Cambridge International Examinations Cambridge Ordinary Level

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1 Cambridge International Examinations Cambridge Ordinary Level * * ADDITIONAL MATHEMATICS 4037/13 Paper1 October/November 016 hours CandidatesanswerontheQuestionPaper. NoAdditionalMaterialsarerequired. READ THESE INSTRUCTIONS FIRST WriteyourCentrenumber,candidatenumberandnameonalltheworkyouhandin. Writeindarkblueorblackpen. YoumayuseanHBpencilforanydiagramsorgraphs. Donotusestaples,paperclips,glueorcorrectionfluid. DONOTWRITEINANYBARCODES. Answerallthequestions. Givenon-exactnumericalanswerscorrectto3significantfigures,or1decimalplaceinthecaseof anglesindegrees,unlessadifferentlevelofaccuracyisspecifiedinthequestion. Theuseofanelectroniccalculatorisexpected,whereappropriate. Youareremindedoftheneedforclearpresentationinyouranswers. Attheendoftheexamination,fastenallyourworksecurelytogether. Thenumberofmarksisgiveninbrackets[]attheendofeachquestionorpartquestion. Thetotalnumberofmarksforthispaperis80. This document consists of 15 printed pages and 1 blank page. DC (ST/AR) [Turn over

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax + bx + c = 0, b b ac x = 4 a Binomial Theorem (a + b) n = a n + ( n 1 ) an 1 b + ( n ) an b + + ( n r ) an r b r + + b n, where n is a positive integer and ( n r ) = n! (n r)!r!. TRIGONOMETRY Identities sin A + cos A = 1 sec A = 1 + tan A cosec A = 1 + cot A Formulae for ABC a sin A = b sin B = c sin C a = b + c bc cos A = 1 bc sin A

3 3 1 On the axes below, sketch the graph of y = cos 3x for 0 G x G 180. [3] y x 3 Express 4m m - m + 9 m 3 m in the form Am + B, where A and B are integers to be found. [3] [Turn over

4 4 3 (i) Given that 3x + p^1 - xh =-3, show that, for x to be real, p - 3p - 9 H 0. [3] (ii) Hence find the set of values of p for which x is real, expressing your answer in exact form. [3]

5 5 4 (i) Find, in ascending powers of x, the first 3 terms in the expansion of J xn 6 K - O. [3] L 4 P (ii) Hence find the term independent of x in the expansion of J 3 NJ xn 6 K4 + + OK - O. [3] L x x PL 4 P [Turn over

6 6 5 5 (i) Given that log9 xy =, show that log x log y =. [3] (ii) Hence solve the equations log 5 log9 xy =, x # log y =- 6. [5] 3 3

7 7 d 6 (i) Find ^ln^3x - 11hh. [] dx (ii) Hence show that y x dx = p ln^3x - 11h + c, where p is a constant to be found, and c is a 3x - 11 constant of integration. [1] (iii) Given that a x y dx = ln, where a, find the value of a. [4] 3x - 11 [Turn over

8 8 7 ln y 5 4 (1.5, 3.5) 3 (4.0, 1.5) x The variables x and y are such that when ln y is plotted against x 1 the straight line graph shown above is obtained. (i) Given that y = Ae xb, find the value of A and of b. [4]

9 9 (ii) Find the value of y when x = [] (iii) Find the value of x when y = 0. [] [Turn over

10 10 8 (a) (i) Show that cosec i = sec i. [3] cosec i - sin i (ii) Hence solve cosec i = 4 for 0 1 i [3] cosec i - sin i

11 11 (b) Solve J rn 3 tankx + O = 1 for 0 1 x 1 r, giving your answers in terms of r. [3] L 4 P [Turn over

12 1 9 (a) A team of 5 students is to be chosen from a class of 10 boys and 8 girls. Find the number of different teams that may be chosen if (i) there are no restrictions, [1] (ii) the team must contain at least one boy and one girl. [4]

13 13 (b) A computer password, which must contain 6 characters, is to be chosen from the following 10 characters: Symbols?! * Numbers Letters W X Y Z Each character may be used once only in any password. Find the number of possible passwords that may be chosen if (i) there are no restrictions, [1] (ii) each password must start with a letter and finish with a number, [] (iii) each password must contain at least one symbol. [3] [Turn over

14 14 e x 10 A curve y = f^xh is such that f l^xh = 6x - 8. (i) Given that the curve passes through the point P ^0, -3h, find the equation of the curve. [5] The normal to the curve y = f^xh at P meets the line y = - 3x at the point Q. (ii) Find the area of the triangle OPQ, where O is the origin. [5]

15 15 11 A particle moving in a straight line has a velocity of v ms 1 such that, t s after leaving a fixed point, v = 4t - 8t + 3. (i) Find the acceleration of the particle when t = 3. [] (ii) Find the values of t for which the particle is momentarily at rest. [] (iii) Find the total distance the particle has travelled when t = [5]

16 16 BLANK PAGE 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.

Cambridge International Examinations Cambridge International General Certificate of Secondary Education

Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 9 2 4 2 8 1 9 0 7 2 * ADDITIONAL MATHEMATICS 0606/12 Paper 1 February/March 2016 2 hours Candidates