1 . 等差数列
的前
项和为
,
(
且
),
.
(1)求
的通项公式与前
项和
;
(2)记
,当
,
时,试比较
与
的大小;
(3)若
,正项等比数列
中,首项
,数列
是公比为4的等比数列
,且
,求
的通项公式与
.
![](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/e7fab51121848ce166035ceab6f4e00b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22dd8b3dc4c609bab82d356a5cc2208d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20849c00c47cbdc43f18d53341b6c4e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec3c9f09e932ad3cc1f3720d93b34c56.png)
(1)求
![](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)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1dcf4e0dd72f254d9eab961b892ae9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15520cf5be7c2685975aac51bc99ac4f.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e258ab9e600435b37465092243d99f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/385275d29d8c8a7841eaeaa3dfab2cdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7afef6271af7462ffa935a1846e3ec90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89ba85f74cda4ddd621278e558bc036f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3ee3a47bf7dfe538ae367cc006a69b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ea2efac902bfb513a224c40faef2516.png)
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解题方法
2 . 多年统计数据表明如果甲、乙两位选手在决赛中相遇,甲每局比赛获胜的概率为
,乙每局比赛获胜的概率为
.本次世界大赛,这两位选手又在决赛中相遇.赛制为五局三胜制(最先获得三局胜利者获得冠军).
(1)现在比赛正在进行,而且乙暂时以
领先,求甲最终获得冠军的概率;
(2)若本次决赛最终甲以
的大比分获得冠军,求甲失分局序号之和
的分布列和数学期望.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
(1)现在比赛正在进行,而且乙暂时以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/124656394b0674aa1266ba4760bc602f.png)
(2)若本次决赛最终甲以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b1dcdac71e394e495d069f64e1f1ce9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
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7日内更新
|
402次组卷
|
4卷引用:河南省濮阳市2024届高三第三次模拟考试数学试题
河南省濮阳市2024届高三第三次模拟考试数学试题2024届河南省名校联盟考前模拟大联考三模数学试题陕西省西安市第一中学2023-2024学年高三下学期高考考前模拟考试理科数学试题(已下线)概率、随机变量及其分布-综合测试卷A卷
名校
解题方法
3 . 定义两组数据
,
的“斯皮尔曼系数”为变量
在该组数据中的排名
和变量
在该组数据中的排名
的样本相关系数,记为
,其中
.
某校15名学生的数学成绩的排名与知识竞赛成绩的排名如下表:
(1)试求这15名学生的数学成绩与知识竞赛成绩的“斯皮尔曼系数”;
(2)已知在这15名学生中有10人数学成绩优秀,现从这15人中随机抽取3人,抽到数学成绩优秀的学生有
人,试求
的分布列和数学期望.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ae7c00ee60437d7056db158a57287f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d5d2984708e34c2f656b8a508d693f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49c3e347cec6bed59c813db91a7f69b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9454ddb2d570f884b15bd3ddf2a4545d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be3069959e6676e603dfa3494c66e6df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bbf83fbb1c900c35ce9a92ba5ed3749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/171102a883b22fe6ca578efc8926f5b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37ce55632072eba85b2741b55fd154c6.png)
某校15名学生的数学成绩的排名与知识竞赛成绩的排名如下表:
![]() | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
![]() | 1 | 5 | 3 | 4 | 9 | 8 | 7 | 6 | 10 | 2 | 12 | 14 | 13 | 11 | 15 |
(1)试求这15名学生的数学成绩与知识竞赛成绩的“斯皮尔曼系数”;
(2)已知在这15名学生中有10人数学成绩优秀,现从这15人中随机抽取3人,抽到数学成绩优秀的学生有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
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2024-05-24更新
|
410次组卷
|
2卷引用:江西省萍乡市2024届高三二模考试数学试卷
4 . 已知
,
,平面内动点P满足
.
(1)求动点P的轨迹C的方程;
(2)动直线
交C于A、B两点,O为坐标原点,直线
和
的倾斜角分别为
和
,若
,求证直线
过定点,并求出该定点坐标;
(3)设(2)中定点为Q,记
与
的面积分别为
和
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48d195911a91d12edd5685f6cd963fa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/743eac5ef7cd9452d9678d797da748ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cb18941b1e55e62e6c3f54a35ccb214.png)
(1)求动点P的轨迹C的方程;
(2)动直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46f478abeeb4da23121b652cf907972d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(3)设(2)中定点为Q,记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/592f84fbbd939b954f52dc6b8c009b53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a51aab5c5e99207337fb64603887579c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/884d40a97fd767e95f34f3b91ab8d84c.png)
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解题方法
5 . 若实数集
对
,均有
,则称
具有Bernoulli型关系.
(1)若集合
,判断
是否具有Bernoulli型关系,并说明理由;
(2)设集合
,若
具有Bernoulli型关系,求非负实数
的取值范围;
(3)当
时,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2df79c96894e48585d810e2d1180b04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de62c03953e609ea331280b1e27ba701.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42acae4bf2a6bead9d904b70d0480fc0.png)
(1)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5055c43ef4c493c056609f617f38e108.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ef4609431a6fc9f2755d8e8ca6617b0.png)
(2)设集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca9d408eb7f234bea73e86bff6a453f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5596a9fe31bffbe73af20f611a9a574d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/953916e76840b10bf27302f42ad98cb9.png)
您最近一年使用:0次
2024-05-12更新
|
1028次组卷
|
3卷引用:福建省福州市2024届高三第三次质量检测数学试题
解题方法
6 . 对于数列
,若存在
,使得对任意
,总有
,则称
为“有界变差数列”.
(1)若各项均为正数的等比数列
为有界变差数列,求其公比q的取值范围;
(2)若数列
满足
,且
,证明:
是有界变差数列;
(3)若
,
均为有界变差数列,且
,证明:
是有界变差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2480f87a11c4cd450bc9454ea7276722.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b12bed9580c9e3efaaae3f234780cef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)若各项均为正数的等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b22febb1e578366695d7628740370bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/385275d29d8c8a7841eaeaa3dfab2cdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e976c0663fa749ca749f99842d21ca03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1165edc23b5782b5942ef7e79130bb94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa7c13436fc942bddb9c562520fb855a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc883e0a2ee951e94f305c807e66010a.png)
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7 . 树人中学高三(1)班某次数学质量检测(满分150分)的统计数据如下表:
在按比例分配分层随机抽样中,已知总体划分为2层,把第一层样本记为
,其平均数记为
,方差记为
;把第二层样本记为
,其平均数记为
,方差记为
;把总样本数据的平均数记为
,方差记为
.
(1)证明:
;
(2)求该班参加考试学生成绩的平均数和标准差(精确到1);
(3)假设全年级学生的考试成绩服从正态分布
,以该班参加考试学生成绩的平均数和标准差分别作为
和
的估计值.如果按照
的比例将考试成绩从高分到低分依次划分为
四个等级,试确定各等级的分数线(精确到1).
附:
.
性别 | 参加考试人数 | 平均成绩 | 标准差 |
男 | 30 | 100 | 16 |
女 | 20 | 90 | 19 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79ae6558e11384a40f3a338b73385ee1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfbe7f95b5d89f9409ec24536da9e826.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfbc29b47b83fdc5368770b7b1acb439.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5b3107354f055c708208a37ab66b30f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb525270c748eddaaecc4a549cca250e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab1295cbd36fdc55a55b549aa2dd5887.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d41042207515dd2e8349c805e6aee400.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/671f43c79d612c93a6d160335e86e177.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/217223e16eb491561c4ca844c0b52f81.png)
(2)求该班参加考试学生成绩的平均数和标准差(精确到1);
(3)假设全年级学生的考试成绩服从正态分布
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29bcc248a7770a16fa10fc4602d71e0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1100379a4385b9ce064847bc21760adc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c0ad7e7853a069537387b5192f73844.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cb65fd949bae6f2d638b4b7a67aaa75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
附:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e53bcf5dca65c16335bc356bcd5a36ef.png)
您最近一年使用:0次
2024-04-26更新
|
2057次组卷
|
5卷引用:广东省广州市广东实验中学2023-2024学年高三下学期教学情况测试(二)数学试卷A
8 . 已知椭圆
的长轴为4,直线
与圆
相切于点
,与
相交于
,
两点,且
,
,
.
(1)记
的离心率为
,证明:
;
(2)若
轴右侧的点
在
上,且
轴,
,
是圆
的两条切线,切点分别为
,
(
在
上方),求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4966973694e3fdf6b63434b7bee5911.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/560adea7b0d4fbe4131fc41f3fcbd871.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/9fbd6303756adb878c1e028f511e51f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b32a5bdae1e1c0082553d71b80dbcd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6270bb08b90f72d5671ab8225f356c43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2fe3251e054fe97089806ba7033f802.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f99df1a7b58018125b99578b779342.png)
(1)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3a16563ac359fda8ac16989aa704b23.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af72fe2d8fdcd6793597db51d070290e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db8305c4ffbf876642440c3d28e91e9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce2790947716b1cfa9c5e7a65db4093.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45b79e16fb149b3ea933cd087f0cc740.png)
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9 . 已知O为坐标原点,椭圆C:
的上、下顶点为A、B,椭圆上的点P位于第二象限,直线PA、PB、PO的斜率分别为
,且
.
(1)求椭圆C的标准方程;
(2)过原点O分别作直线PA、PB的平行线与椭圆相交,得到四个交点,将这四个交点依次连接构成一个四边形,则此四边形的面积是否为定值?若为定值,请求出该定值;否则,请求出其取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b24214f111f7c6d2b64e53ad970438b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dca1726d463bd741c904abd9b6589056.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66cbe6dfc817c97da126729e27978b42.png)
(1)求椭圆C的标准方程;
(2)过原点O分别作直线PA、PB的平行线与椭圆相交,得到四个交点,将这四个交点依次连接构成一个四边形,则此四边形的面积是否为定值?若为定值,请求出该定值;否则,请求出其取值范围.
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2024-04-08更新
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1540次组卷
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4卷引用:重庆市第十一中学校2023-2024学年高三第九次质量检测数学试题
重庆市第十一中学校2023-2024学年高三第九次质量检测数学试题湖南省常德市2024届高三下学期3月模拟考试数学试题 (已下线)数学(九省新高考新结构卷03)(已下线)第30题 几何分析曲径通幽,代数推演水到渠成(优质好题一题多解)
解题方法
10 . 某汽车文化自媒体公司主打对越野车越野能力的测评,为调查车友们对越野车的了解程度,随机抽取了200名车友进行调查,得到如下表的数据:
(1)完成上面的
列联表,根据小概率值
的独立性检验,能否认为车友对越野车的了解程度与性别有关?
(2)该公司组织5名驾驶水平相当的员工在户外场地进行汽车越野活动,他们需要合作闯关,一共有两关,每次由一名员工上场,闯过第一关才能闯第二关,若闯某一关失败,则换下一名员工从失败的这一关开始闯,同一员工不重复上场,当有人闯过第二关时或者5名员工都闯关失败时活动结束.若无论前面的闯关结果如何,每名员工闯过第一关的概率都为
,闯过第二关的概率都为
,求第三名员工闯关后活动恰好结束的概率.
附:
.
女性 | 男性 | 总计 | |
比较了解 | 78 | ||
不太了解 | 38 | ||
总计 | 140 | 200 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b72fcdc709e77910cd36a26369648b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ead9d6ff51996f3ebace6f212e11a9e4.png)
(2)该公司组织5名驾驶水平相当的员工在户外场地进行汽车越野活动,他们需要合作闯关,一共有两关,每次由一名员工上场,闯过第一关才能闯第二关,若闯某一关失败,则换下一名员工从失败的这一关开始闯,同一员工不重复上场,当有人闯过第二关时或者5名员工都闯关失败时活动结束.若无论前面的闯关结果如何,每名员工闯过第一关的概率都为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
附:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2187714e660234f0b72f2b47d3ea685a.png)
0.05 | 0.025 | 0.005 | |
3.841 | 5.024 | 7.879 |
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