解题方法
1 . 拉格朗日中值定理是微分学的基本定理之一,其内容为:如果函数
在闭区间
上的图象连续不断,在开区间
内的导数为
,那么在区间
内存在点
,使得
成立.设
,其中
为自然对数的底数,
.易知,
在实数集
上有唯一零点
,且
.
时,
;
(2)从图形上看,函数
的零点就是函数
的图象与
轴交点的横坐标.直接求解
的零点
是困难的,运用牛顿法,我们可以得到
零点的近似解:先用二分法,可在
中选定一个
作为
的初始近似值,使得
,然后在点
处作曲线
的切线,切线与
轴的交点的横坐标为
,称
是
的一次近似值;在点
处作曲线
的切线,切线与
轴的交点的横坐标为
,称
是
的二次近似值;重复以上过程,得
的近似值序列
.
①当
时,证明:
;
②根据①的结论,运用数学归纳法可以证得:
为递减数列,且
.请以此为前提条件,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63313f7ac7402fcb5a9a840db64c6f08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/090a91e4f3c8930674f98a9fa527709b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63313f7ac7402fcb5a9a840db64c6f08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d59685311c7aa9ca98b1fdbabde40171.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15432e3c4e6c1d9cde98ec9187d162c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6dcd143a57a268a5a8ef486e2a4d5c0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00108fe668a98c905f3f92b720e35a0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8e356055d318b6d336e9e33a1e78aad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70142f9c28dc50c8ab41e71b19d18fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a8488679e2fa13e44ffa5b4d802848d.png)
(2)从图形上看,函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15432e3c4e6c1d9cde98ec9187d162c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15432e3c4e6c1d9cde98ec9187d162c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de261e9b4defbc0be6440397031a87b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168e68d052280fe48e1a3a6de67c6f2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8559f5db9b978cb2bd290dbce7268629.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a24a2c53e3b0b1c08803e95419f909d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87529d4cadc1e84f72d462cb8e3afac0.png)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c1a778faac194e8de4d5178454bd04c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f274881a6ad83e68c9b6652ebf4dc09.png)
②根据①的结论,运用数学归纳法可以证得:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1fd18a909cecbaee7115d6b15631d83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2adb4f1a98a9db3b5d4e4cfc7560fdb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fee28be9d207a3d3eed938484f980195.png)
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2 . 对于整数除以某个正整数的问题,如果只关心余数的情况,就会产生同余的概念.关于同余的概念如下:用给定的正整数
分别除整数
,若所得的余数(小于正整数
的自然数,即0,1,
)相等,则称
对模
同余,记作
.例如:因为
,
,所以
;因为
,所以
.表示对模
同余关系的式子叫做模
的同余式,简称同余式,同余式的记号
是高斯在1800年首创.两个同模的同余式也能够进行加法和减法运算,其运算规则如下:已知整数
,正整数
,若
,则
,
.阅读上述材料,解决下列问题:
(1)若
,且整数
,求
的值;
(2)已知整数
,正整数
,证明:若
,则
;
(3)若
,其中
为正整数,
为非负整数,证明:
能被11整除的充要条件为
能被11整除.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18c18c0cebecdfc0f63f64b98b8618f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bf17f75882ab0a28a78c8c49d1d1255.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/135a1a6b030325a6b417d3d5fecb8778.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0bd5638bfe2f006ab5f707f5039a160.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d62bbd00daf6bbdde9b3d936ab4f2ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a65d0f1fb1b4f913af5741ebe2e98d41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18eae33f07a441a87b75445811e87c27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bf17f75882ab0a28a78c8c49d1d1255.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d10449bc77d692a7270e0f20a68cdf2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfa91f51e5e0650e3fae950da7cbf4a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3113592ea3c033253299a0bdbb619897.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d51c59ce2cd593666329587abed347bf.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f1774978271a3e5a0b970b47de774f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08fc88e26cec31df99dfa1824587ae30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)已知整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d10449bc77d692a7270e0f20a68cdf2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfa91f51e5e0650e3fae950da7cbf4a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce06d8c49a3c57e5cf10e773818a2467.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f966aecd0328697920c0b7a22726cd33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b65a63629464f5a2c90356e367f66be.png)
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名校
3 . 帕德近似是法国数学家亨利·帕德发明的用有理多项式近似特定函数的方法.给定两个正整数
,
,函数
在
处的
阶帕德近似定义为:
,且满足:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a46eaf1cdc0ea6f6b18e8fba22ee7ae2.png)
.(注:
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e51793a343298909a499b0b150660ccb.png)
为
的导数)已知
在
处的
阶帕德近似为
.
(1)求实数
的值;
(2)证明:当
时,
;
(3)设
为实数,讨论方程
的解的个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57b85a97933a1d984f6e484b4021c800.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16563cfb206d0394cac2a0c2595dda6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a46eaf1cdc0ea6f6b18e8fba22ee7ae2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e4baac3118da93995e49b29a5d377e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca214aa6276b96d67a451c3fdbc59b3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e51793a343298909a499b0b150660ccb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/385c9d5f9d6c2c720dd99273021cafd1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eea7fa65b493fc1bdf84e16d39ae07d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35dd621776dee688a0175a1abe39c258.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40765d09390381658d5b4dc0160366cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8de781718020ed3f99538b8e25d6186.png)
(1)求实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/280860dd039e1305a5ccc455f63e8223.png)
(2)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e2e79843faf62dde86bf858d1e0569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/447d6f62c09c1d05346fd16a24159f6e.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cccba081685984454ee4fa955dc4f7ea.png)
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名校
4 . 马尔科夫链是概率统计中的一个重要模型,也是机器学习和人工智能的基石,在强化学习、自然语言处理、金融领域、天气预测等方面都有着极其广泛的应用.其数学定义为:假设我们的序列状态是……
,…,那么
时刻的状态的条件概率仅依赖前一状态
,即
.
现实生活中也存在着许多马尔科夫链,例如著名的赌徒模型.
假如一名赌徒进入赌场参与一个赌博游戏,每一局赌徒赌赢的概率为
,且每局赌赢可以赢得1元,每一局赌徒赌输的概率为
,且赌输就要输掉1元.赌徒会一直玩下去,直到遇到如下两种情况才会结束赌博游戏:记赌徒的本金为
一种是赌金达到预期的B元,赌徒停止赌博;另一种是赌徒输光本金后,赌徒可以向赌场借钱,最多借A元,再次输光后赌场不再借钱给赌徒.赌博过程如图的数轴所示.
时,最终欠债 A元(可以记为该赌徒手中有 ![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c590e4795751a8b932c63e0ad3bc49dd.png)
元)概率为
,请回答下列问题:
(1)请直接写出
与
的数值.
(2)证明
是一个等差数列,并写出公差d.
(3)当
时,分别计算
时,
的数值,论述当B持续增大时,
的统计含义.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02e54fb0a18558ef56d8100f58564c84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4b49fdb5924134bfc54266f0fee35ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2eb150b73ea7c87972a0b57510a99472.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5a27e7e2acb3aef8c7c9b504e8a5ab2.png)
现实生活中也存在着许多马尔科夫链,例如著名的赌徒模型.
假如一名赌徒进入赌场参与一个赌博游戏,每一局赌徒赌赢的概率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b1065ae0947705c7d16a5a86c78f07e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b1065ae0947705c7d16a5a86c78f07e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e9063713e024a66e6daca3ec781a639.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91c4c2fe859ad0805dcc2fc26d6dc537.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c590e4795751a8b932c63e0ad3bc49dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/532084481ae3a67c8208b7783bf22e8e.png)
(1)请直接写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fabb71334b127f1719f2a5e728d5fae1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b459aa38bd06fa9b5b0412c51121dd48.png)
(2)证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaef76a1500c26dc42bd88f89c15dd27.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf47b8e265017c3a85fe62885cfe326.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f2761b0fdb9640f2def02525128c74a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/391c6e33329f5f4ad0c5107520d9a5cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/391c6e33329f5f4ad0c5107520d9a5cf.png)
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2024-04-17更新
|
1189次组卷
|
3卷引用:辽宁省实验中学2023-2024学年高二下学期3月月考数学试题
辽宁省实验中学2023-2024学年高二下学期3月月考数学试题(已下线)专题03 第七章 随机变量及其分布列--高二期末考点大串讲(人教A版2019)江西省南昌市第十九中学2024届高三下学期第四次模拟考试数学试卷
5 . 由椭圆的两个焦点和短轴的一个顶点组成的三角形称为该椭圆的“特征三角形”.如果椭圆
的“特征三角形”为
,椭圆
的“特征三角形”为
,若
,则称椭圆
与
“相似”,并将
与
的相似比称为椭圆
与
的相似比.已知椭圆
:
与椭圆
:
相似.
(1)求椭圆
的离心率;
(2)若椭圆
与椭圆
的相似比为
,设
为
上异于其左、右顶点
,
的一点.
①当
时,过
分别作椭圆
的两条切线
,
,切点分别为
,
,设直线
,
的斜率为
,
,证明:
为定值;
②当
时,若直线
与
交于
,
两点,直线
与
交于
,
两点,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5518f853e3a929edf3dd3cee8ec0760d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8321b4034b3ab70b6cbfa25bca18df2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edaf9a32b79eb97becf706682da7115d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5518f853e3a929edf3dd3cee8ec0760d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8321b4034b3ab70b6cbfa25bca18df2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/271e595c257e4c0ade90a9bbbf0e6b0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(2)若椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5532211b42702f7b281834d500c666d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/249767ae3bf665f1c8db866dbb366940.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24006d28116bc097933cc90bcc0ea69f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2451835b9ad821bc17a317bc0189a38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c01fdc7bc471af0b264a04aef0823e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43a71fc9c0068109dad1382354570665.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24006d28116bc097933cc90bcc0ea69f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2451835b9ad821bc17a317bc0189a38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4757181824e15e0f21e5bdd55448783.png)
②当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e260f5fe6e3637a415344ff137c7a6be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/800c5e266b4ad8462a46970f0a232d52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f46b053f98b1d05a2043e94eeaefea87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f685277f6c178fb1fcd5e8387886721.png)
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3卷引用:河北省石家庄市七县联考2023-2024学年高二下学期3月月考数学试题
6 . 抛物线的弦与过弦的端点的两条切线所围成的三角形常被称为阿基米德三角形,该三角形以其深刻的背景、丰富的性质产生了无穷的魅力.设抛物线
(
),弦
过焦点
,
为其阿基米德三角形,则下列结论一定成立的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fc58c62444bf42a25289c45425a00f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5abd313d4e92a762fb7fb0c1cb65263d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b17f20c25bb16153b5f2d25062ed7a7.png)
A.点![]() ![]() ![]() ![]() |
B.存在点![]() ![]() |
C.![]() |
D.![]() ![]() |
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7 . 高斯是德国著名数学家,近代数学的奠基者之一,享有“数学王子”的称号,用他名字定义的函数
称为高斯函数,其中
表示不超过x的最大整数,如
,
,已知数列
满足
,
,
,若
,
为数列
的前n项和,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1550a97c21c1d71c9e95dde569668be0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d54a0e82778f606d95a486835ac9f56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21f2323cbdf0b1b71092c962ae705102.png)
![](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/d1928c254cfada1f75a5cd1e34db5a63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37d845281cd834068104af1b1aa6027c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7231e303ae39572f6c359c5e83822075.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5735a391a46cfdbd63e171769f8abb38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25c3ac959bdf1b78cb98d92b87c91c46.png)
A.2026 | B.2025 | C.2024 | D.2023 |
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7卷引用:江西省吉安市双校联盟2022-2023学年高二下学期期中考试数学试题
江西省吉安市双校联盟2022-2023学年高二下学期期中考试数学试题(已下线)4.3.2 等比数列的前n项和公式——课后作业(巩固版)云南省曲靖市第一中学2022-2023学年高一下学期7月期末考试数学试题内蒙古赤峰市赤峰二中2024届高三上学期第三次月考数学(理)试题(已下线)第五章 数列 专题8 数列中的递推(已下线)第五章 数列 专题7 有关数列求通项、周期性求和的问题陕西省西安市西安中学2024届高三上学期期末数学(理)试题
名校
8 . 对于三维向量
,定义“
变换”:
,其中,
.记
,
.
(1)若
,求
及
;
(2)证明:对于任意
,经过若干次
变换后,必存在
,使
;
(3)已知
,将
再经过
次
变换后,
最小,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/384c75b6d80b247b341e4d19f231a7dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41bd66e602e9c043218806708e943c2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed50f0b03a7cc5f809e222d283dfc2f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09b05756fbd0f41a4fb35e7379e6b6f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e68d20604666dd9b1be3a5756aa1e06a.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9f42fda276fc8add9ffded503884a0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a5c19921380da55f5f1a00809a34503.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35234a3829d238ea479fef9cec166468.png)
(2)证明:对于任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f389ec068eb1d1aa586b79097d70a7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daac610026ebae0358e9c56d7bf91ff8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03385c625de63ac95bff151de1e2ebe2.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab5d893313655986257eec42d3fcf7ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d344174267f996c7cefecfd6985d380.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6308724fa5b677baf09b81469bf042b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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6卷引用:北京市第十一中学2023-2024学年高二上学期期中练习数学试题
北京市第十一中学2023-2024学年高二上学期期中练习数学试题北京市东城区2022-2023学年高一下学期期末统一检测数学试题广东省东莞市石竹实验学校2023-2024学年高一下学期3月月考数学试卷(已下线)专题02 高一下期末真题精选(1)-期末考点大串讲(人教A版2019必修第二册)【北京专用】专题07平面向量(第三部分)-高一下学期名校期末好题汇编(已下线)专题08 期末必刷解答题专题训练的7种常考题型归类-期末真题分类汇编(北师大版2019必修第二册)
9 . 在平面直角坐标系中,定义
为两点
、
的“切比雪夫距离”,又设点
及
上任意一点
,称
的最小值为点
到直线
的“切比雪夫距离”,记作
,给出下列四个命题,正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32a7ccf5858c4bee028cd4f0c7a8537f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a3efb79f35db8448f3391252ab7d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df332f01628130c084fd46aaca0a4b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32286c3865f06865920816e7685c497a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87ab04028bf648fbb8c9296acdeaaf5a.png)
A.对任意三点![]() ![]() |
B.已知点![]() ![]() ![]() |
C.到定点![]() ![]() |
D.定点![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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2023-06-25更新
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985次组卷
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4卷引用:重庆市西南大学附属中学校2023-2024学年高二上学期期中数学试题
10 . 帕德近似是法国数学家亨利·帕德发明的用有理多项式近似特定函数的方法.给定两个正整数
,
,函数
在
处的
阶帕德近似定义为:
,且满足:
,
,
,
.已知
在
处的
阶帕德近似为
.注:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57986f853e0bfec0e2128309e7d71dad.png)
(1)求实数
,
的值;
(2)求证:
;
(3)求不等式
的解集,其中
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db527571cfd256c515424c6f9d114284.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab984fa2801f780e08903b339c9d041f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d8ef6c18c8edf9f4c781376d5ce400a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa6b902edcff913a34589487e17c9fe6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cf17fbb5f74fa34593ac47a0e8d3269.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/089b65749e52fc6346eab9bb5c49e5b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e96546b3259afe4add331673fb835c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d307aa65d930bc8e51835eb147de513.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96d128f7851b7771f95bffbdbf3ced02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57986f853e0bfec0e2128309e7d71dad.png)
(1)求实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f30a295015a8b1b038076f55f6ec928.png)
(3)求不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5ccd45ddc39488a73ebb0025e517059.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11204e2fb6e560bf7a4ca26eaebfc526.png)
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2023-04-26更新
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2479次组卷
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17卷引用:山东省济南市2022-2023学年高二下学期期中数学试题
山东省济南市2022-2023学年高二下学期期中数学试题(已下线)专题2 导数在研究函数单调性中的应用(B)重庆市璧山来凤中学校2023-2024学年高二下学期3月月考数学试题甘肃省白银市靖远县第四中学2023-2024学年高二下学期4月月考数学试题广东省中山市华辰实验中学2023-2024学年高二下学期第一次月考数学试题(已下线)模块四 期中重组篇(高二下山东) 重庆市巴蜀中学校2023届高三下学期4月月考数学试题吉林省白山市抚松县第一中学2022-2023学年高三第十一次校内模拟数学试题(已下线)重难点突破02 函数的综合应用(九大题型)(已下线)第十章 导数与数学文化 微点2 导数与数学文化(二)(已下线)第六套 九省联考全真模拟(已下线)微考点2-5 新高考新试卷结构19题压轴题新定义导数试题分类汇编(已下线)微考点8-1 新高考新题型19题新定义题型精选(已下线)专题22 新高考新题型第19题新定义压轴解答题归纳(9大核心考点)(讲义)(已下线)模块3 第8套 复盘卷(已下线)模块一 专题2 《导数在研究函数单调性中的应用》 B提升卷(苏教版)(已下线)专题12 帕德逼近与不等式证明【练】