Cambridge International Examinations CambridgeOrdinaryLevel

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1 Cambridge International Examinations CambridgeOrdinaryLevel * * ADDITIONAL MATHEMATICS 4037/12 Paper1 May/June 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. Thisdocumentconsistsof15printedpagesand1blankpage. DC(SJF/CGW)92626/1 UCLES2014 [Turn over

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

3 3 1 Show that cos A 1 sin A sin A cos A can be written in the form p sec A, where p is an integer to be found. [4] [Turn over

4 2 (a) On the Venn diagrams below, draw sets A and B as indicated. 4 (i) (ii) A B = A B [2] (b) The universal set % and sets P and Q are such that n(%) = 20, n ( P, Q) = 15, n ( P) = 13 and n ( P + Q) = 4. Find (i) n ( Q), [1] (ii) n ^( P, Q) lh, [1] (iii) n ( P + Ql ). [1]

5 5 3 (i) Sketch the graph of y = ^2x + 1h^x - 2h for - 2 G x G 3, showing the coordinates of the points where the curve meets the x- and y-axes. [3] (ii) Find the non-zero values of k for which the equation ^2x + 1h^x - 2h = k has two solutions only. [2] [Turn over

6 4 The region enclosed by the curve y = 2 sin 3x, the x-axis and the line x = a, where a 1 1 radian, lies entirely above the x-axis. Given that the area of this region is square unit, 3 find the value of a. [6]

7 7 5 (i) Given that 2 x y 1 # 4 =, show that 5x + 2y = 3. 8 [3] (ii) Solve the simultaneous equations 2 5 x y 1 x 2y # 4 = and 7 # 49 = 1. [4] 8 [Turn over

8 (a) Matrices X, Y and Z are such that X = c 1 2 m, Y = f4 5p and Z = ^1 2 3h. Write 6 7 down all the matrix products which are possible using any two of these matrices. Do not evaluate these products. [2] (b) Matrices A and B are such that A = c m and AB = 9 c m. Find the matrix B. [5] - 6-3

9 9 7 The diagram shows a circle, centre O, radius 8 cm. Points P and Q lie on the circle such that the chord PQ = 12 cm and angle POQ = i radians. Q 12 cm θ rad O 8 cm P (i) Show that i = , correct to 3 decimal places. [2] (ii) Find the perimeter of the shaded region. [3] (iii) Find the area of the shaded region. [3] [Turn over

10 10 8 (a) (i) How many different 5-digit numbers can be formed using the digits 1, 2, 4, 5, 7 and 9 if no digit is repeated? [1] (ii) How many of these numbers are even? [1] (iii) How many of these numbers are less than and even? [3] (b) How many different groups of 6 children can be chosen from a class of 18 children if the class contains one set of twins who must not be separated? [3]

11 11 9 A solid circular cylinder has a base radius of r cm and a volume of 4000 cm 3. (i) Show that the total surface area, A cm , of the cylinder is given by A = + 2rr. [3] r (ii) Given that r can vary, find the minimum total surface area of the cylinder, justifying that this area is a minimum. [6] [Turn over

12 12 10 In this question i is a unit vector due East and j is a unit vector due North. At hours, a ship leaves a port P and travels with a speed of 26 kmh 1 in the direction 5i + 12j. (i) Show that the velocity of the ship is ^10i + 24jh kmh 1. [2] (ii) Write down the position vector of the ship, relative to P, at hours. [1] (iii) Find the position vector of the ship, relative to P, t hours after hours. [2] At hours, a speedboat leaves a lighthouse which has position vector ^120i + 81jh km, relative to P, to intercept the ship. The speedboat has a velocity of ^- 22i + 30jh kmh 1. (iv) Find the position vector, relative to P, of the speedboat t hours after hours. [1]

13 13 (v) Find the time at which the speedboat intercepts the ship and the position vector, relative to P, of the point of interception. [4] [Turn over

14 (a) Solve tan x + 5 tan x = 0 for 0 G x G 180. [3] 2 (b) Solve 2 cos y - sin y - 1 = 0 for 0 G y G 360. [4]

15 15 (c) Solve r sec`2z - j = 2 for 0 G z G r radians. 6 [4]

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 reasonableefforthasbeenmadebythepublisher(ucles)totracecopyrightholders,butifanyitemsrequiringclearancehaveunwittinglybeenincluded,the publisherwillbepleasedtomakeamendsattheearliestpossibleopportunity. UniversityofCambridgeInternationalExaminationsispartoftheCambridgeAssessmentGroup.CambridgeAssessmentisthebrandnameofUniversityof CambridgeLocalExaminationsSyndicate(UCLES),whichisitselfadepartmentoftheUniversityofCambridge.

Cambridge International Examinations Cambridge International General Certificate of Secondary Education

Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 5 7 8 8 7 7 5 7 0 7 * ADDITIONAL MATHEMATICS 0606/23 Paper 2 May/June 2015 2 hours Candidates answer