1 . “让式子丢掉次数”—伯努利不等式(Bernoulli’sInequality),又称贝努利不等式,是高等数学分析不等式中最常见的一种不等式,由瑞士数学家雅各布.伯努利提出,是最早使用“积分”和“极坐标”的数学家之一.贝努利不等式表述为:对实数
,在
时,有不等式
成立;在
时,有不等式
成立.
(1)证明:当
,
时,不等式
成立,并指明取等号的条件;
(2)已知
,…,
(
)是大于
的实数(全部同号),证明:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30cdfc52dbd70827de9e15fffe39c321.png)
(3)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bc98a4d9ae0580aa2c1152ffb770d4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c4fb8df3614557f13bdc68378437e90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3d4045366a437d4003259050718e244.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f75f0daa973c8fc183b7d21bafd7e8cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c78998ba5f2665a1753c3fa84751716.png)
(1)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65a40142c84be68ee2918c3a8303388c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bc98a4d9ae0580aa2c1152ffb770d4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5026dc5ead3b5adf0e5f4b3e7c4eca1d.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a1cc5cfec94bc5686b41b043acdc8ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30cdfc52dbd70827de9e15fffe39c321.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6b29215b2a741c01efc27199e6c6925.png)
您最近一年使用:0次
2024-05-30更新
|
296次组卷
|
3卷引用:2024年海南省海口实验中学高一学科竞赛选拔性考试(自主招生)数学试题
名校
2 . 如图所示,
是圆
的直径,
是圆
上一点,以
为切点的切线交线段
的延长线于点
,作
于
于
.
(1)求证:
;
(2)若
,求
的长.
![](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/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8d2330ed65217dfa342e5aefb58a34f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e5436a255a433d253bb3573016ed579.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/26/c5ba968a-bf09-444b-a720-ebea15e52842.png?resizew=161)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7fa59286f0f6578d21ca854ebd74fe9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/327a6d654db907de3c07cb3a62248364.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
您最近一年使用:0次
3 . 数列
满足:
是大于1的正整数,试证明:在数列
中存在相邻的两项,它们除以
余数相同.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57c4d9c843ed628701f262f3e80ccb62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58e51235780886a13ff7ab8918e97d64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
您最近一年使用:0次
4 . 校乒乓球锦标赛共有
位运动员参加.第一轮,运动员们随机配对,共有
场比赛,胜者进入第二轮,负者淘汰.第二轮在同样的过程中产生
名胜者.如此下去,直到第n轮决出总冠军.实际上,在运动员之间有一个不为比赛组织者所知的水平排序,在这个排序中
最好,
次之,…,
最差.假设任意两场比赛的结果相互独立,不存在平局,且
,当
与
比赛时,
获胜的概率为p,其中![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a998c92f966aae015d3e1e37c967e7b5.png)
(1)求最后一轮比赛在水平最高的两名运动员
与
之间进行的概率.
(2)证明:
,
为总冠军的概率大于
为总冠军的概率.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f31971306914638e5ceb1bbe437535d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9f1ad18371ec533aeac27cf1fad95c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49cc8f06c961b64b15a90b99f7adc604.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/519321dbfc38d9b89948762478f71d0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9454ddb2d570f884b15bd3ddf2a4545d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec6ba141730fd5aae78ada1a8eb17d21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97ea8f47d8d8d9e1832d52b1c7425450.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a998c92f966aae015d3e1e37c967e7b5.png)
(1)求最后一轮比赛在水平最高的两名运动员
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2eae64cb0b1c5e4f556e0ee0ca54fa9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97ea8f47d8d8d9e1832d52b1c7425450.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5654866bd68198db845fb43c6b4c858.png)
您最近一年使用:0次
解题方法
5 . 如图,已知三棱柱
,
平面
.D,E分别是
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/22/03e0cf8e-497b-4d52-a9f4-08f68e020eed.png?resizew=177)
(1)证明:
平面
;
(2)设
与平面
所成角的大小是
,若
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d1d2e0f281222a5f289ea4008370aed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1cc117eb1a2d0ea7123b2ca898547546.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/310e5cf87aa443ca7f0ff80aba6dddc4.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/22/03e0cf8e-497b-4d52-a9f4-08f68e020eed.png?resizew=177)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/063510e3c1fb6a7ccc3b8e3e3c7d660e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b16cff607cdc2d69afc70dc778acbb.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bb49fdc6b6bbb2449fdf8a0de769d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab3e0dba5705e1d749cfb21ebbb2ed93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a8670759c61d785b9a336885df700b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb2cf0e95fdf1fd8a5b01d3dfd905e08.png)
您最近一年使用:0次
解题方法
6 . 在四面体
中,
为
中点,
为
外接球的球心,
.
(1)证明:
;
(2)若
,求四面体
体积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a351136b18bc7d3bd5122332772ab23b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.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/a351136b18bc7d3bd5122332772ab23b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0398ca118304f21b6fc3c36ecf8bf2f4.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab17db0e6518d617247e17afd313a6a2.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/578b12f739ef7fc54c65b8435b3c16aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af286347445bc77ba5dc6efb5fcc5b8f.png)
您最近一年使用:0次
解题方法
7 . 对集合
,定义其特征函数
,考虑集合
和正实数
,定义
为
和式函数.设
,则
为闭区间列;如果集合
对任意
,有
,则称
是无交集合列,设集合
.
(1)证明:L和式函数的值域为有限集合;
(2)设
为闭区间列,
是定义在
上的函数.已知存在唯一的正整数
,各项不同的非零实数
,和无交集合列
使得
,并且
,称
为
和式函数
的典范形式.设
为
的典范数.
(i)设
,证明:
;
(ii)给定正整数
,任取正实数
和闭区间列
,判断
的典范数
最大值的存在性.如果存在,给出最大值;如果不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1304eb00ab95d664dc84385f602a8f09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f69939291758b5eaa19146f76709e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aee6c8ae5004f2ffe7f8392b4d3c39b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/238908949859936af0e109ef684599b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f69939291758b5eaa19146f76709e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f69939291758b5eaa19146f76709e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/937c09d82c480e4d67f8a48d3f66c5f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a02da5d46478a54d279755a295d548f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1b56da93ba7a2dec958070eb2666240.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05386869739fb11a190c637ba8a93174.png)
(1)证明:L和式函数的值域为有限集合;
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f69939291758b5eaa19146f76709e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20b4010030e10725398b64d4dcc09429.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab0fa51de98f090eda3e3f60a26475db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecfcda4333678bafacc4c676c2836977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee06844034f61cab7d421d55179ee367.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/359a16305129aeea0953efd9100f4b9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7b4e32041b54703ade8e8c2cee01f13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ed82555c7d6fc6b449fbdb1f68fef1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20b4010030e10725398b64d4dcc09429.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20b4010030e10725398b64d4dcc09429.png)
(i)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1462612f3654548c39489985987cb67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7870c36161f465fc992534b5fc3777f3.png)
(ii)给定正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f69939291758b5eaa19146f76709e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20b4010030e10725398b64d4dcc09429.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
解题方法
8 . 春节将至,又是一年万家灯火的团圆之时.方方正正的小城里,住着
户人家,恰好构成了坐标平面上集合
的所有点.夜里,小城的人家挂上大红灯笼,交相辉映,将小城的夜晚编织成发光的大网.在坐标平面上看,A中的每个点均独立地以概率p被点亮,或以
的概率保持暗灭.若A中两个点的距离为1,则这两个点被称为是相邻的.若A中的n个被点亮的点
构成一依次相邻的点列
,则称这n个点组成的集合
是长度为n的“相邻灯笼串”.规定空集是长度为0的“相邻灯笼串”.
(1)给定A中3个依次相邻的点
,记随机变量X为集合
包含的“相邻灯笼串”的长度的最大值,试直接写出随机变量X的分布列(用p表示);
(2)若
,证明:存在长度为1000的“相邻灯笼串”的概率小于0.01;
(3)若
,证明:存在长度为1000的“相邻灯笼串”的概率大于0.99.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25ef6c862fe408c21f7779e4e8e82fe9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e0c982180ad66af4330b1a8c43e4c07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ae7fb954b47cb67fdde891c3b9d8295.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce642b73be99b3c1a8c5dd38ec58eb28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3e59f96c54b4a41ed3c5b33b44320b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e69a66f2a66f4cd0fa05f5bbe185b6b.png)
(1)给定A中3个依次相邻的点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e85555fed049f8bb454c7569904bfed8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/718d05f8dd704256c99ac978e1ad5336.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bea306bdd00e500a305816f378060e4.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47938ad49863a8ff60ea48d0820e48f4.png)
您最近一年使用:0次
解题方法
9 . 设
的外接圆半径是
均为锐角,且
.
(1)证明:
不是锐角三角形;
(2)证明:在
的外接圆上存在唯一的一点
,满足对平面上任意一点
,有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ed72a09eb977ca371f5a79262692df4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4da56c7905417250be1d3863e23815c8.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(2)证明:在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6c5e38280a46edd6f123b9f70629d34.png)
您最近一年使用:0次
2024-02-19更新
|
443次组卷
|
2卷引用:2024年2月第二届“鱼塘杯”高考适应性练习数学试题
名校
解题方法
10 . 设数列
满足:
,
,且
,
对
成立.
(1)证明:
是等比数列;
(2)求
和
的通项公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62e567d7e9761951a266953c8d5042ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4995fa0403e013d888c0935ebfe15024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55f19b54e86e33dff4bffda330809a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee20dd197233a0b2399cbd8eb75c861a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/930bc56406e69b785b37a83d48e36724.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5344eadd4711db34e3f935aedd5fb270.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
您最近一年使用:0次
2024-02-19更新
|
279次组卷
|
3卷引用:2024年2月第二届“鱼塘杯”高考适应性练习数学试题
2024年2月第二届“鱼塘杯”高考适应性练习数学试题四川省凉山州安宁河联盟2023-2024学年高二下学期期中联考数学试题(已下线)专题06 等差数列与等比数列(2)--高二期末考点大串讲(人教B版2019选择性必修第二册)