名校
1 . 为应对新一代小型无人机武器,某研发部门开发了甲、乙两种不同的防御武器,现对两种武器的防御效果进行测试.每次测试都是由一种武器向目标无人机发动三次攻击,每次攻击击中目标与否相互独立,每次测试都会使用性能一样的全新无人机.对于甲种武器,每次攻击击中目标无人机的概率均为
,且击中一次目标无人机坠毁的概率为
,击中两次目标无人机必坠毁;对于乙种武器,每次攻击击中目标无人机的概率均为
,且击中一次目标无人机坠毁的概率为
,击中两次目标无人机坠毁的概率为
,击中三次目标无人机必坠毁.
(1)若
,分别使用甲、乙两种武器进行一次测试.
①求甲种武器使目标无人机坠毁的概率;
②记甲、乙两种武器使目标无人机坠毁的数量为
,求
的分布列与数学期望.
(2)若
,且
,试判断在一次测试中选用甲种武器还是乙种武器使得目标无人机坠毁的概率更大?并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44fed1be8b7e50f18cb90077d9fce8e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e5db9fa0bc36e2308bd3eecd5e78351.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f9f0aaaa2695dff4b08d7a52e4c905e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29be23f689eb01e57963495377501257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66577f4cb97c0d2a213ab1a9a02d1324.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f98df968dec4cb1b7e44cb47a5c216.png)
①求甲种武器使目标无人机坠毁的概率;
②记甲、乙两种武器使目标无人机坠毁的数量为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f2beb272e7c3342233f5cb681ac24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b70d4a3fc3e01b5a6358cf4e57578e6.png)
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3卷引用:山东省菏泽第一中学人民路校区2024届高三下学期5月月考数学试题
名校
2 . 在数学中,由
个数
排列成的m行n列的数表
称为
矩阵,其中
称为矩阵A的第i行第j列的元素.矩阵乘法是指对于两个矩阵A和B,如果4的列数等于B的行数,则可以把A和B相乘,具体来说:若
,
,则
,其中
.已知
,函数
.
(1)讨论
的单调性;
(2)若
是
的两个极值点,证明:
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e70354d6ca5ad9f6b4592fac0b5e559.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db452ec3c9e60109fdfe9fae8e456edf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fc970ba32a45946c514e98eac1e80ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e70354d6ca5ad9f6b4592fac0b5e559.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37a14c188b1c9d61aa237b137ba18023.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16a9a7e6a7ff34bb72659677929bf9c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7cf279527982a84842a2d6a4f212892.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b65887e38142a10f30be2296310d1a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9408afcaa76f52987ca43733b828f66a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01f44187cd898fb01a4f8fa76bdc6cdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b8d8e4e3f777270997845f7d9cfe85f.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aca579894dad67bc82cb715fd48e0d70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b3db0fe99d90b9a693562dd988eca5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/def4d3923ea803696106f42140e83bf4.png)
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3卷引用:山东省泰安市2024届高三四轮检测数学试题
山东省泰安市2024届高三四轮检测数学试题江西省宜春市宜丰中学2023-2024学年高二下学期6月月考数学试题(已下线)高二数学期末模拟试卷01【好题汇编】-备战2023-2024学年高二数学下学期期末真题分类汇编(北师大版2019选择性必修第二册)
3 . 已知点
是双曲线
上一点,
在点
处的切线与
轴交于点
.
(1)求双曲线
的方程及点
的坐标;
(2)过
且斜率非负的直线与
的左、右支分别交于
.过
做
垂直于
轴交
于
(当
位于左顶点时认为
与
重合).
为圆
上任意一点,求四边形
的面积
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/177ae4fa30cba0272d338973b8f7bdf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/221820d5f6209f9888cb0965bf99b1d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be452c8bdea6b4e4c7a6d96e9dc6a51c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5845ffc7d1af9053b027cf9e726f5367.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/946dd11e61102ea4ce0772603ae4edf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8da460ebf2fef232e43904aab520cd01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
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名校
4 . 在信息理论中,
和
是两个取值相同的离散型随机变量,分布列分别为:
,
,
,
,
,
.定义随机变量
的信息量
,
和
的“距离”
.
(1)若
,求
;
(2)已知发报台发出信号为0和1,接收台收到信号只有0和1.现发报台发出信号为0的概率为
,由于通信信号受到干扰,发出信号0接收台收到信号为0的概率为
,发出信号1接收台收到信号为1的概率为
.
(ⅰ)若接收台收到信号为0,求发报台发出信号为0的概率;(用
,
表示结果)
(ⅱ)记随机变量
和
分别为发出信号和收到信号,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b08fcbcf19c6ca72cd66c201ef43f9ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f4380cd57f824c5d9df1ca493cbd8cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfe82ce73937d36166659f21492c825e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1a870945a04cd86f2e0026fc53a2b36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12b0e3b00fe47801afb53ec56706c21a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27b4e8e7a49dbe86419e00672d1927c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd67429e1b0f56bc66a547fc9c6eed2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5633fa4fa8837dff506561b7943715fb.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17d0c830d39efe08dad4f2104325b8c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59a8bb9552579e3cd3c7d693ce37b445.png)
(2)已知发报台发出信号为0和1,接收台收到信号只有0和1.现发报台发出信号为0的概率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29c8578f06897aa6fb84aa95c797d3d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47d9b426bcc34a2cca2184dc1310f5e4.png)
(ⅰ)若接收台收到信号为0,求发报台发出信号为0的概率;(用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
(ⅱ)记随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3719852c05eef71dd595791e3dc10de7.png)
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4卷引用:2024届山东省实验中学高三下学期5月高考模拟数学试题
5 . 已知奇函数
对于
满足
,
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2dcc90624f573cd607f18729a761d8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50727bd86875bf1e1f6a2aed398dcb59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e6275d8a8c9f198adb651059c83e305.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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6 . 我们知道,在平面内取定单位正交基底建立坐标系后,任意一个平面向量,都可以用二元有序实数对
表示.平面向量又称为二维向量.一般地,n元有序实数组
称为n维向量,它是二维向量的推广.类似二维向量,对于n维向量,也可定义两个向量的数量积、向量的长度(模)等:设
,
,则
;
.已知向量
满足
,向量
满足
.
(1)求
的值;
(2)若
,其中
,当
且
时,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39c2af42141367e6e9ff0296c31daa7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62b3b354facacd72bc68da6ac07be453.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a48d974578eb15ca117e0cb1b59788d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99aa60676891adca75eac086182a15c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2581496116ddfba6dd03722fd771d5a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5babafd9f4e5c3c222ba25a3de66794.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a48d974578eb15ca117e0cb1b59788d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7cb2f5c0569962cd7c1026f388cb661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99aa60676891adca75eac086182a15c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4492fb816272cd60cf3456c6a064020e.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efa3e5481ce1f11ea4cb1d1ddc71413.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/301fa5679316c282923735aff9285559.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95ac252e9126ab540c0102b941f0ee42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36b98ef143f8159f3a7dafa1fd2f2370.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74cac554f22f3655ef6691b2ef821eac.png)
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解题方法
7 . 法国数学家弗朗索瓦·韦达发现了一元二次方程的根与系数之间的关系,将其推广到高次方程,并在其著作《论方程的识别与订正》中正式发表,后来人们把这个关系称为韦达定理,即如果
是关于x的实系数一元n次方程
在复数集C内的n个根,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/988acbe8533ef50c899650a057717cf5.png)
试运用韦达定理解决下列问题:
(1)已知
,
,
,求
的最小值;
(2)已知
,关于x的方程
有三个实数根,其中至少有一个实效根在区间
内,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44258e924e42ec263b5236499252d4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f44bf708f38a916de0572d8ef1cf45a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/988acbe8533ef50c899650a057717cf5.png)
试运用韦达定理解决下列问题:
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3eb9b6fe8959ae9e71e857b6d6fed49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/751e274e9107d780c39ba9c49d6daefb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/460b68eaa42bc8929edf81e21ad0bca0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bc2818de1c0d7d347718672b0bcec32.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73254f32b6da29ecc32df2e9f87a4c97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a070d41a05c5193153ae18e0235a492.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4a90cfdbfa05577b6ec0b22739e7c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/754826457671db8939098215943e656a.png)
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8 . 定义:若曲线
或函数
的图象上的两个不同点处的切线互相重合,则称该切线为曲线
或函数
的图象的“自公切线”.
(1)设曲线C:
,在直角坐标系中作出曲线C的图象,并判断C是否存在“自公切线”?(给出结论即可,不必说明理由)
时,函数
不存在“自公切线”;
(3)证明:当
,
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b0ee1a614e16f3092d318d74a252775.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e78b9c2b82517c887804b6ad8742a85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b0ee1a614e16f3092d318d74a252775.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e78b9c2b82517c887804b6ad8742a85.png)
(1)设曲线C:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cda51f0c169b59ac826994bebae3bc6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e2e79843faf62dde86bf858d1e0569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88a033e1ff47a23c84900de3c27ef453.png)
(3)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e2e79843faf62dde86bf858d1e0569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/655c46b33730f3a29b9ec3024df71375.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6725fd6db412e3c0caf9987018b43994.png)
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2024-05-30更新
|
433次组卷
|
2卷引用:山东省菏泽市定陶区第一中学2023-2024学年高二下学期5月月考数学试题
9 . 2024年7月26日至8月11日将在法国巴黎举行夏季奥运会.为了普及奥运知识,M大学举办了一次奥运知识竞赛,竞赛分为初赛与决赛,初赛通过后才能参加决赛
(1)初赛从6道题中任选2题作答,2题均答对则进入决赛.已知这6道题中小王能答对其中4道题,记小王在初赛中答对的题目个数为
,求
的数学期望以及小王在已经答对一题的前提下,仍未进入决赛的概率;
(2)
大学为鼓励大学生踊跃参赛并取得佳绩,对进入决赛的参赛大学生给予一定的奖励.奖励规则如下:已进入决赛的参赛大学生允许连续抽奖3次,中奖1次奖励120元,中奖2次奖励180元,中奖3次奖励360元,若3次均未中奖,则只奖励60元.假定每次抽奖中奖的概率均为
,且每次是否中奖相互独立.
(i)记一名进入决赛的大学生恰好中奖1次的概率为
,求
的极大值;
(ii)
大学数学系共有9名大学生进入了决赛,若这9名大学生获得的总奖金的期望值不小于1120元,试求此时
的取值范围.
(1)初赛从6道题中任选2题作答,2题均答对则进入决赛.已知这6道题中小王能答对其中4道题,记小王在初赛中答对的题目个数为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/161cafe3ab0c7b57ed23212f75c407e9.png)
(i)记一名进入决赛的大学生恰好中奖1次的概率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed1f109f79547d6ae0d94339e689e8f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed1f109f79547d6ae0d94339e689e8f7.png)
(ii)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
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2024-05-20更新
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2483次组卷
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3卷引用:山东省泰安市2024届高三下学期高考模拟((三模))数学试题
10 . 已知函数
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7737313c699e9902ee30ea1d4e1db53b.png)
A.曲线![]() ![]() ![]() |
B.方程![]() |
C.曲线![]() ![]() |
D.![]() ![]() |
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