of sinh(ln and sinh( ln

10 Mark Scheme June (i k B B Maimum on LH branch and minimum on RH branch Crossing aes correctly B B Two branches with positive gradient Crossing aes correctly k k. B B Maimum on LH branch and minimum on RH branch Crossing positive y-ais and minimum in first quadrant (ii ( + k( k + k + k y + k k( k + k + + k Straight line when k( k + k, k M A (ag BB Working in either direction For completion (iii(a Hyperbola B (B k y k B B

11 Mark Scheme June (iv B B B B B Asymptotes correctly drawn Curve approaching asymptotes correctly (both branches Intercept on y-ais, and not crossing the -ais Points A and B marked, with minimum point between them Points A and B at the same height ( y

The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72.

The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72. ADVANCED GCE UNIT 7/ MATHEMATICS (MEI Further Methods for Advanced Mathematics (FP THURSDAY 7 JUNE 7 Additional materials: Answer booklet (8 pages Graph paper MEI Examination Formulae and Tables (MF Morning