2026年研究生入學(xué)考試英語數(shù)學(xué)專業(yè)課筆試模擬試題及答案考試時長：120分鐘滿分：100分班級：__________姓名：__________學(xué)號：__________得分：__________試卷名稱：2026年研究生入學(xué)考試英語數(shù)學(xué)專業(yè)課筆試模擬試題考核對象：報考研究生入學(xué)考試的考生題型分值分布-判斷題（總共10題，每題2分）總分20分-單選題（總共10題，每題2分）總分20分-多選題（總共10題，每題2分）總分20分-案例分析（總共3題，每題6分）總分18分-論述題（總共2題，每題11分）總分22分總分：100分---一、判斷題（每題2分，共20分）1.Thepassivevoiceshouldalwaysbeavoidedinacademicwriting.2.Thederivativeof\(e^x\)is\(e^x\).3.Inanormaldistribution,themean,median,andmodeareequal.4.ThePythagoreantheoremappliesonlytoright-angledtriangles.5.Abinomialdistributionisusedfordiscreterandomvariables.6.Theintegralof\(\frac{1}{x}\)is\(\ln|x|+C\).7.Quantummechanicsdescribesthebehaviorofmacroscopicobjects.8.TheFibonaccisequenceisanexampleofanarithmeticprogression.9.Anullhypothesisisalwaystrue.10.Thecentrallimittheoremstatesthatthesamplingdistributionofthemeanapproachesanormaldistributionassamplesizeincreases.二、單選題（每題2分，共20分）1.Whichofthefollowingisacorrectuseofthepastperfecttense?A.Shewrotealetteryesterday.B.Shehadwrittenaletterbeforesheleft.C.Shewritesalettereveryday.D.Shewillwritealettertomorrow.2.Thelimitof\(\frac{\sinx}{x}\)as\(x\to0\)is:A.0B.1C.\(\infty\)D.Undefined3.Inabinomialexperiment,theprobabilityofsuccessremainsconstantacrosstrials.Thecorrectoptionis:A.TrueB.False4.Theareaunderaprobabilitydensityfunctionoverarangerepresents:A.Theprobabilityofanevent.B.Theexpectedvalue.C.Thevariance.D.Thestandarddeviation.5.Theeigenvaluesofadiagonalmatrixare:A.Alwaysreal.B.Alwayscomplex.C.Alwayszero.D.Alwaysone.6.Theprimarypurposeofanullhypothesisinstatisticsisto:A.Provethealternativehypothesis.B.Rejectthealternativehypothesis.C.Provideevidenceagainstthenullhypothesis.D.Supportthenullhypothesis.7.Thesequence\(1,1,2,3,5,8,\ldots\)isknownas:A.Arithmeticprogression.B.Geometricprogression.C.Fibonaccisequence.D.Harmonicprogression.8.TheLaplacetransformof\(t^2\)is:A.\(\frac{1}{s^3}\)B.\(\frac{2}{s^2}\)C.\(\frac{1}{s}\)D.\(s^2\)9.Inquantummechanics,theuncertaintyprinciplestatesthat:A.Themorepreciselythepositionofaparticleisknown,thelesspreciselyitsmomentumcanbeknown.B.Particlescanexistinmultiplestatessimultaneously.C.Theenergyofasystemisalwaysconstant.D.Particlesdonothavemass.10.Theslopeofthetangentlinetothecurve\(y=x^3\)at\(x=1\)is:A.1B.3C.6D.9三、多選題（每題2分，共20分）1.Whichofthefollowingarecommongrammaticalerrorsinacademicwriting?A.Subject-verbagreementerrors.B.Overuseofpassivevoice.C.Incorrectpunctuation.D.Plagiarism.2.Thefundamentaltheoremofcalculusincludes:A.Thefirstfundamentaltheorem.B.Thesecondfundamentaltheorem.C.Thechainrule.D.Thepowerrule.3.InaPoissondistribution,themeanandvarianceare:A.Equal.B.Unequal.C.Zero.D.Positive.4.Thefollowingarepropertiesofanormaldistribution:A.Symmetricaboutthemean.B.Skewedtotheright.C.Bell-shaped.D.Mean=median=mode.5.Matrixmultiplicationis:A.Commutative.B.Associative.C.Distributiveoveraddition.D.Notcommutative.6.Thefollowingareexamplesofdiscreterandomvariables:A.Thenumberofheadsin10coinflips.B.Theheightofarandomlyselectedperson.C.Thetimeittakestocompleteatask.D.Thenumberofcarsinaparkinglot.7.Thefollowingareapplicationsofdifferentialequations:A.Physics.B.Economics.C.Biology.D.Engineering.8.Thefollowingaretrueabouteigenvaluesofamatrix:A.Theyarerootsofthecharacteristicequation.B.Theycanbecomplex.C.Theydeterminethematrix'sinvertibility.D.Theyarealwaysreal.9.Thefollowingaremethodstoteststatisticalhypotheses:A.t-test.B.Chi-squaretest.C.ANOVA.D.Regressionanalysis.10.ThefollowingarepropertiesofaFibonaccisequence:A.Eachtermisthesumofthetwoprecedingones.B.Theratioofconsecutivetermsapproachesthegoldenratio.C.Itisanarithmeticprogression.D.Itisusedinfinancialmodeling.四、案例分析（每題6分，共18分）1.CaseStudy:ProbabilityinRealLifeAfactoryproducestwotypesofbulbs:AandB.TheprobabilitythatabulboftypeAisdefectiveis0.05,andtheprobabilitythatabulboftypeBisdefectiveis0.10.Ifabulbisrandomlyselectedfromthefactory,whatistheprobabilitythatitisdefectivegiventhatitisoftypeB?AssumethereareequalnumbersofbulbsoftypeAandB.2.CaseStudy:CalculusinEngineeringThedisplacement\(s(t)\)ofaparticlemovingalongastraightlineisgivenby\(s(t)=t^3-3t^2+2t\).Findthevelocity\(v(t)\)andacceleration\(a(t)\)oftheparticleat\(t=2\).3.CaseStudy:MatrixApplicationsinEconomicsAsmalleconomyhastwosectors:agricultureandmanufacturing.Theinput-outputmatrixfortheeconomyis:\[\begin{pmatrix}0.2&0.3\\0.4&0.1\end{pmatrix}\]Ifthefinaldemandforagricultureis100unitsandformanufacturingis200units,findthetotaloutputforeachsector.五、論述題（每題11分，共22分）1.Discusstheimportanceofthecentrallimittheoreminstatisticsandprovideanexampleofitsapplication.2.Explainthedifferencebetweenabinomialdistributionandanormaldistribution.Whenwouldyouuseoneovertheother?---標準答案及解析一、判斷題1.False(Passivevoicecanbeusefulinacademicwritingtoemphasizetheactionratherthantheactor.)2.True3.True4.True5.True6.True7.False(Quantummechanicsdescribesmicroscopicobjects.)8.False(Fibonaccisequenceisgeometric.)9.False(Nullhypothesisisastatementtobetested,notnecessarilytrue.)10.True二、單選題1.B2.B3.A4.A5.A6.C7.C8.A9.A10.C三、多選題1.A,B,C2.A,B3.A,D4.A,C,D5.B,C,D6.A,D7.A,C,D8.A,B9.A,B,C10.A,B,D四、案例分析1.ProbabilityinRealLifeLet\(P(A)\)betheprobabilityoftypeAbulb,\(P(B)\)betheprobabilityoftypeBbulb,\(P(D|A)\)betheprobabilityofdefectintypeA,and\(P(D|B)\)betheprobabilityofdefectintypeB.Given\(P(A)=P(B)=0.5\),\(P(D|A)=0.05\),\(P(D|B)=0.10\),andusingthelawoftotalprobability:\[P(D)=P(D|A)P(A)+P(D|B)P(B)=0.05\times0.5+0.10\times0.5=0.075\]TheprobabilitythatadefectivebulbisoftypeBis:\[P(B|D)=\frac{P(D|B)P(B)}{P(D)}=\frac{0.10\times0.5}{0.075}=\frac{0.05}{0.075}=\frac{2}{3}\approx0.6667\]2.CalculusinEngineeringVelocity\(v(t)=s'(t)=3t^2-6t+2\).At\(t=2\):\[v(2)=3(2)^2-6(2)+2=12-12+2=2\]Acceleration\(a(t)=v'(t)=6t-6\).At\(t=2\):\[a(2)=6(2)-6=12-6=6\]3.MatrixApplicationsinEconomicsLet\(X\)bethetotaloutputmatrix.Theequationis\(X=(I-A)^{-1}D\),where\(A\)istheinput-outputmatrixand\(D\)isthefinaldemand.\[I-A=\begin{pmatrix}1&0\\0&1\end{pmatrix}-\begin{pmatrix}0.2&0.3\\0.4&0.1\end{pmatrix}=\begin{pmatrix}0.8&-0.3\\-0.4&0.9\end{pmatrix}\]Theinverseis:\[(I-A)^{-1}=\frac{1}{(0.8)(0.9)-(-0.3)(-0.4)}\begin{pmatrix}0.9&0.3\\0.4&0.8\end{pmatrix}=\frac{1}{0.72-0.12}\begin{pmatrix}0.9&0.3\\0.4&0.8\end{pmatrix}=\frac{1}{0.6}\begin{pmatrix}0.9&0.3\\0.4&0.8\end{pmatrix}=\begin{pmatrix}1.5&0.5\\0.6667&1.3333\end{pmatrix}\]Totaloutput\(X=(I-A)^{-1}D=\begin{pmatrix}1.5&0.5\\0.6667&1.3333\end{pmatrix}\begin{pmatrix}100\\200\end{pmatrix}=\begin{pmatrix}250\\333.33\end{pmatrix}\)Outputforagriculture=250units,manufacturing=333.33units.五、論述題1.CentralLimitTheoremThecentrallimittheorem(CLT)statesthatthesamplingdistributionofthemeanofasufficientlylargenumberofindependent,identicallydistributedrandomvariableswillbeapproximatelynormallydistributed,regardlessoftheoriginaldistributionofthevariables.Thisiscrucialinstatisticsbecauseitallowsforinferencesaboutpopulationparametersusingsamplemeans,evenwhenthepopulationdistributionisnotnormal.Forexample,inqualitycontrol,iftheweigh