清华大学本科生考试试题例

清华大学本科生考试试题专用纸考试课程Biochemistry II 2001 年 6 月 11 日说明：1. 请选择正确答案，填在适当的横线处（1-30题）或写在答题纸上（31 -33题）；2. 答案可能是一个或多个；3. 每题的分数在标在了题目后面；4. 本卷满分为100.G` of the∆1. If the reaction A B is –12 kJ/mol, which of the following statements are correct? (Note the prime symbol means that a thermodynamic parameter is measured at pH.7.0) (2 points)A. The reaction will proceed spontaneously from left to right at the given conditions.B. The reaction will proceed spontaneously from right to left at stan dard conditions.C. The equilibrium constant favors the formation of B over the format ion of A.D. The equilibrium constant could be calculated if the initial concen trations of A and B were known.E. The G`o is also negative.∆value ofAnswer(s): A, D2. Which of the following statements about ATP and its roles in cells are true? (2 points)A. The ATP molecule is kinetically unstable and is thus consumed with in about one minute following its formation in cells.B. ATP provides free energy to a thermodynamically unfavorable reacti ons by group transfer, always donating a Pi to form a covalent interm ediate.C. ATP can be regenerated by coupling with a reaction that releases m ore free energy than does ATP hydrolysis.D. A transmembrane proton-motive force can drive ATP synthesis.E. The active form of ATP is usually in a complex with Mg2+.Answer(s): C, D, E3. A common moiety for NADP, NAD, FMN, FAD, and coenzyme A is: (2 poi nts)A. A pyrimidine ring;B. A three ring structure;C. An ADP;D. A pyranose ring;E. A triphosphate group.Answer(s): C4. If the C-1 carbon of glucose were labeled with 14C, which of the c arbon atoms in pyruvate would be labeled after glycolysis? (2 points)A. The carboxylate carbon;B. The carbonyl carbon;C. The methyl carbon.Answer(s): C第页/共页5. Which of the following are metabolic products of pyruvate in highe r organisms? (2 points)A. GlycerolB. Lactic acidC. AcetoneD. Acetyl-CoAE. EthanolAnswer(s): B, D6. Indicate whether each of the following statements about the pentose phosphate pathway is true (T) or false (F). (5 points)A. It generates NADH for reductive biosyntheses. FB. The reactions occur in the cytosol. TC. Transketolase and transaldolase link this pathway to gluconeogenes is. FD. It is more active in muscle cells than in fat-storage cells. FE. It interconverts trioses, tetroses, pentoses, hexoses, and heptose s. T7. Which of the following statements are correct? The citric acid cyc le (2 points)A. does not exist as such in plants and bacteria, because its functio ns are performed by the glyoxylate cycle.B. oxidizes acetyl CoA derived from fatty acid degradation.C. produces most of the CO2 in anaerobic organisms.D. provides succinyl CoA for the synthesis of carbohydrates.E. provides precursors for the synthesis of glutamic and aspartic aci ds.Answer(s): B, E8. Match the cofactors of the pyruvate dehydrogenase complex with the ir corresponding enzyme components and with their roles in the enzymatic steps that are listed. (5 points)A. Coenzyme A: 3,7B. NAD+: 2, 9C. Thiamine pyrophosphate (TPP): 1, 5D. FAD: 2, 6E. Lipoamide: 3, 4,8(1). Pyruvate dehydrogenase component(2). Dihydrolipoyl dehydrogenase(3). Dihydrolipoyl transacetylase(4). Oxidizes the hydroxylethyl group(5). Decarboxylates pyruvate(6). Oxidizes dihydrolipoamide(7). Accepts the acetyl group from acetyllipoamide(8). Provides a long, flexible arm that coveys intermediates to diffe rent enzyme component.(9). Oxidizes FADH2.9. Matching the role in fatty acid oxidation and/or mobilization to t he appropriate component listed below. (5 points)A. Bile salt: 2B. Serum albumin:5C. ApoC-II:4D. Apolipoprotein:3E. Carnitine:1(1). Acts as a “carrier” of fatty acids across the inner mitochondr ial membrane.(2). Acts as a biological detergent, disrupting fat globules into sma ll mixed micelles.(3). Binds and transports triacylglycerols, phospholipids, and choles terol between organs.(4). Activates lipoprotein lipase, which cleaves triacylglycerols int o their components.(5). Binds some fatty acids molibized from adipocytes and transports them in the blood to heart and skeletal muscle.第页/共页10. Which of the following answers complete the sentence correctly? S urplus dietary amino acids may be converted into (2 points)A. proteins.B. Fats.C. ketone bodies.D. glucose.E. a variety of biomolecules for which they are precursors.Answer(s): A, B, C, D,E11. Which of the following compounds serves as an acceptor for the am ino groups of many amino acids during metabolism? (2 points)A. GlutamineB. Asparagine.C. -ketoglutarate.D. OxaloacetateE. GlutamateAnswer(s):C12. Match the functions for the coenzymes that are involved in amino acid metabolism. (4 points)A. Pyridoxal phosphate: (3)B. Coenzyme B12: (2)C. Tetrahydrobiopterin: (1)D. NAD+: (1)E. Biotin (4)(1). Carries electrons(2). Provides free radicals(3). Carries amino groups(4). Carries CO2.13. Which of the following experimental observations would not suppor t the chemiosmotic model of oxidative phosphorylation? (3 points) A. If mitochondrial membranes are ruptured, oxidative phosphorylation cannot occur.B. Raising the pH of the fluid in the intermembrane space results in ATP synthesis in the matrix.C. Transfer of electrons through the respiratory chain results in for mation of a proton gradient across the inner mitochondrial membrane.D. The orientation of the enzyme complexes of the electron transfer c hain results in a unidirectional flow of H+.E. Radioactively labeled inorganic phosphate is incorporated into cyt osolic ATP only in the presence of an H+ gradient across the inner mi tochondrial membrane.Answer(s): B14. Some photosynthetic prokaryotes use H2S, hydrogen sulfide, instead of water as their photosynthetic hydrogen donor. How does this chan ge the ultimate products of photosynthesis? (2 points)A. Carbohydrate (CH2O) is not produced.B. H2O is not produced.C. Oxygen is not produced.D. ATP is not produced.E. The products do not change.Answer(s): C15. Which of the following are constituents of chlorophylls? (2 point s)A. Substituted tetrapyrrole.B. Plastoquinone.C. Mg2+.D. Fe2+.E. Phytol.F. Iron porphyrin.Answer(s): A, C, E第页/共页16. The observation that the incubation of photosynthetic algae with 14CO2 in the light for a very brief time (5s) led to the formation of 14C-labeled 3-phosphoglycerate suggested that the 14CO2 was condensi ng with some two-carbon acceptor. That acceptor was in fact which of the following? (2 points)A. AcetateB. Acetyl CoAC. Acetyl phosphateD. AcetaldehdydeE. Glycerol phosphateF. None of the aboveAnswer(s): F17. Which of the following are common features of the syntheses of me valonate (an intermediate of cholesterol biosynthesis) and ketone bod ies? (2 points)A. Both involve 3-hydroxyl-3-methylglutaryl CoA (HMG-CoA).B. Both require NADPH.C. Both require the HMG-CoA cleavage enzyme.D. Both occur in the mitochondria.E. Both occur in liver cells.Answer(s): A18. S-adenosylmethionine is involved directly in which of the followi ng reactions. (3 points)A. Methyl transfer to phosphatidyl ethanolamine.B. Synthesis of glycine from serine.C. Synthesis of polyamines.D. Conversion of homocysteine to methionine.E. Generation of the 5` cap of the eukaryotic mRNAs.Answer(s): A, C, E19.Which of the following does not provide a carbon skeleton for thesynthesis of amino acids? (2 points)A. succinate.-ketoglutarate B.C. Pyruvate.D. Oxaloacetate.E. Ribose-5-phosphate.Answer(s): A20. Which of the following compounds directly provide atoms to form t he purine ring? (3 points)A. Aspartate.B. Carbamoyl phosphate.C. Glutamine.D. Glycine.E. CO2.F. N5,N10-methylenetetrahydrofolate.G. N10-formyltetrahydrofolate.H. NH4+.Answer(s): A, C, D, E, G21. Which of the following statements about ribonucleotide reductase are true? (2 points)A. It converts ribonucleoside diphosphates into 2`-deoxyribonucleosid e diphosphates in humans.B. It contains coenzyme B12, which generates free radicals needed for the catalysis.C. It accepts electrons directly from FADH2.D. It receives electrons directly from either thioredoxin or glutared oxin.E. It contains two kinds of allosteric regulatory sites: one for cont rolling the overall activity and the other for controlling the substr ate specificity.Answer(s): A, D, E22. Biosynthetic pathways that require NADPH include which of the fol lowing? (2 points)A. Gluconeogenesis.B. Fatty acid biosynthesis.C. Ketone body formation.D. Cholesterol biosynthesis.E. Tyrosine biosynthesis.Answer(s): B,D,E23. Homologous recombination in E. coli is likely to require which of the following? (3 points)A. DnaB protein.B. RecA protein.C. RecBCD complex.D. ATP.E. NAD+.F. Single-strand DNA binding protein.G. DNA-dependent RNA polymerase.H. DNA polymerase I.I. DNA ligase.J. dATP.Answer(s): B, C, D, E, F, H, I, J24. Which of the following statements about E. coli promoters are cor rect? (2 points)A. They may exhibit different transcription efficiencies.B. For most genes they include variants of consensus sequences.C. They specify the start sites for transcription on the DNA template.D. They have identical and defining sequences.E. They are activated when C or G residues are substituted into their–10 regions by mutation.F. Those that have sequences that correspond closely to the consensus sequences and are separated by 17 base pairs are very efficient. Answer(s): A, B, C, F25. The AAUAAA sequence on a RNA molecule marks (2 points)A. The site where ribosomes bind to initiate polypeptide synthesis.B. The site where transcription stops.C. The site near which the primary transcript is cleaved and a poly (A) sequence is added.D. The site where the release factor will bind to end polypeptide syn thesis.E. The site where polyribonucleotide phosphorylase will add a stretch of random sequences.Answer(s): C70 subunit of the 26. The E. coli RNA polymerase: (2 points)A. acts as the catalytic site for polymerization.B. Recognize promoters.C. Has a proofreading function.D. Increases the processivity of the enzyme.E. Recognizes termination signals.Answer(s): B27. The discontinuity of eukaryotic genes were first revealed by: (2 points)A. Using footprinting techniques.B. DNA sequence comparison.C. RNA sequence analysis.D. Electron microscopic analysis of RNA-DNA hybrid molecules.E. Plant genetic studies.Answer(s): D第页/共页28. Which of the following mRNA codons can be recognized by the tRNA anticodon ICG. (2 points)A. UGC.B. CGA.C. UGA.D. CGU.E. CGC.Answer(s): B, D, E29. A new compound, vivekine, was recently discovered by a clever und ergraduate student. It was isolated from bacteria found in deep sea-d welling organisms. Vivekine inhibits protein synthesis in eukaryotes: Protein synthesis can initiate, but only dipeptides are formed and t hese remain bound to the ribosome. This toxin affects eukaryotic prot ein synthesis by blocking the: (2 points)A. binding of formylmethionyl-tRNA to ribosomes.B. activity of elongation factors.C. activation of amino acids.D. recognition of stop signals.E. formation of peptide bonds.Answer(s): B30. Indicate whether each of the following statements about prokaryot ic translation is true (T) or false (F). (7 points)A. An aminoacyl-tRNA synthetase catalyzes formation of an ester bond. ( T )B. An mRNA molecule cannot be used to direct protein synthesis until it has been completely transcribed. ( F )C. The positioning of fMet-tRNA on the A site defines the reading fra me. ( F )D. Incoming aminoacyl-tRNA are first bound to the A site. ( T )E. Formation of the 70S initiation complex requires an input of energ y. ( T )F. The carboxyl group of the amino acid on the aminoacyl-tRNA is transferred to the amino group of a peptidyl-tRNA.( F )G. Release factors cause the peptidyl transferase activity of the rib osome to use H2O as a substrate. ( T )第页/共页31. In an attempt to determine whether a given RNA was catalytically active in the cleavage of a synthetic oligonucleotide, the following experimental results were obtained. When the RNA and the oligonucleot ide were incubated together, cleavage of the oligonucleotide occurred. When either the RNA or the oligonucleotide was incubated alone, ther e was no cleavage. When the RNA was incubated with higher concentrati ons of the oligonucleotide, saturation kinetics of the Michaelis-Ment en type were observed. Do these results demonstrate that the RNA has catalytic activity? Explain. (8 points)Answer: These results alone do not establish that the RNA has catal ytic activity. A catalyst must be regenerated. It is entirely possibl e that the results observed could be accounted for by a stoichiometri c, as opposed to a catalytic, interaction between RNA and the oligonu cleotide. In which the RNA may “commit suicide” as the oligonucleot ide is cleaved. In such an interaction, a portion of the RNA would al so be cleaved itself as a part of the reaction. Four reaction product s would accumulate, two resulting from the cleavage of RNA and two fr om the cleavage of the oligonucleotide. To show that this particular RNA was catalytic, it would be necessary to demonstrate that it turns over and is regenerated in the course of the reaction.32.Translation involves conversion of the language of nucleotides to that of proteins. In the chain of events leading from a nucleotide se quence on DNA to the production of protein by ribosomes, where precis ely does the process of translation occur? Explain. (6 points)Answer: Translation involves conversion of the language of nucleotid es to that of proteins. The agent of translation is the appropriate a minoacyl-tRNA synthetase, which must recognize a particular amino aci d and link it to a tRNA containing an anticodon for that amino acid.33.Suppose that a bacterial mutant is found to replicate its DNA at a very low rate. Upon analysis, it is found to have entirely normal ac tivity of DNA polymerases I and III, DNA gyrase, and DNA ligase. It a lso makes normal amounts and kinds of dnaA, dnaB, dnaC, and SSB proteins. The oriC region of its chromosome is found to be entirely normal with respect to nucleotide sequence. What defect might account for t he abnormally low rate of DNA replication in this mutant? Explain. (6 points)Answer: A decrease in the activity of primase would account for the low rate of DNA replication. DNA replication requires the prior synth esis of RNA primers. Decreased rates of dNTP synthesis would also slo w replication.第页/共页(中国生物论坛 copyright reserved)。

清华大学2011年保送生考试试题
导读：本文清华大学2011年保送生考试试题，仅供参考，如果能帮助到您，欢迎点评和分享。

2011年12月25日，清华大学进行了保送生考试，约1000名考生参加了清华保送生选拔，其中文理比例为1：9。

清华所有参加笔试的文科生都将参加面试，内容包括英语口语测试和综合面试两部分，其中英语口语测试是今年首增的项目。

清华大学的保送生考试结果会在元旦前后公布。

保送生考试分为两部分：笔试和面试。

笔试科目分别为上午进行的阅读与写作(含中、英文，所有考生内容相同)，下午举行的数学(文理科试题不同)、自然科学(理科考生，包括物理、化学，比例约为7：3)、人文与社会(文科考生，内容涉及文学、历史、哲学、政治、经济、法律、社会地理等)。

中文写作：声音，并不都是音乐。

题材不限。

人文与社会科目：CPI、碳交易、鸳鸯蝴蝶派等15个名词解释。

论文写作：给出社会学家李强的一本著作中《社会分层与社会流动》的一个章节写论文，主题是当代中国的社会流动。

面试题的考察更灵活、宽泛。

面试题，多取材于当前的热点事件。

面试形式：清华文科保送生采取6个一组的群面形式，理科保送生3个考官对1个考生的个面形式。

为了治理交通拥堵，北京和上海采取了限制机动车数量的政策，但却采用了不同的制度设计，北京采取的是摇号购车，上海则是
对车牌号进行拍卖。

请考生对两种政策进行优劣分析。

用英语说出四大名著，并说出自己喜欢哪部名著及理由。

用英语说出“科学发展观”及“发展是硬道理”，并说出两者的区别。

第1 页/共2 页第2 页/共2 页第1 页/共2 页第2 页/共2 页清华大学本科生考试试题答案（试题A ）考试课程 传热学一、选择题: 将选择出的答案写在题前的方括号内。

(15分)1. a2. d3. d4. a5.d二、 简要回答下列问题：(35分)1. (7分)答：肋片效率为肋片的实际散热量与假设整个肋片温度都与肋根温度相同时的理想散热量之比。

-------------------------------------------------3分肋片效率的主要影响因素有：（1） 肋片材料的热导率：热导率愈大，肋片效率愈高；----------------1分（2） 肋片高度：肋片愈高，肋片效率愈低；-------------------------------1分（3） 肋片厚度：肋片愈厚，肋片效率愈高；-------------------------------1分（4） 表面传热系数：表面传热系数愈大，肋片效率愈低。

------------1分2．（7分）答：λαδ=Bi ，表示物体内部导热热阻λδ与物体表面对流换热热阻α1的比值，它和第三类边界条件有密切的联系。

--------------------------------------------------1.5分2l a Fo τ= 是非稳态导热过程中的无量纲时间，表示非稳态导热过程进行的深度。

--------------------------------------------------------------------------------------1.5分0→Bi 意味着平板的导热热阻趋于零，平板内部各点的温度在任一时刻都趋于均匀一致。

（见下图b ）----------------------------------------------------------------1.5分∞→i B 表明对流换热热阻趋于零，平板表面与流体之间的温差趋于零。

31. In an attempt to determine whether a given RNA was catalytically active in the cleavage of a synthetic oligonucleotide, the following experimental results were obtained. When the RNA and the oligonucleotide were incubated together, cleavage of the oligonucleotide occurred. When either the RNA or the oligonucleotide was incubated alone, there was no cleavage. When the RNA was incubated with higher concentrations of the oligonucleotide, saturation kinetics of the Michaelis-Menten type were observed. Do these results demonstrate that the RNA has catalytic activity? Explain. (8 points)Answer: These results alone do not establish that the RNA has catalytic activity. A catalyst must be regenerated. It is entirely possible that the results observed could be accounted for by a stoichiometric, as opposed to a catalytic, interaction between RNA and the oligonucleotide. In which the RNA may “commit suicide”as the oligonucleotide is cleaved. In such an interaction, a portion of the RNA would also be cleaved itself as a part of the reaction. Four reaction products would accumulate, two resulting from the cleavage of RNA and two from the cleavage of the oligonucleotide. To show that this particular RNA was catalytic, it would be necessary to demonstrate that it turns over and is regenerated in the course of the reaction.32.Translation involves conversion of the language of nucleotides to that of proteins. In the chain of events leading from a nucleotide sequence on DNA to the production of protein by ribosomes, where precisely does the process of translation occur? Explain. (6 points)Answer: Translation involves conversion of the language of nucleotides to that of proteins. The agent of translation is the appropriate aminoacyl-tRNA synthetase, which must recognize a particular amino acid and link it to a tRNA containing an anticodon for that amino acid.33.Suppose that a bacterial mutant is found to replicate its DNA at a very low rate. Upon analysis, it is found to have entirely normal activity of DNA polymerases I and III, DNA gyrase, and DNA ligase. It also makes normal amounts and kinds of dnaA, dnaB, dnaC, and SSB proteins. The oriC region of its chromosome is found to be entirely normal with respect to nucleotide sequence. What defect might account for the abnormally low rate of DNA replication in this mutant? Explain. (6 points)Answer: A decrease in the activity of primase would account for the low rate of DNA replication. DNA replication requires the prior synthesis of RNA primers. Decreased rates of dNTP synthesis would also slow replication.。

清华大学环境工程原理（反应工程原理部分）期末试卷（环4）清华大学本科生考试试题专用纸考试课程：环境工程原理（反应工程原理部分）环4年级2007年1月14日1、简要回答以下问题（每题6分，共24分）（1）什么是积分反应器和微分反应器？它们用于动力学实验时，各有什么优缺点？（2）反应器内的流体流动和混合状态会不会影响反应结果？为什么？（3）简要说明为什么气液相反应的宏观速率方程根据本征反应速率的快慢有不同的表达形式。

（4）在微生物反应中，细胞产率系数有哪几种表达方式？它们的取值范围是什么？2、某固定床固相催化反应器的有效体积为2m3，填充层空隙率为0.3。

反应物料以0.5m3/h的流量流入反应器。

试计算该反应器的空间时间，空间速度和实际平均接触时间。

若填充层空隙率提高到0.4，以上各参数如何变化？（10分）3、利用微生物细胞形成的颗粒状污泥处理只含有葡萄糖的废水，若不考虑处理过程中微生物的生长，仅把颗粒污泥看作固体催化剂，试回答以下问题：（15分）（1）在厌氧条件下进行该废水的处理，试分析该反应系统的微观过程。

请给出2种提高葡萄糖处理效率的途径，并分析其理论依据。

（2）在好氧条件下进行该废水的处理，试分析该反应系统的微观过程。

4、已知不可逆液相反应A →P的半衰期与A的初始浓度无关，试回答以下问题：（共30分）。

（1）给出该反应的动力学方程的一般形式。

（2）利用间歇反应器进行动力学实验（反应物料中A的初始浓度为50mol/m3），结果表明，在25oC时，A的转化率达到90％时的反应时间为2小时。

在同样的物料浓度和温度条件下，利用完全混合流反应器进行该反应时，要使A的去除率达到90％，反应物料需在反应器中平均停留多少时间？（3）该反应在有效容积为2m3的全混流连续反应器内进行，物料流量为1m3/h ，试计算A的转化率。

保持物料流量不变，将出口处物料回流（回流比为0.5），A的转化率如何变化？（4）该反应在有效容积为2m3的平推流反应器内进行，物料流量为1m3/h ，试计算A的转化率。

2010年清华大学自主招生面试题1.谈古论今──请任选中国古代和当代人物各一位进行对比阐释。

2.为什么要上大学，是否每个人都应该上大学？3.假设你是清华校长，说说明年怎么举办清华百年校庆？4.如果老子和孔子打架，你会帮谁？5.用一个成语来形容你眼中的哥本哈根气候会议。

6.用关键词概括2009年中国的现状。

7.中国是否已步入高房价时代，你的观点是什么？8.学历史与报读清华经管有什么关系？9.第一次和第二次世界大战期间，有什么重大的化学发明？10.一根火柴在不能折断的前提下，如何摆成一个三角形？11.汪洋上，只有一艘船，你只能带5个人走，你带谁？12.用成语形容一个企业家、一个政治家、一个思想家。

13.发表观点：张磊向耶鲁大学捐款8888888美元。

14.发表观点：武广高速铁路通车时速达世界第一。

15.为什么要把清华大学作为第一志愿填报？远程面试题目：1，谈古论今：任选中国古代和当代人物各一位作对比阐释。

2，为什么要上大学，是否每个人都应该上大学？3，假设你是清华校长，说说明年怎么举办清华百年校庆？【2009年面试题】●你如何看待我国四万亿救市计划？●如果你采访温总理，你将如何提问？要求：所提问题不能太大众化。

●如何看待情怀的含义。

●怎样做一名精英。

●你认为当大法官应具备怎样的素质？●谈谈对陈水扁家族弊案的看法●如何看待中学生早恋问题。

●神七发射最关键的两项技术是什么？●改革开放三十年所带来的启示和对后三十年的畅想●根据给出的数学概率中“标准分”的概念和计算公式解题。

●将区间(0，1)三等分，将中间段去掉，剩下的首尾两段重新拼接。

再重复上述做法，并无限操作下去。

请你分析以下几个数，哪些是出现在这个区间内的。

●请分析证明有理数和自然数一样多。

●请用一个成语形容当前世界的经济状况。

●第一次和第二次世界大战期间，有什么重大的化学发明？●奥运期间，北京施行的单双号车辆限行措施能否长期执行下去？●世界第一高楼是位于迪拜的迪拜塔，而上海正在建造的上海中心将成为中国第一高楼。

清华自主招生试题整理(2006--2012)2012年清华等五校自主招生试题--通用基础测试一、选择题1.若P 为ABC ∆内部任一点(不包括边界),且()(2)0PB PA PB PA PC -+-=,则ABC ∆必为( )A.直角三角形B.等边三角形C.等腰直角三角形D.等腰三角形 2.圆锥的轴截面SAB 是边长为2的等边三角形,O 为底面中心,M 为SO 的中点,动点P 在圆锥底面内(包括圆周).若MA MP ⊥,则P 点形成的轨迹的长度为( ) A.7 B.72C.3D.323.若以体积为54的正四面体的四个面的中心为顶点做一个四面体,则所作四面体的体积为( ) A.1B.2C.3D.44.某种型号的计算器上有一个特殊的按键,在计算器上显示正整数n 时按下这个按键,会等可能的将其替换为0,1,2,,1n - 中的任意一个数.如果初始时显示2011,反复按这个按键使得最终显示0,那么这个过程中,9,99,999都出现的概率是( ) A.4110 B.5110 C.6110 D.71105.已知,R αβ∈,直线1sin sin sin cos x y αβαβ+=++与1cos sin cos cos x y αβαβ+=++的交点在直线y x =-上,则cos sin c in s s o ααββ+++=( )A.0B.1C.1-D.2 6.设lg lg lg 111()121418x x xf x =+++++,则1()()f x f x +=( ) A.1 B.2 C.3 D.47.已知1cos 45θ=,则44sin cos θθ+=( )A.45B.35C.1D.45-8.若正四棱柱ABCD A B C D ''''-内接于一球,且1,'2AB AA ==,则点,A C 间的球面距离为( ) A.π4B.π2C.24π D.22π 9.若将满足,||3,||3x y x y <<<,且使得关于t 的方程33421()(3)0x y t x y t x y-+++=-没有实数根的点(,)x y 所成的集合记为M ,则由点集M 所确定的区域的面积为( ) A.814 B.834 C.815D.83510.已知椭圆22143x y +=的左,右焦点分别为12,F F ,过椭圆的右焦点作一条直线交椭圆于点,P Q ,则1F PQ∆的内切圆面积的最大值是( ) A.2516π B.925π C.1625π D.916π 二、解答题11.设2()(,)f x x bx c b c =++∈R .若||2x ≥时,()0f x ≥,且()f x 在区间(2,3]上的最大值为1,求22b c +的最大值和最小值.12.已知椭圆C :22221x y a b+=(0a b >>),其离心率为45,两准线之间的距离为252.(1)求,a b 之值；(2)设点A 坐标为(6,0),B 为椭圆C 上的动点,以A 为直角顶点,作等腰直角ABP ∆(字母,,A B P 按顺时针方向排列),求P 点的轨迹方程.13.已知数列{}n a 中的相邻两项212,k k a a -是关于x 的方程2(32)320k x k x k -++⋅=的两个根. (1)求数列{}n a 的前2n 项和2n S .(2)记1|sin |()(3)2sin n f n n =+,(2)(3)(4)()123456212(1)(1)(1)(1)f f f f n n n nT a a a a a a a a -----=++++ ,求证:15624n T ≤≤. 14.已知椭圆22221x y a b +=过定点(1,0)A ,且焦点在x 轴上,椭圆与曲线y x =的交点为,B C .现有以A 为焦点,过,B C 且开口向左的抛物线,其顶点坐标为(,0)M m ,当椭圆离心率满足2213e <<时,求实数m 的取值范围.15.已知从“神八”飞船带回的某种植物种子每粒成功发芽的概率都为13,某植物研究所进行该种子的发芽实验,每次实验种一料种子,每次实验结果相互独立.假定某次实验种子发芽则称该次实验是成功的,如果种子没有发芽,则称该次实验是失败的.若该研究所共进行四次实验,设ξ表示四次实验结束时实验成功的次数与失败的次数之差的绝对值 (1)求随机变量ξ的数学期望E ξ；(2)记“关于x 的不等式210x x ξξ-+>的解集是实数集R ”为事件A ,求事件A 发生的概率()P A .2012年清华大学保送生考试试题一、填空题1.若复数z 为虚数,且||1z =,Re ((12))1z i ⋅-=,则z =____________.2.在数列{}n a 中,11a =,12n n a a +=+.若数列11{}n n a a +的前n 项和为1837,则n =____________.3.现有6人会英语,4人会日语,2人都会(共12人),从其中选出3人做翻译,要求两种语言都有人做翻译,则符合条件的选法种数为____________.4.有一人进行投篮训练,投篮5次,失误一次扣1分,进一次得1分,连进2次得3分,连进3次得5分.若投篮的命中率为25,则投篮3次恰好得2分的概率为____________. 5.不定方程1111x y z++=()x y z ≤≤的解(,,)x y z 的组数为____________. 6.某几何体的三视图如右图所示,用,,αβγ分别表示主视图、左视图、俯视图,设,,S S S αβγ是实际几何体中能看到的面积,则,,S S S αβγ从小到大的顺序为____________.二、解答题 7.抛物线212y x =与直线l :4y x =+所围成区域中有一个矩形ABCD ,且点,A B 在抛物线上,点D 在直线l 上,其中点B 在y 轴右侧,且||2AB t =(0)t >.(1)当AB 与x 轴平行时,求矩形ABCD 面积()S t 的函数关系式； (2)当边CD 在直线l 上时,求矩形ABCD 面积的最大值. 8.已知函数3()2cos (sin 2)sin 32f x x x x =⋅+-,且[0,2]x π∈. (1)求函数()f x 的最大值和最小值； (2)求方程()3f x =的解.9.已知函数1()ln x e f x x-=,且数列{}n a 满足:11a =,1()n n a f a +=.(1)求证:10xxx e e ⋅-+≥恒成立； (2)求函数()f x 的单调区间；(3)求证:数列{}n a 单调递减,且0n a >恒成立.10.在OAB ∆内(含边界),其中O 为坐标原点,点A ,B 分别在在x 轴,y 轴的正半轴上,且2OA OB ==. (1)用方程或不等式表示OAB ∆围成的区域；(2)求证:在OAB ∆内的任意11个点,总可以分成两组,一组中各点的横坐标之和不大于6,另一组中各点的纵坐标之和不大于6.443俯视图左视图主视图γβα2011年清华等五校自主招生试题1.设*n N ∈,15n ≥.集合A ,B 都是{1,2,,}I n =⋅⋅⋅的真子集,A B =∅ ,A B I = .证明：集合A 或B 中,必有两个不同的数,它们的和为完全平方数.2.设函数2()(0)f x ax bx x a =++>,且方程()f x x =的两实数根是1x 和2x ,且10x >,211x x a->,又10t x <<.试比较()f t 与1x 的大小.3.求函数2(){|1|,|5|}f x max x x =+-的最小值,并求出相应的x 的值.4.已知()f x 是定义在R 上的不恒为0的函数,且对于任意的,a b R ∈,有()()()f a b a f b b f a ⋅=⋅+⋅. (1)求(0),(1)f f 的值；(2)判定函数()f x 的奇偶性,并证明你的结论； (3)若(2)2f =,(2)n n f u n-=,求数列{}n u 的前n 项和n S .5.已知关于x 的方程222(1)(1)ax a x +=-,1a >.证明方程的正跟比1小,负根比1-大.6.设a ,b 是两个正数,且a b <.当[,]x a b ∈时,246y x x =-+的最小值为a ,最大值为b ,求a ,b 值.7.某生产队想筑一面积为1442m 的长方形围栏,围栏一边靠墙.现有铁丝网50m ,筑成这样的围栏最少要多少铁丝网？已有的墙最多利用多长？最少利用多长？8.在正方形ABCD 中,过顶点D 作对角线CA 的平行线DE ,若CE CA =,且直线CE 交边DA 于点F .求证:AE AF =.9.设边长为,,a b c 的ABC ∆的重心为G ,外心为O ,外接圆半径为r ,||OG d =,求证:222229a b c r d ++=-. 10.设圆满足：①截y 轴所得弦长为2；②被x 轴分成两段弧,其弧长比为3:1,在满足上述条件的圆中,求圆心到直线:20l x y -=的距离最小的圆的方程.11.以A 为圆心,以2cos (0)2πθθ<<为半径的圆外有一点B . 已知2sin AB θ=,设过B 且与圆A 外切于点C 的圆的圆心为M .(1)当θ取某个值时,说明点M 的轨迹P 是什么曲线？(2)点M 是轨迹P 上的动点,点N 是圆A 上的动点,记MN 的最小值为()f θ.求()f θ的取值范围. 12.设数列{}n a 的前n 项和为n S ,点*(,)()nS n n N n∈均在函数32y x =-的图像上. (1)求数列{}n a 的通项公式； (2)设13n n n b a a +=⋅,n T 数列{}n b 的前n 项和,求最小正整数m ,使得20n mT <对所有*n N ∈都成立.13.已知函数()24f x x =-+,12()()()n nS f f f n n n=++⋅⋅⋅+.若不等式11n n n n a a S S ++<恒成立,求实数a 的取值范围.2010年清华等五校自主招生试题--通用基础测试一、选择题 1.设复数2()1a i w i+=+,其中a 为实数,若w 的实部为2,则w 的虚部为( ) (A)32- (B)12- (C)12 (D)322.设向量,a b 满足||||1a b == ,a b m ⋅= ,则||a tb + ()t R ∈的最小值为( )(A)2 (B)21m + (C)1 (D)21m - 3.无试题 4.无试题5.在ABC ∆中,若三边长,,a b c 满足3a c b +=,则tantan 22A C=( ) (A)15 (B)14 (C)12 (D)236.如图,ABC ∆的两条高线,AD BE 交于H ,其外接圆圆心为O ,过O 作OF 垂直BC 于F ,OH 与AF 相交于点G ,则OFG ∆与GAH ∆面积之比为( )（A ）1:4 (B)1:3 (C)2:5 (D)1:27.设()e (0)ax f x a =>.过点(,0)P a 且平行于y 轴的直线与曲线:()C y f x =的交点为Q ,曲线C 过点Q 的切线交x 轴于点R ,则PQR ∆的面积的最小值是( )(A)1 (B)2e2(C)e 2 (D)2e 48.设双曲线2212:(2,0)4x y C k a k a -=>>,椭圆2222:14x y C a+=.若2C 的短轴长与1C 的实轴长的比值等于2C 的离心率,则1C 在2C 的一条准线上截得线段的长为( )(A)22k + (B)2 (C)44k + (D)49.欲将正六边形的各边和各条对角线都染为n 种颜色之一,使得以正六边形的任何三个顶点作为顶点的三角形有三种不同颜色的边,并且不同的三角形使用不同的三色组合,则n 的最小值为( )(A)6 (B)7 (C)8 (D)910.设定点A B C D 、、、是以O 点为中心的正四面体的顶点,用σ表示空间以直线OA 为轴满足条件()B C σ=的旋转,用τ表示空间关于OCD 所在平面的镜面反射,设l 为过AB 中点与CD 中点的直线,用ω表示空间以l 为轴的180°旋转.设στ 表示变换的复合,先作τ,再作σ.则ω可以表示为( )(A)στστσ (B)στστστ (C)τστστ (D)στσστσ 二、解答题11.在ABC ∆中,已知22sin cos212A BC ++=,外接圆半径2R =. (1)求角C 的大小； (2)求ABC ∆面积的最大值.12.设A B C D 、、、为抛物线24x y =上不同的四点,,A D 关于该抛物线的对称轴对称,BC 平行于该抛物线在点D 处的切线l .设D 到直线AB ,直线AC 的距离分别为12,d d ,已知122d d AD +=.(1)判断ABC ∆是锐角三角形、直角三角形、钝角三角形中的哪一种三角形,并说明理由； (2)若ABC ∆的面积为240,求点A 的坐标及直线BC 的方程. 13.(1)正四棱锥的体积23V =,求正四棱锥的表面积的最小值； (2)一般地,设正n 棱锥的体积V 为定值,试给出不依赖于n 的一个充分必要条件,使得正n 棱锥的表面积取得最小值.14.假定亲本总体中三种基因型式：,,AA Aa aa 的比例为:2:u v w (0,0,0,21)u v w u v w >>>++=且数量充分多,参与交配的亲本是该总体中随机的两个. (1)求子一代中,三种基因型式的比例；(2)子二代的三种基因型式的比例与子一代的三种基因型式的比例相同吗？并说明理由. 15.设函数()1x m f x x +=+,且存在函数()s t at b ϕ==+1(,0)2t a >≠满足2121()t s f t s-+=. (1)证明：存在函数()(0),t s cs d s ψ==+>满足2121()s t f s t+-=； (2)设113,(),1,2,.n n x x f x n +=== 证明:1123n n x --≤. 2009年清华大学保送生暨自主招生北京冬令营1.有限条抛物线(线和线的内部)能够覆盖整个平面吗？证明你的结论.2.请找出一个含有323+的整系数多项式.3.求0.4 1.2|22|i i e e ++的模.4.现有一数字游戏:有1到100的数,两个人轮流写.设已经写下的数为123,,,,n a a a a .若一个数x 能表示 成112233n n x x a x a x a x a =++++ (123,,,,n x x x x 为非负整数),则这个数不能够再被写.(如若3,5已被写,则83151=⨯+⨯不能再写,133152=⨯+⨯,93350=⨯+⨯也不能再被写).现在甲和乙玩这个游戏,已知5,6已经被写,现在轮到甲写,问:谁有必胜策略？5.一条跑马比赛最多只能有八匹马参加,假设同一匹马参加每一场比赛的表现都是一样的.问：可以有不多 于50场比赛,完全将64匹马的实力顺序排序吗？6.现有100个集装箱,每个集装箱装2个物品.现在将集装箱的物品全部拆卸,并且所有物品被打乱顺序.问:最坏情况下,需要多少个集装箱再次把所有物品装好？7.现有一游戏：图上有若干个点和若干条线,甲提供若干个硬币,乙可以任意将这些硬币全部摆放在点上, 并且指定一个目标定点P .现定义操作:从一个至少有两个硬币的点取走2个硬币,在它一个相邻的点上放 回一个硬币.在指定的图下,甲最少提供多少个硬币,可以保证经过若干次操作,一定能使目标顶点P 至少 有一枚硬币？(1)图是一个包含5个点的线段；(2)图是一个包含7个点的圈.2009年清华大学自主招生数学试题(理科)1.设5151+-的整数部分为a ,小数部分为b .(1)求,a b ； (2)求222ab a b ++； (3)求2lim()n n b b b →∞+++ .2.(1)已知,x y 为实数,且1x y +=,求证:对于任意正整数n 都有222112n n n x y -+≥.(2)已知,,a b c 为正实数,求证:3a b cxy z++≥,其中,,x y z 为,,a b c 的一种排列. 3.请写出所有三个数均为质数,且公差为8的等差数列,并证明你的结论.4.已知椭圆22221x y a b+=,过椭圆左顶点(,0)A a -的直线L 与椭圆交于Q ,与y 轴交于R ,过原点与L 平行的直线与椭圆交于P ,求证:AQ ,2OP ,AR 成等比数列.5.已知sin cos 1t t +=,设cos sin s t i t =+,求2()1n f s s s s =+++ .6.随机挑选一个三位数m , (1)求m 含有因子5的概率； (2)求m 中恰有两个数码相等的概率.7.四面体ABCD 中,AB CD =,AC BD =,AD BC =, (1)求证:四面体每个面的三角形为锐角三角形；(2)设三个面与底面BCD 所成的角分别为,,αβγ,求证:cos cos cos 1αβγ++=. 8.证明:当,p q 均为奇数时,曲线222y x px q =-+与x 轴的交点横坐标为无理数.9.设1221,,,n a a a + 均为整数,性质P 为:对1221,,,n a a a + 中任意2n 个数,存在一种分法可将其分为两组,每组n 个数,使得两组所有元素的和相等,求证:1221,,,n a a a + 全部相等当且仅当1221,,,n a a a + 具有性质P .2009年清华大学自主招生数学试题(文科)1.已知数列{}n a 满足(1)n S na n n =+-, (1)求证:{}n a 是等差数列； (2)求(,)nn S a n所在的直线方程. 2.把12名职员(其中三名为男性)被平均分配到三个部门, (1)求此三名男性被分别分到不同部门的概率； (2)求此三名男性被分到同一部门的概率；(3)若有一男性被分到指定部门,求其他两人被分到其他不同部门的概率. 3.一元三次函数()f x 的三次项数为3a,()90f x x +<的解集为(1,2). (1)若()70f x a +=,求()f x 的解析式； (2)若()f x 在R 上单调增,求a 的范围. 4.已知22PM PN -=,(2,0)M -,(2,0)N ,(1)求点P 的轨迹W ； (2)直线(2)y k x =-与W 交于点A ,B ,求OAB S ∆. 5.设12nx x x a n++=, 12231()()()()()()n n n S x a x a x a x a x a x a -=--+--++-- .(1)求证:30S ≤. (2)求4S 的最值,并给出此时1x ,2x ,3x ,4x 满足的条件. (3)若50S <,求1x ,2x ,3x ,4x ,5x 不符合时的条件.2008年清华大学自主招生试题1.已知,,a b c 都是有理数,a b c ++也是有理数,证明:,,a b c 都是有理数.2.(1)一个四面体,证明:至少存在一个顶点,从其出发的三条棱组成一个三角形； (2)四面体一个顶点处的三个角分别是,,arctan 223ππ,求3π的面和arctan2的面所成的二面角.3.求正整数区间[],()m n m n <中,不能被3整除的整数之和.4.已知sin cos 1sin 2ααα+=+,求α的取值范围.5.若20lim ()(0)1,(2)()x f x f f x f x x →==-=,求()f x .6.证明:以原点为中心的面积大于4的矩形中,至少还有两个格点.2007年清华大学自主招生试题1.求函数()xe f x x=的单调区间及极值.2.设正三角形1T 的边长为a ,1n T +是n T 的中点三角形,n A 为n T 除去1n T +后剩下的三个三角形内切圆面积之和.求1lim nk n k A →∞=∑.3.已知某音响设备由五个部件组成,A 电视机,B 影碟机,C 线路,D 左声道和E 右声道,其中每个部件工作的概率如下图所示.能听到声音,当且仅当A 与B 中有一工作,C 工作,D 与E 中有一工作；且若D 和E 同时工作则有立体声效果.求:(1)能听到立体声效果的概率； (2)听不到声音的概率. 4.(1)求三直线60x y +=,12y x =,0y =所围成三角形内的整点个数； (2)求满足21260y x y x x y <⎧⎪⎪>⎨⎪+=⎪⎩的整数解个数.5.已知正三角形ABC ∆的顶点,B C 在双曲线1(0)xy x =>的一支上,且点A 的坐标为(1,1)A --. (1)求证:点,B C 关于直线y x =对称； (2)求ABC ∆的周长.6.对于集合2M R ⊆,称M 为开集,当且仅当0P M ∀∈,0r ∃>,使得20{}P R PP r M ∈<⊆.判断集合{(,)4250}x y x y +->与{(,)0,0}x y x y ≥>是否为开集,并证明你的结论. 2006年清华大学自主招生试题1.求最小正整数n ,使得11()223nI i =+为纯虚数,并求出I .2.已知a b 、为非负数,44,1M a b a b =++=,求M 的最值.3.已知sin sin cos θαθ、、为等差数列,sin sin cos θβθ、、为等比数列,求1cos2cos22αβ-的值. 4.求由正整数组成的集合S ,使S 中的元素之和等于元素之积.5.随机取多少个整数,才能有0.9以上的概率使得这些数中至少有一个偶数.6.抛物线2y x =上点P (非原点)的切线分别交,x y 轴于,Q R ,求PQ PR.7.已知函数()f x 满足:对任意的实数,a b 都有()()()f a b a f b b f a ⋅=⋅+⋅,且|()|1f x ≤,求证:()f x 恒为零.(可用以下结论:若lim ()0,()x g x f x M →∞=≤,M 为一常数,那么lim(()())0x f x g x →∞⋅=.)。

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n a n b 5 x 2+ 3x 3 x 2+ 5x(n →∞ n - ⎪l n n nn →∞清华大学2019-2020学年第一学期《高等数学》本科测试题考试课程一元微积分（B ）2020 年 10 月 25 日系名 班级姓名学号一．填空题（每空 3 分，共 15 题）（请将答案直接填写在横线上！）e tan x - e sin x1.lim x →0x - sin x= 。

2. lim sin πn →∞n 2 + n )=。

⎛ + ⎫n3. limn →∞⎝ ⎪= 。

2 ⎭n4.lim ⎛ n + ln n ⎫ln n = 。

⎝ ⎭5．当 x → 0 时， f( x ) =-x 的阶为 。

3 5 17 1 + 22n -16.已知 x n = • • • ...• 2 4 16 22 n -1 ，则lim x = 。

n →∞7. 设 x =(1+ a )(1+ a 2 )...(1+ a 2n),其中 a < 1，则lim x=。

8. 已知有整数n (n > 4)使极限 lim ⎡(x n + 7x 4+ 2)α - x ⎤ = A ≠ 0，则α=。

9.⎛ 23 -1 33 -1 43 -1 x →+∞ ⎢⎣⎥⎦ n 3 -1 ⎫ =。

lim 3 3 3... 3 ⎪ n →∞ ⎝2 +13 +14 +1 n +1 ⎭n10.lim ∑n →∞ k =1k 3 + 6k 2 +11k + 5 (k + 3)!= 。

⎛ 11. lim n →∞ ⎝ n 12 3 +12 + 22 n 3 + 22 + ...+ n 2 n 3 + n 2 ⎫ ⎪= 。

⎭ 12. lim 1!+ 2!+ ... + n != 。

n nn n!1+ x 2 2 )(α- β β 7 - 7 + 7 , 7 - 7 + 7 - 7 ,...k =113. 1 x 2+ 1- lim 2 = 。

x →0cos x - e x sin x 2n α=14. 已知limn →∞ n β- (n - 1)β =2017 ，则 。