电大工程数学（本）期末复习资料考试小抄【最新】.doc

电大工程数学（本）复习资料考试小抄最新一、单项选择题（每小题3分，本题共15分）1. 若，则（A） A. 3 B. 2 C. D. 2. 已知2维向量组，则至多是（B）A B C D 3. 设为阶矩阵，则下列等式成立的是（C）A. B. C. D. 4. 若满足（B），则与是相互独立A. B. C. D. 5. 若随机变量的期望和方差分别为和，则等式（ D ）成立A. B. C. D. 6若是对称矩阵，则等式（B ）成立A. B. C. D. 7（D） A. B. C. D. 8若（A）成立，则元线性方程组有唯一解A. B. C. D. 的行向量线性相关 4. 若条件（C ）成立，则随机事件，互为对立事件A. 或 B. 或C. 且 D. 且 9对来自正态总体（未知）的一个样本，记，则下列各式中（C）不是统计量 A. B. C. D. 10设都是n阶方阵，则下列命题正确的是( A ) A B C D若，则或 11向量组的秩是（ B ） A. 1 B. 3 C. 2 D. 4 12元线性方程组有解的充分必要条件是（A）A. B. 不是行满秩矩阵 C. D. 13. 袋中有3个红球，2个白球，第一次取出一球后放回，第二次再取一球，则两球都是红球的概率是（D ）A. B. C. D. 14设是来自正态总体的样本，则（C）是无偏估计A. B. C. D. 15设为阶矩阵，则下列等式成立的是（ A）A B C D 16方程组相容的充分必要条件是( B )，其中，A B C D 17下列命题中不正确的是（ D ） AA与有相同的特征多项式 B若是A的特征值，则的非零解向量必是A对应于的特征向量 C若=0是A的一个特征值，则必有非零解 DA的特征向量的线性组合仍为A的特征向量 18若事件与互斥，则下列等式中正确的是（ A ）A BC D 19设是来自正态总体的样本，则检验假设采用统计量U =（C ）A B C D 二、填空题（每小题3分，共15分）1. 设均为n阶可逆矩阵，逆矩阵分别为，则2. 向量组线性相关，则-1.3. 已知，则0.64. 已知随机变量，那么2.45. 设是来自正态总体的一个样本，则 6设均为3阶方阵，则8 7设为n阶方阵，若存在数l和非零n维向量，使得 ，则称为相应于特征值l的特征向量 8若，则0.3 9如果随机变量的期望，那么20 10不含未知参数的样本函数称为统计量11设均为3阶方阵，则-18 12设随机变量，则a =0.3 13设为随机变量，已知，此时27 14设是未知参数的一个无偏估计量，则有15设，则的根是1，-1，2，-2 16设4元线性方程组AX=B有解且r（A）=1，那么AX=B的相应齐次方程组的基础解系含有 3 个解向量 17设互不相容，且，则0 18设随机变量X B（n，p），则E（X）= np 19若样本来自总体，且，则 三、计算题（每小题16分，共64分）1设矩阵，求（1），（2）解： （1） （2）利用初等行变换得即 2. 当取何值时，线性方程组有解，在有解的情况下求方程组的全部解解：将方程组的增广矩阵化为阶梯形由此可知当时，方程组无解。当时，方程组有解此时相应齐次方程组的一般解为 （是自由未知量）分别令及，得齐次方程组的一个基础解系令，得非齐次方程组的一个特解由此得原方程组的全部解为（其中为任意常数）3. 设，试求；（已知）解：(1) (2) 4. 已知某种零件重量，采用新技术后，取了9个样品，测得重量（单位：kg）的平均值为14.9，已知方差不变，问平均重量是否仍为15（）？解： 零假设由于已知，故选取样本函数已知，经计算得， 由已知条件，故接受零假设，即零件平均重量仍为155设矩阵，求解：利用初等行变换得即由矩阵乘法得6当取何值时，线性方程组有解，在有解的情况下求方程组的全部解解：将方程组的增广矩阵化为阶梯形由此可知当时，方程组无解。当时，方程组有解。此时齐次方程组化为分别令及，得齐次方程组的一个基础解系 令，得非齐次方程组的一个特解 由此得原方程组的全部解为（其中为任意常数） 7设，试求：(1)；(2)（已知）解：(1)(2) 8某车间生产滚珠，已知滚珠直径服从正态分布今从一批产品里随机取出9个，测得直径平均值为15.1mm，若已知这批滚珠直径的方差为，试找出滚珠直径均值的置信度为0.95的置信区间解：由于已知，故选取样本函数已知，经计算得 滚珠直径均值的置信度为0.95的置信区间为，又由已知条件，故此置信区间为9设矩阵，且有，求解：利用初等行变换得即 由矩阵乘法和转置运算得 10求线性方程组的全部解解： 将方程组的增广矩阵化为阶梯形方程组的一般解为（其中为自由未知量） 令=0，得到方程的一个特解. 方程组相应的齐方程的一般解为（其中为自由未知量）令=1，得到方程的一个基础解系. 于是，方程组的全部解为 （其中为任意常数） 11据资料分析，某厂生产的一批砖，其抗断强度，今从这批砖中随机地抽取了9块，测得抗断强度（单位：kgcm2）的平均值为31.12，问这批砖的抗断强度是否合格（）解： 零假设由于已知，故选取样本函数 已知，经计算得，由已知条件，故拒绝零假设，即这批砖的抗断强度不合格12设矩阵，求 解：由矩阵乘法和转置运算得利用初等行变换得即 14求下列线性方程组的通解解利用初等行变换，将方程组的增广矩阵化成行简化阶梯形矩阵，即®®®方程组的一般解为：，其中，是自由未知量 令，得方程组的一个特解方程组的导出组的一般解为：，其中，是自由未知量令，得导出组的解向量；令，得导出组的解向量 所以方程组的通解为：其中是任意实数 15设随机变量X N（3，4）求：（1）P（1< X < 7）；（2）使P（X < a）=0.9成立的常数a (已知，) 解：（1）P（1< X < 7）= = 0.9773 + 0.8413 1 = 0.8186 （2）因为 P（X < a）= 0.9所以 ，a = 3 + = 5.56 16从正态总体N（，4）中抽取容量为625的样本，计算样本均值得= 2.5，求的置信度为99%的置信区间.(已知 )解：已知，n = 625，且 因为 = 2.5， 所以置信度为99%的的置信区间为： 17某车间生产滚珠，已知滚珠直径服从正态分布今从一批产品里随机取出9个，测得直径平均值为15.1mm，若已知这批滚珠直径的方差为，试找出滚珠直径均值的置信度为0.95的置信区间解：由于已知，故选取样本函数已知，经计算得 滚珠直径均值的置信度为0.95的置信区间为，又由已知条件，故此置信区间为四、证明题（本题6分）1设,是两个随机事件，试证：证明：由事件的关系可知而，故由加法公式和乘法公式可知证毕 2设随机事件,相互独立，试证：也相互独立证明： 所以也相互独立证毕3设是阶对称矩阵，试证：也是对称矩阵证明：是同阶矩阵，由矩阵的运算性质可知已知是对称矩阵，故有，即由此可知也是对称矩阵，证毕4设n阶矩阵A满足，则A为可逆矩阵证明： 因为 ，即所以，A为可逆矩阵5设向量组线性无关，令，证明向量组线性无关。 证明：设，即 因为线性无关，所以 解得k1=0, k2=0, k3=0，从而线性无关6设,为随机事件，试证： 证明：由事件的关系可知而，故由概率的性质可知O(_)O谢谢！【China's 10 must-see animations】The Chinese animation industry has seen considerable growth in the last several years. It went through a golden age in the late 1970s and 1980s when successively brilliant animation work was produced. Here are 10 must-see classics from China's animation outpouring that are not to be missed. Let's recall these colorful images that brought the country great joy. Calabash Brothers Calabash Brothers (Chinese: 葫芦娃) is a Chinese animation TV series produced by Shanghai Animation Film Studio. In the 1980s the series was one of the most popular animations in China. It was released at a point when the Chinese animation industry was in a relatively downed state compared to the rest of the international community. Still, the series was translated into 7 different languages. The episodes were produced with a vast amount of paper-cut animations. Black Cat Detective Black Cat Detective (Chinese: 黑猫警长) is a Chinese animation television series produced by the Shanghai Animation Film Studio. It is sometimes known as Mr. Black. The series was originally aired from 1984 to 1987. In June 2006, a rebroadcasting of the original series was announced. Critics bemoan the series' violence, and lack of suitability for children's education. Proponents of the show claim that it is merely for entertainment. Effendi "Effendi", meaning sir and teacher in Turkish, is the respectful name for people who own wisdom and knowledge. The hero's real name was Nasreddin. He was wise and witty and, more importantly, he had the courage to resist the exploitation of noblemen. He was also full of compassion and tried his best to help poor people. Adventure of Shuke and Beita【舒克与贝塔】 Adventure of Shuke and Beita (Chinese: 舒克和贝塔) is a classic animation by Zheng Yuanjie, who is known as King of Fairy Tales in China. Shuke and Beita are two mice who don't want to steal food like other mice. Shuke became a pilot and Beita became a tank driver, and the pair met accidentally and became good friends. Then they befriended a boy named Pipilu. With the help of PiPilu, they co-founded an airline named Shuke Beita Airlines to help other animals. Although there are only 13 episodes in this series, the content is very compact and attractive. The animation shows the preciousness of friendship and how people should be brave when facing difficulties. Even adults recalling this animation today can still feel touched by some scenes. Secrets of the Heavenly Book Secrets of the Heavenly Book, (Chinese: 天书奇谈) also referred to as "Legend of the Sealed Book" or "Tales about the Heavenly Book", was released in 1983. The film was produced with rigorous dubbing and fluid combination of music and vivid animations. The story is based on the classic literature "Ping Yao Zhuan", meaning "The Suppression of the Demons" by Feng Menglong. Yuangong, the deacon, opened the shrine and exposed the holy book to the human world. He carved the book's contents on the stone wall of a white cloud cave in the mountains. He was then punished with guarding the book for life by the jade emperor for breaking heaven's law. In order to pass this holy book to human beings, he would have to get by the antagonist fox. The whole animation is characterized by charming Chinese painting, including pavilions, ancient architecture, rippling streams and crowded markets, which fully demonstrate the unique beauty of China's natural scenery. Pleasant Goat and Big Big Wolf【喜洋洋与灰太狼】 Pleasant Goat and Big Big Wolf (Chinese:喜羊羊与灰太狼) is a Chinese animated television series. The show is about a group of goats living on the Green Pasture, and the story revolves around a clumsy wolf who wants to eat them. It is a popular domestic animation series and has been adapted into movies. Nezha Conquers the Dragon King（Chinese: 哪吒闹海） is an outstanding animation issued by the Ministry of Culture in 1979 and is based on an episode from the Chinese mythological novel "Fengshen Yanyi". A mother gave birth to a ball of flesh shaped like a lotus bud. The father, Li Jing, chopped open the ball, and beautiful boy, Nezha, sprung out. One day, when Nezha was seven years old, he went to the nearby seashore for a swim and killed the third son of the Dragon King who was persecuting local residents. The story primarily revolves around the Dragon King's feud with Nezha over his son's death. Through bravery and wit, Nezha finally broke into the underwater palace and successfully defeated him. The film shows various kinds of attractive sceneries and the traditional culture of China, such as spectacular mountains, elegant sea waves and exquisite ancient Chinese clothes. It has received a variety of awards. Havoc in Heaven The story of Havoc in Heaven（Chinese: 大闹天宫）is based on the earliest chapters of the classic story Journey to the West. The main character is Sun Wukong, aka the Monkey King, who rebels against the Jade Emperor of heaven. The stylized animation and drums and percussion accompaniment used in this film are heavily influenced by Beijing Opera traditions. The name of the movie became a colloquialism in the Chinese language to describe someone making a mess. Regardless that it was an animated film, it still became one of the most influential films in all of Asia. Countless cartoon adaptations that followed have reused the same classic story Journey to the West, yet many consider this 1964 iteration to be the most original, fitting and memorable, The Golden Monkey Defeats a Demon【金猴降妖】 The Golden Monkey Defeats a Demon (Chinese: 金猴降妖), also referred as "The Monkey King Conquers the Demon", is adapted from chapters of the Chinese classics "Journey to the West," or "Monkey" in the Western world. The five-episode animation series tells the story of Monkey King Sun Wukong, who followed Monk Xuan Zang's trip to the West to take the Buddhistic sutra. They met a white bone evil, and the evil transformed human appearances three times to seduce the monk. Twice Monkey King recognized it and brought it down. The monk was unable to recognize the monster and expelled Sun Wukong. Xuan Zang was then captured by the monster. Fortunately Bajie, another apprentice of Xuan Zang, escaped and persuaded the Monkey King to come rescue the monk. Finally, Sun kills the evil and saves Xuan Zang. The outstanding animation has received a variety of awards, including the 6th Hundred Flowers Festival Award and the Chicago International Children's Film Festival Award in 1989. McDull【麦兜】 McDull is a cartoon pig character that was created in Hong Kong by Alice Mak and Brian Tse. Although McDull made his first appearances as a supporting character in the McMug comics, McDull has since become a central character in his own right, attracting a huge following in Hong Kong. The first McDull movie McMug Story My Life as McDull documented his life and the relationship between him and his mother.The McMug Story My Life as McDull is also being translated into French and shown in France. In this version, Mak Bing is the mother of McDull, not his father.