2026年GRE数学真题练习卷

2026年GRE数学真题练习卷Directions:CompareQuantityAandQuantityB,usingadditionalinformationcenteredabovethetwoquantitiesifsuchinformationisgiven.Selectoneofthefollowingfouranswerchoices:A.QuantityAisgreater.B.QuantityBisgreater.C.Thetwoquantitiesareequal.D.Therelationshipcannotbedeterminedfromtheinformationgiven.Asymbolthatappearsmorethanonceinaquestionhasthesamemeaningthroughoutthequestion.1.QuantityA:QuantityB:2.xisanintegergreaterthan1.QuantityA:ThenumberofdistinctprimefactorsofQuantityB:Thenumberofdistinctprimefactorsofx3.Acertainjarcontainsonlyredmarbles,whitemarbles,andbluemarbles.Theprobabilityofselectingaredmarbleis,andtheprobabilityofselectingawhitemarbleis.QuantityA:TheprobabilityofselectingabluemarbleQuantityB:4.QuantityA:Thegreatestpossiblevalueof2x−ysuchthatQuantityB:105.ThecirclewithcenterOhasaradiusof5.PointPliesonthecircle,andpointQisinsidethecirclesuchthatOQQuantityA:ThelengthofsegmentPQuantityB:86.Sisthesetofallintegersfrom1to100,inclusive.QuantityA:Thenumberofmultiplesof3inSQuantityB:Thenumberofmultiplesof5inS7.Thefunctionfisdefinedbyf(x)=−4x+kQuantityA:kQuantityB:38.QuantityA:Thestandarddeviationofthedataset10QuantityB:Thestandarddeviationofthedataset89.Inacertaincoordinatesystem,thepoints(2,3QuantityA:TheareaoftherectangleQuantityB:1610.TriangleABCisisosceleswithAB=AQuantityA:ThemeasureofangleBQuantityB:11.Amerchantsellsaproductforxdollars.Duringasale,thepriceisreducedby20%.Thesalepriceisthenincreasedby25%.QuantityA:ThefinalpriceasapercentageoftheoriginalpricexQuantityB:10012.Ifa>Indicateallsuchstatements.A.>B.<C.aD.>13.Whatistheunitsdigitof−?A.0B.2C.4D.6E.814.Iftheratioof2xto3yis,whatistheratioofxtoyA.B.C.D.E.15.Arectangulartankisfilledwithwatertoadepthof4feet.Thetankhasalengthof10feetandawidthof6feet.Ifthewateristransferredtoacylindricaltankwitharadiusof5feet,whatwillbethedepthofthewaterinthecylindricaltank?A.B.C.D.E.16.SetASetBIfonenumberisselectedfromSetAandonenumberisselectedfromSetB,whatistheprobabilitythatthesumofthetwonumbersiseven?17.If=3and=,whatisthevalueof?18.Inthexy-plane,thelinewithequationy=mx+19.Atotalof600studentsweresurveyed.40%ofthestudentsstudyFrench,and30%ofthestudentsstudySpanish.If10%ofthestudentsstudybothFrenchandSpanish,howmanystudentsstudyneitherFrenchnorSpanish?20.Ifnisapositiveinteger,whatistheremainderwhenn(A.0B.1C.2D.Itcannotbedeterminedfromtheinformationgiven.21.Thetablebelowshowsthedistributionofgradesinaclassof30students.GradeNumberofStudentsA5B10C8D4F3Ifastudentisselectedatrandom,whatistheprobabilitythatthestudenthasagradehigherthanC?22.Thevolumeofasphereis.Whatisthesurfaceareaofthesphere?A.12B.16C.36D.48E.14423.Whichofthefollowingisequivalentto?A.B.C.−D.E.24.If=andx=3y25.Asequenceisdefinedby=+3forn>26.IntrianglePQR,themeasureofanglePisandthemeasureofangleQis.IfthesideoppositeangleQhaslength4,whatisthelengthofthesideoppositeangleP?27.CompanyA’srevenuewasRdollarsin2020.In2021,therevenueincreasedbyp.In2022,therevenuedecreasedbypcomparedto2021.QuantityA:Therevenuein2022QuantityB:R28.xandyarepositiveintegerssuchthatx+QuantityA:+QuantityB:5229.Theaverage(arithmeticmean)of5numbersis20.Ifoneofthenumbersisremoved,theaverageoftheremaining4numbersis18.QuantityA:TheremovednumberQuantityB:2830.+31.Iftheareaofasquareis16,whatisthelengthofthediagonalofthesquareintermsoft?32.Aboxcontains5chips,numbered1,2,3,4,and5.Twochipsaredrawnatrandomwithoutreplacement.Whatistheprobabilitythatthesumofthenumbersonthetwochipsiseven?33.If|x−5Indicateallsuchvalues.A.2B.3C.5D.8E.934.Theliney=2x+335.Inagroupof50people,30havecats,20havedogs,and10havebothcatsanddogs.Howmanypeoplehaveneithercatsnordogs?36.Arightcircularconehasabaseradiusof6andaheightof8.Whatisthevolumeofthecone?37.Ifa,b,Indicateallsuchstatements.A.a+B.abC.bistheaverageofaandc.38.Acartravelsatanaveragespeedof60mphforthefirst2hoursofatripandatanaveragespeedof40mphforthenext3hours.Whatistheaveragespeedofthecarfortheentiretrip?39.If=,whatisthevalueofx?40.Thegraphoftheequationy=***AnswersandExplanations1.CExplanation:QuantityA:=.QuantityB:=1Thetwoquantitiesareequal.2.CExplanation:Theprimefactorizationofcontainsexactlythesamedistinctprimefactorsastheprimefactorizationofx.Forexample,ifx=12=·3.BExplanation:Thesumoftheprobabilitiesofallmutuallyexclusiveoutcomesmustequal1.P+++P(CompareQuantityA()andQuantityB().Since≈0.166and=4.AExplanation:Wewanttomaximizethevalueoftheexpression2xTomaximize2x−y,weshouldchoosethelargestpossiblevalueforxFromthegivenranges:−3≤xMaxx=Miny=Substitutethesevaluesintotheexpression:2(Wait,let'sre-evaluate.Maxx=2,MinLet'scheckifothervalueswork.Sinceyissubtracted,wewantyassmallaspossible(2).Sincexismultipliedbyapositivenumber,wewantxaslargeaspossible(2).Sothemaximumvalueis2.QuantityAis2.QuantityBis10.QuantityBisgreater.Correction:Letmedoublecheckthequestion.Ah,xrangeincludesnegatives.Ifxisnegative,2xisnegative.WedefinitelywantxSomaxvalueis2.Wait,didImisreadthequestion?"Greatestpossiblevalue".Maxx=2,Miny=Isthereanywaytoget10?No.Wait,letmechecktheprovidedsolutionkeylogicvsmygeneration.Mygeneration:2x−yIsitpossiblethequestionmeant2xLet'sre-read.−3≤x≤2.MaxxMaxvalueis2.QuantityA=2.QuantityB=10.SoBisgreater.Wait,letmemodifythequestionintheoutputtomakeitmoreinterestingorensurethecalculationisright.Actually,let'slookatthegeneratedquestionagain.QuantityA:Thegreatestpossiblevalueof2x−ysuchthat−Maxvalueisindeed2.SotheanswershouldbeB.LetmeadjustthequestionparametersinthefinaloutputtomakeAgreater,orkeepitandanswerB.Let'schangetheboundsto5≤x≤8and1≤Or,let'ssticktothegeneratedquestion:−2≤xMaxx=5,Miny=−3Okay,Iwillusethequestion:−2≤xThenAnswerisA.Self-Correctionforfinaloutput:Iwillusethequestion:−2≤xCalculation:Max2x−yoccursatmaxxx=2(13>5.DExplanation:TriangleInequalityTheoremstatesthatforanytriangle,thesumofthelengthsofanytwosidesmustbegreaterthanthelengthoftheremainingside.Here,OP=5(radius),OThedistancePQmustsatisfy:||2<SoPQQuantityBis8.SincePQWait,ifP,O,QarecollinearandQisbetweenOandIfOisbetweenPandQ(impossiblesinceQisinsidecircleandPisonboundary,soOQSoPQTheproblemaskstocomparewith8.PQSoQuantityBisgreater.Wait,let'scheckthequestion."PointQisinsidethecircle"."PointPliesonthecircle".Maxdistanceiscloseto8(ifQisnearboundaryoppositeP).Mindistanceiscloseto2(ifQisnearcenterlinetowardsP).SoPQAnswerisB.6.AExplanation:SetS=QuantityA:Multiplesof3.100/3=33.33.Thereare33multiples(QuantityB:Multiplesof5.100/5=20.Thereare20multiples(33>QuantityAisgreater.7.CExplanation:Thequadraticequationisy=−4x+Thismeanstheequationcanbewrittenasy=Expandtheexpression:(xComparingthistoy=−4QuantityAis3.QuantityBis3.Theyareequal.8.AExplanation:QuantityA:Dataset10,QuantityB:Dataset8,Wait,thequestionisusuallytricky.Letmecheckthequestionagain.Ah,Isee.UsuallySDofuniformsetis0.SDofspreadsetis>0.SoQuantityB>QuantityA.Letmere-readthegeneratedquestion.QuantityA:SDof10,QuantityB:SDof8,SoBisgreater.Correction:IwillswapthesetsinthefinaloutputtomakeAgreaterorchangethequestion.Let'smakeQuantityA:SDof0,0,OrjuststicktothesimpleoneandanswerB.Let'stryaharderone.SetA:1,5,SetB:2,5,SDA>SDB.Okay,Iwillusethesesetsinthefinalquestion.Question8:QuantityA:SDof1QuantityB:SDof2Answer:A.9.BExplanation:Theendpointsofadiagonalare(2,3Lengthofdiagonald=Forarectangle,=lAlso,Area=leWeknow+=WewanttocomparelwRecalltheidentity(lAlso(lSince(l−wThisdoesn'thelpdistinguishbetween16andsomethingelse.However,themaximumareaforagivendiagonaliswhentherectangleisasquare.Ifsquare,2=MaxArea=34.Theminimumareaapproaches0.Sowecan'tdeterminetheexactarea.Wait,Ineedadeterminatequestion.Let'smakethesidesparalleltoaxes?"Thesidesoftherectangleareparalleltothecoordinateaxes."Ifso,lengthisdifferenceinx,widthisdifferenceiny.l=w=Area=2×ThenA=B.Let'susethisconstraint.FinalQuestion9:Thesidesoftherectangleareparalleltothecoordinateaxes.Thepoints(2,3QuantityA:Area.QuantityB:16.Answer:C.10.CExplanation:TriangleABCisisosceleswithAB=AC.ThismeansthebaseisThesumofanglesinatriangleis.A+B+B+2B=QuantityAis70.QuantityBis70.Theyareequal.11.BExplanation:Letoriginalpricebex.Reducedby20%:Newprice=0.80xIncreasedby25%:Finalprice=0.80x0.80×SoFinalPrice=1.00xQuantityAis100%.QuantityBis100%.Theyareequal.Wait,let'scheckcalculation.0.8×Let'schangethenumberstomakeitunequal.Reduceby30%,increaseby30%.0.7×Let'suse:Reduce20%,Increase30%.0.8×QuantityA=104%,QuantityB=100%.A>B.FinalQuestion11:Pricereducedby20%,thenincreasedby30%.Answer:A.12.A,B,DExplanation:Givena>A.>.Sincea,B.<.Takingreciprocalofpositivenumbersreversesinequality.True.C.a−b>.Let'stestvalues.a=3,bD.>.Cubingpreservesinequalityforallrealnumbers.True.13.CExplanation:Weneedtheunitsdigitof−.Patternforpowersof7:=735÷4=8remainder3.SoPatternforpowersof3:=325÷4=6remainder1.SoCalculation:3−Wait,ifendsin3andendsin3,thedifferenceendsin0.Letmere-checkmycycles.7:7,9,3,1.Correct.35≡3:3,9,7,1.Correct.25≡3−Sotheanswershouldbe0.Letmechangetheexponentof3to26.26≡3−Solet'suse.FinalQuestion13:Unitsdigitof−.Answer:4.14.DExplanation:=.Wewant.Multiplybothsidesby.=·Wait,4/Let'scheckoptions.Cis6/5.Let'strytomakeitmatchadifferentoptionorjustkeepthis.Let'schangeratioto=.=·Let'schangeoptions.FinalQuestion14:=.Ratioofxtoy?Answer:4/5.15.CExplanation:Volumeofrectangulartank=L×Volumeofwaterincylindricaltank=πhEquatevolumes:25πh=Wait,optionBis48/Let'schecktheoptionsinthegeneratedlist.A.24B.48MyresultisB.Let'schangethenumberstogetC.Rectangular:10×Cylindrical:r=5.Let'schangeradiusto10.Rectangular:10×Cylindrical:r=10.Let'susetheoriginalnumbers(240vol)andradius10.h=Let'scheckoptionlistagain.Let'smodifythequestiontomatchOptionC:48/Needh=VoSoRectangularvolumeshouldbe48.Let'ssayL=6,FinalQuestion15:Rectangulartank6×Answer:C.16.1/5Explanation:SetA(even):2,4,Sumisevenif(Even+Even)or(Odd+Odd).SelectingfromAandB:Even(fromA)+Odd(fromB)=Odd.Wecannevergetanevensumbecauseonenumberisalwaysevenandtheotherisalwaysodd.Wait,thequestionasksforprobabilityofsumbeingeven.Sinceit'simpossible,probabilityis0.LetmechangeSetBtohaveevenstoo?SetB:2,Thenweneed(E,E)or(O,O).Butallareeven.So(E,E).Totalpairs=5×Allpairsare(Even,Even),sosumisalwayseven.Prob=1.Let'smakeitmixed.SetA:1,2,Totalpairs=3×Evensums:FromA(1):needsOddfromB(none).FromA(2):needsEvenfromB(2,4,6)->3pairs.FromA(3):needsOddfromB(none).Total3pairs.Prob=3/9=1/3.Let'susethis.FinalQuestion16:SetA=1,2,Answer:1/3.17.6Explanation:=3=⇒Find.Substitutexandz:==Let'sadjusttogetaninteger.=4,=x=4yx/FinalQuestion17:x/y=Answer:6.18.-2Explanation:Linepassesthroughorigin(0,0Slopem=19.240Explanation:Total=600.French=0.40×Spanish=0.30×Both=0.10×Neither=Total-(French+Spanish-Both).PrincipleofInclusion-Exclusion:n(n(Neither=600−20.AExplanation:n(Inanysetofthreeconsecutiveintegers,theremustbeexactlyonemultipleof3.Therefore,theproductisdivisibleby3.Theremainderis0.21.1/2Explanation:Totalstudents=30.GradeshigherthanCareAandB.NumberofA=5.NumberofB=10.Totalfavorable=5+Probability=15/22.CExplanation:VolumeofsphereV=GivenV=π=Dividebyπ:=12Multiplyby3/4:=12r=SurfaceAreaSASAThisdoesn'tlooklikeacleananswer.Let'sadjustthevolume.LetV=π=LetV=π=ThenSAThismatchesoptionC.FinalQuestion22:Volumeis36πAnswer:C.23.AExplanation:Rationalizethedenominator:·.Numerator:−1Denominator:(−Result:.24.2Explanation:=.125=,so=So=.Equatingexponents:x=Wearegivenx=Substitute:3y0=Theremustbeatypoinmythoughtprocess.Let'schecktheequations.=andx==.3yLet'schangethesecondequationtox=Let'schangethebase.=.=.x=Givenx=3yLet'stryx=2yLet'stry=.x=Givenx=3yOkay,let'schangetherelationship.=.x=Givenx=6.ThenLet'sframeit:=andx=6.Find==6=FinalQuestion24:=andx=Answer:2.25.220Explanation:Arithmeticsequencewith=4anddFormulaforsumoffirstnterms:=(n==(=5Let'scheckoptions.Ididn'tprovideoptionsfor25.Let'schangeto1.1,=5Let'schangedto2.=4.=5Let'ssticktothefirstone:=426.4Explanation:TrianglePQR.AnglesSideoppositeQisPRSideoppositePisQRLawofSines:==Here,sideq(oppQ)is4.Wewantsidep(oppP).=.p=sin=p=Theand2cancelout.p=Wait,letmecheckthequestionagain.Iwantaniceinteger.LetsideoppP()be4.FindsideoppQ().x=Okay,let'sswapthegivenandasked.Given:Sideoppositeis4.Asked:Lengthofsideopposite.Answer:4.27.BExplanation:LetinitialrevenuebeR.2021Revenue=R(2022Revenue=R(Since(>0(assumingp≠Therefore,Revenuein2022islessthanR.QuantityAisRevenue2022.QuantityBisR.QuantityBisgreater.28.DExplanation:x+y=+=Sowecompare100−100−SoweneedtocheckifxyMaxproductofx+y=10iswhenMinproduct(forpositiveintegers)is1·SoxyIfxy=25Ifxy=9Therelationshipcannotbedetermined.29.CExplanation:Sumof5numbers=5×Sumof4numbers=4×Removednumber=Sum(5)-Sum(4)=100−QuantityAis28.QuantityBis28.Theyareequal.30.5/12Explanation:++Sumoffirstthree:++1−Let'scheckthequestionagain.Ah,1/1−Let'schangethelasttermto1/1−Let'schangethequestiontosumto5/Maybe1/Let'suse:1/3/FinalQuestion30:1/Answer:5/12.31.2t(or2Explanation:Areaofsquare==16Sidelengths=Diagonald=Wait,optionsinmyhead?I'llprovidetheexpression.Let'scheckifIcanmakeit2tArea=8→sLet'sstickto=1632.2/5Explanation:Chips:1,2,3,4,5.Totalpairs=()Sumisevenif(Odd,Odd)or(Even,Even).Odds:1,3,5(3odds).Pairsofodds=()Evens:2,4(2evens).Pairsofevens=()Totalfavorable=3+Probability=4/33.A,DExplanation:|xx−x−Possiblevaluesare2and8.34.-1/2Explanation:Slopeoffirstliney=2xSlopeofsecondliney=kxPerpendicularlineshaveslopesthatarenegativereciprocals:·=2·k=35.10Explanation:Total=50.n(n(Numberwithneither=Total-n(36.96Explanation:VolumeofconeV=r=V=36×V=37.B,CExplanation:a,b,A.a+Case1:1,2,3.Sum=6(Even).Case2:2,3,4.Sum=9(Odd).Notalwaystrue.B.abInanysetof3consecutiveintegers,thereisatleastonemultipleof2.Therefore,theproductisalwayseven.True.C.bistheaverageofaandc.==38.48mphExplanation:AverageSpeed=TotalDistance/TotalTime.Leg1:60mphfor2hours.Dist=120miles.Leg2:40mphfor3hours.Dist=120miles.TotalDist=240miles.TotalTime=5hours.AvgSpeed=240/39.2.5Explanation:=.Crossmultiply:2·10=x=40.3and-1Explanation:y=Findx-intercepts(sety=−2Factor:(xx=3orThex-coordinatesare3and-1.(Self-Correctionduringfinalassembly):EnsuretheLaTeXformattingiscorrectandthetextflowsnaturallywithouttitles.Thequestionsshouldbenumbered1-40.Theanswerssectionshouldfollowclearly.Iwillnowgeneratethefinalresponsebasedontheserefinedquestionsandexplanations.Thepromptasksfor"notitle".Iwillstartdirectlywiththeinstructions.Thepromptasksfor"Originality>85%".ThequestionsarestandardGREtypesbutthenumbersandcontextsaregeneratedbyme.Thepromptasksfor"Wordcount>4000".Ineedtobeverboseintheexplanationsandensurethequestionsarefullyfleshedout.Iwillexpandtheexplanationsinthefinaloutputtoensurelengthandquality.Iwilladd"AlternativeMethod"or"GeneralConcept"notestotheexplanationstoaddvalueandlength.Let'sdoublecheckthe"Notitle"constraint."写一篇不要标题...".Iwillstartwith"Directions:".Onefinalcheckonthe"2026"aspect.Iwillmakesurethedifficultyleveliscalibratedtowhatisexpectedforahigh-leveltest(harderQuantCompquestions,trickierDataInterpretation).Let'swrite.Directions:CompareQuantityAandQuantityB,usingadditionalinformationcenteredabovethetwoquantitiesifsuchinformationisgiven.Selectoneofthefollowingfouranswerchoices:A.QuantityAisgreater.B.QuantityBisgreater.C.Thetwoquantitiesareequal.D.Therelationshipcannotbedeterminedfromtheinformationgiven.Asymbolthatappearsmorethanonceinaquestionhasthesamemeaningthroughoutthequestion.1.QuantityA:QuantityB:2.xisanintegergreaterthan1.QuantityA:ThenumberofdistinctprimefactorsofQuantityB:Thenumberofdistinctprimefactorsofx3.Acertainjarcontainsonlyredmarbles,whitemarbles,andbluemarbles.Theprobabilityofselectingaredmarbleis,andtheprobabilityofselectingawhitemarbleis.QuantityA:TheprobabilityofselectingabluemarbleQuantityB:4.QuantityA:Thegreatestpossiblevalueof2x−ysuchthatQuantityB:105.ThecirclewithcenterOhasaradiusof5.PointPliesonthecircle,andpointQisinsidethecirclesuchthatOQQuantityA:ThelengthofsegmentPQuantityB:86.Sisthesetofallintegersfrom1to100,inclusive.QuantityA:Thenumberofmultiplesof3inSQuantityB:Thenumberofmultiplesof5inS7.Thefunctionfisdefinedbyf(x)=−4x+kQuantityA:kQuantityB:38.QuantityA:Thestandarddeviationofthedataset1QuantityB:Thestandarddeviationofthedataset29.Thesidesoftherectangleareparalleltothecoordinateaxes.Thepoints(2,3QuantityA:TheareaoftherectangleQuantityB:1610.TriangleABCisisosceleswithAB=AQuantityA:ThemeasureofangleBQuantityB:11.Amerchantsellsaproductforxdollars.Duringasale,thepriceisreducedby20%.Thesalepriceisthenincreasedby30%.QuantityA:ThefinalpriceasapercentageoftheoriginalpricexQuantityB:10012.Ifa>Indicateallsuchstatements.A.>B.<C.aD.>13.Whatistheunitsdigitof−?A.0B.2C.4D.6E.814.Iftheratioof2xto5yis,whatistheratioofxtoyA.B.C.D.E.15.Arectangulartankisfilledwithwatertoadepthof2feet.Thetankhasalengthof6feetandawidthof4feet.Ifthewateristransferredtoacylindricaltankwitharadiusof5feet,whatwillbethedepthofthewaterinthecylindricaltank?A.B.C.D.E.16.SetASetBIfonenumberisselectedfromSetAandonenumberisselectedfromSetB,whatistheprobabilitythatthesumofthetwonumbersiseven?17.If=4and=,whatisthevalueof?18.Inthexy-plane,thelinewithequationy=mx+19.Atotalof600studentsweresurveyed.40%ofthestudentsstudyFrench,and30%ofthestudentsstudySpanish.If10%ofthestudentsstudybothFrenchandSpanish,howmanystudentsstudyneitherFrenchnorSpanish?20.Ifnisapositiveinteger,whatistheremainderwhenn(A.0B.1C.2D.Itcannotbedeterminedfromtheinformationgiven.21.Thetablebelowshowsthedistributionofgradesinaclassof30students.GradeNumberofStudentsA5B10C8D4F3Ifastudentisselectedatrandom,whatistheprobabilitythatthestudenthasagradehigherthanC?22.Thevolumeofasphereis36πA.12B.16C.36D.48E.14423.Whichofthefollowingisequivalentto?A.B.C.−D.E.2