FURTHER MATHEMATICS 9231/13 Paper 1 May/June 2017

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1 Cambridge International Examinations Cambridge International Advanced Level CANDIDATE NAME * * CENTRE NUMBER CANDIDATE NUMBER FURTHER MATHEMATICS 9231/13 Paper 1 May/June 2017 Candidates answer on the Question Paper. Additional Materials: List of Formulae(MF10) 3hours READ THESE INSTRUCTIONS FIRST WriteyourCentrenumber,candidatenumberandnameinthespacesatthetopofthispage. Writeindarkblueorblackpen. YoumayuseanHBpencilforanydiagramsorgraphs. Do not use staples, paper clips, glue or correction fluid. DONOTWRITEINANYBARCODES. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of a calculator is expected, where appropriate. Results obtained solely from a graphic calculator, without supporting working or reasoning, will not receive credit. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. Thenumberofmarksisgiveninbrackets[]attheendofeachquestionorpartquestion. This document consists of 24 printed pages. JC17 06_9231_13/RP UCLES2017 [Turn over

2 1 Theroots ofthe cubicequationx 3 +2x 2 3 = 0are, and. 2 (i) By using thesubstitutiony= 1 x 2,find thecubicequationwith roots 1 1 2, 2 and 1 2. [3] (ii) Hencefindthevalueof [1] (iii) Find also the value of [1] UCLES /13/M/J/17

3 2 (i) Verify that 3 2r +1 r r +1 r +2 = 1 2r +1 2r +3 2 r +1 r +2 2r 1 2r +1. [2] r r +1 (ii) Hence show that n r=1 2r +1 r r +1 r +2 = 1 2n +1 2n [2] 2 n +1 n +2 2 (iii) Deduce the value of r=1 2r +1 r r +1 r +2. [2] UCLES /13/M/J/17 [Turn over

4 3 Prove, by mathematical induction, that n r=1 4 r +1 n +1 n rln =ln for all positive integers n. [6] r n! UCLES /13/M/J/17

5 5 4 AcurveC has equationx 3 3xy +y 2 = 4. Find thevalueof d2 y atthepoint 0, 2 ofc. [7] 2 dx UCLES /13/M/J/17 [Turn over

6 6 5 A curve C has parametric equations x = 2 5 t5 2 2t 1 2, y = 4 3 t3 2, for1 t 4. (i) Findtheexact valueofthearclengthofc. [5] UCLES /13/M/J/17

7 7 (ii) Find also the exact value of the surface area generated when C is rotated through 2 radians about the x-axis. [3] UCLES /13/M/J/17 [Turn over

8 8 2 6 LetI n denote 4 +x 2 n dx. 0 (i) Find d dx x 4 +x 2 n and henceshow that 8nI n+1 = 2n 1 I n +2 8 n. [5] UCLES /13/M/J/17

9 9 1 (ii) Use the result for integrating x 2 with respect to x, in the List of Formulae (MF10), to find 2 +a thevalueofi 1 and deducethat I 3 = [5] UCLES /13/M/J/17 [Turn over

10 10 7 (i) Use de Moivre s theorem to prove that tan4 = 4tan 4tan3 1 6tan 2 +tan 4. [5] UCLES /13/M/J/17

11 11 (ii) Hence find the solutions of the equation t 4 4t 3 6t 2 +4t +1 = 0, givingyour answers in theform tank,wherekis arational number. [5] UCLES /13/M/J/17 [Turn over

12 12 8 Find the solution of the differential equation giventhat,whent = 0,x=3and dx dt d 2 x dt 2 +6dx dt +9x = 18t2 +6t +1, = 0. [10] UCLES /13/M/J/17

13 UCLES /13/M/J/17 [Turn over

14 14 9 Theplane 1 passesthroughthepoints 1, 2, 1 and 5, 2, 9 andisparalleltothevectori+2j +3k. (i) Findthecartesian equation of 1. [4] Theplane 2 containsthelines r = 2i 3j +k+ i 2j k and r = 2i 3j +k+ 2i +3j k. (ii) Findthecartesian equation of 2. [4] UCLES /13/M/J/17

15 15 (iii) Findtheacuteanglebetween 1 and 2. [3] UCLES /13/M/J/17 [Turn over

16 16 10 Thematrix Ais givenby A = (i) Given that is an eigenvector of A, find the corresponding eigenvalue. [2] (ii) Given also that 1 is an eigenvalue of A, find a corresponding eigenvector. [2] UCLES /13/M/J/17

17 17 (iii) It is given that the determinant of A is equal to the product of the eigenvalues of A. Use this result to find the third eigenvalue of A, and find also a corresponding eigenvector. [3] UCLES /13/M/J/17 [Turn over

18 18 (iv) Write down matrices P and D such that P 1 AP = D, where D is a diagonal matrix, and hence findthe matrix A n in termsofn,wherenis apositive integer. [6] UCLES /13/M/J/17

19 19 11 Answer only one of the following two alternatives. EITHER AcurveC has polarequationr = 2acos for0 <2,whereaisapositiveconstant. (i) Show thatr = 2asin2 andsketchc. [4] (ii) Deduce that the cartesian equation of C is x 2 +y = 4axy. [2] UCLES /13/M/J/17 [Turn over

20 20 (iii) Findthearea ofoneloop ofc. [5] UCLES /13/M/J/17

21 21 (iv) Showthat,atthepoints(otherthanthepole)atwhichatangenttoC isparalleltotheinitialline, 2tan = tan2. [3] UCLES /13/M/J/17 [Turn over

22 22 OR Thematrix A, given by A = represents atransformation from 4 to , (i) Find the rank of A and show that, is a basis for the null space of the transformation. [6] UCLES /13/M/J/17

23 23 (ii) Show that if Ax = p q 1 1, 8 3 wherepandqaregiven real numbers, then p +2 + x = q +2 +3, where and arerealnumbers. [2] UCLES /13/M/J/17 [Turn over

24 (iii) Findthevaluesofpandqsuch that 1 3 p +q = [3] (iv) Find the solution of the equation Ax = of the form x = 4 9, where and are positive integers to be found. [3] 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(ucles) 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. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate(UCLES), which is itself a department of the University of Cambridge. UCLES /13/M/J/17

MATHEMATICS 9709/31 Paper 3 Pure Mathematics 3 (P3) May/June 2018

Cambridge International Examinations Cambridge International Advanced Level CANDIDATE NAME *7084725728* CENTRE NUMBER CANDIDATE NUMBER MATHEMATICS 9709/31 Paper 3 Pure Mathematics 3 (P3) May/June 2018