复旦大学2020年自主招生真题

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复旦自主招生试题及答案

复旦自主招生试题及答案一、语文部分阅读理解：1. 阅读下面短文，回答问题。

（1）张謇（1854-1926）是中国现代教育家、政治家、社会活动家和思想家。

张謇曾于光绪十九年（1893年）中策划创办私立南洋公学，为中国留学生出国深造提供了机会，也为中国教育事业的发展做出了巨大贡献。

（2）南洋公学的创建者张謇，生于一般家庭，自小勤奋好学。

他上学时就常常帮助同学，与人为善，仗义疏财。

他在学习中总是能将分散的学习知识联系起来，形成系统地知识体系，且能应用灵活。

在上学期间，他还考取了举人资格。

（3）1875年，张謇考入了南洋公学，他将自己多年的积累展示了出来，并在广大师生的面前，发表了知名的《自力更生》演讲。

这个演讲使他备受赞誉，也为他未来的事业奠定了基础。

（4）张謇毕业后曾出任江西体育局局长，任内积极推行体育运动，提倡健身，改变了当时青年学生体育锻炼不足的状况。

后来，他还出任过官员、教育家等多个职位，努力改革中国教育制度，致力于提高教育的普及率和质量。

（5）张謇还通过创办南京高级工业学堂（复旦大学前身），来培养工科人才。

这一举措对中国的现代工业化进程起到了积极作用。

张謇还参与了辛亥革命，成为众多反清护国的活动家之一。

问题：（1）张謇是中国的哪位教育家和思想家？（2）张謇曾任职的机构有哪些？（3）南京高级工业学堂现在的名称是什么？答案：（1）张謇是中国现代教育家、政治家、社会活动家和思想家。

（2）张謇曾任江西体育局局长，官员和教育家等多个职位。

（3）南京高级工业学堂现在的名称是复旦大学。

二、数学部分选择题：1. 已知函数 f(x) = 2x + 3，求 f(4) 的值。

A. 7B. 9C. 11D. 132. 若 a + b = 20，且 a:b = 2:3，则 a 的值为多少？A. 8B. 10C. 12D. 16填空题：1. C = πd 的公式中，若 d = 10cm，则 C = ______ cm。

(复旦大学)自主招生与三位一体面试真题及答案分析

复旦大学自主招生与三位一体面试真题及答案分析1.你对当前房价上涨有何看法？【参考答案】现在的房子不是用来住的，是用来炒的。

真正想住房子的人住不上房子，房子只是那些商家用来赚钱的工具。

中国楼市这样下去总有一天崩盘。

投机倒把，没有任何理性，房价显示出人性最丑陋的一面，正确认识这些问题，对症下药、加大力度，遏制房价上涨势头，这既是广大群众的强烈呼声，也是关乎国民经济又好又快发展的重大问题。

还有人认为改革开放以来居民收入增加是房价上涨的首要原因，其次宽松的信贷政策是房价上涨的重要支撑，所以房价上涨也是理所当然的。

当然，每个人的年龄和他所处的环境不同，对房价上涨都会有不同的看法。

【专家点评】考查学生对时事现象的关注和思考情况，言之有理自圆其说即可。

2.金融危机对农村有何影响？【参考答案】金融危机不断向各国各个领域蔓延。

目前已经对农村和农民也有了非常直接的影响。

首先是由于出口需求的缩减，导致相关企业经营不景气，首当其冲的就是农民工失去就业机会，造成农民工返乡潮，既影响农民收入，也影响农村稳定，造成一些混乱局面；其次是以农副产品为原材料的加工业也很不景气，从而导致对农副产品需求的减少和价格回落，直接影响农民收入；第三，由于整体上预期收入和购买力增长的不协调，导致农产品价格合理上涨的势头受到抑制，甚至可能出现一定程度的价格回落；第四，由于各级财政收入可能因金融危机的影响而增幅减缓，而用于涉农的财政支出必将受到较大的影响，而直接影响农村基础设施和公共品的供给。

这些问题应当引起相关部门的重视。

换言之，应对金融危机，农村经济和农民生活是不可忽视的一个重要领域，国家要大力重视这个问题。

【专家点评】考查学生对于社会热点问题的理解。

3.如何对待文理分科？【参考答案】现在的社会是一个讲求专业人才、讲究分工协作的社会，人们主要从各自的专长和兴趣出发进行工作。

经过初中以及高一的文理科学习，相信大部分学生已经能够发现自己的能力和兴趣所在，此时进行文理分科有利于各类专才的早日培养，因为到了高中阶段学生的学习特长、兴趣倾向已经呈现，要根据自身的特长和今后的发展方向，自主选择侧重文科还是理科，有利于他们集中精力，为今后的发展奠定基础。

2020年复旦大学强基计划数学试题及参考解答(回忆版)

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满足 ∠AOB = 2 arctan 1 的概率为
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要 tan ∠COB = 1 , 利用到角公式可得 1 = 1 − k , 解出 k = 1 . 点 (1, 2), (2, 4), (3, 6), (4, 8)
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(A) m 和 n 可能重合
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练习:
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考虑 f (x) = x3 + sin x, 显然 f (x) 为奇函数, 且当 x > 0 时, 求导后容易验证 f (x) 为单调增函数, 又 f (x) + f (2y) = 0, 则 x + 2y = 0, cos(x + 2y) = 1.

2020年复旦大学强基计划数学试题及其详解

２０２０年复旦大学强基计划数学试题及其详解甘志国(北京市丰台二中㊀１０００７１)摘㊀要:２０２０年复旦大学强基计划数学试题共计３３道ꎬ全部是单项选择题.本文中的试题均是由参加考试的学生回忆得出的ꎬ因而回忆出的题目可能不准确(没有回忆出选项的题目均改成了填空题)ꎬ题号也不准确.㊀关键词:复旦大学ꎻ强基计划ꎻ数学试题ꎻ详解ꎻ华清园教育ꎻ整理中图分类号:Ｇ６３２㊀㊀㊀㊀㊀㊀文献标识码:Ａ㊀㊀㊀㊀㊀㊀文章编号:１００８－０３３３(２０２０)２８－００５１－０３收稿日期:２０２０－０７－０５作者简介:甘志国(１９７１－)ꎬ湖北省竹溪人ꎬ研究生ꎬ正高级教师ꎬ特级教师ꎬ从事高中数学教学研究.基金项目:本文系北京市教育学会十三五教育科研滚动立项课题数学文化与高考研究 (课题编号ＦＴ２０１７ＧＤ００３ꎬ课题负责人:甘志国)阶段性研究成果之一.㊀㊀试题及其解答是笔者由华清园教育(ｈｔｔｐ://ｇｋ.ｑｈｙｅｄｕ.ｃｏｍ/ｑｈｙ/２０２００７１４/９８１８５.ｈｔｍｌ)等公开的内容整理而成的.１.已知直线ｍ:ｘｃｏｓα－ｙ＝０和直线ｎ:３ｘ＋ｙ－ｃ＝０ꎬ则(㊀㊀).Ａ.ｍ和ｎ可能重合Ｂ.ｍ和ｎ不可能垂直Ｃ.直线ｍ上存在点Ｐꎬ使得直线ｎ绕点Ｐ旋转后与直线ｍ重合Ｄ.以上都不对２.Ｇｉｖｅｎ㊀ｔｗｏ㊀ｓｅｔｓ㊀Ａ:１ꎬ２ꎬ３ꎬ４ꎬ５{}㊀ａｎｄ㊀Ｂ:３ꎬ４ꎬ５ꎬ６ꎬ７{}ꎬｔｈｅｎ㊀ｔｈｅ㊀ｉｎｔｅｒｓｅｃｔｉｏｎ㊀ｓｅｔ㊀ｏｆ㊀Ａ㊀ａｎｄ㊀Ｂ㊀ｉｓ(㊀㊀).Ａ.{１ꎬ２}㊀㊀㊀㊀㊀㊀Ｂ.{３ꎬ４ꎬ５}Ｃ.{１ꎬ２ꎬ３ꎬ４ꎬ５ꎬ６ꎬ７}㊀㊀Ｄ.{６ꎬ７}３.若实数ｘꎬｙ满足ｘ２＋２ｘｙ－１＝０ꎬ则ｘ２＋ｙ２最小值为.４.若点Ｏ是әＡＢＣ的内心ꎬｃｏｓøＢＡＣ＝１３ꎬ且满足ＡＯң＝ｘＡＢң＋ｙＡＣңꎬ则ｘ＋ｙ的最大值是.Ａ.３＋３２㊀㊀Ｂ.３－３２㊀㊀Ｃ.３４㊀㊀Ｄ.４３５.Ｗｈｉｃｈ㊀ｎｕｍｂｅｒ㊀ｔｈａｔ㊀ｎｕｍｂｅｒ㊀５㊀ｉｓ㊀ｔｈｅ㊀ｃｕｂｉｃ㊀ｒｏｏｔ㊀ｏｆ?Ａ.３㊀㊀Ｂ.５㊀㊀Ｃ.２５㊀㊀Ｄ.１２５６.在抛物线ｙ２＝２ｐｘ(ｐ>０)中ꎬ过焦点Ｆ作直线交抛物线于两点ＡꎬＢꎬ且有ＡＦң＝３ＦＢң.再过点Ａ作抛物线准线的垂线ꎬ垂足为Ａᶄꎬ准线与ｘ轴交于点Ｃ.若四边形ＣＦＡＡᶄ的面积是１２３ꎬ则ｐ＝.７.已知抛物线ｘ＝３ｙ２的焦点为Ｆꎬ若该抛物线在点Ａ处的切线与直线ＡＦ的夹角为３０ʎꎬ则点Ａ的横坐标为.Ａ.１９㊀㊀Ｂ.１３６㊀㊀Ｃ.１４㊀㊀Ｄ.１１６８.已知点Ｐ在直线ｘｙ－６－１４＝０上ꎬ且点Ｐ到点Ａ２ꎬ５()和点Ｂ(４ꎬ３)的距离相等ꎬ则点Ｐ的坐标为.９.已知两点Ａ(ｘꎬｙ)ꎬＢ(ｙꎬｘ)ꎬ其中ｘꎬｙɪ{１ꎬ２ꎬ３ꎬ４ꎬ５ꎬ６ꎬ７ꎬ８}且ｘʂｙꎬ连结ＯＡꎬＯＢ(其中Ｏ是坐标原点)ꎬ则øＡＯＢ＝２ａｒｃｔａｎ１３的概率为.１０.ａｒｃｓｉｎ１４＋３２８＋ａｒｃｓｉｎ３４＝.Ａ.π３㊀㊀Ｂ.π２㊀㊀Ｃ.２π３㊀㊀Ｄ.３π４１１.已知三棱锥Ｐ－ＡＢＣ的体积为２１２ꎬ且ＡＢ＝６ꎬＡＣ＝ＢＣ＝４ꎬＡＰ＝ＢＰ＝１０ꎬ则ＣＰ＝.１２.在әＡＢＣ中ꎬ已知ＡＢ＝９ꎬＢＣ＝６ꎬＣＡ＝７ꎬ则ＢＣ边上中线长为.１３.已知ｆｘ()＝ａｓｉｎ(２πｘ)＋ｂｃｏｓ(２πｘ)＋ｃｓｉｎ(４πｘ)１５＋ｄｃｏｓ(４πｘ)ꎬ若ｆ１２＋ｘæèçöø÷＋ｆｘ()＝ｆ２ｘ()ꎬ则在ａꎬｂꎬｃꎬｄ中能确定的参数是.１４.若关于ｘ的实系数一元三次方程ｘ３＋ａｘ２＋４ｘ＋５＝０有一个根是纯虚数ꎬ则ａ＝.１５.ｘ２＋１ｘ＋ｙ３＋１ｙæèçöø÷１０的展开式中的常数项为１６.ｌｉｍｎң¥１１ˑ４＋１２ˑ５＋＋１ｎｎ＋３()[]＝１７.已知ｘꎬｙɪ[－π４ꎬπ４]ꎬ若ｘ２＋ｃｏｓ(ｘ＋３π２)－２ａ＝０ꎬ４ｙ２＋ｓｉｎｙｃｏｓｙ＋ａ＝０ꎬ{则ｃｏｓ(ｘ＋２ｙ)的值是.Ａ.０㊀㊀Ｂ.１㊀㊀Ｃ.－１㊀㊀Ｄ.与ａ有关图１１８.如图１所示ꎬ在凸四边形ＡＢＣＤ中ꎬøＢＡＣ＝øＢＤＣ是øＤＡＣ＝øＤＢＣ的.Ａ.充分不必要条件Ｂ.必要不充分条件Ｃ.充要条件Ｄ.既不充分也不必要条件１９.如图２所示ꎬ平面内两条直线ｌ１ꎬｌ２交于点ＯꎬＭ为该平面内的任意一点.若点Ｍ到直线ｌ１ꎬｌ２的距离分别为ｐꎬｑꎬ则称(ｐꎬｑ)是点Ｍ的距离坐标 .ｐꎬｑ是已知的非负常数ꎬ给出下列三个结论:图２(１)若ｐ＝ｑ＝０ꎬ则距离坐标为(０ꎬ０)的点有且仅有１个ꎻ(２)若ｐｑ＝０ꎬ且ｐ＋ｑʂ０ꎬ则距离坐标为ｐꎬｑ()的点有且仅有２个ꎻ(３)若ｐｑʂ０ꎬ则距离坐标为ｐꎬｑ()的点有且仅有４个.其中正确结论的个数是.Ａ.０㊀㊀Ｂ.１㊀㊀Ｃ.２㊀㊀Ｄ.３２０.若函数ｆ(ｘ)＝３ｘ－３－ｘ的反函数为ｙ＝ｆ－１(ｘ)ꎬ则ｇ(ｘ)＝ｆ－１(ｘ－１)＋１在[－３ꎬ５]上的最大值和最小值的和为.Ａ.０㊀㊀Ｂ.１㊀㊀Ｃ.２㊀㊀Ｄ.４２１.若ｋ>４ꎬ则直线ｋｘ－２ｙ－２ｋ＋８＝０与２ｘ＋ｋ２ｙ－４ｋ２－４＝０与两条坐标轴围成的四边形面积的取值范围是.２２.如图３所示ꎬ已知ＡꎬＢꎬＣꎬＤ四点共圆ꎬ且ＡＢ＝１ꎬ图３ＣＤ＝２ꎬＡＤ＝４ꎬＢＣ＝５ꎬ则ＰＡ＝.Ａ.１３３㊀㊀Ｂ.１４３㊀㊀Ｃ.５㊀㊀Ｄ.１６３２３.已知向量数列ａｎң{}满足ａｎ＋１ң＝ａｎң＋ｄң(ｎɪＮ∗)ꎬ且ａ１ң＝３ꎬａ１ңｄң＝－３２.若Ｓｎ＝ａ１ңðｎｉ＝１ａｉңꎬ则当Ｓｎ取最大值时ꎬｎ＝.Ａ.８㊀㊀Ｂ.７㊀㊀Ｃ.６㊀㊀Ｄ.６或７２４.给定５个函数ꎬ其中３个是奇函数但不是偶函数ꎬ２个是偶函数但不是奇函数ꎬ则在这５个函数中任意取３个ꎬ其中既有奇函数又有偶函数的概率为.２５.方程５ρｃｏｓθ＝４ρ＋３ρｃｏｓ２θ所表示的曲线形状是.２６.在平面直角坐标系ｘＯｙ中ꎬ把点(４ꎬ５)绕点(１ꎬ１)顺时针旋转６０ʎꎬ所得的点的坐标为.２７.已知实数ｘꎬｙ满足ｘ２＋ｙ２＝１ꎬ若ｘ＋２ｙ－ａ＋ａ＋６－ｘ－２ｙ的值与ｘꎬｙ的取值无关ꎬ则ａ的取值范围是.２８.某公司安排甲㊁乙㊁丙等７人完成除夕到大年初六共７天的值班任务ꎬ每人值班一天.已知甲不值第一图４天ꎬ乙不值第二天ꎬ甲和丙在相邻两天值班ꎬ则不同的安排方式共有种.２９.若函数ｆ(ｘ)的图象如图４所示ꎬ则函数ｆ(ｆ(ｘ))的图象大致为.２９.解法１㊀Ｂ.设ｇ(ｘ)＝ｆｆｘ()().由ｆ(ｘ)的图象关于ｙ轴对称不关于坐标原点对称ꎬ可得ｆ(ｘ)是偶函数不是奇函数ꎬ所以可得ｇ(ｘ)也是偶函数不是奇函数ꎬ从而可排除选项ＡꎬＤ.可得ｇ(１)＝ｆｆ１()()＝ｆ(０)＝－１<０ꎬ可排除选项Ｃ.３０.定义ｆＭ(ｘ)＝１ꎬｘɪＭꎬ－１ꎬｘ∉Ｍꎬ{ＭＮ＝ｘ｜ｆＭ(ｘ)ｆＮ(ｘ)＝－１{}ꎬ已知集合Ａ＝ｘ｜ｘ<２－ｘ{}ꎬＢ＝２５ｘ｜ｘ(ｘ＋３)(ｘ－３)>０{}ꎬ则ＡＢ＝.３１.方程３ｘ＋４ｙ＋１２ｚ＝２０２０的自然数解的组数为.Ａ.Ｃ２１６８㊀㊀Ｂ.Ｃ２１６９㊀㊀Ｃ.Ｃ２１７０㊀㊀Ｄ.Ｃ２１７１３２.已知ｍꎬｎɪＺꎬ且０ɤｎɤ１１.若２２０２０＋３２０２１＝１２ｍ＋ｎꎬ则ｎ＝.Ａ.４㊀㊀Ｂ.７㊀㊀Ｃ.１０㊀㊀Ｄ.１３３３.下列不等式恒成立的是(㊀㊀).Ａ.ｘ２＋１ｘ２ȡｘ＋１ｘ㊀㊀㊀Ｂ.｜ｘ－ｙ｜＋１ｘ－ｙȡ２Ｃ.｜ｘ－ｙ｜－１ｘ－ｙȡ２Ｄ.｜ｘ－ｙ｜ȡ｜ｘ－ｚ｜＋｜ｙ－ｚ｜参考答案１.Ｃ㊀２.Ｂ㊀３.㊀５－１２㊀４.Ｂ㊀５.Ｂ㊀６.２２㊀７.Ｃ８.(１ꎬ２)㊀９.１９㊀１０.ｄ㊀１１.９８ʃ７４３㊀１２.２１４㊀１３.ａꎬｂꎬｃꎬｄ㊀１４.５４㊀１５.１２６００㊀１６.１１１８㊀１７.Ｂ㊀１８.Ｃ㊀１９.Ｄ㊀２０.Ｃ㊀２１.(１７４ꎬ８)㊀２２.Ｂ㊀２３.Ｄ㊀２４.９１０㊀２６.(５２＋２３ꎬ３－３２３)㊀２７.[５－６ꎬ－５]㊀２８.１１２８㊀２９.Ｂ㊀３０.(－ɕꎬ－３]ɣ[０ꎬ１)ɣ(３ꎬ＋ɕ)㊀３１.Ｃ㊀３２.Ｂ㊀３３.Ａ㊀㊀参考文献:[责任编辑:李㊀璟]函数与不等式齐驱并驾㊀多角度解决最值问题２０２０年全国Ⅱ卷第２１题一题多解探讨张培杰(云南省大理大学教师教育学院㊀６７１０００)摘㊀要:最值问题能考查学生推理㊁转换㊁归纳等综合数学能力ꎬ每年高考都会出现.在高中数学教学中ꎬ最值问题的有两个主要的解决策略ꎬ一是转换成函数ꎬ利用函数性质求解ꎬ二是利用不等式求解.２０２０年全国Ⅱ卷第２１题第(２)问是典型的最值问题ꎬ本文分别从函数性质和不等式的角度给出不同的解答ꎬ以总结出一般的思路步骤ꎬ供复习参考.关键词:最值问题ꎻ函数ꎻ不等式ꎻ一题多解中图分类号:Ｇ６３２㊀㊀㊀㊀㊀㊀文献标识码:Ａ㊀㊀㊀㊀㊀㊀文章编号:１００８－０３３３(２０２０)２８－００５３－０３收稿日期:２０２０－０７－０５作者简介:张培杰(１９９３.９－)ꎬ男ꎬ研究生ꎬ从事数学教学研究.㊀㊀一㊁真题再现(２０２０年全国Ⅱ卷第２１题)已知函数ｆ(ｘ)＝ｓｉｎ２ｘｓｉｎ２ｘ.(１)讨论ｆ(ｘ)在(０ꎬπ)的单调性ꎻ(２)证明:ｆ(ｘ)ɤ３３８ꎻ(３)证明:ｓｉｎ２ｘｓｉｎ２２ｘｓｉｎ２４ｘｓｉｎ２２ｎｘɤ３ｎ４ｎ.通过观察题目发现ꎬ该题以三角函数为背景ꎬ考查判断函数在区间内的单调性㊁求函数值域㊁不等式证明等多个知识点.题目综合性强ꎬ难度较大ꎬ对考生的逻辑推理能力和运算能力有较高的要求ꎬ很好地体现了课程标准要求的核心素养导向ꎬ具有高考命题需要的区分度.下面重点给出第(２)问的一题多解ꎬ对于第(１)㊁(３)问仅给出一种可行的解答.３５。

上海复旦附中自主招生英语试题(3)含答案(20200802193610).pdf

上海复旦附中自主招生英语试题(3)1.A.increasinglyB.loweringmonD.solutionsE.individuallyF. callingG.dependent H.shortcomings I.consumption J.introduced K.sensesA United Nations panel on climate change says global warming is causing extreme weather all over the world and is__41__on governments to come up with new ideas to curb gas emissions. Since much of the air pollution comes from coal-fired power plants,scientists in Denmark areenergy needs with new types of smart streetlights.trying to lower their nation’ｓOur__42__industrialized world needs huge amounts of electrical energy,but our powerplants, plants still rely mostly on technology__43__as early as the19th century—coal—powered hydroelectric dams or somewhat never nuclear energy,all of which have their own__44__.But scientists say we can slow down global warming by__45__energy demands for street lights.urban areasrequire a lot of electricity.These__46__features of the world’ｓThe U.S.Energy Information Administration estimates that in2012,lighting for buildings, streets and highways in the ed about274billion kilowatt hours.first carbon-neutral city by2025,scientists in the In a bid to make Copenhagen the world’ｓDanish capital are testing a variety of new street lighting technologies.Kim Brostrom is the chief technical officer at the Danish Outdoor Lighting Lab.“We h ave installed nine kilometers of streets,we have280masts placed here,we have50 different__47__,we have10different management systems,and we have a lot of differentsaid Brostrom.sensorsand things out in the open area,”Chief science officer Jakob Andersen says the lights can be managed__48__from a tablet computer or a smart phone.The main goal is to lower energy__49__when the light is not needed.So the lamp brightens up only when it__50__an approaching pedestrian,cyclist or vehicle.Some of them even have a backup wind generator or a solar cell.2.Diet Coke,diet Pepsi,diet pills,on-fat diet,vegetable diet…We are__51__by the word everywhere we look and listen.We have so easily been attracted by the promise and__52__“diet”of diet products that we have stopped thinking about what diet products are doing to us.We are paying for products that__53__us psychologically and physically(身体上).Diet products significantly weaken us psychologically.On one level,we are not allowing our brain to__54__that our weight problems lie not in actually losing the weight,but in controlling the__55__of fatty,high-calorie,unhealthy foods.Diet products allow us to jump over the thinking stage and go straight for the scale(秤)instead.All we have to do is to swallow or__56__ in food labels.the word“diet”On another level,diet products have greater psychological effects.Every time we have azero-calorie drink,we are telling ourselves without our__57__that we don’ｔhave to work to get results.Diet products make people believe that gain comes without pain,and that life can be without resistance a(n)__58__.The danger of diet products lies not only in the psychological effects they have on us,but also in the physical harm that they cause.Diet foods can__59__harm our bodies because consuming them instead of healthy foods means we are preventing our bodies from having__60__ nutrients(营养成分).Diet foods and diet pills__61__zero calorie only becausethe diet industry has created chemicals to produce these wonder products.Diet products may not be__62__,and the chemical that go into diet products are potentially dangerous.Now that we are aware of the effects that diet products have on us,it is time to__63__think about buying them.Losing weight lies in the power of minds,not in the power of chemicals. __64__we realize this,we will be much better able to resist diet products,and__65__prevent the psychological harm that comes from using them.51. A.confused B.surrounded C.mixed D.fascinated52. A.potential B.affection C.obligation D.calculation53. A.weaken B.harm C.enforce D.inspire54. A.admire B.argue C.deny D.admit55. A.assumption B.digestion C.dissolution D.consumption56. A.recognize B.foresee C.overlook D.research57. A.frustration B.ignorance C.awareness D.emphasis58. A.maintenance B.efficiency C.struggle D.setback59. A.indirectly B.inevitably C.incredibly D.innocently60. A.extra B.basic anic D.delicate61. A.contain B.persist C.exclude D.melt62. A.nutritional B.poisonous C.ridiculous D.sensitive63. A.strongly B.enthusiastically C.eagerly D.seriously64. A.Unless B.While C.Though D.Once65. A.therefore B.otherwise C.meanwhile D.furthermore3.beard was It was a bitter cold evening in northern Virginia many years ago.The old man’ｓfrost while he waited for a ride across the river.The wait seemed endless.His glazed by winter’ｓbody became numb and stiff from the frigid north wind.It was long before he heard the faint,steady rhythm of hooves(马蹄)along the frozen path. Anxiously watching several horsemen coming up,he let the first one passby without any effort to get his attention,then another passed by,and another.Finally,the last rider drew near where the old man sat like a snow statue when old man caught his eyes and said,“Sir,would you mind giving an old man a ride to the other side?”Seeing the old man unable to lift Stopping his horse,the rider replied,“Sure!Jump aboard.”his half-frozen body from the ground,the horseman got down to help him.The horseman took him not just across the river,but to his destination just a few miles away.curiosity causedhim to inquire, As they approached the tiny but cozy cottage,the horseman’ｓ“Sir,I noticed you let several other riders passby without making an effort to secure a ride.Then I came up and you immediately asked me for a ride.I am curious why,on such a bitter night,you would wait and ask the last rider.What if I had refused and left you here?”The old man looked the rider straight in the eyes and replied,“Ｉreckon I know people pretty good.I looked into the eyes of the other riders and immediately found no concern for my situation. It would have been useless even to ask them for a ride.But when I looked into your eyes,your gentle spirit would welcome the opportunity to give me assistancein my time of need.”so grateful for what you Those heart-warming comments touched the horseman deeply.“I’ｍhe told the old man.“May I never get too busy in my own affairs that I fail to respond have said,”to the needs of others with kindness and compassion.”With that,Thomas Jefferson turned his horse around and made his way back to the White House.the old man ask the previous riders for a ride?66.Why didn’ｔA.Because he was waiting for the last rider.B.Because his efforts failed to get their attention.C.Because he assumed they had no concern for him.D.Because he was too numb to lift his half-frozen body.67.What caused the last rider to stop and help the old man?warm comments.poor situation. B.The old man’ｓA.The old man’ｓC.His own affairs.D.Kindness and sympathy.68.What do you know about the old man?A.He judged people by looking into their eyes.B.He lived in a cozy cottage a few miles away.C.He gave the last rider an opportunity to help.D.He recognized the president at first.69.Which can be the best title of this text?A.Look into your eyes.passion of a great man.C.The last rider to help.D.Return to the White House.4.International Book ClubHave you ever thought of joining a book club and buying new books through the post?Here at the International Book Club,we already have many members buying books from us by mail. Immediate benefits：As a special offer,you may choose any reduced price books from our new members’book list, to the value of$6in total(plus postage and packing).By doing this,you will save pounds on prices.the publishers’Tick the box on your form to order a free watch.If you reply within seven days,we will send you another free gift carefully chosen from our book list by our staff.Order aDVD from the many on offer in our list,at half the recommended retail price.joined：When you’veprice on As a member,you’ll enjoy savings of between30%and50%off the publisher’ｓevery book you buy,and what’ｓcome straight to your door.Your free club magazinemore,they’llarrives once a month,to keep you up to date with the latest bestsellers.This means that every year we offer over1,000books to choose from.On the Internet,you can find all our titles for the yearwebsite.on our exclusive members’Being a member：All we are asking you to do while you are a member is to choose four books during your first year.After that,you can decide on the number of books you wish to take.In each of our monthly club magazines,our experienced staff choose a Club Choice book—a work of fiction or a reference title which they feel is particularly worth buying,and which is offered at an extra special price.However,if you do not want this book,just say so in the space provided on the form.We will always send the book if we do not receive this.not every day we can make you an So,return your application form today,but hurry—it’ｓoffer like this.To apply to become a member,all you need to do is simply fill in the enclosed form and return it in the postage paid envelope supplied.send any money now,as we Before you know it,your books will be with you.Please don’ｔwill send you your bill with the books.And remember,you have up to a fortnight to decide if you wish to keep the books you have ordered.You should then either return the books or send your payment.belong to the benefits of being a club member?70.Which of the following doesn’ｔA.Get four books for free in the first year.B.Order a free watch.C.Get a gift for nothing.D.Order a DVD at a low price.71.Every month the club provides a free club magazine in order to_______.A.let readers know the number of books to be soldB.keep readers well informed of recent bestselling booksdevelopmentC.make readers know the club’ｓD.attract more and more readers to join the club72.If a book is worth$100,how much will a member probably save if he buys one?A.$20B.$40C.$60D.$7073.What should you do if you want to join the book club?A.Just fill in a form with personal information and send the club an email.B.Ask other club members to recommend you to the club.C.Fill in an application form with neededinformation and return it to the club.D.Apply for a membership card.5.In1789the ernment passeda law which said that the land of the American Indians could never be taken from them without their agreement.One hundred years later,however,the Indians only had a very small part of the land that originally belonged to them.How did this great injustice occur?After1812white settlers began to move west across North America.At first,the settlers and the Indians lived in peace.However,the number of settlers increased greatly every year,and slowly the Indians began to see the white settlers as a danger to their survival.To feed themselves,the settlers killed more and more wild animals.The Indians,who depended on these animals for food,had to struggle against starvation.The settlers also brought with them many diseaseswhich were common in white society,but which were new for the Indians.Great numbers of Indians became sick and died.Between1843and1854the Indian population in one area of the country went down from100,000to30,000.More land was needed for the increasing number of white settlers.In Washington,the old respect for the rights of the Indians disappeared.The old promises to the Indians were broken;the government began to move groups of Indians from their original homelands to other poorer parts of the country.Some Indians reacted angrily and violently to this treatment.They began to attack white settlers,and the Indian war began.For30years,until the late1880s,different groups of Indians fought against the injustices of the white man.They had a few famous successes,but the result of the struggle was never in doubt.There were too many white soldiers,and they were too powerful.Many Indians were killed;the survivors were moved from their homelands to different area of the country.It was a terrible chapter in the history of a country that promised freedom and equality to everyone.74.It can be inferred from the passagethat______.A.in the U.S.there were many laws that provided the rights of American IndiansB.the law which was passed in1789by the ernment was not successfully carried outC.in the19th century no injustices were done against the Indians by the ernmentD.the majority of white settlers were openly opposed to the law passedin178975.According to the passagewhich of the following is TRUE?A.The Indians believed that killing too many wild animals had disturbed the balance of nature.B.The government began to have a better understanding of the Indians in the fifties of the 19th century.C.Between1843and1854about70,000Indians were killed in the battle.D.The whites carried serious diseasesinto where the Indians lived.76.It is implied in the passagethat______.A.the Indians had many great successes in the Indians warsB.the Indians had no doubt that they would win the warsC.after the war the Indians stayed where they were beforeD.the Indians were too weak to win the struggle77.What is the writer’ｓopinion about the treatment that the Indians received from the U.S. government?A.He believed that the government always respected the rights of the Indians.be criticized for its treatment to the Indians.B.He believed that the government can’ｔC.He believed that the government treated the Indians unjustly.unfair treatment against the Indians was not onD.He believed that the government’ｓpurpose.6.The question of whether our government should promote science and technology or thean either/or proposition(命题),although the current emphasis liberal arts in higher education isn’ｔon preparing young Americans for STEM(science,technology,engineering,maths)-related fieldscan make it seemthat way.The latest congressional report acknowledges the critical importance of technical training,butalso asserts that the study of the humanities(人文学科)and social sciences must remain centraleducational system at all levels.Both are critical to producing citizens components of America’ｓwho can participate effectively in our democratic society,become innovative(创新的)leaders,and benefit from the spiritual enrichment that the reflection on the great ideas of mankind over time provides.Parents and students who have invested heavily in higher education worry about graduates’job prospects as technological advances and changes in domestic and global markets transformnatural to look professions in ways that reduce wages and cut jobs.Under these circumstances,it’ｓway out of the problem“Major in a subjectfor what may appear to be the most“practical”designed to get you a job”seems the obvious answer to some,though this ignores the fact thatoften,in fact,lead to employment andmany disciplines in the humanities characterized as“soft”success in the long run.Indeed,according to surveys,employers have expressed a preference for students who have received a broadly-based education that has taught them to write well,think critically,research creatively,and communicate easily.Moreover,students should be prepared not just for their first job,but for their4th and5th jobs, little reason to doubt that people entering the workforce today will be called upon toas there’ｓplay many different roles over the course of their careers.The ones who will do the best in thisnew environment will be those whose educations have prepared them to be flexible.The ability toup from science,arts,and technology—to draw upon every available tool and insight—pickedsolve the problems of the future,and take advantage of the opportunities that present themselves,will be helpful to them and the United States.(Note:Answer the questions or complete the statements in NO MORE THAN TWELVE WORDS.)78.The latest congressional report suggeststhat________________________________.79.What is the main concern of students when they choosea major?80.What kind of job applicants do employers look for?81.What suggestions does the author give to college students?7.In order to study enzymes(酶),a small piece of potato is cut into very thin pieces.These thin pieces are placed in a little glass container and covered with water.Then a little of the proper chemical is added.Immediately,because of the enzyme present in the potato,the chemical starts joining with the oxygen in the air above the water.If the container is entirely closed,a partial vacuum(真空)is framed inside.If the container is now connected with a fine tube which is dippedin water,the water is sucked part way up the tube becauseof the vacuum.The scientists doing the experiment carefully notice how far up the tube is when water is sucked in a certain time.The higher it is sucked,the more of that particular enzyme and evenabout the way in which it operates.And all the time we are working with amounts far too small to see or weigh.1.What do the scientists want to find out in the experiment?a.Whether enzyme is present in the potato.b.What particular enzyme there is in the potato.c.How much enzyme there is in the potato.d.How enzyme works in the potato.A.a and bB.c and dC.b and dD.a and c2.The proper title for the passageis______.A.How a Vacuum Is FormedB.How Enzyme Is StudiedC.How Enzyme Works in a PotatoD.How Much Enzyme There Is in the Potato3.Which of the following correctly shows how the experiment is done?A.Put very thin pieces or potato in a vacuum container.B.Connect the glass container with a tube through which oxygen is sent in.C.Add some chemical to the water before thin pieces of potato are put in.D.Place thin pieces of potato in a glass container,cover them with water and then close it.4.The result of the study shows that______.A.the higher water is sucked,the less of enzyme in the potato isB.the more water is put into,the more of enzyme in the potato isC.the higher water is sucked,the more of enzyme in the potato isD.the lower water is sucked,the more of enzyme in the potato is上海复旦附中自主招生英语试题(3)1.FAJHB CDEIK2.BABDD ACCAB AADDA3.CDAB4.ABBC5.BDDC6.78.The humanities and STEM subjects should be given equal importance.79.Their chances of getting a good job.80.Those who have received a well-rounded education.81.Preparing themselves to be flexible;achieving every available tool and insight.7.BBDC。

复旦大学自主招生考试数学试题及答案

1、设函数y=f（x）=e x+1，则反函数OyxOyxO x答案：A2、设f（x）是区间[a，b]f（x）是[a，b]上的递增函数，那么，f（xA．存在满足x<y的x，y∈[a，b]B．不存在x，y∈[a，b]满足x<y且fC．对任意满足x<y的x，y∈[a，b]D．存在满足x<y的x，y∈[a，b]答案：A3、设]2,2[,ππβα-∈，且满足sinαA． [−2，2] B． [答案：D4、设实数0,≥yx，且满足2=+yxA．97/8 B．答案：C5则该多面体的体积为______________。

A．2/3 B．3/4答案：D6、在一个底面半径为1/2，高为1的圆柱内放入一个直径为1的实心球后，在圆柱内空余的地方放入和实心球、侧面以及两个底面之一都相切的小球，最多可以放入这样的小球个数是___________。

A ．32个；B ．30个；C ．28个；D ．26个答案：B7、给定平面向量（1，1），那么，平面向量（231-，231+）是将向量（1，1）经过________． A ．顺时针旋转60°所得； B ．顺时针旋转120°所得； C ．逆时针旋转60°所得；D ．逆时针旋转120°所得；答案：C8、在直角坐标系O xy 中已知点A 1（1，0），A 2（1/2，3/2），A 4（−1，0），A 5（−1/2，−3/2）和A6（1/2， −3/2）．问在向量−−→−ji A A （i ，j=1，2，3，4，5，6，i≠j）中，不同向量的个数有_____． A ．9个； B ．15个； C ．18个； D ．30个答案：C9、对函数f:[0，1]→[0，1]，定义f 1（x ）=f （x ），……，f n（x ） =f （f n −1（x ）），n=1，2，3，……．满足f n （x ）=x 的点x ∈[0，1]称为f 的一个n −周期点．现设⎪⎪⎩⎪⎪⎨⎧≤≤-≤≤=121,22,210,2)(x x x x x f 问f 的n −周期点的个数是___________．A ．2n 个；B ．2n 2个；C ．2n个；D ．2（2n−1）个．答案：C10、已知复数z 1=1+3i ，z 2=−3+3i ，则复数z 1z 2的幅角__________． A ．13π/12 B ．11π/12 C ．−π/4 D ．−7π/12答案：A11、设复数βαβαcos sin ,sin cos i w i z +=+=满足z w =3/2，则sin （β−α）=______． A ．±3/2B ．3/2，−1/2C ．±1/2D ．1/2，−3/2答案：D12、已知常数k 1，k 2满足0<k 1<k 2，k 1k 2=1．设C 1和C 2分别是以y =±k 1（x −1）+1和y =±k 2（x −1）+1为渐近线且通过原点的双曲线．则C 1和C 2的离心率之比e 1/e 等于_______．A ．222111k k ++ B ．212211k k ++ C ．1 D ．k 1/k 2答案：C13、参数方程0,)cos 1()sin (>⎩⎨⎧-=-=a t a y t t a x 所表示的函数y=f （x ）是____________．A ．图像关于原点对称；B ．图像关于直线x =π对称；C ．周期为2a π的周期函数D ．周期为2π的周期函数．答案：C14、将同时满足不等式x −k y −2≤0，2x +3y −6≥0，x +6y −10≤0 （k>0）的点（x ，y ）组成集合D 称为可行域，将函数（y +1）/x 称为目标函数，所谓规划问题就是求解可行域中的点（x ，y ）使目标函数达到在可行域上的最小值．如果这个规划问题有无穷多个解（x ，y ），则k 的取值为_____．A ．k≥1;B ．k≤2C ．k=2D ．k=1．答案：C15、某校有一个班级，设变量x 是该班同学的姓名，变量y 是该班同学的学号，变量z 是该班同学的身高，变量w 是该班同学某一门课程的考试成绩．则下列选项中正确的是________．A ．y 是x 的函数；B ．z 是y 的函数；C ．w 是z 的函数；D ．w 是x 的函数．答案：B16、对于原命题“单调函数不是周期函数”，下列陈述正确的是________． A ．逆命题为“周期函数不是单调函数”； B ．否命题为“单调函数是周期函数”； C ．逆否命题为“周期函数是单调函数”； D ．以上三者都不正确答案：D17、设集合A={（x ，y ）|log a x +log a y >0}，B={（x ，y ）|y +x <a}．如果A∩B=∅，则a 的取值范围是_______ A ．∅ B ．a>0，a≠1 C ．0<a≤2， a≠1 D ．1<a≤2答案：D18、设计和X 是实数集R 的子集，如果点x 0∈R 满足：对任意a>0，都存在x ∈X 使得0<|x −x 0|<a ，则称x 0为集合X 的聚点．用Z 表示整数集，则在下列集合（1）{n/（n+1）|n ∈Z ， n≥0}，（2） R\{0}，（3）{1/n|n ∈Z ， n≠0}，（4）整数集Z 中，以0为聚点的集合有_____． A ．（2），（3）B ．（1），（4）C ．（1），（3）D ．（1），（2），（4）答案：A19、已知点A （−2，0），B （1，0），C （0，1），如果直线kx y =将三角形△ABC 分割为两个部分，则当k =______时，这两个部分得面积之积最大？A ．23-B ．43-C ．34-D ．32-答案：A20、已知x x x x f 2cos 3cos sin )(+=，定义域⎥⎦⎤⎢⎣⎡=ππ127,121)(f D ，则=-)(1x f_____A ．π12123arccos 21+⎪⎪⎭⎫ ⎝⎛-x B ．π6123arccos 21-⎪⎪⎭⎫ ⎝⎛-x C ．π12123arcsin 21+⎪⎪⎭⎫ ⎝⎛--x D ．π6123arcsin 21-⎪⎪⎭⎫ ⎝⎛-x 答案：A21、设1l ，2l 是两条异面直线，则直线l 和1l ，2l 都垂直的必要不充分条件是______ A ．l 是过点11l P ∈和点22l P ∈的直线，这里21P P 等于直线1l 和2l 间的距离 B ．l 上的每一点到1l 和2l 的距离都相等 C ．垂直于l 的平面平行于1l 和2lD ．存在与1l 和2l 都相交的直线与l 平行答案：D22、设ABC −A’B’C’是正三棱柱，底面边长和高都为1，P 是侧面ABB’A’的中心，则P 到侧面ACC’A’的对角线的距离是_____A ．21B ．43C ．814D ．823答案：C23、在一个球面上画一组三个互不相交的圆，成为球面上的一个三圆组．如果可以在球面上通过移动和缩放将一个三圆组移动到另外一个三圆组，并且在移动过程中三个圆保持互不相交，则称这两个三圆组有相同的位置关系，否则就称有不同的位置关系．那么，球面上具有不同的位置关系的三圆组有______A ．2种B ．3种C ．4种D ．5种答案：A24、设非零向量()()()321321321,,,,,,,,c c c c b b b b a a a a ===为共面向量，),,(31x x x x x = 是未知向量，则满足0,0,0=⋅=⋅=⋅x c x b x a的向量x 的个数为_____A ．1个B ．无穷多个C ．0个D ．不能确定答案：B25、在Oxy 坐标平面上给定点)1,2(),3,2(),2,1(C B A ，矩阵⎪⎪⎭⎫⎝⎛-112k 将向量OC OB OA ,,分别变换成向量,,，如果它们的终点',','C B A 连线构成直角三角形，斜边为''C B ，则k 的取值为______A ．2±B ．2C ．0D ．0，−2 答案：B26、设集合A ，B ，C ，D 是全集X 的子集，A∩B≠∅，A∩C≠∅．则下列选项中正确的是______． A ．如果B D ⊂或C D ⊂，则D∩A≠∅； B ．如果A D ⊂，则C x D∩B≠∅，C x D∩C≠∅； C ．如果A D ⊃，则C x D∩B=∅，C x D∩C=∅； D ．上述各项都不正确．27、已知数列{}n a 满足21=a 且n a n ⎧⎫⎨⎬⎩⎭是公比为2的等比数列，则∑==nk k a 1______A ．221-+n nB ．22)1(1+-+n n C ．)1(22-+n n n D ．n n n 22)1(+-28、复平面上圆周2211=+--iz z 的圆心是_______ A ．3+i B ．3−iC ．1+iD ．1−i29．已知C 是以O 为圆心、r 为半径的圆周，两点P 、P *在以O 为起点的射线上，且满足|OP|∙|OP *|=r 2，则称P 、P *关于圆周C 对称．那么，双曲线22x y -=1上的点P （x ，y ）关于单位圆周C':x 2＋y 2=1的对称点P *所满足的方程是（A ）2244x y x y -=+（B ）()22222x y x y-=+（C ）()22442x y x y-=+（D ）()222222x y x y-=+30、经过坐标变换⎩⎨⎧+-=+=θθθθcos sin 'sin cos 'y x y y x x 将二次曲线06532322=-+-y xy x 转化为形如1''2222=±b y a x 的标准方程，求θ的取值并判断二次曲线的类型_______ A ．)(6Z k k ∈+=ππθ，为椭圆 B ．)(62Z k k ∈+=ππθ，为椭圆C ．)(6Z k k ∈-=ππθ，为双曲线D ．)(62Z k k ∈-=ππθ，为双曲线31、设k ， m ， n 是整数，不定方程mx+ny=k 有整数解的必要条件是____________ A ．m ，n 都整除kB ．m ，n 的最大公因子整除kC ．m ，n ，k 两两互素D ．m ，n ，k 除1外没有其它共因子。

2020年《高校自主招生考试》数学真题分类解析之1、不等式

专题之10、不等式一、选择题。

1．(2009年复旦大学)若实数x满足对任意实数a>0,均有x2<1+a,则x的取值范围是( ) A.(-1,1) B.[-1,1]C.(-,)D.不能确定2．(2010年复旦大学)已知点A(-2,0),B(1,0),C(0,1),如果直线y=kx将△ABC分割为两个部分,则当k= 时,这两个部分的面积之积最大. ( )A.-B.-C.-D.-3．(2010年复旦大学)将同时满足不等式x-ky-2≤0(k>0),2x+3y-6≥0,x+6y-10≤0的点(x,y)组成的集合D称为可行域,将函数z=称为目标函数,所谓规划问题就是求解可行域内的点(x,y),使目标函数达到在可行域内的最小值.如果这个规划问题有无穷多个解,则( ) A.k≥1 B.k≤2 C.k=2 D.k=14．(2011年复旦大学)设n是一个正整数,则函数y=x+在正实半轴上的最小值是( )A. B. C. D.5．(2011年复旦大学)若对一切实数x,都有|x-5|+|x-7|>a,则实数a的取值范围是( ) A.a<12 B.a<7 C.a<5 D.a<26．(2011年清华大学等七校联考)已知向量a=(0,1),b=(-,-),c=(,-),xa+yb+zc=(1,1),则x2+y2+z2的最小值为( )A.1B.C.D.2二、填空题。

7．(2010年中南财经政法大学)已知实数a,b满足a>b,ab=1,则的最小值是 . 8．(2009年华中科技大学) 对任意的a>0,b>0,的取值范围是 .三、解答题。

9．(2009年中国科技大学)求证:∀x,y∈R,不等式x2+xy+y2≥3(x+y-1)恒成立.10．(2009年南京大学)P为△ABC内一点,它到三边BC,CA,AB的距离分别为d1,d2,d3,S为△ABC的面积,求证:++≥.11．(2010年南京大学)(a+b)2+3a+2b=(c+d)2+3c+2d. (*)证明:(1)a=c,b=d的充分必要条件是a+b=c+d;(2)若a,b,c,d∈N*,则(*)式成立的充要条件是a=c,b=d.12．(2010年浙江大学)有小于1的n(n≥2 )个正数:x1,x2,x3,…,x n,且x1+x2+x3+…+x n=1.求证:+++…+>4.13．(2009年清华大学)设a=(n∈N*),S n=(x1-a)(x2-a)+(x2-a)(x3-a)+…+(x n-1-a)(x n-a),求证:S3≤0.14．(2009年清华大学)(1)x,y为正实数,且x+y=1,求证:对于任意正整数n,x n+y n≥;(2)a,b,c为正实数,求证:++≥3,其中x,y,z为a,b,c的一种排列.15．(2009年北京大学)∀x∈R都有acos x+bcos 2x≥-1恒成立,求a+b的最大值. 16．(2011年北京大学等十三校联考)求f(x)=|x-1|+|2x-1|+…+|2 011x-1|的最小值. 17．(2012年北京大学等十一校联考)求+=1的实数根的个数.1.B【解析】对任意实数a>0,函数f(a)=1+a的值域是(1,+∞),因此只要x2≤1即可.由x2≤1,解得x∈[-1,1].3.C【解析】可行域如图中阴影部分所示,目标函数z=的几何意义是可行域内的点与点(0,-1)连线的斜率,如果要使其取得最小值的点有无穷多个,则直线x-ky-2=0必过点(0,-1),即k=2.选C. 在解含有参数的平面区域问题时要注意含有参数的直线系的特点,本题的突破点是直线系x-ky-2=0过定点(2,0).4.C【解析】题中函数为非常规函数,可利用导数求其最值.因为y=x+=x+x-n,所以y'=1-x-n-1=1-,令y'=0得x=1,且函数y在(0,1)上递减,在(1,+∞)上递增,故函数y在正实半轴上的最小值为1+=.5.D【解析】可先求出函数y=|x-5|+|x-7|的最小值,然后根据不等式恒成立的条件求得a的取值范围.由于|x-5|+|x-7|≥|5-7|=2,即函数y=|x-5|+|x-7|的最小值等于2,所以要使|x-5|+|x-7|>a恒成立,应有a<2.方法二∵xa+yb+zc=(1,1),∴-y+z=1,x-y-z=1,∴-y+z=,y+z=2x-2,∴z=+x-1,y=-+x-1,∴x2+(-+x-1)2+(+x-1)2=3x2-2(+1)x+(+1)2+2(-1)x+(-1)2=3x2-4x++2=3(x2-x+)++2-=3(x-)2+≥,当且仅当x=,z=,y=时等号成立.9.x2+xy+y2-3(x+y-1)=(x+y)2+x2+y2-3x-3y+3=(x+y)2+(x-3)2+(y-3)2-6≥(x+y)2+(x+y-6)2-6=(x+y)2-3(x+y)+3=[(x+y)-]2≥0,故∀x,y∈R,不等式x2+xy+y2≥3(x+y-1)恒成立.10.2S=2(S△PBC+S△PCA+S△PAB),2S=ad1+bd2+cd3.要证++≥成立,即证(ad1+bd2+cd3)(++)≥(a+b+c)2成立.由柯西不等式可得上面不等式成立,当且仅当d1=d2=d3时等号成立.11.(1)由a=c,b=d得到a+b=c+d是显然的;反之,把a+b=c+d代入(*)式可得a=c,于是b=d.因此,a=c,b=d的充要条件是a+b=c+d.(2)充分性是显然的,下面证明必要性.当a+b=c+d时,由(1)可知:a=c,b=d,即必要性成立.当a+b>c+d时,有a-c>d-b,设a-c=d-b+p(p≥1),由(*)式得(a+b+1)2+a=(c+d+1)2+c,∴(a+b-c-d)(a+b+c+d+2)+a-c=0,∴[(a-c)-(d-b)](a+b+c+d+2)+a-c=0.∴a-c+p(a+b+c+d+2)=0,∴(1+p)a+pb+(p-1)c+pd+2p=0,这与p≥1相矛盾,于是a+b>c+d不能成立. 同理可证a+b<c+d也不能成立.综上可知:必要性成立.12.∵0<x i<1,∴>(i=1,2,3,…,n).∴+++…+>+++…+≥,又∵1=x 1+x2+x3+…+x n≥n,∴≥n,又∵n≥2,∴+++…+>n2≥4.13.S3=(x1-)(x2-)+(x2-)(x3-)=(x2-)(x1-+x3-)=·=-(x1+x3-2x2)2≤0.14.(1)设x=+a,则y=-a,其中-<a<,于是x n+y n=(+a)n+(-a)n=()n+()n-1·a+()n-2·a2+…+a n+()n-()n-1·a+()n-2·a2-…+(-a)n=2[()n+()n-2·a2+()n-4·a4+…]≥2×()n=.(2)不妨设a≥b≥c>0,即0<≤≤,且{,,}={,,},由排序不等式得++≥++=3.15.2【解析】方法一令cos x=t,则-1≤t≤1,f(t)=2bt2+at+1-b≥0恒成立.(1)当b<0时,,利用线性规划知识，如下图，可以解得:-1≤a+b<1.(2)当b=0时,at+1≥0,由-1≤t≤1,得-1≤a+b≤1.(3)当b>0时,(i),利用线性规划知识，如下图，可以解得:0<a+b<;(ii),即,⇒9b2-(2k+8)b+k2≤0,Δ≥0⇒-1≤k≤2,∴(a+b)max=2;(iii),即,利用线性规划知识，如图，可以解得:-1≤a+b<0.综上,(a+b)max=2.方法二2bcos2x+acos x-b+1≥0,令cos x=-,得+≤1,即a+b≤2,又当a=,b=时,cos2x+cos x+=(2cos x+1)2≥0成立,∴(a+b)max=2.16.【解析】解法一由绝对值的几何意义联想到求距离的最小值,如|x-a|+|x-b|的最小值应该是在数轴上a,b两点之间取得,为|a-b|,所以将函数f(x)的右边整理为|x-1|+|x-|+|x-|+|x-|+|x-|+|x-|+…+|x-|+|x-|+…+|x-|,共有1+2+3+…+2 011=1 006×2 011项,则f(x)可以理解为x到这1 006×2 011个零点的距离之和.从两端开始向中间靠拢,每两个绝对值的和的最小值都是在相应的零点之间取得,而且范围是包含关系,比如|x-1|+|x-|的最小值是在x∈[,1]上取得,|x-|+|x-|的最小值是在x∈[,]上取得,…,所以f(x)的最小值应该在正中间的零点或正中间的相邻两个零点之间取得.由=503×2 011可知,f(x)取得最小值的范围在第503×2 011个零点和第503×2 011+1个零点之间(这两个零点也可能相等).由<503×2 011算得n≤1 421,所以第503×2 011个零点和第503×2 011+1个零点均为,则[f(x)]min=f()=.解法二由零点分区间法讨论去绝对值:当x∈(-∞,]时,f(x)=(1-x)+(1-2x)+…+(1-2 011x),此函数图象是一条直线中的一部分,斜率k1=-1-2-…-2 011.当x∈(,]时,f(x)=(1-x)+(1-2x)+…+(1-2 010x)+(2 011x-1),此函数图象是一条直线中的一部分,斜率k2=-1-2-…-2 010+2 011.当x∈(,]时,f(x)=(1-x)+…+(1-2 009x)+(2 010x-1)+(2 011x-1),此函数图象是一条直线中的一部分,斜率k3=-1-2-…-2 009+2 010+2 011. ……当x∈(,]时,f(x)=(1-x)+…+(1-mx)+[(m+1)x-1]+…+(2 011x-1),此函数图象是一条直线中的一部分,斜率k2 012-m=-1-2-…-m+(m+1)+…+2 011. 当x∈(,]时,f(x)=(1-x)+…+[1-(m-1)x]+(mx-1)+…+(2 011x-1),此函数图象是一条直线,斜率k2 013-m=-1-2-…-(m-1)+m+…+2 011.令,即,即,由于m∈N*,解得m=1 422.所以当x∈(,]时,f(x)=(1-x)+…+(1-1 422x)+(1 423x-1)+…+(2 011x-1)=833-711×1 423x+1 717×589x, [f(x)]min=f()=.。

复旦大学自主选拔录取申请资格测试试卷.doc

复旦大学自主选拔录取申请资格测试试卷复旦大学自主选拔录取申请资格测试试卷本试卷27页，满分为1000分；每题5分；考试时间为180分钟。

考生注意1、答卷前，考生务必在试卷和答题卡上都用钢笔或圆珠笔填写姓名、准考证号，并用2B铅笔在答题卡上正确涂写试卷类型（A卷或B卷）和准考证号。

2、本卷为单选题，由机器阅卷，答案必须全部涂在答题卡上。

在答题卡上，考生应将代表正确答案的小方格用铅笔涂黑。

注意试题题号和答题卡编号一一对应，不能错位。

答案需要更改时，必须将原选项用橡皮擦去，重新选择并填涂。

答案不能写在试卷上，写在试卷上一律不给分。

1．杜甫旅夜书怀的前四句是细草微风岸，桅樯独夜舟。

A．穷年忧黎元，叹息肠内热。

B．飘飘何所以，天地一沙鸥。

C．星垂平野阔，月涌大江流。

D．山随平野尽，江入大荒流。

2．下面表达中，哪一种体现了意象的虚伪性特征。

A．“燕山雪花大如席” B．“春来江水绿如蓝” C．“半江瑟瑟半江红” D．“雾失楼兮，月迷津度” 3．原始舞蹈和原始绘画中所表现的。

A．只有原始宗教的观念B．只是物质匮乏和技术落后的形象化体现C．已有审美意识的反映D．只是闲来无事的涂鸦4．阿Q头上有疤，而忌讳“亮”、“光”，这属于。

A．相似联想B．接近联想C．对比联想D．对立联想5．中国古代文论中的“兴”，它被提出来主要用于阐释。

A．诗经B．乐府C．古诗十九首D．楚辞6．孙子曰凡用兵之法，全国为上，破国次之；全军为上，破军次之。

（孙子兵法·谋次）与“全国为上”的“全”用法不同的一项是。

A．全军为上，破军次之B．故兵不顿而利可全C．故今墓中全乎为王人也D．毁人之国而非久也，必以全争天下7．杜牧在阿房宫赋里描写秦人的奢华、浮靡时有这样几句话“鼎铛玉石，金块珠砾，弃之逦迤，秦人视之，亦不甚惜。

”下列与“金块珠砾”结构完全相同的一项是A．金玉良言B．蓬户瓮牖C．瓮牖绳枢D．渔夫樵父8．孟尝君曰“先生老矣，春秋高矣，何以教之”（汉·刘向新序·杂事第五）句中的“春秋”一词的含义是。

复旦大学自主招生试题

复旦大学自主招生试题（正文）复旦大学自主招生试题自主招生，作为一种独特的选拔方式，给予了高中生更多展示自己的机会，而复旦大学作为一所顶尖的综合性大学，其自主招生试题更是备受考生关注。

本文将通过介绍复旦大学自主招生试题的一些例子，分析其考查内容和要求。

一、数学试题1. 已知函数f(x) = 2x^3 - 3x^2 - 12x + 5，求函数f(x)在区间[-2, 3]上的最小值和最大值。

分析：首先，我们需要先求出函数f(x)的导函数f'(x)，然后再通过导函数的零点来找出函数f(x)的极值点。

根据极值的定义，我们可以通过求解f'(x) = 0来得到。

2. 某商店商品价格打9折，然后再减去10元，最后的价格是原价的40%。

求该商品的原价。

分析：假设原价为x元，那么根据题意，我们可以得到以下等式：0.9x - 10 = 0.4x。

通过解这个方程，我们可以求出该商品的原价x。

二、英语试题1. 阅读下面短文，并根据短文内容完成后面的题目。

Most people know that exercise is good for their health. Regular physical activity can prevent a multitude of diseases and improve one’s overall well-being. However, it is essential to find an exercise routine that suits your lifestyle and preferences. In this regard, yoga is a great option for many.Yoga combines physical poses, breathing exercises, and meditation to promote a healthy mind and body. The slow and controlled movements help build flexibility, strength, and balance. Additionally, the focus on deep breathing and mindfulness promotes relaxation and stress reduction.Furthermore, yoga can be practiced by people of all ages and fitness levels. From beginner classes to advanced poses, there are variations suitable for everyone. It is a versatile practice that can be adapted to individual needs and goals.Based on the information provided in the passage, answer the following questions:a. What are the benefits of regular exercise?b. What aspects does yoga combine?c. Why is yoga suitable for people of all ages and fitness levels?三、文学试题阅读下面的《Active Learning》一文，根据文章内容回答问题。

2020年上海市复旦附中自招模拟物理试题(教师版)

2020上海复旦附中物理自招模拟试卷（二）单项选择题（一）1.甲、乙两个同学各用200 N 的力沿相反的方向拉测力计，则测力计的示数为A. 100NB. 200 NC. 0 ND. 400 N 【答案】B【解析】【详解】弹簧测力计指示的力为：在平衡状态(静止或匀速直线运动)下，一端的拉力数值，两位同学沿相反方向拉弹簧测力计，各用200N 的拉力，弹簧测力计示数应为200N.故B 正确．.2.一个物体同时受到两个力的作用，这两个力的三要素完全相同，那么这两个力A. 可能是平衡力B. 肯定不是平衡力C. 肯定是平衡力D. 无法确定【答案】B【解析】【详解】两个力的三要素完全相同，说明了这两个力的方向也相同．而平衡力的方向是相反的，由此可知这两个力一定不是平衡力．故B 正确.3.如图所示，物体A 在水平力F 的作用下，静止在竖直墙壁上。

当水平力减小为2F 时，物体A 恰好沿竖直墙壁匀速下滑，此时物体A 所受摩擦力的大小（）A. 减小为原来的12B. 和原来一样C. 增大为原来的2倍D. 无法判断【答案】B【解析】【详解】物体静止和匀速下滑，都是平衡状态，所受平衡力为重力和摩擦力，根据平衡力特点可知：摩擦力等于重力，而重力不变，所以两次摩擦力不变。

故选B 。

4.一物体做匀速直线运动，由公式v=s t可知（）A. v与s成正比B. v与t成反比C. s与t正比D. 以上说法都不对【答案】C【解析】【详解】A．已知物体做匀速直线运动，所以v是确定的，与s的大小无关，A错误；B．已知物体做匀速直线运动，所以v是确定的，与t的大小无关，B错误；C．已知物体做匀速直线运动，由公式s=vt知，v一定，t越大，s越大，即s与t成正比，C正确；D．因为选项C正确，D错误。

故选C。

5.重约3N的物体可能是（）A. 一枚1元硬币B. 一本练习本C. 一只鸡蛋D. 一听可乐【答案】D【解析】【详解】3N物体的质量m=3N10N/kgGg=0.3kg=300gA．一枚1元硬币的质量大约5g，不符合题意；B．一本练习本的质量大约40g，不符合题意；C．一只鸡蛋的质量大约50g，不符合题意；D．一听可乐质量大约300g，符合题意。

自主招生英语试卷解读及应试策略试题

第一讲：近年自主招生英语试卷解读及应试策略————————————————————————————————————————第一局部题型分析〔一〕复旦大学复旦大学的自主招生考题由200个选择题组成，一共1000分，英语一共30题左右，包括语法与词汇、完形填空和阅读理解三种题型。

⏹特点是：词汇量、阅读量、知识量较大〔二〕交通大学HY的一张卷，内容更加丰富，题型更加多变。

最初的题型包括听力理解〔短对话、短文和做笔记填空〕、阅读理解、语法与词汇、翻译和作文。

最近几年取消了语法、听力和翻译局部，加大了对阅读和词汇的考察，增加了完形填空。

其最显著的特点是两篇作文，一小一大，往往占一半分数。

⏹特点：词汇量大、作文是大餐，四、六级特点明显。

第二局部复旦大学英语自主招生考题分析〔一〕语法与词汇此局部考察语法、固定搭配、短语、形近词辨析、同义词辨析以及生词。

06、07年各8题，08年16题。

⏹语法题一般只有一个题，考察的重点是定语从句和非谓语动词等。

如There are fewerworking hours in the executive jobs, the job stress is comparatively higher.〔A．that B．where C．which D．what〕。

⏹固定搭配题主要考短语中的介词，每年至少考一题，如be identified with〔2021〕，be fed up with〔2021〕，rest on one’s laurels〔2021〕。

这里的第二个短语比拟根底，第一个短语稍有难度，最后一个短语有点偏，意思是“满足于已获得的成绩、不思进取〞。

固定搭配还可能考一些其他短语，如raw score〔原始分数〕〔2021〕。

在以前的大学英语四级考试词汇局部中经常出现这类搭配，如prevailing winds〔季风〕，intensive care〔重病护理〕。

固定搭配在最近几年的大学英语教学中越来越受到重视，势必也会在考试中得以表达。

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复旦大学2020年自主招生真题
出国留学高考网为大家提供复旦大学2017年自主招生真题，更
多高考资讯请关注我们网站的更新!
复旦大学2017年自主招生真题
2017年自主招生分望道计划、博雅计划、奥赛决赛(丘赛决赛)
三种，不同的招生类别考核形式不同，2017年望道计划以体验营的
形势来进行，体验营设有笔试、面试、名师讲座、校园体验等内容。

【笔试】
笔试考核方式笔试科目：数学、物理、化学、生物，全部都是选择题。

笔试时间：3小时
科目设置：数学50题，物理40题，化学22题，生物8题。

满
分600，共120道单项选择题，每题5分，选对得5分，不选得0
分，选错倒扣2分。

文综：
120道题里，考了5道英文数学题和六七道英文历史题。

英文数
学题，比较简单。

英文历史题，要对国家名、专有名词等有些了解，才好做。

例如有一个历史题，给了一段英文的题干，然后问你这是
历史上的哪次国际会议?选项有波茨坦会议、开罗会议等。

历史考了
很多史料文本，比如给一段文字，问出自哪本书，还考查了文言文。

理综：
理科是用英文考物理题和化学题。

其中，物理题是关于热力学的。

【面试】
五对一，每人15分钟，教授自由提问，学生回答。

(主要是物理、化学组老师)
首先做1分钟的自我介绍，接着，老师会根据考生的自荐信、自我介绍及报考专业相关内容进行提问。

据部分参加考生反应，面试主要主要集中在三个方向：
1、考生的竞赛/科创/课外活动经历的。

根据考生的学习或者考试等方面进行提问，如：竞赛学习方法、看过的书籍、对竞赛的认知等等。

2、高中学习和对复旦的看法。

如有考生被问到：高考估计，高考体验，为什么选择复旦，未来专业方向等等。

3、学术性问题。

该部分在整个面试过程中时间最长，也是考核最为细致的一部分。

等到时间差不多了，老师便会宣布面试结束。