解题方法
1 . 已知
,不等式
恒成立,则实数
的取值范围为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28d3815052a73da51fc087867c0c725c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
2 . 对于定义在R上的连续函数
,若存在常数t(
),使得
对任意的实数x都成立,则称
是阶数为t的回旋函数.
(1)试判断函数
是否是一个阶数为
的回旋函数,并说明理由;
(2)若
是回旋函数,求实数ω的值;
(3)若回旋函数
(
)在[0,1]上恰有2024个零点,求ω的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/476d332663b8fc357c1a3fc85f9fa5cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4e71ab4caeea9e300aa3886ff2ef8ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(1)试判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5c7fcef9e4a32491be482939d21ceca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/798af57938408f6e1fa1493c05242aa9.png)
(3)若回旋函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d1989dff229887fdd3fdda4a9a05c8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4456675a5dbe545462a22cef9aca8fe.png)
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3 . 在平面直角坐标系xOy中,A,B点的坐标分别为
和
,设
的面积为S,内切圆半径为r,当
时,记顶点M的轨迹为曲线C.
(1)求C的方程;
(2)已知点E,F,P,Q在C上,且直线EF与PQ相交于点A,记EF,PQ的斜率分别为
,
.
(i)设EF的中点为G,PQ的中点为H,证明:存在唯一常数
,使得当
时,
;
(ii)若
,当
最大时,求四边形EPFQ的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d06d31c9a2298b02664a86ddd91b1121.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cee9e51b1dd360d8e141e7fa8f4224d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a5e0a51c9e14fb246b0ba0b231c1e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b722e2e83c2d125453ee2d80a5e64d26.png)
(1)求C的方程;
(2)已知点E,F,P,Q在C上,且直线EF与PQ相交于点A,记EF,PQ的斜率分别为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
(i)设EF的中点为G,PQ的中点为H,证明:存在唯一常数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f132407bf1cc9d1f460d50f1b0547993.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b3fa41635da8da11d6c04287ff7513.png)
(ii)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/129fa211eb0cfb3968d38c3c90249842.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7560c67fac3a45cf25ec33aa3ca52519.png)
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4 . 下列不等式中正确的是( )
A.![]() | B.![]() | C.![]() | D.![]() |
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名校
5 . 设
为正整数,已知函数
,
,
. 当
时,记
,其中
. 则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e19024088b481805fd372b7b2ffdd8e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95372397621e3a2fc70a6e198e29f44d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02511beb97ace697703ada6473ed5d4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dd600b451b2b7f1680cbbcf36a49703.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be308b9f7b2b213abcbcea79f0d42f7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9464ddd75dd13f2019e6a868301ff629.png)
A.![]() ![]() |
B.![]() ![]() |
C.若![]() ![]() |
D.若![]() ![]() |
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解题方法
6 . 抛掷一枚不均匀的硬币,正面向上的概率为
,反面向上的概率为
,记
次抛掷后得到偶数次正面向上的概率为
,则数列
的通项公式![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f7fda69e2b32b9ced2239f915fa59b.png)
____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d266a04f3dc7483eddbc26c5e487db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b2a698891d42c70b597f0da4f215f09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f7fda69e2b32b9ced2239f915fa59b.png)
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5卷引用:河南省郑州市2024届高三第三次质量预测数学试题
河南省郑州市2024届高三第三次质量预测数学试题河南省许昌市许昌高级中学2024届高三下学期三模数学试题(已下线)第四套 艺体生新高考全真模拟 (三模重组卷)(已下线)第4套 新高考全真模拟卷(三模重组)云南省昆明市第三中学2024届高三下学期高考考前检测数学试卷
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解题方法
7 . 数列
满足
则称数列
为下凸数列.
(1)证明:任意一个正项等比数列均为下凸数列;
(2)设
,其中
,
分别是公比为
,
的两个正项等比数列,且
,证明:
是下凸数列且不是等比数列;
(3)若正项下凸数列的前
项和为
,且
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0bee75d4d83c0b76421fd87113e4dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)证明:任意一个正项等比数列均为下凸数列;
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f67fc95a626251da11649acb5e1706f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c340d7d093dd4a275ffea4b87cd26827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6268630d5e5288048d32f4aa5c8bc02d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c171ff5c2728e7cf00a88f88de14f308.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3755d7aa870e2f199d6c12264fc9be86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
(3)若正项下凸数列的前
![](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/0002f427eded1721f43d60dd0fd3ffe0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd419dc0a6580ab97777b2cb8fd7cded.png)
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解题方法
8 . 设函数
在区间
上的导函数为
,且
在
上存在导函数
(其中
).定义:若在区间
上
恒成立,则称函数
在区间
上为凸函数.已知
,
(
).
(1)判断函数
在区间
上是否为凸函数,说明理由;
(2)已知函数
为
上的凸函数,求
的取值范围,并证明:函数
图象上任意一点的切线总在
的图象的上方;
(3)若
,求函数
(
)的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10acd6d864583617dd3e71240bf0c857.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca214aa6276b96d67a451c3fdbc59b3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/683d7c81a47aeaae60c0dde61164ab7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/882d17c122eb8008105e85d55bf55587.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95267538729f438f1ceb509860efe3b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10bbdef421c976962a270a2beabbad91.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d01dc2d99655cf7598837cb0886166ed.png)
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d562dc22dfb3b81d0c3f88b54d063c2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10ede78fd7ac619ea597856254bb5d75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a716dc950dbf17b8241ee8f4c6d6c494.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/636289ad84b4a3a51095dd32ca201f94.png)
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解题方法
9 . 关于x的方程
有实根,则
的最小值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4636917935cd25f6dc22891165ee9fb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c925be255ca736a53b24d13ddede1a86.png)
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2卷引用:河南省南阳市社旗县第一高级中学2024届高三下学期三模理科数学试题
10 . 数列极限理论是数学中重要的理论之一,它研究的是数列中数值的变化趋势和性质.数列极限概念作为微积分的基础概念,它的产生与建立对微积分理论的创立有着重要的意义.请认真理解下述3个概念.
概念1:对无穷数列
,称
为数列
的各项和.
概念2:对一个定义域为正整数集的函数
,如果当
趋于正无穷大时,
的值无限趋近于一个常数
,即当
时,
,就说常数
是
的极限值,记为
.如:
,当
时,由反比例函数的性质可知
,即记为
.当
(
为常数)时,
.
概念3:对无穷数列
,其各项和为
,若当
时,
(
为常数),即
,则称该数列的和是收敛的,
为其各项和的极限;若当
时,其各项和
的极限不存在,则称该数列的和是发散的,其各项和的极限不存在.
试根据以上概念,解决下列问题:
(1)在无穷数列
中,
,求数列
的各项和
的极限值;
(2)在数列
中,
,讨论数列
的和是收敛的还是发散的;
(3)在数列
中,
,求证:数列
的和是发散的.
概念1:对无穷数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88c434a9e76de70c0af36c324e1fd48d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
概念2:对一个定义域为正整数集的函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4136968179e01108272af01324034127.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e345e86daf74312a6992e5d1c3f47f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a107eb946e0fe41629c644b7628d5cba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6784211a2342d9d829bd95e15b549b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e345e86daf74312a6992e5d1c3f47f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0057f1742dc20e867bcbc29e6475773a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40cd74412213ddb92f6b4637888cf3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a107eb946e0fe41629c644b7628d5cba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5cfc53624067d3c8e01f09361295dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc76422aeaa304648c34cd1c6c0674e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f4eb29a351c1efa18e8e45d083491df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/961ea9a98e63ba37f650fde96c774026.png)
概念3:对无穷数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b06ea1200f943f6eb160b49e584b4335.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a107eb946e0fe41629c644b7628d5cba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f614310a33734a2d82f0d84c627028e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2cb2108952d47acb4f0a9518cbef443.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a107eb946e0fe41629c644b7628d5cba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b06ea1200f943f6eb160b49e584b4335.png)
试根据以上概念,解决下列问题:
(1)在无穷数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5beb1d3014af78f347ea9cf3661881cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccecde965d7557d5ee35dea8ae7164a3.png)
(2)在数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1111d85a7c8b1842e38b5d59da90954.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
(3)在数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ab0309e2cd35585ea9fb2cc3017abf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f34f1354aaa4fa27de5215098e0b1e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ab0309e2cd35585ea9fb2cc3017abf.png)
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