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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:重庆市乌江新高考协作体2023-2024学年高二下学期第二阶段性学业质量联合调研抽测(5月)数学试题
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解题方法
2 . 如图,在多面体
中,平面
与平面
均为矩形且相互平行,
,设
.
平面
;
(2)若多面体
的体积为
:
(i)求
;
(ii)求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d17d4a6cf11cda87b3dfafaecdec683f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/611f100dcfa7803db6eb233e2e7f2dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dff64de03b0302dbc12f2fc207b70d1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/336e0a8f5fbc1c44a02adab5a1fffb60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dc99203b785fbdbd399bb03c7556fbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/611f100dcfa7803db6eb233e2e7f2dab.png)
(2)若多面体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d17d4a6cf11cda87b3dfafaecdec683f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a391005600bdd69c96750589f9adb048.png)
(i)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
(ii)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b03428a8f91a5674cb8f54766c165f7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffee8b7eff437080a0936d837ceabe95.png)
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2卷引用:重庆市乌江新高考协作体2023-2024学年高二下学期第二阶段性学业质量联合调研抽测(5月)数学试题
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3 . 在棱长为 1 的正方体
中,已知
分别为线段
的中点,点
满足
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cf9b288c48c73463a2f214f02b6952a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85789b7d63712c81dcc0fb60014bbb8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b829ac65651fac7a19a0b837939c3ff.png)
A.当![]() ![]() |
B.当![]() ![]() ![]() |
C.![]() ![]() |
D.若![]() ![]() ![]() |
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2卷引用:重庆市乌江新高考协作体2023-2024学年高二下学期第二阶段性学业质量联合调研抽测(5月)数学试题
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4 . 代数基本定理:任何一个
次复系数多项式方程
至少有一个复根.由此可得如下推论:
推论一:任何一元
次复系数多项式
在复数集中可以分解为
个一次因式的乘积;
推论二:一元
次多项式方程有
个复数根,最多有
个不同的根.即一元一次方程最多有1个实根,一元二次方程最多有2个实根等.
推论三:若一个
次方程有不少于
个不同的根,则必有各项的系数均为0.
已知
.请利用代数基本定理及其推论解决以下问题:
(1)求
的复根;
(2)若
,使得关于
的方程
至少有四个不同的实根,求
的值;
(3)若
的图像上有四个不同的点
,以此为顶点构成菱形
,设
,
,求代数式
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ab009a153dfcc13ba9eb4916c76f8ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b7bff9b2431134f7683a9cc4e68acd.png)
推论一:任何一元
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ab009a153dfcc13ba9eb4916c76f8ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
推论二:一元
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
推论三:若一个
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0876215b2fd463d151523cd3c6b447.png)
已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b14c686bfce270ec65d068555d1866ff.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dadabea3f5008d97a32382752e62bdd8.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ec4e65c4c043edef8084b292675395c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcecb855c13987b207aec2db73c9ec5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fc04eee630e386f7be4ac709ff4e16c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df74fc4cedb204eb6dcce64b706e99c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed0c942fae0e9dd2d219ad8269511898.png)
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5 . 已知椭圆
:
的右焦点为
,过点
作
轴的垂线交椭圆
于点
.过点
作椭圆
的切线,交
轴于点
.
(1)求点
的坐标;
(2)过点
的直线(非
轴)交椭圆
于
、
两点,过点
作
轴的垂线与直线
交于点
,求证:线段
的中点在定直线上.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0dfb290b1a84f670549554a0c988593.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdba1337ec85fa9722cb4b320a82ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
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解题方法
6 . 有
个编号分别为1,2,…,
的盒子,第1个盒子中有3个白球1个黑球,其余盒子中均为1个白球1个黑球,现从第1个盒子中任取一球放入第2个盒子,再从第2个盒子中任取一球放入第3个盒子,以此类推,从第
个盒子中取到白球的概率是_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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7 . 伯努利双纽线最早于1694年被瑞士数学家雅各布·伯努利用来描述他所发现的曲线.在平面直角坐标系
中,把到定点
,
距离之积等于
的点的轨迹称为双纽线,已知点
是
的双纽线
上一点,下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcb162568cb923c31c7209c8a22e4674.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cca4d2dd6a806193dfd4d66991a48a82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be45dd63a0db0b7ab458f30ee6a67881.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aee82283f06cedef32eb15b87964f5d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
A.若直线![]() ![]() ![]() ![]() ![]() ![]() ![]() |
B.双纽线![]() ![]() |
C.![]() ![]() |
D.![]() ![]() |
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解题方法
8 . 设集合
(
),
为
的非空子集,随机变量
,
分别表示取到子集
中得最大元素和最小元素的数值.
(1)若
的概率为
,求
;
(2)若
,求
且
的概率;
(3)已知:对于随机变量
,
,有
.求随机变量
的均值
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9dfe86bf99f7bd82b3ea703febf26ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.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/5963abe8f421bd99a2aaa94831a951e9.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6c4b25a0b76fba785d5769c08714b15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41ab109ec88d6f3d24b2f01ca77e7038.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe08722cf9300fe188dbbb71989c06c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e32a2f594955e456f0fddad1e090bb04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8b3576b4d98a5b4ddc380ddaa0fa281.png)
(3)已知:对于随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/096b1ece1dcd29c59a46a4b3e02cb548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5031a3a951c4a1d1c5e9f80a5e26513.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bed5c625495d0ae6d4c3c476aa73c80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec9f6ea6346066054b5c722763d6b026.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4e2517ab0c7decdfd0f90c79dc3cb16.png)
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9 . 已知函数
是定义域为
的可导函数,若
,且
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee0b94eef7875086f8aff56d4e1de81e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a1d88f08209350ff227f0f4ddba626b.png)
A.![]() | B.![]() |
C.![]() | D.![]() ![]() |
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10 . 已知函数
.
(1)当
时,讨论
的单调性;
(2)若曲线
在点
处的切线与曲线
也相切,求实数
的值;
(3)若不等式
对任意的
恒成立,求
的取值范围.
为自然对数的底数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d21981615f871746645b1c97031b771.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5c4b27524cee9197557b528bcf536b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4ce5dbad6b45921e407123f4a7acefa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)若不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef849152f5509a13bdb8c2d5b0694c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66692ec49a458f9e48c7315d03dfc37b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d8010f9d55e091cac9c543defc9faa9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04582116cd765fcc5a52f44279ad6c94.png)
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