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TEST OF MATHEMATICS FOR UNIVERSITY ADMISSION SPECIMEN 60 minutes PAPER 2 INSTRUCTIONS TO CANDIDATES Please read these instructions carefully, but do not open the question paper until you are told that you may do so. A separate answer sheet is provided for this paper. Please check you have one. You also require a soft pencil and an eraser. This paper is the second of two papers. There are 20 questions on this paper. For each question, choose the one answer you consider correct and record your choice on the separate answer sheet. If you make a mistake, erase thoroughly and try again. There are no penalties for incorrect responses, only points for correct answers, so you should attempt all 20 questions. Each question is worth one mark. Any rough work should be done on this question paper. No extra paper is allowed. Please complete the answer sheet with your candidate number, centre number, date of birth, and full name. Calculators must NOT be used. There is no formulae booklet for this test. Please wait to be told you may begin before turning this page. This question paper consists of 15 printed pages and 5 blank pages. ©UCLES 2016 2 BLANK PAGE ©UCLES 2016 3 1. Theradiusofthecircle2 2 8 A 12 15 0is B C D √37 E √67 2. Thegradientofthecurve A atthepointwhere √ 2is √2 B 3√2 C 4√2 D √2 E 6√2 ©UCLES 2016 [Turn over 4 Considerthefollowingattempttosolveanequation.Thestepshavebeennumberedfor reference. 3. x+5 = x+3 (1) 2 x + 5 = x + 6x + 9 2 (2) x + 5x + 4 = 0 (3) ( x + 4) ( x + 1) = 0 x = −4 or x = −1 Whichoneofthefollowingstatementsistrue? A Both 4and 1aresolutionsoftheequation. B Neither 4nor 1aresolutionsoftheequation. C Onesolutioniscorrectandtheincorrectsolutionarisesasaresultofstep 1 . D Onesolutioniscorrectandtheincorrectsolutionarisesasaresultofstep 2 . E Onesolutioniscorrectandtheincorrectsolutionarisesasaresultofstep 3 . ©UCLES 2016 5 4. Asetoffivecardseachhavealetterprintedontheirfrontandanumberprinted ontheirback,asfollows: Whichoneofthefivecards A,B,C,DorE providesacounterexampletothe followingstatement? Everycardthathasavowelonitsfronthasanevennumberonitsback. 5. Usingtheobservationthat2 approximately A B C D E F 2 ©UCLES 2016 3 ,itispossibletodeducethatlog 2is [Turn over 6 6. Theareaofarectangleismeasuredtobe5600cm correctto2significant figures. Thewidthoftherectangleismeasuredtobe80cmcorrecttothenearest centimetre. Whichoneofthefollowingexpressionsgivesthegreatestpossibleheightofthe rectangle? A 70.5cm B 75cm C D E F ©UCLES 2016 cm . cm cm . cm 7 7.Whichoneofthefollowingisasketchofthegraph 1? 4 4 3 3 2 2 1 1 0 0 –4 –3 –2 –1 0 –4 –3 –2 –1 0 1 2 3 4 5 6 –1 –1 –2 –2 –3 –3 –4 –4 A B 4 4 3 3 2 2 1 1 0 0 –4 –3 –2 –1 0 0 1 2 3 4 5 6 –4 –3 –2 –1 –1 –1 –2 –2 –3 –3 –4 –4 D C ©UCLES 2016 1 1 2 3 4 5 6 2 3 4 5 6 [Turn over 8 8. Considerthefollowingstatementaboutthepositiveinteger : Statement * :Thesumofthefourconsecutiveintegers,thesmallestofwhichis , isamultipleof6. Whichoneofthefollowingistrue? A B C D E Statement * istrueforallvaluesof . Statement * istrueforallvaluesof whichareodd,butnotforany othervaluesof . Statement * istrueforallvaluesof whicharemultiplesof3,butnot foranyothervaluesof . Statement * istrueforallvaluesof whicharemultiplesof6,butnot foranyothervaluesof . Statement * isnottrueforanyvalueof . 9. ConsiderthestatementaboutFred: * Everydaynextweek,Fredwilldoatleastonemathsproblem. Ifstatement * isnottrue,whichofthefollowingiscertainlytrue? A Everydaynextweek,Fredwilldomorethanonemathsproblem. B Somedaynextweek,Fredwilldomorethanonemathsproblem. C OnnodaynextweekwillFreddomorethanonemathsproblem. D Everydaynextweek,Fredwilldonomathsproblems. E Somedaynextweek,Fredwilldonomathsproblems. F OnnodaynextweekwillFreddonomathsproblems. ©UCLES 2016 9 10. Whichoneofthefollowingisasketchofthegraphof 4 –1 1 0 –1 –2 0 1 2 3 4 5 6 7 8 9 10 –2 5 6 7 8 9 10 01 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 –3 –4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 –1 –2 –3 –3 –4 C D 3 5 4 2 1 0 3 2 –1 3 4 4 3 1 0 2 B 5 4 –1 –2 0 1 –5 A 2 1 0 1? 3 2 5 3 2 1 0 log 2for –1 01 2 3 4 5 6 7 8 9 10 –2 –3 –2 –4 –3 –5 E F ©UCLES 2016 [Turn over 10 11. A tan B log 100 C sin D log 10 E 12. Whichoneofthefollowingnumbersislargestinvalue? Allanglesaregiveninradians. √2 Apolynomial 1 hasthepropertythat 1 2. Whichoneofthefollowingcanbededucedfromthis? A 1 2forsomepolynomial . B 1 2forsomepolynomial . C 1 2forsomepolynomial . D 1 2forsomepolynomial . E 2 1forsomepolynomial . F 2 1forsomepolynomial . G 2 1forsomepolynomial . H 2 1forsomepolynomial . ©UCLES 2016 11 13. Fiverunnerscompetedinarace:Fred,George,Hermione,Lavender,andRon. FredbeatGeorge. HermionebeatLavender. LavenderbeatGeorge. RonbeatGeorge. Assumingtherewerenoties,howmanypossiblefinishingorderscouldthere havebeen,givenonlythisinformation? A 1 B 6 C 12 D 18 E 24 F 120 14. Thegraphofthepolynomialfunction , issketched,where , , , , ,and arerealconstantswith 0. Whichoneofthefollowingisnotpossible? A Thegraphhastwolocalminimaandtwolocalmaxima. B Thegraphhasonelocalminimumandtwolocalmaxima. C Thegraphhasonelocalminimumandonelocalmaximum. D Thegraphhasnolocalminimaorlocalmaxima. ©UCLES 2016 [Turn over 12 15. Foranyrealnumbers , ,and where 1 2 3 ,considerthesethreestatements: 2 Whichofthestatements1,2,and3mustbetrue? A none B 1only C 2only D 3only E 1and2only F 1and3only G 2and3only H 1,2and3 16. Thesequence isgivenbytherule: 2 1 for Whatis A 150 B 250 C D 5150 E 4 1 F 4 ©UCLES 2016 4750 1 1 13 17. Let beasetofpositiveintegers,forexample couldconsistof3,4,and8. Apositiveinteger iscalledan ‐numberifandonlyifforeveryfactor of with 1,thenumber isamultipleofsomenumberin . Sointheaboveexample,9isan ‐number;thisisbecausethefactorsof9greater than1are3and9,andeachoftheseisamultipleof3. Positiveinteger isthereforenotan ‐numberifandonlyif A forevery positive factor of with 1, thereisanumberin whichisnotafactorof . B forevery positive factor of withm 1, thereisnonumberin whichisafactorof . C forevery positive factor of with 1, everynumberin isafactorof . D forsome positive factor of with 1, thereisanumberin whichisnotafactorof . E forsome positive factor of with 1, thereisnonumberin whichisafactorof . F forsome positive factor of with 1, everynumberin isafactorof . ©UCLES 2016 [Turn over 14 18. Agroupoffivenumbersaresuchthat: theirmeanis0 theirrangeis20 Whatisthelargestpossiblemedianofthefivenumbers? A 0 B 4 C 4 D 6 E 8 F 20 ©UCLES 2016 15 19. Thepositiverealnumbers , ,and aresuchthattheequation hasthreerealroots,onepositiveandtwonegative. Whichoneofthefollowingcorrectlydescribestherealrootsoftheequation ? A Ithasthreerealroots,onepositiveandtwonegative. B Ithasthreerealroots,twopositiveandonenegative. C Ithasthreerealroots,buttheirsignsdifferdependingon , ,and . D Ithasexactlyonerealroot,whichispositive. E Ithasexactlyonerealroot,whichisnegative. F Ithasexactlyonerealroot,whosesigndiffersdependingon , ,and . G Thenumberofrealrootscanbeoneorthree,butthenumberofroots differsdependingon , ,and . ©UCLES 2016 [Turn over 16 20. Fivelogicianseachmakeastatement,asfollows: MrP: Ofthesefivestatements,anoddnumberaretrue. MsQ: Bothstatementsmadebywomenaretrue. MrR: MyfirstnameisRobertandMrP’sstatementistrue. MsS: Exactlyonestatementmadebyamanistrue. MrT: Neitherstatementmadebyawomanistrue. Howmanyofthefivestatementscanbesimultaneouslytrue? A none B 1only C 2only D 3only E 4only F noneor1only G 1or2only H 2or3only END OF TEST ©UCLES 2016 17 BLANK PAGE ©UCLES 2016 18 BLANK PAGE ©UCLES 2016 19 BLANK PAGE ©UCLES 2016