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电大经济数学基础考试小抄
一、单项选择题(每小题3分,共15分)1.设A为3x2矩阵,B为2x3矩阵,则下列运算中(AB)可以进行.2.设AB为同阶可逆矩阵,则下列等式成立的是()3设为同阶可逆方阵,则下列说法正确的是( ).4.设AB阶方阵,在下列情况下能推出A是单位矩阵的是(D).7.设下面矩阵ABC能进行乘法运算,那么(AB=AC,A可逆,则B=C成立.9.设,则rA=
(1).10.设线性方程组的增广矩阵通过初等行变换化为,则此线性方程组的一般解中自由未知量的个数为
(1).11.线性方程组解的情况是(无解).12.若线性方程组的增广矩阵为,则当=)时线性方程组无解. 13.线性方程组只有零解,则(可能无解).14.设线性方程组AX=b中,若rAb=4,rA=3,则该线性方程组(无解).
1、下列函数中为偶函数的是(A).A.
2、下列函数在区间上是单调下降的是(D).D.
3、下列定积分计算正确的是(D).D.
4、设A=则r=(C)C.
35、设线性方程组的增广矩阵为,则此线性方程组的一般解中自由未知量的个数为(B).B.
21、函数的定义域是(A)A.(-2,4)解答
2、曲线在点(0,1)处的切线斜率为(A)A.解答
3、若是的一个原函数,则下列等式成立的是(B).B.解答
4、设A,B为同阶可逆矩阵,则下列等式成立的是(D).D.解答
5、设线性方程组有唯一解,则相应的齐次方程组(C).C.只有零解解答只有零解即C
1、各函数对中的两个函数相等的是(C).C.解答∵∴选
2、已知,当(A )时为无穷小量A.解答∵∴选
3、下列函数中,(B)是的原函数.B.解答∵∴选
4、设为矩阵,为矩阵,且乘积矩阵有意义则为(B)矩阵.B.解答∴选
5、若线性方程组的增广矩阵为,则当=(B)时线性方程组无解.B.-3解答当时∴线性方程组无解∴选
二、填空题(每小题3分,共15分)1.两个矩阵既可相加又可相乘的充分必要条件是与是同阶矩阵2.计算矩阵乘积=
[4].3.若矩阵A=,B=,则ATB=.4.设为矩阵,为矩阵,若AB与BA都可进行运算,则有关系式5.设,当0 时,A称矩阵.6.当a时,矩阵可逆.7.设AB个已知矩阵,且1-B则方程的解.8.设为阶可逆矩阵,则A=n9.若矩阵A=,则rA=2.10.若rAb=4,rA=3,则线性方程组AX=b无解.11.若线性方程组有非零解,则-1.12.设齐次线性方程组,且秩A=rn,则其一般解中的自由未知量的个数等于n–r.13.齐次线性方程组的系数矩阵为则此方程组的一般解为.14.线性方程组的增广矩阵化成阶梯形矩阵后为则当d-1组AX=b解.15.若线性方程组有唯一解,则只有0解.
6、函数的图形关于原点对称.
7、.函数y=x-2的驻点是x=
28、
4.
9、矩阵的秩为
210、已知齐次线性方程组中的为3×5矩阵,且该方程组有非0解,则
3.
6、若函数,则解答令则即
7、曲线在点42处的切线方程为解答即即
8、0解答是奇函数
9、设A=,当=1时,A是对称矩阵.解答当时,矩阵为对称矩阵
10、线性方程组的增广矩阵化成阶梯形矩阵后为,则当-5时,方程组有无穷多解.解答时,有无穷多解
6、若函数,则解答∴=
7、已知,若在内连续,则 2 .解答∴
8、若,则=解答
9、设矩阵A=,I为单位矩阵,则(I-A)=解答∴
10、齐次线性方程组只有零解的充分必要条件是m=n=rA.
三、微积分计算题(每小题10分,共20分)
11、设,求.解
12、计算积分.解原式
11、已知,求.解
12、计算.
1.解原式
11、设y=,求解
12、计算.解原式=
四、代数计算题(每小题15分,共30分)1设矩阵,,求解因为===所以==2设矩阵,,计.解===3设矩阵A=,求解因为AI=所以A-1=4设矩阵A=,求逆矩阵因为AI=所以A-1=5设矩阵A=,B=,计算AB-1解因为AB==ABI=所以AB-1=7解矩阵方程.解因为即所以,X==8解矩阵方程解因为即所以,X===10设线性方程组,求其系数矩阵和增广矩阵的并.解因为所以rA=2,r=
3.又因为rAr,所以方程组无解.11求下列线性方程组的一般解解因为系数矩所以一般解为(其中,是自由未知量)12.求下列线性方程组的一般解解因为增广矩阵所以一般解为(其中是自由未知量)13设齐次线性方程组问取何值时方程组有非零解,并求一般解.13.解因为系数矩阵A=所以当=5时,方程组有非零解.且一般解为(其中是自由未知量)14当取何值时,线性方程组有解?并求一解因为增广矩阵所以当=0时,线性方程组有无穷多解,且一般解为是自由未知量〕
13、已知AX=B其中A=B=求X解
14、讨论为何值时,齐次线性方程组有非零解,并求其一般解.解
13、设矩阵A=,,求
14、求当取何值时,线性方程组有解,并求一般解.
14.即时线性方程组有无穷多解一般解为
13、设矩阵=,=,求解矩阵方程.解∴
14、设齐次线性方程组,问取何值时方程组有非0解,并求一般解.解当即时方程组有非零解一般解为(为自由未知量)五.应用题(本题20分)
15、某厂生产某种产品件时的总成本函数为(元),单位销售价格为(元/件),试求
(1)产量为多少时可使利润达到最大?
(2)最大利润是多少?解
(1)
(2)最大利润是1855元
15、设生产某产品的总成本函数为(万元),其中为产量,单位百吨.销售百吨时的边际收入为(万元/百吨),求
(1)利润最大时的产量;
(2)在利润最大时的产量的基础上再生产1百吨,利润会发生什么变化?
15.解
(1)令即检验知百吨时利润最大
(2)在利润最大的基础上再生产1百吨利润将减少1万元
15、投产某产品的固定成本为36(万元),且边际成本为(万元/百台),试求产量有4百台增至6百台时总成本的增量,及产量为多少时,可使平均成本达到最低解(万元)即当产量从4百台增加至6百台时,总成本增加140万元令即(百台)检验知当产量为352百台时平均成本最小Visa-freepolicybringsChengdubiztourismboost.MakingnationalheadlinesseveraltimesChengdus72-hourvisa-freepolicyhasattractedwideattentionfrombothChineseandforeignexpertsandbusinessmensinceittookeffectonSept1lastyear.Theprogrampermitscitizensfrom51countriesandregionsincludingtheUnitedStatesAustraliaCanadaandJapanwhohavevalidvisasandflightticketstoathirdcountrytospendthreedaysinthecity.ThecapitalofSichuanprovinceisthefirstcityinthewesternregionofChinatoofferforeigntouristsathree-dayvisaandthefourthnationwidetoadoptthepolicyfollowingShanghaiBeijingandGuangzhou.LiZhiyongdeputydeanofthetourisminstituteatSichuanUniversitysaidthemovecontributestoalargeincreaseinthenumberofoverseastouristsandraisesthecityslevelofinternationalization.ThepolicywillalsobringdirecteconomicrevenueLisaid.Chengduhasmanyculturallegaciesandisalsoaparadiseforpandaloverswiththeworldslargestbreedingandresearchcenter.Threedaysarelongenoughforforeignvisitorstovisitthoseiconictouristspotshenoted.ThecityishometotheremainsoftheJinshacivilizationthatdatesbackmorethan3000yearsaswellastheQingchengMountainsandtheDujiangyanirrigationsystem.QingchenghaslongbeenrecognizedasthebirthplaceofTaoismChinasancientindigenousreligionwhileDujiangyanisconsideredtobetheoldestfunctioningwater-controlprojectintheworld.Chengdurankedthirdintouristfacilitiesmanagementandservicesamong60Chinesecitiesinacustomersatisfactionsurveyreleasedlastyear.ButLiaddedthateffortsarestillneededtodevelopmoretourismproductsimproveEnglishservicesandprovideaccuratetranslationoftrafficsignsandscenicbillboards.ZhaoYunchairwomanofBritishChamberofCommerceSouthwestChinatoldChinaDailythathiscolleaguesfoundthepolicyveryconvenient.ABritishclientonceflewhereandstayedforjustonedaytocheckherorderedgoodsshesaid.ZhaowasborninShanxiprovincebutshehaslivedinChengduformorethan10years.Mylifewaslikearunningracemovingfromplacetoplace.IalsolivedinBeijingandShanghaibeforeshesaid.ButChengduisaplacethatyouneverwanttoleaveoncesettlingdown.Itisnowmysecondhometownshesaid.Iftheenvironmentisfurtherimprovedthecitywillattractmorepeopletovisitandlivewiththe72-hourvisa-freepolicyandcompellingconditionsintransportationcultureclimateandcuisinehesaid.Foreignersalsogavepositivefeedbackonthepolicy.AspokesmanfromDellIncsaidthecompanyhasaglobalhubofoperationinChengdusothethree-dayvisahasanimmediateandpositiveinfluenceonthecompanysbusinessdevelopment.RudyButtignolpresidentofthepublicbroadcastingcompanyinBritishColumbiaCanadasaidhisworkrequiresfrequenttraveltoChengduandthepolicymakesthetripseasier.Datafromthecityspublicsecuritybureaushowssome100foreignvisitorsenjoyedthe72-hourpolicybytheendofMarchmostofthemfromtheUnitedStatestheUnitedKingdomandGermany.Chengdualsoreportedrobustgrowthinitsoveralltouristindustrylastyear.Officialstatisticsshowthatitreceivedsome150milliontouristslastyearanincreaseof28percentfrom
2012.Around
1.7millioncamefromabroadanincreaseof12percent.Totalrevenuefromtourismsurpassed133billionyuan$
21.7billion.DuringhisvisittoKazakhstaninSeptemberChinesePresidentXiJinpingproposedthatChinaandCentralAsiajoinhandstobuildaSilkRoadeconomicbelttoboostcooperation.TheideahasbeenwidelyechoedinCentralAsiancountriesbecominganencouragingblueprintforChineseareasalongtheSilkRoadthathaslinkedAsiaandEuropeformorethan2000years.InthenextthreeweeksChinaDailyreporterswilltravelthroughthebeltinChinaandinKazakhstanUzbekistanandTurkey.Theywillshowtheprogressandexpectationsofthecountriesbusinessesandpeoplesontheroute.Shaanxi-thestartoftheancientSilkRoad-haspositioneditselfasthenewstartingpointforthedevelopmentoftheSilkRoadEconomicBeltwhichwillstrengthenChinascooperationwithCentralAsiancountriesaseniorofficialsaid.ShaanxiGovernorLouQinjiansaidtheprovinceisfreshrichanduniqueasitwaswhenitanchoredoneendoftheancientSilkRoad.ItisthebestoptionforaccommodatingindustrialtransfersfromEastChinaortheworldatlargehesaidonWednesdayinXian.Louheldajointinterviewwith27mediaincludingChinaDailythefirstinaseriesofinterviewsentitledChineseMediaAlongtheSilkRoad.TheinterviewswillbeinShaanxiandGansuprovincesandtheXinjiangUygurautonomousregionaswellasKazakhstanUzbekistanandTurkey.ThemediagroupheldthefirstinterviewonWednesdaymorninginXianthestartingpointoftheancientSilkRoadatradechannelestablishedmorethan2000yearsagolinkingChinaCentralAsiaandEurope.。