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1、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，则下列说法对的的是（ ）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A, B, C能进行乘法运算，那么（AB = AC，A可逆，则B = C 成立. 9设，则r(A) =（ 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
2、 线性方程组只有零解，则（也许无解）.14设线性方程组AX=b中，若r(A, b) = 4，r(A) = 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、下列定积分计算对的的是（ D ）. D. 4、设A=,则r=（ C ）。 C.3 5、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.2 1、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
3、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C1、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，且乘积矩阵故意义,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足必要条件是与是同阶矩阵2计算矩阵乘积=4
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r(A) = 2 10若r(A, b) = 4，r(A) = 3，则线性方程组AX = b无解11若线性方程组有非零解，则-112设齐次线性方程组，且秩(A) = r n，则其一般解中的自由未知量的个数等于n r13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、
5、 函数的图形关于 原点 对称.7、.函数y=(x-2) 的驻点是 x=2 8、 4 . 9、矩阵的秩为 2 。 10、已知齐次线性方程组中的为35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(4, 2)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
6、）=解答： 10、 齐次线性方程组只有零解的充足必要条件是 m=n=r(A) . 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 由于 = =所以 = 2设矩阵 ，计解：= = = 3设矩阵A =，求解 由于 (A I )= 所以 A-1 = 4设矩阵A =，求逆矩阵由于(A I ) = 所以 A-1= 5设矩阵 A =，B =，计算(AB)-1解 由于AB = (AB I ) = 所以 (AB)-1
7、= 7解矩阵方程解 由于 即 所以，X = 8解矩阵方程解：由于 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 由于 所以 r(A) = 2，r() = 3. 又由于r(A) r()，所以方程组无解. 11求下列线性方程组的一般解： 解由于系数矩 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 由于增广矩阵 所以一般解为 （其中是自由未知量） 13设齐次线性方程组问l取何值时方程组有非零解，并求一般解.13解 由于系数矩阵A = 所以当l = 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 由
8、于增广矩阵 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、讨论为什么值时，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
9、达成最大？（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 检查知 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，可使平均成本达成最低。解： （万元） 即当产量从4百台增长至6百台时，总成本增长140万元
10、 令 即 （百台）检查知当产量为352百台时平均成本最小。Visa-free policy brings Chengdu biz, tourism boost. Making national headlines several times, Chengdus 72-hour visa-free policy has attracted wide attention from both Chinese and foreign experts and businessmen since it took effect on Sept 1 last year. The program permits
11、 citizens from 51 countries and regions including the United States, Australia, Canada and Japan who have valid visas and flight tickets to a third country to spend three days in the city. The capital of Sichuan province is the first city in the western region of China to offer foreign tourists a th
12、ree-day visa and the fourth nationwide to adopt the policy following Shanghai, Beijing and Guangzhou. Li Zhiyong, deputy dean of the tourism institute at Sichuan University, said the move contributes to a large increase in the number of overseas tourists and raises the citys level of internationaliz
13、ation. The policy will also bring direct economic revenue, Li said. Chengdu has many cultural legacies and is also a paradise for panda lovers with the worlds largest breeding and research center. Three days are long enough for foreign visitors to visit those iconic tourist spots, he noted. The city
14、 is home to the remains of the Jin sha civilization that dates back more than 3,000 years as well as the Qing cheng Mountains and the Du jiang yan irrigation system. Qing cheng has long been recognized as the birthplace of Taoism, Chinas ancient indigenous religion, while Du jiang yan is considered
15、to be the oldest functioning water-control project in the world. Chengdu ranked third in tourist facilities, management and services among 60 Chinese cities in a customer satisfaction survey released last year. But, Li added that efforts are still needed to develop more tourism products, improve Eng
16、lish services and provide accurate translation of traffic signs and scenic billboards. Zhao Yun, chairwoman of British Chamber of Commerce Southwest China, told China Daily that his colleagues found the policy very convenient. A British client once flew here and stayed for just one day to check her
17、ordered goods, she said. Zhao was born in Shanxi province, but she has lived in Chengdu for more than 10 years. My life was like a running race moving from place to place. I also lived in Beijing and Shanghai before, she said. But Chengdu is a place that you never want to leave once settling down. I
18、t is now my second hometown, she said. If the environment is further improved, the city will attract more people to visit and live, with the 72-hour visa-free policy and compelling conditions in transportation, culture, climate and cuisine, he said. Foreigners also gave positive feedback on the poli
19、cy. A spokesman from Dell Inc said the company has a global hub of operation in Chengdu, so the three-day visa has an immediate and positive influence on the companys business development. Rudy Buttignol, president of the public broadcasting company in British Columbia, Canada, said his work require
20、s frequent travel to Chengdu and the policy makes the trips easier. Data from the citys public security bureau shows some 100 foreign visitors enjoyed the 72-hour policy by the end of March, most of them from the United States, the United Kingdom and Germany. Chengdu also reported robust growth in i
21、ts overall tourist industry last year. Official statistics show that it received some 150 million tourists last year, an increase of 28 percent from 2023. Around 1.7 million came from abroad, an increase of 12 percent. Total revenue from tourism surpassed 133 billion yuan ($21.7billion). During his
22、visit to Kazakhstan in September, Chinese President Xi Jinping proposed that China and Central Asia join hands to build a Silk Road economic belt to boost cooperation. The idea has been widely echoed in Central Asian countries, becoming an encouraging blueprint for Chinese areas along the Silk Road
23、that has linked Asia and Europe for more than 2,000 years. In the next three weeks, China Daily reporters will travel through the belt in China and in Kazakhstan, Uzbekistan and Turkey. They will show the progress and expectations of the countries, businesses and peoples on the route. Shaanxi - the
24、start of the ancient Silk Road - has positioned itself as the new starting point for the development of the Silk Road Economic Belt, which will strengthen Chinas cooperation with Central Asian countries, a senior official said. Shaanxi Governor Lou Qinjian said the province is fresh, rich and unique
25、, as it was when it anchored one end of the ancient Silk Road. It is the best option for accommodating industrial transfers from East China or the world at large, he said on Wednesday in Xian.Lou held a joint interview with 27 media, including China Daily, the first in a series of interviews entitle
26、d Chinese Media Along the Silk Road. The interviews will be in Shaanxi and Gansu provinces and the Xinjiang Uygur autonomous region, as well as Kazakhstan, Uzbekistan and Turkey.The media group held the first interview on Wednesday morning in Xian, the starting point of the ancient Silk Road, a trade channel established more than 2,000 years ago linking China, Central Asia and Europe.