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1 . 用反证法证明命题“已知x、
,且
,求证:
或
”时,应首先假设“______ ”.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91792ac4262a83e082aa03d6d66c437a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f9e131cdd242d56b6dba05ab3363ef3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fde64f4d3c38e43fbdee24eadc4b0dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eec336faee8689281a6f6b465e7fcff9.png)
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2023-03-10更新
|
252次组卷
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8卷引用:上海市崇明区2022-2023学年高一上学期期末数学试题
上海市崇明区2022-2023学年高一上学期期末数学试题青海省海南藏族自治州高级中学2022-2023学年高二下学期期末考试数学(文)试题上海市松江区2023-2024学年高一上学期期末质量监控数学试卷 上海市嘉定区2022-2023学年高一下学期3月调研数学试题陕西省宝鸡市金台区2022-2023学年高二下学期期中文科数学试题(已下线)1.2 常用逻辑用语-高一数学同步精品课堂(沪教版2020必修第一册)(已下线)专题04常用逻辑用语-【倍速学习法】(沪教版2020必修第一册)上海市上海外国语大学附属浦东外国语学校2023-2024学年高一上学期期中考试数学试卷
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2 . 新教材人教B版必修第二册课后习题:“求证方程
只有一个解”.证明如下:“化为
,设
,则
在
上单调递减,且
,所以原方程只有一个解
”.解题思想是转化为函数.类比上述思想,不等式
的解集是__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c18c032d75893db45e61e6c4eb0d4e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49cfb1e9557770560280b5248ae2d0d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/856491b01dab707170d83a1bc4b1f257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dec65a2bec3d4296c613a80b3ae41d5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/707ea658f3a9359f5740d5aab48f7948.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5eb24655f40cd3200323b4f920c9f473.png)
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2020-11-04更新
|
706次组卷
|
7卷引用:湖北省黄冈市麻城一中2019-2020学年高三上学期期末数学(理)试题
湖北省黄冈市麻城一中2019-2020学年高三上学期期末数学(理)试题辽宁省抚顺市二中、旅顺中学2019-2020年高三上学期期末考试数学试题辽宁省辽南协作体2019-2020学年高三上学期期末考试数学文试题辽宁省辽南协作体2019-2020学年高三上学期期末考试数学理试题安徽省六安市舒城中学2020-2021学年高二下学期开学考试数学(理)试题(已下线)第18讲 数学思想选讲(二)-【提高班精讲课】2021-2022学年高一数学重点专题18讲(沪教版2020必修第一册,上海专用)内蒙古海拉尔第二中学2021-2022学年高三上学期第一次阶段考数学(文科)试题
名校
3 . 用反证法证明“设
,求证
”时,第一步的假设是______________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/127a0d8c1c7d15ed40ec4b8bca0ebdf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/485a2d99320384a0857b00ce9ab9e990.png)
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2020-03-20更新
|
474次组卷
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7卷引用:上海市浦东新区2021-2022学年高一上学期期末数学试题
名校
4 . 设
,
,
,…,
,希望证明
,在应用数学归纳法求证上式时,第二步从
到
应添的项是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d7d41cdc17d1d73868a0eafb5621a2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f395576519def6a4df88b8fa4e524767.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0598b97e0d061dd458626a080bd1ec6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8be7b032a433583d2414f9f504b8630.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff0ad23f8781ebb49107aa5dbf5fa9fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b00f4eb7f1bd2ccefbabf0c1dfa8f69.png)
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2020-01-30更新
|
234次组卷
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2卷引用:上海市交大附中2019-2020学年高一下学期期末数学试题
解题方法
5 . 如图,在正方体
中,
是
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/4/6c0d410b-9639-4583-88dd-cf29de37c0da.png?resizew=176)
(1)求证:
平面
;
(2)求证:平面
平面
.(只需在下面横线上填写给出的如下结论的序号 :①
平面
,②
平面
,③
,④
,⑤
)
证明:(1)设
,连接
.因为底面
是正方形,所以
为
的中点,又
是
的中点,所以_________.因为
平面
,____________,所以
平面
.
(2)因为
平面
平面
,所以___________,因为底面
是正方形,所以_______,又因为
平面
平面
,所以_________.又
平面
,所以平面
平面
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22adbc0da438220f9cace11b629d799b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/4/6c0d410b-9639-4583-88dd-cf29de37c0da.png?resizew=176)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f11f1840eb8b17e7b07c3fe7e987a9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca48c18021e7be4bbb3e95576e1c1b5f.png)
(2)求证:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/077dcbb7a11d1ab7c9644bd1fe9b4368.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca48c18021e7be4bbb3e95576e1c1b5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e56fdf217165748fafe938b64fa08179.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc78a86b12ba0b4553135a3a635fc418.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cffb30e92f06a10b82da994fa304d2c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca48c18021e7be4bbb3e95576e1c1b5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0d7af0ab753d5017928560ae47de106.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70734a8e672376bb0bd1522e229f86a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/105cdc325328b683456c1443f6296f8b.png)
证明:(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a23f01af749100e1888bba06268843db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f90c780dac29ff8b7df5881d3b33abab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d97dc3b752832906de41447bb58a341.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22adbc0da438220f9cace11b629d799b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b2104bb78c7d7d58c1f51a25daa8086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca48c18021e7be4bbb3e95576e1c1b5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f11f1840eb8b17e7b07c3fe7e987a9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca48c18021e7be4bbb3e95576e1c1b5f.png)
(2)因为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ddbb0422a136f45653c8c369f2d75fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9263a4c1054ec924e900a1544180a83b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22bc82cd42e79c932abe14fa79265ed3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a03f5a663501d8fa265fdb593ed4bfab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc78a86b12ba0b4553135a3a635fc418.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f196748dc6a0d0bd9e9e4dd30ac4ed0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca48c18021e7be4bbb3e95576e1c1b5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/077dcbb7a11d1ab7c9644bd1fe9b4368.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca48c18021e7be4bbb3e95576e1c1b5f.png)
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解题方法
6 . 某物理学家用数学方法证明数学对物理是有用的:把物理世界G(现实世界)看作时空点(四元数
),找到一个函数
,若存在实数
,使对任意的
均有不等式
(
是与物理世界G的时空点
有关的另一个函数)成立.则称物理世界G与函数
在区间
上“拟同态”,函数
叫物理世界G在区间
上的“拟同态函数”,通过研究“拟同态函数”
,可以获得物理世界G(现实世界)的相关信息.现在知道某具体物理现象G,在s的区间
上的“拟同态函数”:
,且
,则实数n的取值范围是________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58a7efc92ffeb95718510efaf46799b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4dafff83fd807d0010d1805d9f4552e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/451dd67f383f1db987303734f5c84406.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c51159984b2cb00f30b3986315019623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd92eb9352850c673e3bd415ca79c004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/057c8c9b02e580e97ddcd78e8f8bf82c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58a7efc92ffeb95718510efaf46799b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2709ca478fb15ea08e8aa55328eae8e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2709ca478fb15ea08e8aa55328eae8e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6c1756b564bf1d998d8179637011c88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab65bd428ba508dadc9a4c4ea56e63d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37327dfa46306d1e571ae5742ed969e2.png)
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解题方法
7 . 假设视网膜为一个平面,光在空气中不折射,眼球的成像原理为小孔成像. 思考如下成像原理: 如图,地面内有圆
,其圆心在线段
上,且与线段
交于不与
重合的点
,
地面,且
,
点为人眼所在处,视网膜平面与直线
垂直. 过
点作平面
平行于视网膜平面. 科学家已经证明,这种情况下圆
上任意一点到
点的直线与平面
交点的轨迹(令为曲线
)为椭圆或圆,且由于小孔成像,曲线
与圆
在视网膜平面上的影像是相似的,则当视网膜平面上的圆
的影像为圆时,圆
的半径
为____________ . 当圆
的半径
满足
时,视网膜平面上的圆
的影像的离心率的取值范围为____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e5c62f22d7afc5627fcb86599faa8e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e5c62f22d7afc5627fcb86599faa8e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/137d33c8575602cb3480ba3825dece9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a7442b64b37f685bc3ae88ff450c1a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70e92e4810c9461c39fae1acde95e489.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3dc6893e52bbca0d011ac46845334d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
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2024-05-09更新
|
99次组卷
|
2卷引用:四川省成都市实验外国语学校2023-2024学年高二上学期期末能力测评数学试题
8 . 勾股定理,是几何学中一颗光彩夺目的明珠,被称为“几何学的基石”. 中国是发现和研究勾股定理最古老的国家之一. 据记载,在公元前1120年,商高答周公曰“故折矩,以为勾广三,股修四,径隅五,既方之,外半其一矩,环而共盘,得成三四五,两矩共长二十有五,是谓积矩. ”因此,勾股定理在中国又称“商高定理”. 数百年后,希腊数学家毕达哥拉斯发现并证明了这个定理,因此“勾股定理”在西方被称为“毕达哥拉斯定理”. 三国时期,吴国的数学家赵爽创制了一幅“勾股圆方图”,用数形结合的方法给出了勾股定理的详细证明. 如图所示的勾股圆方图中,四个全等的直角三角形与中间的小正方形拼成一个大正方形. 若中间小正方形面积(阴影部分)是大正方形面积一半,则直角三角形中较小的锐角
的大小为_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
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9 . 《几何原本》卷2的几何代数法(以几何方法研究代数问题)成了后世西方数学家处理问题的重要依据.通过这一原理,很多代数的公理或定理都能够通过图形实现证明,也称之为无字证明.现有如图所示的图形,点
在以
为直径的半圆上,
为圆心,点
在半径
上(不与
点重合),且
.设
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1bc69e886c7afedf1c9233e9a2a6870.png)
__________ (用
表示),由
可以得出的关于
的不等式为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ebef5bab02280cdc99cc7f689135cd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294726f8e596ce099d050ebcd538e421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1bc69e886c7afedf1c9233e9a2a6870.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76edbc800f52f6f8f710b1d7179fb31f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/26/82b41fab-4a23-4fdb-8191-a5a97d6b0134.png?resizew=160)
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10 . 德国数学家高斯在证明“二次互反律”的过程中首次定义了取整函数
,其中
表示“不超过x的最大整数”,如
,
,
,则
________ .
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