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1 . 一个车间有3台机床,它们各自独立工作,其中
型机床2台,
型机床1台.
型机床每天发生故障的概率为0.1,B型机床每天发生故障的概率为0.2.
(1)记X为每天发生故障的机床数,求
的分布列及期望
;
(2)规定:若某一天有2台或2台以上的机床发生故障,则这一天车间停工进行检修.求某一天在车间停工的条件下,B型机床发生故障的概率.
![](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)
(1)记X为每天发生故障的机床数,求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fc79c66ebaacd709ec9965b90a22b14.png)
(2)规定:若某一天有2台或2台以上的机床发生故障,则这一天车间停工进行检修.求某一天在车间停工的条件下,B型机床发生故障的概率.
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解题方法
2 . 已知数列
的前
项和为
,若存在常数
,使得
对任意
都成立,则称数列
具有性质
.
(1)若数列
为等差数列,且
,求证:数列
具有性质
;
(2)设数列
的各项均为正数,且
具有性质
.
①若数列
是公比为
的等比数列,且
,求
的值;
②求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc983f1bad03411ae64d84ff7bdf2551.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64a548095fa134cb2b52721af225cbbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ee7ed704a954d0414be6c3148bd566d.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/193a0efaa1aa835eb3e061bb25dad4dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7470297de40027847c5c73fc5d1719c.png)
(2)设数列
![](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/1ee7ed704a954d0414be6c3148bd566d.png)
①若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4338dd5d6ac02dbb9d5069eb98f436d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
②求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
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4卷引用:江苏省泰州市2024届高三下学期四模数学试题
江苏省泰州市2024届高三下学期四模数学试题河南师范大学附属中学2024届高三下学期最后一卷数学试题江西省临川第二中学2023-2024学年高二下学期6月月考数学试题(已下线)高二下期末考前押题卷01--高二期末考点大串讲(人教B版2019选择性必修)
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解题方法
3 . 为培养学生的阅读习惯,某学校规定所有学生每天在校阅读时长不得少于1小时.若认为每天在校阅读的时长不少于1小时为达标,达到2小时的学生为“阅读之星”.假设该校学生每天在校阅读时长
(
的单位:小时),达标学生是“阅读之星”的概率为
.
(1)从该校学生中随机选出1人,求达标的概率;
(2)为进一步了解该校学生不达标是否与性别有关,随机调查了90名学生,其中男生占
,已知不达标的人数恰是期望值,且不达标的学生中男生占
,是否有99%的把握认为不达标与性别有关?
附:参考公式:
,其中
.
参考数据:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33b9b2211d82dac614b099eb52c67c68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d266a04f3dc7483eddbc26c5e487db.png)
(1)从该校学生中随机选出1人,求达标的概率;
(2)为进一步了解该校学生不达标是否与性别有关,随机调查了90名学生,其中男生占
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d86ab7c97cd8a0b15ba5efc1be94230.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e6486784415f3537c9a13556c05d893.png)
附:参考公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2187714e660234f0b72f2b47d3ea685a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/356b05e46b10ee51c3e43546d73ec96c.png)
参考数据:
3.841 | 5.024 | 6.635 | 10.828 | |
0.050 | 0.025 | 0.010 | 0.001 |
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解题方法
4 . 为某商品设计一个“H”型商标,如图所示,“H”型商标由两竖一横三个等宽的矩形组成.设计要求“H”型商标关于点O中心对称,两个竖直矩形全等且它们的长边是横向矩形长边的2倍,点O到点A的距离为4cm.若记“H”型商标的面积为S,则S的最大值为________
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88ce13774b09ff2edddaf21a072cf60a.png)
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5 . 已知
,有一组数据为
,3,
,
,8,10,
,12,13,若在这组数据中去除第5个数8,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7569cd7e9b31ad838230133b9bc8314.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d91df89feab98e01e76cf66a4043879c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3143ddb1813fa9a0ad3786132686908.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b34fd89f8dd6a08a40651903832b854.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8fb9a8167ae69c721b9143914d6bc23.png)
A.平均数不变 | B.中位数不变 | C.方差不变 | D.极差不变 |
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6 . (1)已知各项均为正数的无穷数列
满足:对于
,都有
,
,求数列
的通项公式;
(2)已知各项均为正数的无穷数列
满足:对于
,都有
,其中
为常数.
①若
,
,记
,数列
的前
项和
满足
,求数列
的通项公式:
②记
,证明:数列
中存在小于1的项.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bde2576b383ae3c851529435805b3adc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ea27f3bbd9ff6515c7d957889202c8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09ed1465c2469cd518a13802bf6044fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)已知各项均为正数的无穷数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bde2576b383ae3c851529435805b3adc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86bb4325524dcb935bfd167cab6fb09f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09ed1465c2469cd518a13802bf6044fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f66884efff7400f92b530d69d029778d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/472b92ebc99baaa71adf06ce85df434c.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/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a4c9e626304cd58d8a995a0e4813ba2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
②记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41d3db5570a5ab31ff7468c0d64d0f42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
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7 . 从棱长为
的正方体的八个顶点中任意取四个点
,则
值的不同种数为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec92d556c107f92dddb25625d3560309.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
8 . 在正三棱柱
中
,
的重心为
,以
为球心的球与平面
相切.若点
在该球面上,则下列说法正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/141ef400af3ec09829c4a640867acea9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c510b85dfbca0e3ab0744655d77e8c93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e168672b47d7e64dc1b404f8882c7dcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
A.存在点![]() ![]() ![]() |
B.三棱锥![]() ![]() |
C.若直线![]() ![]() ![]() ![]() ![]() |
D.若![]() ![]() ![]() |
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9 . 假定射手甲每次射击命中目标的概率为
其中
.
(1)当
时,若甲射击
次,命中目标的次数为
.
①求
;
②若
其中
求
的值.
(2)射击积分规则如下:单次未命中目标得
分,单次命中目标得
分,若连续命中目标
次,则其中第一次命中目标得1分,后一次命中目标的得分为前一次得分的2倍.记射手甲射击4次的总得分为
,若对任意
有
成立,求所有满足上述条件的有序实数对
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aac95e741a0cda29e2e907e4f114d2e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20c11f6c800b8e0410674a0c6d307d26.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e74b0aa7a6f6dcab7d9101b98504ae2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf3baba074e8aeb6f3ea117865bbd1b.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02af71919e09c05e34b95ba014bf07ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6f594e2141ce84a0b5d36f50b66b7e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
(2)射击积分规则如下:单次未命中目标得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c95b6be4554f03bf496092f1acdfbb89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/841f8c6051dff35b6f44c89ff3117267.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ca1e2f8e87b1b90a214cd6aad99cb7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb4d5ca4e251ff0e503a26f9a7375326.png)
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10 . 甲、乙两名篮球运动员连续10场比赛的得分如下表所示,则下列说法正确的有( )
场次 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
甲 | 18 | 20 | 22 | 13 | 20 | 27 | 10 | 21 | 19 | 30 |
乙 | 3 | 10 | 20 | 9 | 24 | 27 | 13 | 28 | 9 | 17 |
A.甲的众数大于乙的众数 |
B.甲的平均数大于乙的平均数 |
C.甲的极差大于乙的极差 |
D.甲的60百分位数大于乙的60百分位数 |
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