1 . 设p为任意给定的大于1的整数,每个正整数n均可以唯一地表示成
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9065ce45fd23edbaf85a45ff15c2fc6d.png)
,我们将
称为n的p进制表示,将
称为n在p进制下的数字和.例如:由
可知
,
.
(1)请给出2024的三进制表示;
(2)若
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec3f3539f44a967ded37d695b8c5a754.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9065ce45fd23edbaf85a45ff15c2fc6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebb7c6bb11637246e20a186de268f0c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d451ea758732eb3e8530a0e510efd835.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cce766f5a782ee1123eaaf4ec9773f3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efa82a1ee441389e563ed32c2c660b5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cf0d4f1ae3c405709ffc9215063ee21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d10a86dea76b9b21e2c221fab3a6a167.png)
(1)请给出2024的三进制表示;
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8f8cbdf6bfa897019ce1eb963a869f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32c363c1fc5f6c82234385e8e96fb644.png)
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2 . 在概率较难计算但数据量相当大、误差允许的情况下,可以使用UnionBound(布尔不等式)进行估计概率.已知UnionBound不等式为:记随机事件
,则
.其误差允许下可将左右两边视为近似相等.据此解决以下问题:
(1)有
个不同的球,其中
个有数字标号.每次等概率随机抽取
个球中的一个球.抽完后放回.记抽取
次球后
个有数字标号的球每个都至少抽了一次的概率为
,现在给定常数
,则满足
的
的最小值为多少?请用UnionBound估计其近似的最小值,结果不用取整.这里
相当大且远大于
;
(2)然而实际情况中,UnionBound精度往往不够,因此需要用容斥原理求出精确值.已知概率容斥原理:记随机事件
,则
.试问在(1)的情况下,用容斥原理求出的精确的
的最小值是多少(结果不用取整)?
相当大且远大于
.
(1)(2)问参考数据:当
相当大时,取
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54262f37f86a8b1320d22ffc3f5d3477.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd780f6da9abba35cb0d9ad56ce2bd2c.png)
(1)有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be8f69402300f6ed932697689212e91c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81bb4a9294276b027fecd5dd7f848412.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(2)然而实际情况中,UnionBound精度往往不够,因此需要用容斥原理求出精确值.已知概率容斥原理:记随机事件
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54262f37f86a8b1320d22ffc3f5d3477.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2833ccb3e3d658fa090f7bc327abd34b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(1)(2)问参考数据:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e875164c06cd47489aee8c9f77af495.png)
您最近一年使用:0次
2024-05-16更新
|
1367次组卷
|
3卷引用:浙江省杭州学军中学2024届高三下学期4月适应性测试数学试题
3 . 对给定的在定义域内连续且存在导函数的函数
,若对在
定义域内的给定常数
,存在数列
满足
在
的定义域内且
,且对
在区间
的图象上有且仅有在
一个点处的切线平行于
和
的连线,则称数列
为函数
的“
关联切线伴随数列”.
(1)若函数
,证明:
都存在“
关联切线伴随数列”;
(2)若函数
,数列
为函数
的“1关联切线伴随数列”,且
,求
的通项公式;
(3)若函数
,数列
为函数
的“
关联切线伴随数列”,记数列
的前
项和为
,证明:当
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3c6c201ef006e571184386147529e95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62aff86e290c8874efbb4a7bc197da13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99f3472211834b02fde7f1741b0e6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a462f40a65837da43de04d8b7630f25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ba4f2dd0d53bd7024bf98cbfdcb9fd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3301e5c2891c6d025ab66982e91c5875.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c64ab61f03db328b8860ff20c6b9b51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bc07cb6fd30f25f0f8ca0dd7ef7919a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c501e683a4cf517c61f2aec4c990b187.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b593315a098b5310825524dd1834af9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdd3ca60aab0148b2c3d0570c2195378.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a4f62b56f3a05848417a247e5f0e200.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45c4de9fcfc43eed1df21b52d4896403.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.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/f915157c69267722e3cb47a7a2471ee8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcda70ff15071f59a5fb53ba4b00bf76.png)
您最近一年使用:0次
名校
解题方法
4 . 已知椭圆
的上、下顶点分别为
椭圆
上的点到直线
的距离和其与
的左焦点的距离之比始终为
为
上一点,直线
分别交
于
记
,
的面积分别为
.
(1)求
;
(2)若
和
的横坐标异号,
,求
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/533b93dd6eb6b474481247736699c76c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e31c3268f0a8fd1d76041c9abb8f0367.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02d61b78d69cfd82cf0d69792c977dc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a334d1b952cd569ce5593227a94f1546.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/790ef3382b1c731f2885eecfd92c2a86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e88f79da28a2878b27e3742dc8c92fd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1ed4c4e8edbd179f3fc38a6653f18c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/160bd4db735d9499328407971b7a4bad.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f49bc994e2a224886f665f699ed196d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98013a5042685a1db94249e70c62c09a.png)
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5 . 设双曲线
,直线
与
交于
两点.
(1)求
的取值范围;
(2)已知
上存在异于
的
两点,使得
.
(i)当
时,求
到点
的距离(用含
的代数式表示);
(ii)当
时,记原点到直线
的距离为
,若直线
经过点
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5bef93a53a2004910a8cac32f93c4b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d00a5df9d281dd4e1e45bf6a4d6fb27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8521b3f195753c88de1e2c12fbf310b2.png)
(i)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a15a13b4190ac3d5feaee27a4c97b21b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/521f998124b18a71397ef9374a494aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(ii)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cbeede118c407a800b05757b9a1393e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f02108f31a77114bd68cf0477ea506d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
您最近一年使用:0次
2024-05-08更新
|
1167次组卷
|
2卷引用:浙江省杭州学军中学2024届高三下学期4月适应性测试数学试题
名校
解题方法
6 . 已知
是方程
的两根,数列
满足
,
,
.
满足
,其中
. 则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d98919b1335a8b7ca020636d1494ad0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/073c95775e8c6c15c7f2f8a4a2ad050b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19a31ae70f96dc4aef6e1ca3ef9fed38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf6746bfcdf694447215a11f5b677d40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/140b5428dace7d41a3967db2f60d633e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c071ed6bc646439b162e096e64fbcd50.png)
A.![]() |
B.![]() |
C.存在实数![]() ![]() ![]() |
D.不存在实数![]() ![]() ![]() |
您最近一年使用:0次
2024-05-08更新
|
950次组卷
|
2卷引用:浙江省杭州学军中学2024届高三下学期4月适应性测试数学试题
名校
解题方法
7 . 若函数
在
上有定义,且对于任意不同的
,都有
,则称
为
上的“
类函数”.
(1)若
,判断
是否为
上的“2类函数”;
(2)若
,为
上的“2类函数”,求实数a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32f0e1ac411fd3a260a5c71df178bd7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b62f64bce0222f01a519ab1b26236bcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee4613910bb8aa030db2fc5d2768e533.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12f248318141e0016d38f9f5a692797f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a3616e4a7268ef41b750fe22afbcd74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12f248318141e0016d38f9f5a692797f.png)
您最近一年使用:0次
2024-05-08更新
|
246次组卷
|
3卷引用:广西南宁市第二中学·柳州高级中学2023-2024学年高二下学期5月联考数学试题
名校
解题方法
8 . 已知曲线
为坐标原点.给出下列四个结论:
①曲线
关于直线
成轴对称图形;
②经过坐标原点
的直线
与曲线
有且仅有一个公共点;
③直线
与曲线
所围成的图形的面积为
;
④设直线
,当
时,直线
与曲线
恰有三个公共点.其中所有正确结论的序号是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad59807e7700def79ad947b88d8c4dcd.png)
①曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
②经过坐标原点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
③直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3070c679fb57d37e2224c5205fd3812.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/015c9ed38ba5b709c5d51102857cd905.png)
④设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94a8d6991873e79b298984a95b8954b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d69fa853794ba03bf379140ecccf59c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
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3卷引用:广东省江门市新会第一中学2024届高三下学期高考热身考试数学试题
名校
9 . 已知函数
,其中
.
(1)讨论
的单调性;
(2)若
有两个极值点
,且
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab1009335ff006785dc8acaf2b955f34.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26d8dafc71b106f39f4e15442220897b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de11c4103b6b4f702ff0728f2a6161fe.png)
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10 . 设函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff675ee3434b2efa4dd19e8c57451f96.png)
(1)若
,求
极小值.
(2)讨论函数
的单调性;
(3)若
,
是函数
的两个零点,且
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff675ee3434b2efa4dd19e8c57451f96.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e258ab9e600435b37465092243d99f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)讨论函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)若
![](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/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d02b706dfb1e60e5ba6488558034484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/450398974b1561ca801e102e16df6789.png)
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2024-05-04更新
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2卷引用:四川省眉山市彭山区第一中学2023-2024学年高二下学期4月月考数学试题