2023高三·全国·专题练习
名校
1 . 已知由实数组成的数组
满足下面两个条件:
①
;②
.
(1)当
时,求
,
的值;
(2)当
时,求证
;
(3)设
,且
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0766a9dd0058b52abd9ad17ddb04fc2c.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8126d64e2becb117d8d42af22a5919b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c12927cb41f08b3fd18f338623db8d8d.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc2d3df37e73a8abea815f37dbb3fff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24819c61a0a42291903e3c2f5e1c6e41.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf7310dc844ccb33e0ff0b62aeb47b8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/818dfdff56f53b3ecfb1096a692e914d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/beb74763554fd459d736ed9c7b387e01.png)
您最近一年使用:0次
名校
解题方法
2 . 在
中,
对应的边分别为
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41cc48b9017b4828713efe931111e782.png)
(1)求
;
(2)奥古斯丁.路易斯.柯西(Augustin Louis Cauchy,1789年-1857年),法国著名数学家.柯西在数学领域有非常高的造诣.很多数学的定理和公式都以他的名字来命名,如柯西不等式、柯西积分公式.其中柯西不等式在解决不等式证明的有关问题中有着广泛的应用.现在,在(1)的条件下,若
是
内一点,过
作
垂线,垂足分别为
,借助于三维分式型柯西不等式:
当且仅当
时等号成立.求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9c1e84aaa7e1b5c1283075b36c72fb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41cc48b9017b4828713efe931111e782.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)奥古斯丁.路易斯.柯西(Augustin Louis Cauchy,1789年-1857年),法国著名数学家.柯西在数学领域有非常高的造诣.很多数学的定理和公式都以他的名字来命名,如柯西不等式、柯西积分公式.其中柯西不等式在解决不等式证明的有关问题中有着广泛的应用.现在,在(1)的条件下,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c98b702a52b5262939995dd9f77d1bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0e08a39c6619123557148d195abfbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/927456b0989846a2f1573844bbaa2105.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cde96534c28492e563efd72f941bed5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb5ba135022def1bcc1cddea66496706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ebbd1d0e4d44a11d9b0d65e73eef212.png)
您最近一年使用:0次
2023-06-11更新
|
1701次组卷
|
8卷引用:福建省部分优质高中2023-2024学年高一下学期第一次阶段性检测数学试卷
名校
解题方法
3 . 已知
,
,函数
.
(1)若函数
在
上有两个不同的零点,求
的取值范围;
(2)求证:当
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dd0914dc4d4c7f75710ff460a286fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a48015d2df8d9fd21d576f4381e65ddd.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/417ab20883d799aaf311371393fa7d7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c6ce02259a85ea191541f4a708738f1.png)
(2)求证:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66f0ca536621ec8db02707ba65917029.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c93f19320977aeecaa8801a82bb2b4d5.png)
您最近一年使用:0次
2021-01-30更新
|
853次组卷
|
5卷引用:湖北省襄阳市第五中学2022-2023学年高一上学期12月月考数学试题
湖北省襄阳市第五中学2022-2023学年高一上学期12月月考数学试题浙江省嘉兴市2020-2021学年高一上学期期末数学试题(已下线)【新东方】双师149高一下(已下线)【新东方】在线数学102高一上浙江省东阳市外国语学校2022-2023学年高一上学期期末数学试题
名校
解题方法
4 . 如果数列
满足“对任意正整数i,j,
,都存在正整数k,使得
”,则称数列
具有“性质P”.已知数列是无穷项的等差数列,公差为d.
(1)若
,
,判断数列
是否具有“性质P”,并说明理由;
(2)若数列
具有“性质P”,求证:
且
;
(3)若数列
具有“性质P”,且存在正整数k,使得
,这样的数列共有多少个?并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b71af6590f0f369c164a054a8b63bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab406d94b4907ab8a20ae3214628b045.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86fc336b4a83bf6d66c4afcc431597f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d3b18d23d41e5f456dfd6485feed523.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3068733ef2ceda9f1620d5c9bcdfa542.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f8e68eb4ade6e22982d2df5102d8894.png)
(3)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c0af74258551ca3f28b2c6ce54bffd1.png)
您最近一年使用:0次
5 . 已知集合
,且
中的元素个数
大于等于5.若集合
中存在四个不同的元素
,使得
,则称集合
是“关联的”,并称集合
是集合
的“关联子集”;若集合
不存在“关联子集”,则称集合
是“独立的”.
分别判断集合
和集合
是“关联的”还是“独立的”?若是“关联的”,写出其所有 的关联子集;
已知集合
是“关联的”,且任取集合
,总存在
的关联子集
,使得
.若
,求证:
是等差数列;
集合
是“独立的”,求证:存在
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10cdcb0e77b3ae3e701c6b51e15e2346.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d10449bc77d692a7270e0f20a68cdf2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8eeda5cef4846ef829069fe27f64e34e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9bf4032eb5a9ba68131b15182aa3491.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bf6c84731e5e1bd335ecfc2d36c3d81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9859aa908844a32c0e1e069a046727.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/918f1c94368c3a41177ff42cfedc0eb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f53190d6ead827a6338b9de847aeaf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6e0ad51c5541ec3dcca4a9845f8b7db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/498f92bf2e605cdbc91973e29b047566.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4d89801d24aa43f47d6a366aad0571.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e1ccce8225324817b0577551956464f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/874781ab5711bff6ee8c9cbad5b3b3dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f62295c36d2e2174908c2bec0eb5b30f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acfc595518cf752e1c7903dfff93dbda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8021f4f4c253a00360bf8f9425610e1.png)
您最近一年使用:0次
2020-02-09更新
|
1547次组卷
|
10卷引用:北京市清华大学附属中学朝阳学校2021-2022学年高二5月月考数学试题
北京市清华大学附属中学朝阳学校2021-2022学年高二5月月考数学试题北京市第八中学2023届高三上学期12月测试数学试题北京市日坛中学2023-2024学年高二下学期第三次月考(6月)数学试卷2020届北京市海淀区高三上学期期中数学试题(已下线)专题02 拿高分题目强化卷(第三篇)-备战2021年新高考数学分层强化训练(北京专版)北京市海淀区2021届高三模拟试题(一)(已下线)考点47 推理与证明-备战2022年高考数学(文)一轮复习考点帮北京市第五十七中学2021-2022学年高二下学期期末考试数学试题上海市上海中学2022届高三下学期高考模拟3数学试题北京市朝阳区中国人民大学朝阳分校2021-2022学年高三上学期开学考数学试题
6 . 已知集合
(
).对于
,
,定义
;
(
);
与
之间的距离为
.
(Ⅰ)当
时,设
,
.若
,求
;
(Ⅱ)(ⅰ)证明:若
,且
,使
,则
;
(ⅱ)设
,且
.是否一定
,使
?说明理由;
(Ⅲ)记
.若
,
,且
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e423803a7eceeb306d9020fdb86ddc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecfb4290cab93c0521d2596031625448.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9d22720c8b8fbbf1b8e4406400b135f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5200656a5ed2197cabde9c99afcf33ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c83919f9b83559bd5d1db0b9256a2524.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13e87088da41685cc8d433fbbe0e18d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/badd0969b43deabb1e8f3fcca73ce1c5.png)
(Ⅰ)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e45cf86650443d1b86c79b1e3edc7e5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3af388e3a6185f4e6e2a7db5dba6e1d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88f68a89451ca87d57d88d786c23d484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49f6e0e6a7cd9b3ec681407e10b44901.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f65fc200f10b97588a0c9896277c9c64.png)
(Ⅱ)(ⅰ)证明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81345ca73b711411e665820b5672913d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/559305e1e35d369f1d056bb4191a23aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a47d9b859a3ca077f7fc1a4cdc5b5d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3153bbcedc215adf208c82b65c8e6eff.png)
(ⅱ)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81345ca73b711411e665820b5672913d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3153bbcedc215adf208c82b65c8e6eff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/559305e1e35d369f1d056bb4191a23aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a47d9b859a3ca077f7fc1a4cdc5b5d6.png)
(Ⅲ)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/003fbb029cdb6d5d7f93e29dca371f7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81a3b20462ba83086d0711a25ed83bad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b39ca902e466d5b24d13846b3bc4a8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4a2681390214200443ae07c01a4abe.png)
您最近一年使用:0次
2020-05-19更新
|
928次组卷
|
5卷引用:2020届北京市第四中学高三第二学期数学统练1试题
(已下线)2020届北京市第四中学高三第二学期数学统练1试题北京市第二中学2020~2021学年高一下学期第四学段考试数学试题北京市第二中学2021-2022学年高一下学期第四学段考试数学试题(已下线)重难点01平面向量的实际应用与新定义(3)北京景山学校2023-2024学年高一(1,2,3班)下学期期中考试数学试题
名校
解题方法
7 . 已知数列
满足:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b5663d5502a1c6e510c18380aa592d4.png)
(1)证明:
(2) 证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b5663d5502a1c6e510c18380aa592d4.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bc481825c0ad50169ac3363a1214d14.png)
(2) 证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aebef360b3a16caf948450dafa522aff.png)
您最近一年使用:0次
名校
8 . 若存在实数
使得
则称
是区间
的
一内点.
(1)求证:
的充要条件是存在
使得
是区间
的
一内点;
(2)若实数
满足:
求证:存在
,使得
是区间
的
一内点;
(3)给定实数
,若对于任意区间
,
是区间的
一内点,
是区间的
一内点,且不等式
和不等式
对于任意
都恒成立,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2d2919e5bd26b5e9be672a3ff7604cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/220070d9bef1244a81af87a13885d817.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1add6ea18092e4db74ff941591d8950.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85b7e150af2052a1664cde963273d905.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96dfaeb8281858eafc652223d9abecab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30277e0be448b4955903e81e8795e45d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(2)若实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/663a61ad241d5d874c9a9362f0ee917c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a727f8c2f098bc3bce6719a9616c013e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2d2919e5bd26b5e9be672a3ff7604cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2f89a8b5cf6996a6455375e405bfb9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fe5f1f33b2085760378b8abc0f1a582.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(3)给定实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/958ca60e8f470ce1b747c7e6a8c5cc69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1add6ea18092e4db74ff941591d8950.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75b5dc876d7dcd3c971b36d26668b1e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28643dbf049ac0eab51b51f6c1c64646.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cd882dced6d048a704bfc678b8e7791.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cb0aa8c434bdadb3725591e5e49099d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83eb829e3338a9e4be598124855685e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a85ea4968343b0d94ed2fe01b535.png)
您最近一年使用:0次
2019-10-23更新
|
1274次组卷
|
4卷引用:上海市南模中学2019-2020学年高三上学期10月月考数学试题
名校
9 . 已知数列
满足
.
(1)证明:当
时,
;
(2)证明:
(
);
(3)证明:
为自然常数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fdc794580cc85e899e42cd2fd6e846a.png)
(1)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab6f8a982ee922f792173ab5e4cf10ad.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1581f06f862dbb41f4dcfcec29b658e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e01c755ad6b4c4288b9663ad59cccbe.png)
您最近一年使用:0次
2019-10-15更新
|
931次组卷
|
7卷引用:2018届浙江省杭州市第二中学高三上学期市统测模拟数学试题
2018届浙江省杭州市第二中学高三上学期市统测模拟数学试题【全国百强校】浙江省余姚中学2018届高三选考科目模拟卷(二)数学试题1浙江省余姚中学2018届高三选考科目模拟考试(一)数学试题(已下线)专题6.6 数学归纳法 (练)-浙江版《2020年高考一轮复习讲练测》2018届浙江省宁波市余姚中学高三下学期6月高考适应性考试数学试题(已下线)专题12不等式的证明技巧的求解策略解题模板(已下线)专题7.6 数学归纳法(练)-2021年新高考数学一轮复习讲练测