Cambridge International Examinations Cambridge Ordinary Level * 3 9 5 8 3 8 6 4 7 * 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) 16436 [Turn over
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 1 On the axes below, sketch the graph of y = cos 3x for 0 G x G 180. [3] y 3 1 0 1 30 60 90 10 150 180 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 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 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 5 5 (i) Given that log9 xy =, show that log x log y 5 3 + 3 =. [3] (ii) Hence solve the equations log 5 log9 xy =, x # log y =- 6. [5] 3 3
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 7 ln y 5 4 (1.5, 3.5) 3 (4.0, 1.5) 1 0 1 3 4 5 1 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 (ii) Find the value of y when x = 0. 3. [] (iii) Find the value of x when y = 0. [] [Turn over
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 1 360. [3] cosec i - sin i
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
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 (b) A computer password, which must contain 6 characters, is to be chosen from the following 10 characters: Symbols?! * Numbers 3 5 7 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 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 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 = 1. 5. [5]
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