1 . 某批零件一级品的比例约为
,其余均为二级品.每次使用一级品零件时肯定不会发生故障,而在每次使用二级品零件时发生故障的概率为
.某项任务需要使用该零件
次(若使用期间出现故障则换一件使用).
(1)某零件在连续使用3次没有发生故障的条件下,求该零件为一级品的概率;
(2)当
时,求发生故障次数
的分布列及期望.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbb00d558e456638de8ff1788db5a8d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a87796ee30e6c5d5e6b6285b32abe10c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)某零件在连续使用3次没有发生故障的条件下,求该零件为一级品的概率;
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc2d3df37e73a8abea815f37dbb3fff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
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名校
2 . 有甲乙两个骰子,甲骰子正常且均匀,乙骰子不正常且不均匀,经测试,投掷乙骰子得到6点朝上的概率为
,若投掷乙骰子共6次,设恰有3次得到6点朝上的概率为
,
是
的极大值点.
(1)求
;
(2)若
且等可能地选择甲乙其中的一个骰子,连续投掷3次,在得到都是6点朝上的结果的前提下,求这个骰子是乙骰子的概率;
(3)若
且每次都等可能地选择其中一个骰子,共投掷了10次,在得到都是6点朝上的结果的前提下,设这10次中有
次用了乙骰子的概率为
,试问当
取何值时
最大?并求
的最大值(精确到0.01).(参考数据
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed1f109f79547d6ae0d94339e689e8f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3606c4a853a6a34cb7f33bea81b15a1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed1f109f79547d6ae0d94339e689e8f7.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3606c4a853a6a34cb7f33bea81b15a1f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/098e663b79254b0a2e0e00f92bd14b8d.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/098e663b79254b0a2e0e00f92bd14b8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e97dd472fe7779d5c729aa8dedd99190.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e97dd472fe7779d5c729aa8dedd99190.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e97dd472fe7779d5c729aa8dedd99190.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdcbd28fefa404513768b10747e2291a.png)
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解题方法
3 . 在数列
的第
项与第
项之间插入
个1,称为变换
.数列
通过变换
所得数列记为
,数列
通过变换
所得数列记为
,以此类推,数列
通过变换
所得数列记为
(其中
).
(1)已知等比数列
的首项为1,项数为
,其前
项和为
,若
,求数列
的项数;
(2)若数列
的项数为3,
的项数记为
.
①当
时,试用
表示
;
②求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b00f4eb7f1bd2ccefbabf0c1dfa8f69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fd6f81b87ebc6a949329520a6e42ed8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fd6f81b87ebc6a949329520a6e42ed8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b536225194ade4db8e99df253041e034.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/819f6ef3827f73ee7dea7883be3bdfdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3788785078f4cbc29eb4b3c202d1587.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
(1)已知等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.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/b1752474698cd5466dd180df0a00ba9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e5577ef4b9b5764af5e21eef60ebca6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fd6f81b87ebc6a949329520a6e42ed8.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3788785078f4cbc29eb4b3c202d1587.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e9d41db805f5235ec4b69e436690222.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
②求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/147a33b715da5466c2d4f9b18cb925d7.png)
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解题方法
4 . 已知圆
,直线
,直线
和圆交于A,B两点,过A,B分别做直线
的垂线,垂足为C,D.
(1)求实数b的取值范围;
(2)若
,求四边形ABDC的面积取最大值时,对应实数
的值;
(3)若直线AD和直线BC交于点
,问是否存在实数
,使得点
在一条平行于
轴的直线上?若存在,求出实数
的值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79382ba44ba669b5d43fdd5427adf16c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13b673b7961af5c4b51e67d6ffe00912.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aa941aa5d428f5d78ec3f281464f826.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
(1)求实数b的取值范围;
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdbf6506e9de40da0f3c51b81b35a901.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
(3)若直线AD和直线BC交于点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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解题方法
5 . 对于有穷数列
,从数列
中选取第
项、第
项、
、第
项
,顺次排列构成数列
,其中
,则称新数列
为
的一个子列,称
各项之和为
的一个子列和.规定:数列
的任意一项都是
的子列.则数列
的所有子列和的和为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df5f59bc23cf55f56312c9ed9806371f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6af9e7b1c23db5584ad8521d4444d5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/608d034715f9b1dfb306f9c89d383582.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9e636879245230dd00a3ab3cbcfbfd1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dcb1ddb73e4087f8cfcc40eead8b893.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/533484c347e86e6bb635006a94530dc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dcb1ddb73e4087f8cfcc40eead8b893.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dcb1ddb73e4087f8cfcc40eead8b893.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa301d4a86341fd749f1f8c72c970565.png)
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6 . 某高校统计的连续5天入校参观的人数(单位:千人)如下:
并计算得,
.
(1)求
关于
的回归直线方程,并预测第10天入校参观的人数;
(2)已知该校开放1号,2号门供参观者进出,参观者从这两处门进校的概率相同,且从进校处的门离校的概率为
,从另一处门离校的概率为
.假设甲、乙两名参观者进出该校互不影响,已知甲、乙两名参观者从1号门离校,求他们从不同门进校的概率.
附:回归直线方程
,其中
.
样本号 | 1 | 2 | 3 | 4 | 5 |
第 | 1 | 2 | 3 | 4 | 5 |
参观人数 | 2.4 | 2.7 | 4.1 | 6.4 | 7.9 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/985f1673ddf7ed5872c91f01ba8eef01.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)已知该校开放1号,2号门供参观者进出,参观者从这两处门进校的概率相同,且从进校处的门离校的概率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
附:回归直线方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1db6103cb0f1d2bd6b19235d53ee7e98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/446c21b8025405469a473aa0b32f9373.png)
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解题方法
7 . 截面惯性矩
是衡量截面抗弯能力的一个几何参数,若截面图形为矩形,则
,其中
为矩形的宽,
为矩形的高.某木器厂要加工如图所示的长方体实木梁,已知该实木梁的截面图形为矩形
,且矩形
外接圆的直径为
,要使该截面的惯性矩最大,则矩形
对应的高应为______
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e105760638b22b26ff8bec4354255e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/910c6cc1a6e5f6f601f9336d5f08c7a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eabd5f3a86afe49dcd70571e2b96cfd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbe9f7abf7bcf4e1aa2579cd191d7761.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efa9fbcfb9595e2f031aa691db4564b.png)
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解题方法
8 . 已知
是函数
有四个零点,记
的导函数为
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8ccd22fd0ca1a8e1468329284f91b6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51fb3ef65a6ee809219a3f9f78d127fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a4b04824a308519a61318a82aa97a05.png)
A.![]() | B.![]() |
C.![]() ![]() ![]() | D.存在![]() ![]() |
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解题方法
9 . 甲、乙、丙、丁四位同学坐在一排
个座位上,由于某种原因,甲旁边要留一个空座位,则共有______ 种坐法.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d91e07104b699c4012be2d26160976a2.png)
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解题方法
10 . 已知函数
.
①当
时,
,记
前
项积为
,若
恒成立,整数
的最小值是______________ ;
②对所有n都有
成立,则
的最小值是_____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/227c3613c9bdafd6f258c53dbf49776b.png)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e258ab9e600435b37465092243d99f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bef2639f7d9950f9481a5ddd92334be7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40c16b5df55ae677b721c5401ca1a8f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
②对所有n都有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f791e32e2606e39c024f26c7a02feb08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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