解题方法
1 . 信息熵是信息论中的一个重要概念.设随机变量X所有可能的取值为1,2,
,n,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0da467b6613d9d20676bc9bddbe6f5ca.png)
,
,定义X的信息熵
,则下列判断中正确的是( )
①若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee2c054f4510895b4c526fedbe3d4f64.png)
②若
,则
;
③若
,则当
时,
取得最大值
④若
,随机变量Y所有可能的取值为1,2,
,m,且
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7802794b6d58c5755eb4a471f270533.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0da467b6613d9d20676bc9bddbe6f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85719767bc8b764fcde16731c1ea45c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25c94a17b49550283be4ec1a348c8534.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae49e5608e5b61ac710f93955af5798e.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba47fb63d946d478fcc51769657eeada.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee2c054f4510895b4c526fedbe3d4f64.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7947b04dfd9793eefe588ae1696f32eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c87b351f16728b0023fd63678f8103c7.png)
③若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc2d3df37e73a8abea815f37dbb3fff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e5f84b0617cbfa27da74b62ef8aae4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5790d5181783c15fd46d95bf18b796f0.png)
④若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6b21c7729887167e605d912861339bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5baa3c2e56ea2aca943b9b2a5b938b46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7802794b6d58c5755eb4a471f270533.png)
A.①② | B.②③ | C.①②④ | D.①②③④ |
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解题方法
2 . 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)
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3 . 已知函数
.
(1)讨论函数
在
上的单调性;
(2)若
,函数
有两个极值点
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8ec7c787929d13c7161528f61f3343.png)
(1)讨论函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/879234adbae93aa72b7e101b3738d4e0.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b140e221ddf537b8964fff8557cca0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/050b3e1a7bf3fb787b79cf0cdbb2f2f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ac1deb6fd4bb982996f4a80cb1ed221.png)
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名校
4 . (1)已知
为
中点,过点
作
于
,交
于点
,求
.
(2)已知
,过点
作
于
,交
于点
,求
.
(3)在(2)的条件下,
为常数,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0c4c49765e33742b4204d6904dc3a98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e39b13d187b25461d85a3b8d10c7b678.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5378a3867bf1a7386b1330aa8b36f0a7.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/24/1e748517-2524-4763-aeb1-30d7a2bd7e1d.png?resizew=155)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df89cefacfd826262825276a739ca4f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e39b13d187b25461d85a3b8d10c7b678.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5378a3867bf1a7386b1330aa8b36f0a7.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/24/7e2e9fd4-ce48-414d-8682-dee6f813a104.png?resizew=154)
(3)在(2)的条件下,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f03cab451843012fd80fa6cc698c648.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbe857997fa13837880f520673db7809.png)
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解题方法
5 . 已知椭圆
的离心率为
,依次连接四个顶点得到的图形的面积为
.
(1)求椭圆C的方程;
(2)过直线
上一点P作椭圆C的两条切线,切点分别为M,N,求证:直线
过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41322821ce31416fdac8dd6e0aa41c71.png)
(1)求椭圆C的方程;
(2)过直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f23d29646155e27b172ecdf263e2d702.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
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名校
解题方法
6 . 已知函数
.若
,不等式
恒成立,则实数
的取值范围是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9b919a59954cf503f515e45573deba6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d96b743603ab1c10330622f16db78dbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b9f109b7ce6ec37e69d54ec70643c55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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名校
7 . 某制药公司研制了一款针对某种病毒的新疫苗.该病毒一般通过病鼠与白鼠之间的接触传染,现有
只白鼠,每只白鼠在接触病鼠后被感染的概率为
,被感染的白鼠数用随机变量X表示,假设每只白鼠是否被感染之间相互独立
(1)若
,求数学期望
;
(2)接种疫苗后的白鼠被病鼠感染的概率为
,现有两个不同的研究团队理论研究发现概率
与参数
的取值有关.团队A提出函数模型为
,团队B提出函数模型为
.现将100只接种疫苗后的白鼠分成10组,每组10只,进行实验,随机变量
表示第
组被感染的白鼠数,将随机变量
的实验结果
绘制成频数分布图,如图所示.
”发生的概率表达式(用
表示,组合数不必计算);
(ⅱ)在统计学中,若参数
时使得概率
最大,称
是
的最大似然估计.根据这一原理和团队A,B提出的函数模型,判断哪个团队的函数模型可以求出
的最大似然估计,并求出最大似然估计.参考数据:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/907e780479f1a9e1371b491538ace976.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b0919cf56a1b743189a019551b2d5a0.png)
(2)接种疫苗后的白鼠被病鼠感染的概率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1daba45353250e5d756cd483c681dbce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a2bc9afcbcb1fa2f356beb0fadecf8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7d0178b72c60a4fa59a815c1d9fa995.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd30ac5421bb71b65ec79d08c04ed686.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd30ac5421bb71b65ec79d08c04ed686.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa4355ff47d4d0206d9ab17d450c5a95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a52d31cef5bdd028641ae0ed4f3a2464.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
(ⅱ)在统计学中,若参数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41ba7a71c56fe7355a2b3ad5bade55ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aedd1de682115fdc8b5107bf8f9dc60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a4438bae1705c0f26beddf41322c087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07bf99e2f6d5c0c024c3e26a65a61b63.png)
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2024-02-23更新
|
1506次组卷
|
6卷引用:浙江省9+1高中联盟2022-2023学年高二下学期期中数学试题
解题方法
8 . 数列
满足
且
,
,
,
构成等差数列.
(1)试求出所有三元实数组(α,β,γ),使得
为等比数列.
(2)若
,求
的通项公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9f390f47fa5678c9a165c50fb9dec58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be536a2097ded867adac5edebb79906b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75e6820c50fa2aa589de5331d7d5f950.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13739ca823d61005798cc3298400c6b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad28237c0f9ca65341101d9d7e73e73e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cee766a75ae9ee290e403b42b3569db6.png)
(1)试求出所有三元实数组(α,β,γ),使得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cee766a75ae9ee290e403b42b3569db6.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4623bc660145c6ff98af7b1753d5357a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cee766a75ae9ee290e403b42b3569db6.png)
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9 . 设![](https://staticzujuan.xkw.com/quesimg/Upload/formula/176bf1f7402b32bddd0c38e081fc79f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c90282d4a37c9a20620d4bbb0c263cae.png)
(1)若
,讨论
的单调性;
(2)若
,求
的最大值(用
表示);
(3)若
恰有三个极值点,直接写出
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/176bf1f7402b32bddd0c38e081fc79f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c90282d4a37c9a20620d4bbb0c263cae.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b4d795709b0abcf47bceec2250f2f9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce3a34d6f60032718820c3da2b07786b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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10 . 设双曲线Γ:
,
,B,C在Γ上且直线
经过A.设
分别为Γ在B,C处的切线,点D满足
,则D的轨迹方程是___________ ;若D的横纵坐标均为正整数,且二者之和大于2024,则D可以是_____ .(写出1个即可).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af3674d64026a371b001453280477ba5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40630a669f4eedf626bc24851df10c85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86de0e038b0304e34ca8c9b117fec917.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04e6aa92080d4e1f43a1ac2f9c2965fe.png)
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