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Rez Imprimare Bac m2 2009

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.Seobtinesumaegalacu3+6=9.2.Conditia4340,;3xx+>...-8.....ecuatiadevine434257;3xx+=.=..-8......3.121212111.2

    xxxxxx+=+=-4.f(1)=-1,V(0;0)punctdemaxim.f(x)=f(0)=0.f(x).[-1,0].5.AB=(-1-2)i+(3+1)j=-3i+4j.a=-3,b=4...................6.SeaplicateoremacosinusuluintriunghiulABC()2223:cos30.22ABBCACBmBABBC

    =+-=.=....

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.Deoarecef(3)=0rezultacaprodusulesteegalcu0.2.Conditiix+2>0six>0.x.(0,8).Ecuatiadevinex2+2x=8cusolutiax=2.3.Inecuatiasescriex2-5x+4=0.x.[1,4]n..={1,2,3,4}.4.13153132xx-++=x+,decinumerelesuntnprogresiearitmetica,pentru.x....5.OA+OB=(4i-8j)+(6i+3j)=10i-5j.............................VectorulOA+OB................arecoordonatele(10,-5).6.Ariasin2.2ACABA

    ABC.==

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.Sirulesteoprogresiearitmeticaderatier=6.a10=a1+9r=55.2.Exista23numerenaturaledetreicifrescrisecuelementedinmultimea{1,2}.Dintreacesteasuntdivizibilecu3numerele111si222.Probabilitateaesteegalacu0,25.3.Conditiax.[0,8).Ecuatiadevinex2-x-2=0.x=2.4.f(-2)+f(-1)+f(0)+f(1)=-3-1+1+3=0.5.EcuatiadrepteiAB:x-y-3=0.6.Ariasin1.22ACABAABC.==

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.Inecuatiasescriex2-x-6

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.A=[-3;1]n..adicanumarulelementelormultimiieste5.2.nmultimea{1,2,3,...,30}singurelecuburiperfectesunt1,8si27,deciprobabilitateaeste30,130=.3.Ecuatiadevine8x+8=0.x=-1.4.x=400.5.5u+3v=-15i+10j+15i-3j=7j...............Coordonatelevectoruluisunt(0,7).6.BC=2AD=10.SeaplicateoremaluiPitagoran..ABC:AB2=BC2-AC2=64.AB=8.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.a2+b2=(a+b)2-2ab=16-6=10.2.Serezolvasistemul214yxxyx..=-+....=+2x-x+1=x+4.x1=-1,x2=3.y1=3,y2=7.Coordonatelecerutesunt(-1,3)si(3,7).3.Deoarece()333lglg2lg310101002x+x=.xx=.x=.x=.x=.3.131531232xx

    -++=x+,decinumerelesuntnprogresiearitmetica.4.SingurelenumererationaledinmultimeaAsunt4si9.Probabilitateaesteegalacu2.95.Dinconditiadeparalelismadreptelor213a25=-.rezultaa=-4.6.DeoareceAB2+AC2=BC2...ABCdreptunghicnA,deci5cos

    5ABBBC==.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.Deoarecex1+x2=2,x1x2=-2.x1+x2+x1x2=0.2.Inecuatiasescrie1280,.4xx-=...-8.....3.Ecuatiadevine3x-2=3-x.x-2=-x,darx=0six=2,adicax.[0;2],decix=1.4.3-3=0.5.AB..CD.mAB=mCD.mAB=-a-1,mCD=2.a=-3.6.Seaplicateoremacosinusuluin2221cos.25ABACBCABCAABAC.=+-=

    ..

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.(113)7492S+==.2.f(x)=x.2x+1=x.x=-1.Punctulcerutarecoordonatele(-1,-1).3.Ecuatiasescrie2x+82x=36.2x=4.x=2.4.4!+1=25.5.Pantadrepteiesteegalacu-2,deciecuatiadrepteiestey-1=-2(x-1).2x+y-3=0.6.Deoarecesin130..=sin(180..-50..)=sin50...sin250..+cos250..=1.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.log39=2,log28=3si41log14=-,deaiciconcluzia.2..=0.m2-4m=0.m.(-8,0].[4,8).3.x2-x-2=0.x.{-1;2}.4.Ratadobnziieste8%.5.xB-3=1.xB=4,yB-4=1.yB=5.6.92AABCD=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.341312713a=aq==.2.Ecuatiadevine4x2-4=0.x.{-1,1}.3.Senoteaza2x=t>0siserezolvaecuatiant,t2-3t+2=0;obtinemt1=1,t2=2.AtunciS={0,1}.4.a=b=8.5.w=2(3i+4j)-3(2i-3j).w(0,17).................6.Ariasin115sin22ABACAABCA

    ==.=...

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.5+120=125.2.Sumaesteegalacu12181.3.Seobtine3ax+3b+2=3x+5,.x....a=b=1.4.Conditii:x2-2x>0si2x-3>0.x.(2,+8).Serezolvaecuatiax2-4x+3=0.x=3.5.AB..CD.mAB=mCD,15,622ABCDamma==-.=.6.Seaplicateoremasinusurilorn242sinBCABCRR

    A...=.=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.Deoarecef(-5)=0,produsuldinenuntesteegalcu0.2.Ecuatiadevinen2-n-56=0.n=8.3.32log32+log32+log325-log325=5a.4.Deoarecex2+x+1>0,.x...,dupaaducerealaacelasinumitorsiefectuareacalculelor,inecuatiadevinex2-x-2=0.x.[-1,2].5.PantadrepteiABesteegalacu1.EcuatiadrepteiABestey-3=x-2.x-y+1=0.6.Ariasin16sin.22ABBCBABCB==.=..

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.Numarultuturorsubmultimilorde2elementecesepotformacuelementedinmultimea{1,2,3,4,5}esteegalcu2C5=10.2.Ecuatiasescrie22()()03200,3fxgxxxx+=.-=.........3.Conditie:()2{}x.2.x-2=9.x.5,-1.4.Conditie:.=0.m2-4m+4=0.m=2.5.AB=5.Aria2325344l..ABC==.6.3cos

    5x=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.Ecuatiaaredouasolutiirealedistinctedeoarece.=1>0.2.Deoarecef(6)=0produsulesteegalcu0.3.72x=28.x=2.4.6565202-=.5.LungimeasegmentuluiAB=(5-2)2+(-1-3)2=5.6.SeaplicateoremacosinusuluintriunghiulABC.AC2=12.Perimetrul..ABC=6+23.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.Numarulsubmultimilorcuctekelementealeuneimultimifinitecunelemente,0=k=neste246kCn.C=.2.Ecuatiasescrie31155313x=-.x=-.x=-.3.Conditie:110140,.44mmm..=.-=.=..8.....4.24x=32x+2.Notnd2x=t>0,serezolvaecuatia2t2-3t-2=0.t=2.Decix=1.5.AB+BC+CA=AC+CA=0..........................................

    .6.SeaplicateoremacosinusuluintriunghiulABC.BC2=21.Perimetrul..ABC=9+21.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.33C8-C8=0.2.Conditiex+5>0.x.(-5,8);x+5=8.x=3.3.Senoteazax1+x2=S,x1x2=P.Deoarecex2-Sx+P=0.x2-x-2=0.4.Deoarecef(0)=2.f(2)-f(2)=0.5.PunctulBestemijloculsegmentuluiAC522xC+.-=si()419,22yCC+=.--.6.TriunghiulABCestedreptunghicnA,decilungimeanaltimiidinAesteegalacu

    12.5ABACBC=

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.2log34-4log32=4log32-4log32=0.2.Ecuatiasescrie2212283.2x+x=.x=.x=3.Ecuatiasescrie2n=10.n=5.4.Functiafestedescrescatoarepe[0,2],f(0)=3,f(2)=-5.f(x).[-5,3].5.FieDmijloculsegmentuluiBC,atunciOB+OC=2OD=AO=-OA.OA+OB+OC=OA-OA=0...................................................................................6.()2sin135sin18045sin452=-==.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.22221log3loglog3log30.3+=-=2.Deoarece5!=120,4!=24.probabilitateaesteegalacu56.3.Senoteaza2x=t>0.Ecuatiadevinet2+5t-14=0.t=2,decix=1.4.Deoarece.4sin2a-4(1-cos2a)=4sin2a-4sin2a=0.ecuatiaadmitesolutiirealeegale,.a....5.3OA-5OB=6i-9j-5i+10j=i+j.a==1.............................6.SeaplicateoremasinusurilorntriunghiulABC2sin1.sin2BCBCRAAR

    .=.==

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.log624-log64=log66=1.2.f(1)=0,deciprodusulesteegalcu0.3.Conditie:x-5=0.x.[5,8).Dinx-5=4.x=9.4.()()()()()()25!43643676065!nnnnnnnnn---=.--=.-+=.=-.5.AB=(5-a)2+(2+a)2=5.a2-3a+2=0,deciS={1,2}.6.sin135..=sin(180..-45..)=sin45...cos245..+sin245..=1.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.333312log6log2log4log1.4+-==2.Conditie:x2-x-2=0.x.(-8,-1].[2,8).Ecuatiadevinex2-x-6=0.S={-2,3}.3.Senoteazax1+x2=S,x1x2=P.Deoarecex2-Sx+P=0.x2-2x-3=0.4.Conditie:m-1>0.m.(1,+8).Din()222.21mmm+=.=-5.AB=16+9=5.

    6.Conditie:x.(0,8).ConformteoremeiluiPitagora(x+8)2=x2+(x+7)2.x2-2x-15=0.x=5.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.Conditii:x+1=0si5-x=0.x.[-1,5].x2-11x+24=0.x=3.2.f(0)+f(1)++f(5)=2(1+2+...+5)+63=48.3.Scaznd2dinfiecaremembrualinegalitatiisiapoimpartindcu3,seobtine22,.3x..-.....4.Distantaesteegalacux1-x2,undex1six2suntsolutiileecuatiei()fx=0.x1-x2=-2-4=6.5.22.ABABBCBC=.=................6.PrinreciprocateoremeiluiPitagora,triunghiulABCestedreptunghicA.Aria24.2

    ABACABC==..

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.2x=8.x=4.2.Distantaesteegalacux1-x2,undex1six2suntsolutiileecuatieif(x)=0.x1-x2=7-1=6.3.1+3+5++21estesumaa11termeninprogresiearitmeticaderatieegalacu2,deciE=112=11.4.Cuelementelemultimii{1,2,3,4}sepotforma3A4=24denumeredectetreicifredistincte.5.DinCA=2CBsiC.(AB).AC=2CB.................DinAC(xC-2,yC-1)........si()551,20,0,33CBxCyCxCyCC---.==...............

    .6.SeaplicateoremasinusurilorntriunghiulABC423sinsinsin3sin42ABBCACAA.=.=.=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.Deoarece796218729,4.22xxxx=-=.=3.Deoarecex-3=1.x=4.4.Cuelementelemultimii{1,2,3,4}sepotforma42=16numerededouacifre.5.3312222OAOBijOMij=+=+=+...................................

    CoordonatelevectoruluiOM..........sunt31,.22......6.()3sin120sin18060sin60.2=-..=..=

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.1+3+5++19estesumaa10termeninprogresiearitmeticaderatieegalacu2,deciesteegalacu100.2..=-4a2

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2REZOLVARE1.60lg20lg3lg6lg1.6+-==2.619015p==.3.Conditie:7-x=0.x.(-8,7].Din7-x=1.x=6.S={6}.4.x1+x2=2m+1,x1x2=3m.5m+1=11,decim=2.5.AC=asinB,AB=asinC.2S=ABAC=a2sinBsinC.6.sin170..=sin(180..-10..)=sin10...sin10..-sin10..=0.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.311615251115112aaraaarr.=.+=.=......=.+=.=;a9=a1+8r=17.2.(322)20(1)(2)...(20)34...22250.2fff++++=+++==3.242522x+=2x+.2x+4=x+5.x=1.4.n+2=2.n=0.5.

    23312mm=.=--.6.cosx+cos(180..-x)=0,.x...;cos30+cos60+cos120+cos150=0

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.2x-1=1.-1=2x-1=1.x.[0,1]n...A={0,1}.2.121235xxxx+=-...=-;22()2x1+x2=x1+x2-2x1x2=19.3.x2-25=0.x.(-8,-5].[5,8);x2-25=144.x=13.(-8,-5].[5,8).4.Princalculseobtine0.5.FieMmijloculluiAB.M(3,4);CM:3x-4y+7=0.6.sinAria62MNNPN

    MNP.==.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.festedescrescatoarepe[-2,1].ceamaimicavaloareestef(1)=-2.2.f(1)+f(2)+...+f(6)=1+3+...+11=36.3.22505,3302xxxx+>.....-8......++>..;2125525331,,2,22xxxxx+=++.=..-8.=-..-8.....

    .....4.22345416,10,43C=C=C=.p=.5.FieMmijloculsegmentuluiBC57,22M

    .......;2257223222AM=.-.+.-.=.........6.33sin60cos300.22

    -=-=

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.22C5-A4+6=10-12+6=4.2.f(-6)+f(0)+f(6)+f(12)=0.3.x2-1>0.x.(-8,-1).(1,8);x2-1=3.x=2.(-8,-1).(1,8).4.{()()}2223232,1;2,7274xyyxSxxyx..-=..=-....=--..+-=..=.5.301102mnmmnn

    .-+=.=....++=.=-

    .6.sin120=sin60..,cos150=-cos30...produsuleste0.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.82721122...2125521++++=-=-.2.x2-3x+2>x-3.(x-2)2+1>0,.x....3.[)2300,0xxx.+=...8.=;2[)[)2x+3=x.x1=3.0,8,x2=-1.0,8.x=3.

    4.Inegalitateaesteverificatapentrun.{1,2,4,5}45.p=.5.23215mmm-=-.-.=.6.123123

    sin30cos45sin602222-+=-+=-+.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.11511133aaar.=.=....=.=;a2009=a1+2008r=6025.2.12122xxmxx+=-...=;m2-4=5.m=3.3.222222012,21x-x=.x-x-=.x=x=-.

    4.22()211(1)114410210,44f=-.m-+m+=-.m+m+=.m+=.m....5.Fie{M}=ADnBC.MestemijloculsegmentelorADsiBC5,22M.......siD(6,5).6.cosx+cos(180..-x)=0.cos100+cos80=0.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.a10-a2=16.8r=16.r=2.2.f(2)+f(22)+...+f(27)=2+3+22+3+...+27+3=275.3.[)101,10xxx.+=...8.-=;[)[)21201,12131,xxxx

    x..=.8+=-+....=.8.Decisolutiaestex=3.4.Inegalitateaesteverificatapentrun=1sin=42142.p==.5.()2202,2380xyAxy.--=

    ...+-=;d=22+22=22.6.222222222sinsin1ACABACABBCBCBCBC+=+=+=

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.r=a2-a1=2;a10=a1+9r=20;(110)10101102aaS+==.2.()277373222Vbxmfxxxa=.-=.=.=-+.3.32x-1=35-x.2x-1=5-x.x=2.4.2A5-P3=20-6=14.

    5.4-3+m=0.m=-1.6.2sin46Aria26222MNNPNMNP.===.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.()2[]2x-1=9.-3=2x-1=3.x.-1,2.2.f(0)+f(1)+...+f(10)=1+2+...+11=66.3.Conditii:240440xxx..+>..>-..+>;2x+4=x+4.x1=0,x2=1.4.1333426,3,43P=A=C=.p=.

    5.AB:x+y+1=0.6.3sin56153Aria2222ABACAABC.===.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.5555103log10log3log6log16+-==.2.f(1)+f(2)+...+f(6)=3+5+...+13=48.3.2552555511,25x-x=x-.x-x=x-.x=x=.4.Senoteazacuxpretulinitial.Seobtineecuatia110120660500100100x=.x=lei.5.()2()2AB=-2-2+2+1=5.6.2222NP=MN+MP-MNMPcosM.NP=19.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.22x=24.x2=4.x=2.2.f(2)=0.produsuleste0.3.Conditii:[)2202,20xxxx..--=...8..-=;x2-x-2=x2-4x+4.x=2.[2,8).4.Inegalitateaesteverificatapentrun=5sin=612.p=.5.FieCsimetriculluiAfatadeB.Bestemijlocullui(AC).C(0,0).6.sin10=cos(90-10)=cos80;sin280+sin210=sin280+cos280=1.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.213bqb==;4b5=b1q=162.2.2181344mma-.=-.-=.=.3.22x-5=x-8.x1=3,x2=-1.4.()!21

    2!2!nn=-;-6nuconvine.n=7.5.d:y=x+n;A(1,1).d.n=0.d:y=x.6.2223cos22ABBCACB

    ABBC=+-=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.1321log4822222-+..-=+-=.....2.f(0)+f(1)+...+f(6)=3+1-...-9=-21.3.Conditii:169-x2=0.x.[-13,13];169-x2=144.x=5.[-13,13].4.3A4=24.5.FieDmijlocullui(BC).D(2,0);()2()2AD=2-2+4-0=4.6.Catetelesunt4si43.Aria=83.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.x2-Sx+P=0.x2-5x+6=0.2.{()()}221;1;2;0220yxSxxx=-....=..-+-=.3.Conditii:9-x2>0.x.(-3,3);9-x2=5.x=2.(-3,3).4.Inegalitateaesteverificatapentrun=1sin=2,deci12p=.5.sin(180..-x)=sinx.sin135=sin45;2

    sin13521cos4522==.6.2sin84282.22ABCABACAA..===

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.x2-9=0.S=[-3,3].2.201022009f..=.....punctulapartinegraficului.3.3x=t>0;t2-4t+3=0.t=1.x=0,t=3.x=1.4.192122xx+=+.=.5.MN:x+y-3=0.6.2214tg30ctg451

    33+=+=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.r=a2-a1=-1;a7=a1+6r=0.2.x2+3=12.x.[-3,3].3.2x=t>0.t2-6t+8=0.t=2.x=1,t=4.x=2.4.4A5=120.5.AB=22,AC=2,BC=10.AB2+AC2=BC2..ABCestedreptunghicnA.6.cosx+cos(180..-x)=0;(cos10+cos170)+(cos20+cos160)=0.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.3211xyxxyy.+=.=....-=.=.2.f(2)+f(22)+...+f(25)=2+5+22+5+...+25+5=87.3.2232321252223231,2x+x-=.x+x-=.x=x=-.4.Inegalitateaesteverificatapentrun=2sin=3,deci12p=.

    5.AB:(a+1)x+4y-2a+2=0;cumO(0,0).AB.a=1.6.224sincos1cos5x+x=.x=si4cos0cos5x>.x=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.a8=a2+6r=23.2.f(3)+f(32)+...+f(35)=3+2+32+2+...+35+2=373.3.1210,2xx+>...-8.....;12152,2xx+=.=..-8......4.2C6=15.

    5.FieCmijlocullui(AB).SeobtineC(1,1).6.sin(180..-x)=sinx.sin150=sin30;sin2150+cos230=sin230+cos230

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.22Vbxa=-=-;94yVa=-.=-.2.f(1)+f(2)+...+f(10)=-1+2+...+26=125.3.10-x>0.x.(-8,10);10-x=9.x=1.(-8,10).4.n(n-1)=12.n=4,n=-3.n=4.5.AB=4,AC=13,BC=13;P=4+213.6.123sin30,sin45,sin60222===;1

    3p=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.213bqb==;3b4=b1q=27.2.1212xx1xxm+=...=;1211333611424m

    xxm+=-.=-.=-+++.3.Conditii:[)2402,20xxx..-=...8..-=

    ;x2-4=x-2=0.x=2.[2,8).4.Inegalitateaesteverificatapentrun.{1,2,4}34.p=.5.FieCsimetriculluiAfatadeB.Bestemijlocullui(AC).SeobtineC(1,3).6.3sin104Aria2103.22MNNPNMNP.===

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.11101077752295.375aaaSar..=.=....=.=..=.=2.f(7)=0.produsuleste0.3.x=1;x-1=2.x=5.[1,8).4.554C7-C6-C6=21-6-15=0.5.()2()22+1+a+1=5.a1=3,a2=-5.a=3.6.3236;93

    24llh=.l=A==.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.1133372aaar.=.=....=.=;(110)10101202aaS+==.2.f(m)=-1.m1=2,m2=1.3.

    3230,2xx+>...-8.....;3232511,2xx+=.=..-8.....

    .4.3C5=10.5.FieMmijloculluiAB.M(0,0);CM=5.6.1sin88Aria21622ABACAABC.===

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.(1111)12111...1116722++++==.2.2f(m)=4.m-2m+4=4.m1=0,m2=2.3.2132221311,22x+x+=.x+x+=.x=x=-.4.Inegalitateaesteverificatadoarde414.p=.5.2m+m+m=0.m1=0,m2=-2.6.3sin66Aria293

    22MNNPNMNP.===.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Rezolvare1.3x+2=4x-1.x=3,x....A={0,1,2,3}.2.GfnOy:A(0,-3);3:,02GfOxBn.......3.x2-4=0.x.(-8,-2].[2,8);S={22}.4.850040100=lei.5.OA=2i+3j;OB=-i+5j.OA+OB=i+8j.v(1,8)...............................................6.l=2.P=6,deci23

    4lA=.A=3.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Serezolvaecuatia2(2x-3)=x+1+x-3siseobtinex=2.2.Dacaxestepretulinitialalprodusului,atunci1099100x-x=,deundex=110lei.3.Numaruleste0deoarececombinarilesuntcomplementare.4.Dacaf(x)=ax2+bx+ccua,b,c...,a.0,atuncif(1)=3,f(0)=5sif(-1)=11,deundea=2,b=-4sic=5.5.MN,MP,PNliniimijlocii;deciAMNPparalelogram.ObtinemAN=AM+AP...........................6.Dinteoremacosinusuluiseobtine2cos2A=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Numaruleste1deoarece223loglog312=-.2.Serezolvasistemul24030xyxy+-=...+-=siseobtinepunctulcomunA(1;2).3.Senlocuiestexcu5siseobtine71;4S=.-...

    ...4.Dinconditiirezultax.[-2;8).Seobtineecuatia3x2+2x-1=0cusolutiile-1si13.5.AB=10;AC=1;BC=13.DeciP=1+10+13.6.SeaplicateoremasinusurilorsiseobtineAC=2.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.1291lglg...lglg1231010+++==-.2.Numaruleste0deoarececombinarilesuntcomplementare.3.Senoteaza3xcutsiserezolvaecuatia1103tt+=.Seobtine13;3t.......six.{1;-1}.

    4.Trebuiesaavem.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Seaplicaproprietatilelogaritmilorsiseobtinerezultatul1.2.Deoarecefunctiaestedescrescatoareavemmaxf=f(-1)=5.3.Dinx1x2=3six1=3x2seobtinex2=1.Dacax2=1,atuncim=-3;dacax2=-1,atuncim=5.4.Sefolosesteformulacombinarilorcomplementare:111nCn+=Cn+sirezultatuleste0.5.sin10..-cos80..=cos80..-cos80..=0.6.MijloculsegmentuluiABarecoordonatele(3,3).

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.22=4silog232=5,deci22

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Deoarecelog28=3,seobtinenumarul2....2.Serezolvasistemul46202370xyxy--=...+-=siseobtinepunctulcomunA(2;1).3.Deoarece2x1+x2=m+3six1x2=3,seobtineecuatiam2+6=7carearesolutiile1.4.Dupasimplificariseobtineecuatian2+3n-54=0;convinedoarsolutia6.5.22222222coscos1ABACBCBC

    BCBCBC+=+==.6.sinria432ABACAA==.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Ratiaeste3si()62176572S+==.2.Sepuneconditia.=0siseobtinem2-36=0,deundem=6.3.Conditiax2+3x-10>0.x.(-8;-5).(2;8).Ecuatiadevinex2+3x-10=8.Seobtinex.{3,-6}.4.Elementeledivizibilecu5sunt15si35,deci14p=.5.Ecuatiaeste1042xy+-=,adicax+2y-4=06.Seaplicateoremacosinusuluisiseobtine

    3cos5B=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Numaruleste0deoarecelog525=2silog39=2.2.Serezolvasistemul2a+b=7si-a+b=-2siseobtinea=3,b=1.3.22()2x1+x2=x1+x2-2x1x2=1+2=x1+x2+2.4.Din10-3n=0sin...seobtinen.{0;1;2;3}.5.MijloculsegmentuluiBCesteM(3;0)siAM=5.6.Deoarececos30ABBC..=rezultaBC=8.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.()2()()x+7=5-x3x+11siseobtine13;2x..-......2.Dacasenoteazacuxpretulnleialprodusuluilaproducator,seobtine19238100x+x=,deundex=200sivaloareaTVAeste38lei.3.log24=2,log39=2,deci2+2

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Seobtinex2+x=2,deundex.{1;-2}.2.Sepuneconditia2x-3>0siseobtine3;2D=.8......3.2fmin=3m-m.fmin=2.m.{1;2}.4.20091004-20071004-2008=0.5.SeaplicateoremacosinusuluisiseobtineAC=57.6.MestemijloculsegmentuluiAB,deciM(3;-2).

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Numaruleste630log15=.2.Sepuneconditia.=0siseobtine()2m-1=0,adicam=1.3.Seobtine(x+1)(x-5)=0adicax.[-1;5].4.8!563!5!=,9!362!7!=decinumaruleste20.5.sinxsinx+(-cosx)2=sin2x+cos2x=1.6.Aria1010sin30

    252==...

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.x+2>0.x.(-2;+8).x+2=9.x=7.(-2;+8).2.maxf=4-m=10.m=-63.Conditia:11;;214924;22xxx..-8.+=.=..-8..........4.n(n-1)=n+8,n...,n=2.n.{2;3;4}.5.()2{}AB=13.25+a-1=169.a.-11;13.6.SeaplicateoremasinusurilorsiseobtineR=20.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a1+a1+4=10.a1=3.2.x1+x2=m,x1x2=m+2;2m+4=m.m=-4.3.Conditia:()21;;201xxxx.-8+=.=+.4.42105p==.5.MijloculsegmentuluiBCeste3;22

    M......,iarsimetriculesteD(0;4).6.1016sin60ria4032A==...

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Ratiaeste2si132b=.2.121212112009xxxxxx+=+=.3.Conditia:x2-x-2>0.Seobtinex2-x-2=4,deundex.{-2;3}.4.Seobtineinecuatia(19-n)(18-n)=n(n-1),n...,n=2,n=17,deunden.{10;11;...;17}.5.SerezolvasistemulformatdinceledouaecuatiisiseobtineA(-2;1).6.SefolosesteteoremacosinusuluisiseobtineAB=32.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Numaruleste3....2.Seobtineecuatiax2-4x=-3.x.{1,3}.3.Seobtineecuatia4m-m-6=0,deundem=2.4.219045p==.5.PuncteledeintersectiecuaxelesuntA(0;-5)si5;03B......,iarariaeste256.

    6.2231sin120cos60144..+..=+=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Searataca12C3=log22+5.2.PunctelesuntA(0;2)siB(-1;0).3.12616-=m-.m=-4.Numaruleste10.5.AB=BCcos60..=4,AC=BCsin60..=43.6.Ariaeste4.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.5+1=6=3!2.PunctelesuntA(1;0),B(-1;0)siC(0;-1).3..=5m2+4>0,.m....4.Serezolvasistemulb1+b2=8sib2-b1=4.Seobtineb1=2,apoiratiaq=3sisuma26.5.SefolosesteteoremasinusurilorsiseobtineAC=102.6.Seobtine5;22M......siAM=6,5.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Serezolvasistemuldeinecuatii3x+2>-4si3x+2

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Rezultatuleste02.Serezolvainecuatiax2-2x-15=0siseobtinex.[-3;5].3.Sepuneconditia.=0,deundem=-1.4.A=2....5.Deoarecesin10..=cos(90..-10..),rezultasin10..-cos80..=0.6.SegmenteleMPsiNQauacelasimijlocsiMP=NQ=210,deciMNPQestedreptunghi.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.x.[2;3].2.42min124mmfm=-=.=.3.Conditia:x.0.Serezolvaecuatiax2=4siseobtinex=2.4.Rezultatuleste10.5.SeobtineM(1;-2)siapoiAM=3.6.()1cos180cos2..-x=-x=-.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.5+10+1=16=24.2.Seobtine6x=62,deundex=2.3.Sefolosescrelatiilex1+x2=2msi2x1x2=m-1si()()2x1x2-x1+x2+2=m-1=0.4.Seobtinex2+2x-3=5six2+2x-3>0,deundex.{-4;2}.5.23a=-deoarece23AG=AM..................,deci23AG=-MA................

    .6.Avemm(..ABC)=30..siariaesteegalacu40.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Rezultatuleste6.2.xV=1,yV=1.3.Seridicalaputereaatreiasiseobtinex=-1.4.719113p==.5.SeobtineAC=20,AO=OD=10siseaplicateoremacosinusuluintriunghiulAOD:()7cos25..AOD=.6.111442+=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Seobtineratia2sia5=7+24=15.2.Seobtineecuatian(n-1)=12,n...,n=2cusolutian=4.3.Avem.=1>0,.m....4.Conditia:31;22x..-......Seobtineecuatia(x+4)(2x+3)=1-2x,deundex=-1.5.SeobtineAC=CB.................6.AC=BCsinB=9siAB=12.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Rezultatuleste0.2.b1(q-1)=3.q=2.3.Conditia:x.(-1;8).Seobtineecuatiax+1=2,deundex=3.4.x2-11x+30=0.5.x+y-7=0.6.BC=5,AB=53,Aria=253.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Ratiaprogresieieste2six=3.2.f(x)=g(x).x2-4x-5=0.DeciA(5;9)siB(-1;3).3.x3+x2-x-2=x3.x.{-1;2}.4.8150015001620100+=lei.5.OM+ON+OP=PO+OP=0............................................6.Aria=123.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Avem2log39=log33=2log33=21=2si364=343=4;cum2=14,rezultacatermeniisuntnprogresiegeometricaderatie2.2.Cumf(2)=2-2=0,rezultaprodusul0.3.Conditii:x2+2x-3=0,prinridicarelapatratobtinemx2+2x-15=0,ecuatiedegradul2cusolutiilex=3six=-5,careverificaconditiiledeexistenta,deciS={-5;3}.4.215120;2520,222xtttttt=>+=.-+=........,adicaS={-1,1}.5.Coordonatelepunctuluisunt350(2)4,1

    22xy=+==+-=-.6.2222sin135cos45sin45cos451+=+=.........

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.22222512log5log12log30loglog2130+-===.2..=-3m2-40siavem2t2+2=t+4t.2t2-5t+2=0,cusolutiile12sau2t=t=,decia=1saua=-1.4.122Cn+1=n-1.n+1=n-1.n=2.5.MN-MQ=....................PN-PQ.

    ................QN=QN

    ................,adevarat.6.cos(90..-x)=sinxsicos(180..-x)=-cosx;obtinemsin2x+cos2x=1,adevaratpentruoricarex..

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.141322CA+=.2.(1)(21)12xxx+=-+-,deci2x+2=3x-2,x=4.3.()()()012344325

    4111112222121310142222221616fff+++=..+..+..+..+..=++++=-=.....................4.UtilizamrelatiileluiVite:S=x1+x2=m-1;P=x1x2=-m,decim-1=2(-m+4).m=3.5.12136,

    2112yxyx-=-.-=----obtinemAB:y=3x-5.6.sin,sin,ACABABACBCADBCBCBC===;obtinem22ACABABAC

    ADABACBCBCBC==......,adevarat.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.5555518log18log2loglog92log32-===,deci55555log18log22log32log3log3-==.2.a(f(x)+h(x))=a(4x+4)=2a(2x+2)=2ag(x),()()122agx=gx.a=.3.Cum4x=(22)x=2x,dinegalitateadataobtinem23x=8=23sidininjectivitateafunctieiexponentialerezultax=1.4.P4=4!=24.

    5.212532Cmxm==-+.=sau3m=-.6.DC=AC-BC=AC+CB........................................,deciDC=AB.................RezultacaABCDparalelogram.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.182!3!261C8+=+=.2.Echivalentcuaaratacaf2(0)=f(1)f(-3),adica32=19,adevarat.3.Dinprimaecuatieavemy=3-x,siadouaecuatiedevinex2+2x-3=0.x1=1,x2=-3,careimplicay1=2,y2=6;deciS={(1,2);(-3,6)}.4.Conditii:3x+1>0,x-1>0.decix>1;()(())log53x+1=log55x-1sivem3x+1=5x-5.x=3solutiecareverificaconditiiledeexistenta,deciS={3}.5.ON=OM=(-2-0)2+(3-0)2=13,deciMN=2OM=213.6.2sinBCRA

    =,adica633sin243232BCAR====,decim(..A)=60...

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.23221loglog22;824=-=--=-,rezultatuleste0.2.Seobtineinecuatia:2x2+2x-12=0,echivalentacux2+x-6=0,deciS=[-3,2].3.422xV=-=-,decif(2)estemaximulfunctiei,decif(x)=f(2),oricarearfix....4.101025540800100100100xxxxx

    .-.-.-.=.=........

    .5.OM2=xM2+yM2=22+m2=5,decim2=5-4=1,m=1.6.BC2=AB2+AC2-2ABACcosA,deciBC2=16+36-246cos600=52=28,BC=27.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.3332333333933===,deciobtinem0.2.DinrelatiileluiViteavemx1+x2=-a;x1x2=-a-1,decix1+x2-x1x2=1.3.Obtinem12233x-..=..........six=-1.4.C.(AB)siCA=2CB;deci14

    3CB=AB=.5.EcuatialuiAB:y=2x+1.EcuatialuiCD:y=2x-1.ABesteparalelacuCDpentrucaaupanteleegale,mAB=mCD=2siordonatelelaoriginediferite(-1.1).6.Utilizamproprietatileunghiurilorsuplementare:sin(180..-x)=sinx,cos(180..-x)=-cosx,decisin100..+cos100..-a=sin(180..-80..)+cos(180..-80..)-a=sin80..-cos80..-a=a-a=0.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.122C3-A3=23-32=0.2.22222214376log14log3log6logloglog766+-===.3.Conditii:x+1=0,x2-x-2=0;obtinemx+1=x2-x-2,adicax2-2x-3=0carearesolutiilex1=-1,x2=3.4.DinrelatiileluiVite,1212(1)1;11mmxxmxxm+=--+=+==,decix1+x2-x1x2=(m+1)-m=1.5.2

    46sin()26222ABCABACBACA===.6.Cumsin(180)sinxx-=..,obtinemsin135tg45cos451+-=.......

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Cum2>1si3-2

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.34111616228b=..==.....2.Sistemulesteechivalentcurezolvareaecuatieit2-St+P=0,undeS=x+y=-6,P=xy=8,decit2+6t+8=0.SistemularesolutiileS={(-2,-4);(-4,-2)}.3.Ecuatiaesteechivalentacu2-x=22siobtinem-x=2,x=-2.4.Numarulcazurilorposibileesteegalcu32=9.Numarulcazurilorfavorabileeste3;3193P==.

    5.AC-AD=AD-BD.DC=AB.................................................6.sin(180..-x)=sinx,deci()4sin180.5..-x=

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Sistemulesteechivalentcurezolvareaecuatieit2-St+P=0,undeS=x+y=5,P=xy=6,decit2-5t+6=0,t1=2,t2=3;decisistemularesolutiileS={(2,3);(3,2)}.2.f(-1)=5-(-1)=5;f(0)=50=1;5f(1)=55-1=50=1,obtinemf(-1)+f(0)+5f(1)=5+1+1=7.3.Cum(1+2)2=1+22+2=3+22obtinem(3+22)x=(3+22)1siobtinemx=1.4.2665152C==.5.FieM(x;y)mijloculsegmentuluiAB,241(3)3;122

    xy=+==+-=-,M(3;-1).6.Avemproprietateacos(180..-x)=-cosx,deci()1cos1803..-x=-.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a4=2+33=11.2.Dinconditiile.>0,P0sim0;2x-4>0.x.(2;8);dinproprietatilelogaritmilorobtinem:(2)log2x-x-2=log2(2x-4)+1=log2(2x-4)+log22=log22(2x-4)siavemx2-x-2=4x-8.x2-5x+6=0,cusolutiile2si3,dardoar3verificaconditiileimpuse,deciS={3}.4.n+n(n-1)=4.n2=4,darn=2,decin=2.5.122sin2122ABCABACAA===.6.Cumsin(180o-x)=sinx,obtinem

    222212sin1352sin4522122o..==....==.....

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a1=10,a4=19=a1+3r.10+3r=19.r=3.2.fdescrescatoare.fmin=f(1)=0.3.Conditie:x>0.Notamlgx=t.2t-3t+2=0.t1=1,t2=2.S={10;100}.4.15460400100x+x=.x=5.OA=3i+4j.B(3,4);OB=7i+2j.A(7,2)........................,deciM(5,3).6.Cumsin100..=sin80..;cos100..=-cos80..,rezultatuleste0.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Obtinem1+2+22++26=27-1=127.2.(x-1)(x+1)2=0;cum(x+1)2=0,pentrux.-1avemx-1=0,decix=1.Pentrux=-1,inecuatiaseverifica,decisolutiaesteS={-1}.[1,+8).3.Cum.=20092+4m2>0,existasolutiirealesidinrelatiileluiVite,1mPm=-=-,constant.4.Cn0+Cn1=1+n=8,decin=7.5.DO=OB.AO+DO=AO+OB=AB=DC.................................................................6.Cumprintrefactoriiprodusuluiseaflasilg(tg45..)=lg1=0,produsuleste0.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.251174-+=,deci7(125)7912S=+=.2.x.(-2;1)n..={-1;0}.3.36x=108.6x=62.x=2.4.Cerintaechivalentacuadeterminanumarulfunctiilorf:{a,b,c}.{1,2},adica23=8.5.RelatiaAB+CD=0..................serescrieAB=DC................,ceeaceimplicafaptulcaAB||CD,AB=CD.ABCD

    paralelogram.6.Dinteoremasinusuriloravem2sinBCRA=,deci101sin202A==.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.ElementelemultimiiAsunttermeninprogresiearitmeticaderatie3,decisunt4011143-+=elemente..2.f(-3)f(3)=f(-2)f(2)=f(-1)f(1)=f(0)=1,deciprodusuleste1.3.Conditiideexistenta:x>0;obtinem3x=2.x=23=8.4.Cerintaechivalentacunumarulpermutarilorde3elemente,adica3!=6.5.DinconditiadeapartenentaaluiBladreaptarezultaa-1+4-5=0,decia=2.DinconditiadeapartenentaaluiAladreaptarezulta2+b-5=0,decib=3.6.Printrefactoriiprodusuluiseaflasicos50-cos50=0;produsuleste0.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Avem:b1=2,b2=-2,b3=22,decib1b2b3=-8.2.f(x)+2g(x)=4x2-4x+1+2(2x-1)=4x2-1.4x2-1=-1.x=0.3.32x+23x-3=32x-3x+33x-3=3x(3x-1)+3(3x-1)=(3x-1)(3x+3)=0.x=0.4.24433!3!660.2C-=-=-=5.AO=(-6)2+82=10.6.sin,cossincosACABABACBBBBBCBCBC==.+=+.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Cumf(1)=0,produsuleste0.2.Minimulfunctieieste2,4a-.=-deci22m-8=8.m=16.m=4.3.x>0.Utilizamproprietateaalogax=x,decix=4.4.n+2+n+2=n2+5.n=1.5.ObtinemB(2,-3)siC(-2,3),deciBC=(2-(-2))2+(-3-3)2=52=213.6.Dinteoremasinusurilor,2sinBCRA=,deci12442BC==.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.(2)log218=log232=2log23+log22=2a+1.2.f(1)+f(2)+f(3)=(a+b)+(2a+b)+(3a+b)=6a+3b,decib=0.Cumf(4)=8,obtinem4a=8.a=2,adicaf(x)=2x.3.IntersectiacuOyeste(0,f(0))=(0,6).PentruintersectiacuOxrezolvamecuatiaf(x)=0.2x+3=2.x=-2;punctuldeintersectieeste(-2,0).4.5400-4860=540;540054010100x=.x=,deci10%.5.Conditia:2284aa=..Dina2=16rezultaa=4,darpentrua=4cele3fractiidevinegale,a=-4.

    6.DacaM(x,y)estemijloculluiBC,atunci20021,122xy=+==+=,deci(1,1)M.AM=(2-1)2+(3-1)2=5.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.(1+2)2+(1-2)2=1+22+2+1-22+2=6....2.Cerintaesteechivalentacux2-4x+3=-1,adicax2-4x+4=(x-2)2=0(A)pentruoricarexreal.3.mpartimprimaecuatieprin2sinotamS=x+y=8,P=xy=12,decix,ysuntsolutiialeecuatieit2-St+P=0,adica2t-8t+12=0.t1=2,t2=6,decisistemularesolutiile(2,6)si(6,2).4.mpartimecuatiaprin(n-2)!;seobtinen(n-1)=12=34,decisingurasolutieesten=4.5.OA=i-j;OB=3i+5j,OC=OA+OB=4i+4j....................................................,deciC(4,4).6.UtilizamteoremacosinusuluipentruunghiulA:222416911cos21616ABACBCA

    ABAC=+-=+-=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.(1)(3)222xxx-=-++;obtinem4x-4=2x+2.2x=6.x=3.2.Cum.=m2+4>0,conditiacasolutiilesafieopuseestecasumalorsafie0,deciS=m.m=0.3.2-x=2x-2,-x=x-2.2x=2.x=1.4.9811C10-C9=C10-C9=10-9=1.5.AB:y=-x+6siC.AB.5=-m+6.m=1.6.Cumtriunghiulestedreptunghicavemproprietateaca3sincossin5C=B.C=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.(1)(25)12xxx+=-++,deundeavem2x+2=3x+4.x=-2.2..=9-4m>0,deci94m0;(lgx-1)(lgx-3)=0,decix=10saux=1000.4.15680800100x-x=.x=lei.5.AB2=(-m-2)2+(m+2)2=(42)2,deci2(m+2)2=32.(m+2)2=16.m=2saum=-6.

    6.2222575100cos02503ABACBCAABAC=+-=+-=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.3log324=log3(32)=log33+3log32=1+3a.2.-a+b=-b+a.2a=2b.a=b,decif(x)=g(x)=ax+a,.x....f=g.3.4x-1=4-1rezultax-1=-1,adicax=0.4.2(1)66(1)12432nnnCnn=.-=.-==,decisolutiaeste4n=.5.Obtinemecuatia14312034xy+=.x+y-=.6.TriunghiulMONestedreptunghicnO,arecateteledelungimi3si4,deciipotenuzaeste5sinaltimeadinOeste3412

    55=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.Din2x+1=3x-1,obtinemx=2,x...,deciA={0,1,2}.2.f(1)+f(4)-f(2)=log21+log24-log22=0+2-1=1.3..=9-4m>0,deci94m

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009-MATEMATICA-ProbaD,tipulsubiectuluiMT2,programaM2Solutie1.b1b2b3=1(-2)4=-8.2.()()3f1+f3=2+log31+2+log33=2+0+8+1=11.3.33;0222VVbxyfa=-==..=.....4.1+5-10=-4.5.DacaM(x,y),atunci325235;2222xy

    =+==+=,rezultaca225552222OM=..+..=.........6.Aplicamteoremasinusurilor,2sinBCRA=,deci

    44122R==.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)UtilizndrelatiileluiViteseobtinex1+x2+x3=0.b)Senlocuiescsolutiilex1,x2,x3...nrelatie.Adunndrelatiileobtinuteseajungelarezultatul333x1+x2+x3=-6.c)SeutilizeazarelatiileVitesiseobtined=0.2.a)Egalitateasedemonstreazapringrupareatermenilorsauefectuareacalculelor.b)x..(-4)=-4,oricarearfix....c)Dinpunctulb)rezultaa..(-4)=-4,oricarearfia...,decirezultatulcompuneriieste4.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)Seobtinerezultatuld=14.b)3333d=a+b+c-abcsiapoiseverificaprincalculdirect.c)Ecuatiasescrienforma(2x)3+(3x)3+(5x)3-32x3x5x=0,seutilizeaadescompunereanprodusdelapunctulb)siseobtineunicasolutiex=0.2.a)x..y=2x(y-3)-6(y-3)+3=2(x-3)(y-3)+3.b)x..3=-18+21=3..x=3.c)Siruldecompuneri1..2..3....2009contineelementul9,pentrucareavema..9=3,.a....Seobtinerezultatul3.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)DinrelatiileluiViteseobtinex1+x2+x3=0.b)Utilizndformula()()2222x1+x2+x3=x1+x2+x3-2x1x2+x2x3+x1x3sirelatiileluiVieteseobtine2221234xxx++=.c)Seobtined=0.2.a)Princalculdirectseobtineh=X4+2X3-28X2-8X+96.b)Seobtinvalorilea=2sib=-8.c)Ecuatiasescrienforma(2x)4+2(2x)3-28(2x)2-82x+96=0.Utilizrezultateledelapuncteleanterioareseobtinsolutiilex1=2six2=1.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)EfectundnmultireaA2=AA=AsededuceegalitateaA3=A.b)Seutilizeazaegalitatea2X(a)X(b)=(I2+aA)(I2+bA)=I2+aA+bA+abA=I2+(a+b+ab)A.c)AvemX(1)+X(2)+X(3)+...+X(2009)=I2+A+I2+2A+...+I2+2009A,deundeseobtineX(1)+X(2)+...+X(2009)=2009I2+10052009A.2.a)Seobtinsolutiilex1=1..si..x2=4.b)Seobtined=0...c)Solutiilesistemuluisuntperechileordonate(1..,2..),(3..,4..)(5..,0..).

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)Dinegalitateadet(A)=(x-3)2-1=0seobtinvalorilex1=2six2=4.b)Severificaprincalculdirect.c)Seobtinex=4.2.a)x..y=xy-2x-2y+6=x(y-2)-2(y-2)+2=(x-2)(y-2)+2.b)Seutilizeazarezultatuldelapunctula).c)Utilzndproprietateadeasociativitateaoperatieisiegalitateax..2=2,.x...,seobtinerezultatulE=2.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)Conditiadecoliniaritate0011210241=esteverificata.b)DincoliniaritateapunctelorO;A1;A2rezultacanumaruldedreptecaretrecprincelputindouadintrepuncteleO;A0;A1;A2este4.c)Seutilizeazaformula12S=.siseobtinerezultatul11222S=n=n-.2.a)Severificaprinefectuareacalculelor.b)EgalitateaAxAe=Ax+e=Ax,.x...esteverificatapentrue=0,decielementulneutrudingrupul(G,)estematriceaA0=I3.

    c)Severificaegalitateaf(x+y)=f(x)f(y),oricarearfix,y....

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)SeobtineB2=A.b)Severificaegalitatea1AAI2-=.c)Seutilizeazaegalitatea211CBAAA6I2=+-=+-=.2.a)Seutilizeazaconditiag|f.f(2..)=0..,seobtinea=1..b)Severificaprincalculdirect.c)Ecuatiasescrienforma2(1)(1)0XX++=......,carearesolutiile..x1=4,..x2=2six3=3...

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)Yt=(12-3);()121233A..=..-=......123246369.-..-....-....b)Seobtinedet(A)=0c)Avemdet(B(a))=1-4a,1

    \4a.........,deci()1det()0()BBa-....2.a)()()00fg=si()()11fg=...2a=si..2b=.b)Seobtinef(0..)+f(1..)+f(2..)+f(3..)+f(4..)=0..c)Serezolvaecuatia2..x2+2..x=0..n..5siseobtinsolutiilex1=0..si..x2=4.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)2220022,0,0,2200abAIOabcdcd.+...+=...=...=-===-.+....b)()20det0tbcBAABcbcb.-.=-=...=-.-..

    c)222021,202tabcAAIadbcbcd.+...+=...=...===-.+....det(A-At)=4b2..4.2.a)e=5.b)()3()3()()()x..x..x=x-4+4;x-4+4=x.x-4x-5x-3=0,adicax.{3,4,5}.

    c)Luam,deexemplu,243a-=si341452b-=.a..b=+=....

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)Seobtinedet(A)=0.b)Seobtine23A+A=-A+A=O2.c)SededucrelatiileA2k=-AsiA2k-1=A,.k...*siseobtinerezultatulA+2A2+...+10A10=-5A.2.a)Seobtinedescompunereag=(X-1)(X-2).b)gnudividefpentrucaf(1).0.c)Seutilizeazateoremampartiriicurest.Seobtinerestulr=1.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)Seobtinesistemuldeecuatii090vxyxvy+=...+=,deunderezultax(v2-9)=0.b)detV.0.v...\{-3,3}.c)Sistemulestenedeterminat,avndoinfinitatedesolutiideforna(a,-3a),a....Deexemplu,treisolutiidistinctealesistemuluisunt(0,0),(-1,3),(1,-3).2.a)x..(-x)=3x3+(-x)3-1=-1.b)(x..y)..z=3x3+y3+z3-2=x..(y..z).legeaesteasociativa.c)Dina)sib)rezulta(-4)..(-3).......3..4=(-4)..4..(-3)..3..(-2)..2..(-1)..1..0==(-1)..(-1)..(-1)..(-1)..0=-2.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)Severificaprincalculdirect.b)Seobtine22124012001AB..+=........c)Seobtine()21121012001A-.-.=.-.....

    ..2.a)Seobtine2..(2)=40b)x..y=xy+7x+7y+42=x(y+7)+7(y+7)-7=(x+7)(y+7)-7.c)Ecuatiasescrienforma(x+7)3-7=xsiseobtinsolutiilex1=-8,x2=-7,x3=-6.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)ValoareadeterminantuluiesteD(9)=96.b)Serezolvaecuatia2a2-8a+6=0siseobtinsolutiilea1=1sia2=3.c)AvemD(3x)=0.2(3x-1)(3x-3)=0deundeseobtinsolutiilex1=0six2=1.2.a)Avem2*3=2.k2-4k+4=0,decik=2.b)Pentruk=2avemM=[2,8)six*y=xy-2(x+y)+6.Avemderezolvatecuatiax2-4x=0,cux.[2,8).Ecuatiaaresolutiilex1=0six2=4,darcumx.[2,8),rezultacaecuatiaareosingurasolutienM=[2,8),adicax=4.c)Avemdedemonstratinegalitateaxy-k(x+y)+k2+k=k,pentruoricex,y.M.Inegalitateasescrienforma(x-k)(y-k)=0,careesteadevaratapentruoricex,y.M.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)Seobtinerezultatul230002AA..+=.....b)Avemderezolvatecuatia25n-125=5n.Seobtinesolutian=3.c)Seobtine220092009200955...505150011...14deori02009B.+++..-.=..=....+++....................siBt=B.2.a)Dinconditiilef(0)=0sif(1)=0seobtinvalorilem=-1sin=0.

    b)Avem()()22222x1+x2+x3+x4=x1+x2+x3+x4-2x1x2++x3x4=2sifolosindrelatiileluiVieteseobtinevaloaream=-1.c)Avem()()()4222222f=X+X+1=X+1-X=X-X+1X+X+1factoricaresuntireductibilipentrucaau.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)Severificaprincalculdirect.b)Seobtinerezultatul2222A+B=5I.c)SefolosesterelatiaC=A+B=5I2.2.a)Seobtinvalorilea=-3sib=1.b)Seobtinef=(X-3)(X-1)(X2+X+2).c)Ecuatiasescrienforma(3x)4-3(3x)3+(3x)2-53x+6=0.Folosindrezultatuldelapunctulprecedent,seobtinsolutiilex1=1six2=0.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)Seobtinem=2.b)Seobtinesistemul220(1)492mmm.--=....++-=-,carearesolutiam=2.c)Pentrum=-1solutiasistemuluiestetripletul(4;8;-6).2.a)SeobtinectuldeX-9sirestulegalcu0.b)Senlocuiescradacinilex1,x2,x3...necuatiaf(x)=0siseadunarelatiileobtinute.c)Folosindrezultatuldelapunctula)ecuatiasescrienforma((3x)2-1)(3x-9)=0siseobtinsolutiilerealex1=0six2=2.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)EcuatiadrepteiA1A2este2x-y+1=0b)Folosindformuladecalcul12S=.pentruariatriunghiuluiOA1A2seobtinerezultatul12S=.c)SeconsideratreipuncteAm(m,2m+1);Ap(p,2p+1);Aq(q,2q+1),m,p,q...,pentrucareseverificaconditiadecoliniaritate.2.a)SeverificaprinefectuareanmultiriiA(a)A(b)b)Arelocegalitatea111()()2()222AaAAAaAaAa..=..=..=............

    ,deci12A......esteelementulneutrufatadeoperatiadenmultireamatricelordinmultimeaM.c)Arelocegalitatea11(1)()(2)24

    AAxAxAx==...=....,decisimetriculmatriceiA(1)fatadenmultireamatricelordinmultimeaMeste14A.......

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)Pentruvalorile22101,0101abaIG..==.=.=.....,iarpentruvalorile220000100baaOG..=.=...=.....

    .b)Seobtinematricea1111B..=...--.c)det(A)=a2=1.0..A-1,sedeterminamatriceleAt,A*siseobtineA-1=A*.G,oricarearfimatriceaA.G.2.a)Dinconditiaf(-2)=0seobtinevaloareaa=-4.b)Seobtinsolutiilex1=-2,x2=3+2,x3=3-2.c)Senlocuiescsolutiilex1,x2,x3necuatiaf(x)=0siseadunarelatiileob

    tinute.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)EcuatiadrepteiB1B2este2x+y=0.b)SeverificaegalitateacoordonatelorpunctelorAnsiBn,oricarearfin...*.c)SeverificaconditiadecoliniaritateapunctelorA1,A2,An,oricarearfin=3.2.a)Princalculseobtinectulq=X2+2X+4sirestulr=7X+5.b)Dinegalitateay2-y-1=0seobtiney2=y+1,adicay3=y2+y.y3=2y+1.c)Dinteoremampartiriicurestrezultaf(y)=(y2-y-1)(y2+2y+4)+7y+5.Dinpuncteleanterioareavemy2-y-1=0si7y+5...,decif(y)....

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)EcuatiadrepteicaretreceprinpuncteleA1siA2este-3x+y+8=0.b)Folosindformula12A..=.seobtinerezultatul1842A==.c)SeverificaconditiadecoliniaritateapunctelorA1,A2,An,oricarearfin=3.2.a)Seobtinef(0..)+f(1..)=2...b)Solutiileecuatieif(x)=0..n..5[X]sunt..x1=2si..x2=4c)Seobtinectul..243qXX=++..sirestulr=0...

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)Seobtinerezultatuldet(I3+B)=1.b)Seobtine2f(A)=A-3A+I3=I3+B.c)Dinpunctulb)rezulta()()3323f(A)=I3+B=I3+3B+3B+Bsi3B=O3.2.a)Serezolvaecuatia(x-3)2-2(x-3)=0,carearesolutiilenumerentregix1=3six2=5.b)Seobtinevaloareaa=3.c)Solutiasistemuluideecuatiiesteperecheaordonata(4,2).

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)Pentruvalorile22a=1;b=0.a-3b=1.I2.G,iarpentruvalorile22a=b=0.a-3b.1.O2.Gb)Severificaegalitateaprincalcul.c)Folosindegalitatea1AAI2-=,seobtine1,3abAGba-.-.=....-.oricarearfimatriceaA.G.2.a)Dinconditiaf(1)=0seobtinem=-9.b)()()72222272227222f=m+++m....m++m+....m=-.

    c)Sefolosescrelatiilentresolutiilesicoeficientiiecuatieif(x)=0siseobtinerezultatul22212324781x+x+x=.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)Folosindformula12S=.,seobtineS=13.b)Pentrua=-2ecuatiadrepteicaretreceprinpuncteleBsiCestey+2=0c)ConditiadecoliniaritateapunctelorB,C,Mesteverificatadacaarelocegalitatea(a+2)x-3(a+2)=0,.x....Seobtinevaloareadea=-2.2.a)DinrelatiileluiViteseobtinea=-3.b)Dinconditiaf(2)=0seobtinea=-3.c)Seobtinedescompunereaf=(X-1)(X-2)(X-2)(X+2).

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)Senlocuiestetripletului(2,1,-1)nsistemuldeecuatiisiseobtinem=3.b)Seobtineecuatiam2+2m-15=0,carearesolutiilerealem1=3sim2=-5.c)Pentrum=-5solutiasistemuluideecuatiiestetripletul(0,3,1)2.a)DinrelatiileluiViterezultam=0.b)Dinconditiaf(3)=0seobtinem=0.c)Seobtinedescompunereaf=(X-1)(X2-3).

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)Seobtinerezultatuldet(A(4))=6.b)Dinconditiadeexistentaamatriceiinversedet(A(a)).0rezulta(a-2)(a-1).0.DecimatriceaA(a)esteinversabilapentruoricenumara...\{1,2}c)Solutiasistemuluiestetripletul(1,0,0)oricarearfinumarula...\{1,2}.2.a)DinrelatiileluiViterezultaa=2.b)Seobtinea=2.c)Radacinilerationaleposibilesuntprintredivizoriitermenuluiliber.Conformcerinteiavemdeverificatdivizoriipozitiviailui-4.Polinomulfanuadmiteradacinax=1oricarearfia...Pentrux=2seobtinea=-2,pentrux=4seobtinea=-5.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutie1.a)22abAabab..=.....+..b)200aaIa..=....,20b

    bAabb..=.......c)FiexyXzt..=....

    ;ayxbyXAatzbt.+.=...+.;ztAXaxbzaybt..=...++..

    Obtinemz=aysit=x+by,deci2xyXxIyAayxby..=..=+.+.;m=xsin=y.2.a)f(1)=a-1;f(1)=0,a=1.b)f=X4+X3-X-1;f=(X+1)(X-1)(X2+X+1);x=1.c)Radacinilerationalesuntprintredivizoriitermenuluiliber.Singureleradac

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    inirationalesunt1,caresuntntregi.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)22222A..=.....b)2222AB.-.=...-.;22222B.-.

    =...-..c)xzytAXxzyt.++.=...++.;xyztxyztAXBxyztxyzt.+++----.

    =...+++----.;AXB=(x+y+z+t)B.x+y+z+t=0.2.a)221g=X+X+X+;221gXf=+=.b)f+g=X2+X;c=1,r=X+1.c)Numarulfunctiilordelaomultimecu3elementelaunacu2elemente:23=8.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)a=1,b=0.I2.M.b)2abbAaIbVbab.+-.=+=...-.;detA=a2.Aesteinversabila.detA.0.a.0.c)2V=O2;AB=(a1I2+b1V)(a2I2+b2V)=a1a2I2+(a1b2+a2b1)V,careestedinM,deoarecea1a2,a1b2+a2b1.R.2.a)Princalcul.b)Legeaestecomutativasix*e=x.(x-5)(e-5)+5=x,.x.R.(x-5)(e-6)=0,.x.R.e=6.c)()2x*x=x-5+5;()3x*x*x=x-5+5.Solutiilesunt:4,5,6.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)22It=I;22220202IItI..+==.....b)mambmAmcmd..=....;()tmamcmA

    mbmd..=....;tmamcmAmbmd..=.....c)2

    2tabcAAbcd.+.+=...+..a=d=0,c=-b,deci00bAb..

    =...-..2.a)()2x*x=x.x-2=x-2.x=2saux=2+1.b)(x*y)*z=x*(y*z)=(x-2)(y-2)(z-2)+2c)Legeaestecomutativasix*e=x,.x.R.e=2+1.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)01242xxyzxyz=..++=...++=;x=0,y=1,z=0.b)detA=(a-b)(b-c)(c-a).c).x=.z=0,.y=detA;x=0,y=1,z=0.2.a)(x*y)*z=x*(y*z)=x+y+z+2m,.x,y,z....b)m=6.c)m=2.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)()111,110A..=.-...;detA(1,1)=1.b)()()1212121212aabbABbbaabb.++.+=...-++-+.,cuelementedin...A+B.M.c)()00,b

    Abbb..=...--.,2()10,1bIAbbb.-.-=...+.

    ;(())2detI2-A0,b=1+b+b221310,24bbbb++=.+.+>......R.2.a)g(0)=1.b)()22f=XX+;12=22=1,f(0)=f(1)=f(2)=0.c)h=aX3+bX2+cX+d,h(0)=d.d=0,h(1)=a+b+c=0,h(2)=2a

    2c=0.b=0si0a+c=.Solutiile:a=2,b=0,c=1,h=2X3+Xsaua=1,b=0,c=2.h=X3+2

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)y=x.b)001111112241A==.c)(p-n)(p-m)(n-m)estenumarpardeoarecedintreinumerenaturale,celputinoperecheauaceeasiparitate.Atunci()()()12p-np-mn-mestentreg,deciariaenumarnatural.2.a)f(1)=m2+8m+15;f(1)=0.m.{-5,-3}.b)Sumaradacinilorpolinomuluieste404m

    -=-m.m=.c)()2211fX4XX5X6XX......=..+.-.+.+.........;{}1YXy2,3X=+..,123,435

    1,2xxx===.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)156015001AB..=.........b)12121212101001xxzzxyXYyy.+++.

    =.+.......;XY.M,deoareceareelementeledinZsiformamatricelordinM.c)101001mpUn..=....

    ....;101001amcpanVUbn.+++.=.+.......,

    101001amcpanUVbn.+++.=.+.......;

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    VU=UV.an=bm,.a,b.Z.m=0,n=0,p...,p....2.a)()()224211fXXX=-+=-;12341x=x=x=x=.b)Seefectueazanmultirilesausedescompunecuformuladiferenteidepatrate.c).1=-4a,.2=4a;.1=0.a=0,.2=0.a=0.Decia=0.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)121351536261545AtA......=....=.........b)2222tacabacabcdXXbdcdabcdbd.....++.=....=.......++....det(XXt)=(a2+c2)(b2+d2)-(ab+cd)2=(ad-bc)2.c)det(XXt)=0.ad=bc;obtinem

    acbd=.2.a)(x..y)..z=xyz-xy-xz-yz+x+y+z;x..(y..z)=xyz-xy-xz-yz+x+y+z.b)x..y>1.(x-1)(y-1)>0.Eadevaratpentrucax>1,y>1.c)x..a=a.x(a-1)-2(a-1)=0,.x....a=1.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)21001It..=....;(2)2002fI..=.....b)()ttaebfaecgAB

    cgdhbfdh.++..++.+=..=...++..++.;ttacegaecgABbdfhbfdh.....++.+=..+..=.......++..c)2222tabc

    AAOObcd.+.+=...=.+..a=d=0,b+c=0;0,0bAbb..=....-.

    R.detA=ad-bc=1.b2=1.b=1,deci10110A=...-...si20110A=.-.....

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    .2.a)x1+x2+x3+x4=a,a=5.b)x4-x3-x+1=0.x3(x-1)-(x-1)=0.(x-1)2(x2+x+1)=0.Decisolutiilerealesunt:x1=x2=1.c)Solutiilentregisuntprintredivizoriitermenuluiliber,adica1.Pentrux=1,a=1;pentrux=-1,a=-1.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)2333333333B..=........;B2=3B.b)3mnnnmInBnmnnnnmn.+.+=.+....+...,n,m+n.Z.c)

    2222222222222222222222ababbabbAabbababbabbabbab.+++...=.+++...

    .+++...,222a+2b=0.a=b=0.2ab+b=0.A=O3.2.a)Princalculdirect.b)f=(X2-7)(X2-5).Radacinile5,7nusuntntregi.c)f=(X-5)(X+5)(X-7)(X+7).

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)2102002abAFc.+.+=........;a=3,b=3,c=5.b)detF=1.0,deciFesteinversabila;1101010001F-F*.-.==......

    ...c)1123666456456789789XF-...---.=..=...............2.a)Seefectueazanmultirilesisereductermeniiasemenea.

    b)x*(y*z)=4xyz-2xy-2xz-2yz+x+y+z;(x*y)*z=4xyz-2xy-2xz-2yz+x+y+z.c)x*(1-x)=0.2x(1-x)=0.x.{0,1}.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)()1321227114.=-a=a-.b).=-16,.x=-60,.y=12,.z=-4;1531,,444x=y=-z=.c)Dinultimaecuatie.z0=00000001134,25,33.x+y=x-y=.y=-x=;103b=.2.a)f(5)=0,g(5)=-2.f(5)+g(5)=-2.

    b)()(2009)a0+a1+...+a2009=g1=-5+5

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)a=c=1,b=0;0,1....b)12121212aabbABbbcc.++.+=...++..Elementelesuntnumerereale.c)12211221122112210()0ababbcbcABBAababbcbc.-+-.-=...--+-.;

    ()2detAB-BA=(a1b2-a2b1+b1c2-b2c1)=0.2.a)x*4=10.-2x+6=10.x=-2.b)x*a=a.x(2-a)=2-a,.x....a=2.Legeaestecomutativa,decia*x=x*a.c)Unelementalcompuneriieste401822009=.Decix*2*y=2*y=2.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)1144det355322A=.detA1=0(are2coloaneegale).b)Adouaecuatiearesolutiadatapentrua=-8,iaratreiapentrua=10.c)Dacay+z=3.x=3(dinprimaecuatie);scazndultimele2ecuatii.2(y+z)+a(y+z)=6,decia=0;y+z=3,2y+3z=7.y=2,z=1.2.a)x.(-1)=x+(-1)+1=x,.x.Z;(-1).x=-1+x+1=x,.x.Z.b)(x..y)..z=a2x+aby+bz-a-1;x..(y..z)=ax+aby+b2z-b-1.x,y,z.Z.a=b=1saua=b=0.c)f(x.y)=x+y+3;f(x)..f(y)=x+y+3.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a).=-7.b).x=-7,.y=-7,.z=0;x=1,y=1,z=0.c)x=y+z=1;y=1,z=0.a=0.2.a)f(2)=0.a=4.b)f=X(X2-2X+4)-8.ctulestec(X)=Xsirestulester=-8.c)1232X+X+X=si222X1X2+X1X3+X2X3=a.X1+X2+X3=4-2a.Dacaa>2.2a>4.4-2a

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)22002A..=....;220202I..=.....b)21111x

    AxIx.-.-=...--.;()2detA-xI2=x-2;x2-2=0.x=2.c)()()4222AX=AX=2I2=4XsilafelXA4=4X.2.a)3,2...si32-222=1,deci3+22.G.b)x=a1+b12,y=a2+b22,a1,b1,a2,b2...si2222a1-2b1=1,a2-2b2=1;xy=(a1+b12)(a2+b22)=(a1a2+2b1b2)+(a1b2+a2b1)2;a1a2+2b1b2,a1b2+a2b1...;

    ()2()2a1a2+2b1b2-2a1b2+a2b1=()()2222a1-2b1a2-2b2=1.c)()x=a+b2,a2-2b2=1.x.0;x-1=a-b2,a2-2-b2=1.x-1.G.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)a=b=c=d=0;0....O3.M.b)1212121212121212121212000aaabbaaccabdAAaaaddaaa.+++.=.+.......;areelementelenR,celedepediagonalaegalesi0subdiagonala,decimatriceaprodusestedinM.c)detA=a3;a3=0.a=0;000

    000bcAd..=........,200000000bdA

    ..=....

    ..

    ..;3A=O3.2.a)f=X4-X3+X2-X+1;q=X2-X,r=1.b)f(1)=-1.c=-a-b-1.RestulmpartiriilaX2+1este(b+1)X-2a-b;a=b=0,c=-1.c)12341234x+x+x+x=1,xx+...+xx=a;22221234.x+x+x+x=1-2a;1-2a

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)det(A)=ad-bc=1.b)tacAbd..=....;2222tabacbdAAacbdcd.++.=...++....c)S=a2+b2+c2+d2+2ac+2bd=(a+c)2+(b+d)2.S=0.a+c=0sib+d=0;detA=0.2.a)x1+x2+x3+x4=-2.

    b)f=X4+2X3-X2-2X;f=X(X+2)(X+1)(X-1);1234x=0,x=-2,x=1,x=-1.c)Dinprimarelatiex1+x4=x2+x3=-1;(x1+x4)(x2+x3)+x1x4+x2x3=a.x1x4+x2x3=a-1x1x4(x2+x3)+x2x3(x1+x4)=-b.x1x4+x2x3=b.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)222abcabbdAaccdbcd.++.=...++....b)()22aadabbdadAaccdadd.++.+=...++...;()()

    222abcabbdadAadbcIaccdbcd.++.+--=...++....c)A2M=(a+d)AM-(ad-bc)M;MA2=(a+d)MA-(ad-bc)M;a+d.0.AM=MA.2.a)f=X3-2X2+X;f=X3-2X2+X;()2f=XX-1;123x=0,x=x=1.b)x1+x2+x3=2,x1x2+x1x3+x2x3=a;222

    x1+x2+x3=2.4-2a=2.a=1.c)2222x1x2x3=b=-b.b=0saub=-1;222x1+x2+x3=x1+x2+x3.a=1;a=1sib=0.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)20000A=.......b)()311,141M.-.=...-..det(M(1,1))=1,(())1111,143M-.-.=...-..

    c)()2,42xyyMxyyxy.+-.=...-..det(M)=x2.x.R..*.2.a)f(-p)=1.b)f(1)=0.2+p=0.p=-2.c)2222123121332123123x+x+x=-p,xx+xx+xx=0.x+x+x=

    p,xxx=-1.222222()()121323.xx+xx+xx=0-2-1-p=-2p.4444123x+x+x=p+4p.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)2400010021A..=.........b)3800010031A..=........

    ;23800452010031AAI..-+=.........c)detA=2.0.11

    002010011A-......=...-.....,23

    42000002mnpmAnApImnpmnmnp.++.++=.++....+++....

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    Identificndelementeleobtinem15,2,22m=n=-p=.2.a)Adouaecuatie12233112312xxxxxxxxx++.=;x1x2x3=-4.b)s1=x1+x2+x3,s2=x1x2+x1x3+x2x3,s3=x1x2x3;ecuatia321230x-sx+sx-s=ecuatiacerutax3-2x2-2x+4=0,decia=-2,b=-2,c=4.c)f=(x-2)(x2-2)=(x-2)(x-2)(x+2).

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)2123012001X..=........b)()1100110det1,110,011111001tXXX-...-.==..=.-..............

    c)3136013001X..=........,234693046

    004rrrXrXIrrr.+++.++=.++....+....Identificndelementele,.a,b,c.R,obtinemr=-3.2.a)2008..(-2008)=20=1.b)x..x2=64.x+x2=6.x.{-3,2}.c)()2xy22;222z12xy10xyzxyzxyz+z++zzxyxy++++....=..==.+=+.+=.=-.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)122302MM=..+....;det(M1+M2)=4.b)21201aaM=.......c),xyXzt

    ..=..

    ..axazyatMXzt.++.=....,axaxyXM

    zazt.+.=...+.;x+az=x,y+at=ax+y,z=z,t=az+t,.a.R;obtinemz=0;t=x,deci0xyXx..=....

    ,pentruoricarex,y.R.2.a)x*0=x.b)x*(y*z)=x*3y3+z3=3x3+y3+z3;(x*y)*z=3x3+y3*z=3x3+y3+z3.c)333x1=2x0=2x0.Prininductie:301nx=n+x;3x7=8x0=2x0...,pentrux0....33333333

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    x2=x0*x1=x0+2x0=x03;x3=x0*x2=x0+3x0=x04,x0...si34....x3....

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)a=1,b=c=0;0si1....b)12121212aabbABccaa.++.+=...++..Elementelesuntnumerereale.c)1221211200bcbcABBAbcbc.-.-=...-.;()()2

    1221detAB-BA=-bc-bc=0.2.a)f=X2+X+1;f(1)=0.b)b=1,1+a+1=1.a=2.c)9=numarulfunctiilordelaomultimecu2elementelaunacu3elemente.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)Seobtinedet(H(a))=a.b)()()()1ln0010,,0.00abHaHbHababab..=..=.>......c)()()()()2009ln2009!0123200902009020092010002HHHH

    ....

    ..++++=........Calcululdeterminantului.=200931005.2.a)x..y=xy-2(x+y)+6=xy-2x-2y+6=(x-2)(y-2)+2.b)Pentrux,y.G.(x-2)(y-2)>0.x..y>2.Decix..y.G.c)Sedeterminae.Obtinemxe-2x-2e+6=x.e=3.Obtinem231

    226322,.22xxxxxxxGxx'--'+=.'=-=+>..--

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)SeobtineA2=3A.b)A10=39A,decidet(A10)=0.c)21.det40.23BB..=..=...DeciBesteinversabila.Princalcul1311422B--..=...-..2.a)x3lne=8.x3=8.x=2.b)Pentrux,y.(0,8)\{1}.x3lny>0six3lny.1.x..y.G.c)x..(y..z)=x9lnylnz=(x..y)..z..x,y,z.G.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)SedeterminapuncteleA0(0,2)siA1(1,3).011:0210131xyAA=.Finalizarex-y+2=0.b)A2(2,4).Atunci0120211310,,coliniare.241=.AAAc)0011unde2121.2131AnnAnn

    =..=+=-.=++2.a)137.28f.-.=-....b)Relatiasescrief(a)=-5.Seobtinea.{-1,0,1}.c).=0pentrucax1+x2+x3=0.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)Severifica,nlocuindnfiecareecuatieasistemului,x=0,y=3siz=1.b)Sistemuladmitesolutieunicadacadeterminantulmatriceisistemuluiestenenul;det(A)=-5m+15.0pentrum...\{3}.c)Pentrum.3avemdetA.0.SeaplicaregulaluiCramer.Obtinemsolutiax=0,y=3,z=1.2.a)x*y=2xy-6x-6y+21=2x(y-3)-6(y-3)+3=2(x-3)(y-3)+3.b)Ecuatiaesteechivalentacu:()225x-3=8.5x-3=2cusolutiilex=0six=1.c)Sedeterminae.Obtinem726621.2xe-x-e+=x.e=Obtinem(){}7123526621,\3.243x

    xxxxxxx'--'+=.'=-..-..

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)A2=A.A+A2=2A.b)246det()23xXAxxx+-+==+-.Seobtinex1=-2,x2=1.Atunci2010sau.0201XX.-...=..=...-...c)Parteastngasemaiscrie()()1

    23123,2nnAAAnAnAA+++++=++++=.n...*.2.a)Avemf(2)=0.Obtinem13.2m=-b)xi,cui.{1,2,3}suntradacinialepolinomuluif.nlocuimsiobtinemxi3+xi2+mxi+3=0.

    AdunamceletreirelatiisiobtinemS3+S2+mS1+3=0.c)Radacinilerationalesegasescprintredivizoriitermenuluiliber.Deci1si1suntposibileleradacinirationale.nlocuimnpolinomulfsiobtinemm=1saum=3,decipentrumnumarparfnuareradacinirationale.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)det(A)=-4-3=-7.b)232A=7I.A=7A.c)Seefectueazacalculul(2)3AB=AA-6I2=A-6A=7A-6A=A.2.a)Xg+1=X4+X3+X2+X+1=fb)Polinomulgsescrieg=(X2+1)(X+1).x=-1esteunicaradacinareala.c)Dinpunctula).f(a)=ag(a)+1.Stiindcag(a)=0.f(a)=1.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)A(1)A(-1)=A(-1).b)(())(())222222510124111020481xxxxAxAx,x.xxxx.++--.=..=+-...+--+.....c)detA(1)=2.0.DecimatriceaA(1)esteinversabila.()11321.2106A-.-.=...-.2.a)0=0+03.Gpentruca02-302.1.

    1=1+03,0,1...si12-302=1.Deci1.G.b)Fiex,y.Gx=a+b3,a,b...,a2-3b2=1,y=c+d3,c,d...,c2-3d2=1()()()22xy=ac+3bd+ad+bc3.Avemac+3bd,ad+bc...siac+3bd-3ad+bc=1.Deunderezultaconcluzia.()22133.3abxGabGxab

    ..=-=+-.-c)

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)()254det3115.201A-=-=--b)det(A).0.SistemularesolutieunicasiaplicamRegulaluiCramer.Obtinemsolutiile:443,,.555x=y=z=c)Rezolvamsistemulsiobtinem9514101310,,5.555aaa

    xyza=-=-=-.=...2.a)2008..2009=4018.b)Inecuatiasescriex2+x-2=0.x.[-2,1].c)Cn0..C1n..Cn2=n+6.n2-n-6=0.n=-2.Asin=3.A.MultimeaAareunsingurelement.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)detA=0.b)2010110.100A..=.--.......c)3(3)010110det10.011IAIA.-.+=...+=.......

    MatriceaI3+Aesteinversabila.()13110100.101IA-..+=.-.......2.a)f(0)-f(1)=p-q-1.b)(1-x1)(1-x2)(1-x3)=f(1)=1-p+q-r.

    c)g=(X-1)(X2+2X+3)carearedoaroradacinareala.

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    MinisterulEducatiei,CercetariisiInovariiCentrulNationalpentruCurriculumsiEvaluarennvatamntulPreuniversitarBACALAUREAT2009MATEMATICAProbaD,tipulsubiectuluiMT2,programaM2Solutii1.a)Seobtine11,.3a=b=b)2A=3I2.B=I2.DeciAB=A.c)Fie(),atuncidinxyXCAXXAAXzt...=..=..obtinem:x=t,y=3z.Notam3,.ab

    xazbXba..=.=..=........2.a)2224252015xxxxx

    x*==.-+=+,cusolutiile1212si.2x=.Gx=.Gb)Severificausorprincalculdirect.c)Pentruoricex,y.(-1,1).1+xy>0.Sedemonstreazadublainegalitate:11.1xy

    xyGxy-

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