名校
1 . 在概率统计中,常常用频率估计概率.已知袋中有若干个红球和白球,有放回地随机摸球
次,红球出现
次.假设每次摸出红球的概率为
,根据频率估计概率的思想,则每次摸出红球的概率
的估计值为
.
(1)若袋中这两种颜色球的个数之比为
,不知道哪种颜色的球多.有放回地随机摸取3个球,设摸出的球为红球的次数为
,则
.
(注:
表示当每次摸出红球的概率为
时,摸出红球次数为
的概率)
(ⅰ)完成下表,并写出计算过程;
(ⅱ)在统计理论中,把使得
的取值达到最大时的
,作为
的估计值,记为
,请写出
的值.
(2)把(1)中“使得
的取值达到最大时的
作为
的估计值
”的思想称为最大似然原理.基于最大似然原理的最大似然参数估计方法称为最大似然估计.具体步骤:先对参数
构建对数似然函数
,再对其关于参数
求导,得到似然方程
,最后求解参数
的估计值.已知
的参数
的对数似然函数为
,其中
.求参数
的估计值,并且说明频率估计概率的合理性.
![](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/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/613f6de938db4bb3a7f98226d3a4c793.png)
(1)若袋中这两种颜色球的个数之比为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc5881f1ce9b4172ca346032d0fd1e3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fadbd1d2d0294d04834dde31e0e4caaf.png)
(注:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c74de541a96a252ca6b4bf05381a03ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(ⅰ)完成下表,并写出计算过程;
0 | 1 | 2 | 3 | |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c74de541a96a252ca6b4bf05381a03ec.png)
![](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/0cf2e58249dd993ae42a7bd6d9ba0005.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cf2e58249dd993ae42a7bd6d9ba0005.png)
(2)把(1)中“使得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c74de541a96a252ca6b4bf05381a03ec.png)
![](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/0cf2e58249dd993ae42a7bd6d9ba0005.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0807dbbfdeeaeffd987c4de037b892f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acb13cf58c2aa7591391cfa8d515dc64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b1aecbef5ad07da9949972dbcb9d659.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c21d19789d426d0ed871d45ac6175f66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/889b80977780bb8caec3c90954b91a21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
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7卷引用:吉林省长春市实验中学2024届高三下学期对位演练考试数学试卷(一)
吉林省长春市实验中学2024届高三下学期对位演练考试数学试卷(一)浙江省杭州市2024届高三下学期4月教学质量检测数学试题重庆市七校联盟2024届高三下学期三诊考试数学试题贵州省贵阳市第一中学等校2024届高三下学期三模数学试题(已下线)压轴题08计数原理、二项式定理、概率统计压轴题6题型汇总山东省青岛第一中学2023-2024学年高二下学期第一次模块考试数学试题(已下线)专题02 高二下期末真题精选(压轴题 )-高二期末考点大串讲(人教A版2019)
名校
解题方法
2 . 已知函数
.
(1)若
,求
的值域;
(2)若关于x的方程
有三个连续的实数根
,
,
,且
,
,求a的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3a8af8cae82d4d35ae1c3fe57f55ef8.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b88ab2328fdf6dd3a53429345bba99f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若关于x的方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/577ae7344fd256ae4c8034a4c5fc83fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/291c25fc6a69d6d0ccfb8d839b9b4462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1310a7a80d1f8751a3f8cafe7f8c8b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/087470ba11dd0290e600f165fc19367f.png)
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名校
解题方法
3 . 已知函数
.
(1)若
,求
的极值;
(2)若
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccb0501cffbdfd96e56b8a2de2e59c4c.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48adb8a59b5c02fad5eada1b35171cf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e9c599e8d420006448905acec2b8234.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18f0281e6bbdbe08beeccb55adf84536.png)
您最近一年使用:0次
2024-06-14更新
|
504次组卷
|
2卷引用:吉林省长春市实验中学2023-2024学年高三下学期对位演练考试数学试卷(七)
名校
解题方法
4 . 已知双曲线
的左、右顶点分别为
,
,渐近线方程为
,过左焦点
的直线
与
交于
,
两点.
(1)设直线
,
的斜率分别为
,
,求
的值;
(2)若直线
与直线
的交点为
,试问双曲线
上是否存在定点
,使得
的面积为定值?若存在,求出定点
的坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a2cfa22139b3e9c9a73500e1ba19f52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b6bb019e2d7c6d17d15ec4d9043f5e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/030700126fb012f13935f57780b96677.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
(1)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77a7e4a6765ce78b05ee97764771e01f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/826c728050e3378921442ace20269ef6.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)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77a7e4a6765ce78b05ee97764771e01f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcdae78f4d3b8d8213ac3ac9a9567eb5.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/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11e29303195c563855aee4c14cbcb9bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
您最近一年使用:0次
2024-06-14更新
|
391次组卷
|
3卷引用:吉林省长春市实验中学2023-2024学年高三下学期对位演练考试数学试卷(七)
名校
5 . 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)
您最近一年使用:0次
2024-06-14更新
|
665次组卷
|
4卷引用:吉林省长春市实验中学2023-2024学年高三下学期对位演练考试数学试卷(七)
名校
解题方法
6 . 某单位招聘大学应届毕业生,已知共有6名学生进入最后面试环节,分别是来自A校的3人,
校的2人和
校的1人.该单位要求所有面试人员面试前到场,并随机给每人安排一个面试号码
,按面试号码
由小到大依次进行面试,每人面试时长5分钟.
(1)分别求面试号码为6号的学生来自A校、B校、C校的概率;
(2)记随机变量
表示最后一名A校学生完成面试所用的时长(从第1名学生开始面试到最后一名A校学生完成面试所用的时间),求
的分布列与数学期望.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6067693bd0eca1a1bc4940b990a867f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(1)分别求面试号码为6号的学生来自A校、B校、C校的概率;
(2)记随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
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解题方法
7 . 在四棱锥
中,底面ABCD是边长为2的正方形,
,直线PA与BC所成的角的正切值等于
、N分别是PB、PC的中点.
(2)证明:平面
平面ABCD;
(3)求平面MPD与平面APD夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62974d34de3a12418d6b700420afd1b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b8759906b81c12158e202cff8fd8b90.png)
(2)证明:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93edc7bb513f40a89173121c8570cd65.png)
(3)求平面MPD与平面APD夹角的余弦值.
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8 . 记集合
无穷数列
中存在有限项不为零,
,对任意
,设
.定义运算
若
,则
,且
.
(1)设
,用
表示
;
(2)若
,证明:
:
(3)若数列
满足
,数列
满足
,设
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/100d76814e366c60298ea21aad6ddea4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b21aa3e2f0c8de96d08195e5f66b725a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ba8cfb33f75f570c4d9cab8b522be30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e8b70b7fbc19242014383f0ee8621dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff83cd0e7e7b17d6f90cd29b3fe7a19b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a083253cd5a7df93f553e5e71b4aa7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87adb7b83f14cc809c1b7161e83c171f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77c5afa350510e7a8b3b27b5fa7803ad.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/557d484154f5ff1194d22e1b02fff5dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/202bc33cd714c241671d6d4457c5637f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/342d0252c1b2f7d2a84b5c985d19d547.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9660167870a1eed0a0d19edc430c8180.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1196e9280fbc7cbd6a01694af1dd42c.png)
(3)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64ed1a5374a245c7cc789dd17c2f9be0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20c103c41bc5f744916b1aa6e0b38c23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/557d484154f5ff1194d22e1b02fff5dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b597616902954c408ef4d86b25016c98.png)
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名校
9 . 已知双曲线
的一条渐近线方程为
,焦距为6,左顶点为
,点
是双曲线
的右支上相异的两点,直线AB,AC分别与直线
交于点
,且以线段
为直径的圆恰过双曲线
的右焦点
.
(1)求双曲线
的标准方程;
(2)求
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ef66f4832adc43902055a7e6d258037.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25869dd14f3e7412beda491bb83f982d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab609a6574633ebabcff3e73fa862081.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40460e97733a56b0b9963f8c641c47c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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解题方法
10 . 已知函数
,其中
.
(1)若曲线
在
处的切线在两坐标轴上的截距相等,求
的值;
(2)是否存在实数
,使得
在
(
为自然对数的底数)上的最大值是
?若存在,求出
的值;若不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffc4839b64009be5e58cd24fd22f7a40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10bbdef421c976962a270a2beabbad91.png)
(1)若曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b211dbdc592cb219df3f4b779755231.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041a7c8fc017f596542c5e6ec7d1c40b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274a9dc37509f01c2606fb3086a46f4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次