2024高三·全国·专题练习
名校
解题方法
1 . 欧拉函数
的函数值等于所有不超过
且与
互质的正整数的个数(公约数只有1的两个整数称为互质整数),例如:
,
.记
,数列
的前
项和为
,若
恒成立,则实数
的取值范围为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f435d0e2319eb04b19bd4037129c470.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ea1d22420e844884025655b0893066e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39de1bc04496b97dcf401c669e6ab44e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28f02d9917e72ed162b272d9f2090cdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f69ee393a7b89f76ea10a9647bb29bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
您最近一年使用:0次
7日内更新
|
201次组卷
|
3卷引用:艺体生押题卷三
2 . 在不大于
的正整数中,所有既不能被2整除也不能被3整除的个数记为
.
(1)求
,
的值;
(2)对于
,
,是否存在m,n,p,使得
?若存在,求出m,n,p的值;若不存在,请说明理由;
(3)记
表示不超过
的最大整数,且
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bc1e9444e6cbbcccfb19bef934fda45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c581f06adc031bd163f98c461300d862.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f0f3595c506dd94a3399da87f0b33ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/985ea7ad3004613e28dd691829437c11.png)
(2)对于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5510ef06b326f131933224473550d41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acf45fc1d20ec9adb3b25794ac938855.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b80b43936d042aae836465212e716964.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19bbe68c798af91a4f5fbf939c4ed315.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a3651b3fedba1f0e9998fa88acefd08.png)
您最近一年使用:0次
2024-06-07更新
|
476次组卷
|
3卷引用:情境10 存在性探索命题
3 . 已知集合
,定义:当
时,把集合
中所有的数从小到大排列成数列
,数列
的前
项和为
.例如:
时,
,
.
(1)写出
,并求
;
(2)判断88是否为数列
中的项.若是,求出是第几项;若不是,请说明理由;
(3)若2024是数列
中的某一项
,求
及
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bc6c83969f0c67473049709952b50e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d47710bdf547780bc9c29c42423cce2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4b33a624f32310f1ef43686ea593ae2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4b33a624f32310f1ef43686ea593ae2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8bd390fa43014ff48549a6ca941d38c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cbeede118c407a800b05757b9a1393e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2046390ed7657e860458b026a4ced115.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88ac0dba37b8eb22670b24c350af0b54.png)
(1)写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ae09a97dd40d6b317a448664bf3816.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd5f4584826926dbc15fae9fb75d36ec.png)
(2)判断88是否为数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f59038077a6db85e9b790b14eecf717a.png)
(3)若2024是数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4b33a624f32310f1ef43686ea593ae2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1180a64221e78248cb691ecc21ec18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64e05ea594b5b86bcbbad940b46000f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9c79f3f90e6ed44e479b2e4ff16f05.png)
您最近一年使用:0次
2024-04-17更新
|
1439次组卷
|
5卷引用:压轴题08计数原理、二项式定理、概率统计压轴题6题型汇总
(已下线)压轴题08计数原理、二项式定理、概率统计压轴题6题型汇总(已下线)压轴题01集合新定义、函数与导数13题型汇总-22024届浙江省嘉兴市二模数学试题(已下线)第四套 艺体生新高考全真模拟 (二模重组卷)天津市南开中学2024届高三下学期模拟检测数学试题
名校
4 . 已知函数
.
(1)当
时,求曲线
在点
处的切线方程;
(2)当
时,若在
的图象上有一点列
,若直线
的斜率为
,
(ⅰ)求证:
;
(ⅱ)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c9e6940a234deb9afdcbc45a450800a.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5828873f8369183faf71181cda5b61d2.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7038739c261870bd71d9df8db016025.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f810cd01b6c3aeb01b488f31506bd61f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/425f3ce645095842006c80a509268f85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53a3dcb9f3022b912345c5460653f5e0.png)
(ⅰ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89f1a4d0fb65e5a7521d49839106e4d6.png)
(ⅱ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/210f476a7490aea439b89218b121df8d.png)
您最近一年使用:0次
2024-03-21更新
|
1808次组卷
|
4卷引用:专题1 数列不等式 与导数结合 练(经典好题母题)
(已下线)专题1 数列不等式 与导数结合 练(经典好题母题)2024届天津市十二区县重点学校一模模拟考试数学试卷山东省济宁市第一中学2024届高三下学期3月定时检测数学试题山东省济宁市第一中学2024届高三下学期4月质量检测数学试卷
5 . 若数列
满足对任意
,数列
的前
项至少有
项大于
,且
,则称数列
具有性质
.若存在具有性质
的数列
,使得其前n项和
恒成立,则整数
的最小值是_____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b4a1fb2354b4716afb9191c880fcc11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ceef1abeeef220b4fe5f7d96feedd90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a400b5443af7580aa8f0fb7499fe362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e040f089ea13dbd9d9133d6934537f32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e040f089ea13dbd9d9133d6934537f32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0be2256f1cda4db7aa1e92b7372ee8ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
您最近一年使用:0次
名校
解题方法
6 . 2023年10月11日,中国科学技术大学潘建伟团队成功构建255个光子的量子计算机原型机“九章三号”,求解高斯玻色取样数学问题比目前全球是快的超级计算机快一亿亿倍.相较传统计算机的经典比特只能处于0态或1态,量子计算机的量子比特(qubit)可同时处于0与1的叠加态,故每个量子比特处于0态或1态是基于概率进行计算的.现假设某台量子计算机以每个粒子的自旋状态作为是子比特,且自旋状态只有上旋与下旋两种状态,其中下旋表示“0”,上旋表示“1”,粒子间的自旋状态相互独立.现将两个初始状态均为叠加态的粒子输入第一道逻辑门后,粒子自旋状态等可能的变为上旋或下旋,再输入第二道逻辑门后,粒子的自旋状态有
的概率发生改变,记通过第二道逻辑门后的两个粒子中上旋粒子的个数为
.
(1)若通过第二道逻辑门后的两个粒子中上旋粒子的个数为2,且
,求两个粒子通过第一道逻辑门后上旋粒子个数为2的概率;
(2)若一条信息有
种可能的情况且各种情况互斥,记这些情况发生的概率分别为
,
,…,
,则称
(其中
)为这条信息的信息熵.试求两个粒子通过第二道逻辑门后上旋粒子个数为
的信息熵
;
(3)将一个下旋粒子输入第二道逻辑门,当粒子输出后变为上旋粒子时则停止输入,否则重复输入第二道逻辑门直至其变为上旋粒子,设停止输入时该粒子通过第二道逻辑门的次数为
(
,2,3,⋯,
,⋯).证明:当
无限增大时,
的数学期望趋近于一个常数.
参考公式:
时,
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(1)若通过第二道逻辑门后的两个粒子中上旋粒子的个数为2,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79a18d2bd429301b5478dcd26c572266.png)
(2)若一条信息有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ef66ba6d5421383f47b4783db53bf7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8be646cd52d7f2f1714e7542e75810f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adad9633b73dfbbb3d84b4f15979e99e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ffb021aa7d5a5c2f0691e337caad624.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b930a98ed7eb5ae313050f7c97d2a16c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33c5a2ba6cfa94756ac1a0f74ac9e4f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
(3)将一个下旋粒子输入第二道逻辑门,当粒子输出后变为上旋粒子时则停止输入,否则重复输入第二道逻辑门直至其变为上旋粒子,设停止输入时该粒子通过第二道逻辑门的次数为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f157de581046dc6a6002f771b60ad61c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
参考公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ca664b1e82da6f50064a76fe118aa80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d71b352414c4a600fc4ea827a0c64f22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c0aceee7cba466e6bf17f43d15bf25f.png)
您最近一年使用:0次
2024-03-04更新
|
1797次组卷
|
4卷引用:专题09 计数原理与随机变量及分布列(讲义)
(已下线)专题09 计数原理与随机变量及分布列(讲义)湖南省新高考十八校联盟2024届高三下学期3月月考数学试题(已下线)第2套 重组模拟卷(模块二 2月开学)湖北省襄阳市第五中学2024届高三第二次适应性测试数学试题
7 . 已知数列
的前n项和为
,且
,
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e31fb9cdc11d44c3dcab0488366ad458.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6691617159667fcfc3f6ce9146cd075a.png)
A.当![]() ![]() | B.![]() |
C.数列![]() ![]() | D.当![]() ![]() |
您最近一年使用:0次
名校
解题方法
8 . 已知
为有穷正整数数列,且
,集合
.若存在
,使得
,则称
为
可表数,称集合
为
可表集.
(1)若
,判定31,1024是否为
可表数,并说明理由;
(2)若
,证明:
;
(3)设
,若
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67905ad53186bb2908b603bc14005d10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/702dcfe2523f774f6bc4f075f3d24fe6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80566aaf96db9c785cda10dc0935c1c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84076d0854ef7c1a99a937fd50b25843.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c6985405452b5d04bd0d3305544cc2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c93e3391890fc877c761121b68cb927.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54119668d2f6cbc9ce0cb92310037713.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c93e3391890fc877c761121b68cb927.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b83efe191fb8adaf89737c03ef34d1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c93e3391890fc877c761121b68cb927.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2ebfe653088b1a534d0731947db43d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/562441c2767a65f3671afa93b190126b.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffceb52b543819898a9a6fc96d7337e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7eab142f716f69be57d3f4ca2197894.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
您最近一年使用:0次
2024-01-20更新
|
1487次组卷
|
7卷引用:专题1 集合新定义题(九省联考第19题模式)练
9 . 设数列
的前
项和为
.若对任意的正整数
,总存在正整数
,使得
,则称
是“
数列”.
(1)若数列
,
,判断
和
是否是“
数列”;
(2)设
是等差数列,其首项
,公差
.若
是“
数列”,求
的值;
(3)证明:对任意的等差数列
,总存在两个“
数列”
和
,使得
成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3693c7c942afef5517a3c18997c878df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e903c3dc6bdb559fd173f8d4e930f78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15ff259bff098430a6512d0e4f6fb2d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2d8bbb4a09e0ac86bbae46222a90841.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
(3)证明:对任意的等差数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50a13fcd18316e035cdc08901073672e.png)
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2023-12-25更新
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761次组卷
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4卷引用:专题06 信息迁移型【练】【北京版】
解题方法
10 . 已知正项数列
满足:
,
,则以下结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/798b81ff16b427407e6a47ba4e452498.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145b6046bc80d0ffecc61ac67c87ca1.png)
A.若![]() ![]() |
B.若![]() ![]() |
C.若![]() ![]() |
D.若![]() ![]() ![]() ![]() ![]() |
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