1 . 已知集合
中的元素都是正整数,且
.若对任意
,且
,都有
成立,则称集合A具有性质
.
(1)判断集合
是否具有性质
;
(2)已知集合A具有性质
,求证:
;
(3)证明:
是无理数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ea7fcdb5423c1c8c032a3efcf245682.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c3301017a56b4427b6fab492f63b86d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e13a814f8e081078dcf3788177affcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a11a069688e4c797fcf527eab15afa82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d13266f62539701a58bbcf895de46b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(1)判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ab146eb4208985dfe60ae3b41ba2bdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)已知集合A具有性质
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30ff2bdedce1d88ef6f2607f0a05c1cd.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
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2 . 若集合A具有以下性质:①
,
;②若x、
,则
,且
时,
.则称集合A是“好集”.
(1)分别判断集合
是否是“好集”,并说明理由;
(2)设集合A是“好集”,求证:若x、
,则
;
(3)对任意的一个“好集”A,证明:若x、
,则必有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2faf3937abcb6a59071c17bc6bb10f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35a2410ce34b36954ed4923e600d42f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e63c91626ffa91e590925e6f206c3c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c46de01c5104b9112a688df37eadb000.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38f0e9c04402a0ffdaa25c3e3c82c7dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cd77104cc745d1e0e262122da34482d.png)
(1)分别判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc51474b54120493b300a14e566cbc0e.png)
(2)设集合A是“好集”,求证:若x、
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e63c91626ffa91e590925e6f206c3c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/957d41dbe52b49c3a7339e3519a3fe84.png)
(3)对任意的一个“好集”A,证明:若x、
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e63c91626ffa91e590925e6f206c3c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09db4828342fb0a79e0ea8d132b76e6f.png)
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解题方法
3 . 已知集合
为非空数集,定义:
,
(1)若集合
,直接写出集合
(无需写计算过程);
(2)若集合
,且
,求证:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb1868c8a0db983c9cc2695294fa03b1.png)
(3)若集合
,记
为集合
中的元素个数,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/312ab0127457545c7bac2193505e0a6b.png)
(1)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16e8d171c3d2d86dc5594cffb51096fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0a44fa352255b81702d306cb32cf468.png)
(2)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77239c98c78a026cc03336edca067ea3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c76ad1e03a6ba59e8164e37c5e7e063e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb1868c8a0db983c9cc2695294fa03b1.png)
(3)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df87128f2e2850f12dc88216def10cc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc8279d9dd0b7750953cb9e2098b3b90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc8279d9dd0b7750953cb9e2098b3b90.png)
您最近一年使用:0次
2023-09-17更新
|
361次组卷
|
3卷引用:上海市高桥中学2023-2024学年高一上学期月考(一)数学试题
上海市高桥中学2023-2024学年高一上学期月考(一)数学试题(已下线)高一上学期期中考试解答题压轴题50题专练-举一反三系列北京市中央民族大学附属中学(朝阳)2023-2024学年高一上学期期中考试数学试题
4 . 已知集合
,设
是
的至少含有两个元素的子集,对于
的任意两个不同的元素
,若
都不能整除
,则称集合
是
的“好子集”.
(1)判断数集
与
是否是集合
的“好子集”,并说明理由;
(2)证明:若
是
的“好子集”,则对于
中的任意两个不同的元素
,都有
;
(3)求集合
的“好子集”
所含元素个数的最大值,并写出取到元素个数最大值时的
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c54730b349603779705381ecfaa3d114.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5e2f1ee19bfd9953867afa426375688.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/059d6df06a5b85848dc4fa33327f8e07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b88584cf1df43e28d03592c7998b1653.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
(1)判断数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef27cd09f4ecc055fd7e72b3b368e5df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c91edb18f49d6acc68a3d8d1a1be6a5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
(2)证明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5e2f1ee19bfd9953867afa426375688.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d4be7ed8ae7440b7b7efae8889cc510.png)
(3)求集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
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5 . 对任意给定的不小于3的正整数
,
元集合
均为正整数集的子集, 若满足:
①
;
②
;
③
,则称
互为等矩集.
(1)若集合
与
互为等矩集,求
的值;
(2)证明: 如果集合
互为等矩集,那么对于任意的正整数
,集合
也互为等矩集;
![](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/5debbaded2b2b268512d53339e460349.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b04f7ed829546d2b2260985f507f3a8.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b7bde3e3d8155e79ab1fa1fa9ee19f1.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dea9a4259cca10c1f5af28e621ebafd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
(1)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12adcda385580201a896d40562dd497f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd0dc1dc5f1c10b956f04abde185490a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
(2)证明: 如果集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5debbaded2b2b268512d53339e460349.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b454a8f5d20d6962b47c1c2508b1c16f.png)
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2023-10-17更新
|
160次组卷
|
2卷引用:上海市朱家角中学2023-2024学年高一上学期第一阶段质量检测数学试题
6 . 已知数集
(
,
)具有性质
:对任意的
,
(
),
与
两数中至少有一个属于
,(如
与
中至少有一个属于
).
(1)分别判断数集
和
是否具有性质
,并说明理由;
(2)求
的值;
(3)设正整数集合
(
,
)具有性质
,证明:对任意
(
为正整数),
都是
的因数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3a3f24673b6e954db3a8b229d8c4564.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fec089f9d4b5d8a70cb2ecfd8f1678c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7600d2cfbdc6146db96cc545706004f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31ae331839bce8f3c14d7efd7f9d8915.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f52783e7a39f438adf08ef7d05d8c78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abf9fc9e8c9940547678ff7934363f52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3c00b1aaa1b31dbfc8f8b904f6f25dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34fb3b69403fe5307601f073d8b271a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(1)分别判断数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cd5556ed59bb2c223833259a42890ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5038b4f82da68c008ade7275ad241993.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
(3)设正整数集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bacfbe332324e9d2447feb74a869aeb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05bc5d045516472a12d9dcdab97a3f0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ef835c9ad2636a9662fb6c99e3abc78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50a272adba0f1120109824440f0e252c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
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解题方法
7 . 对于实数构成的集合
.若对任意
都有
(其中“
”表示普通的乘法运算),则称集合
对“
”是封闭的.
(1)已知集合
,判断
是否属于集合
;
(2)在(1)的条件下,若
,证明
的充要条件是
;
(3)若集合
对“
”都是封闭的,试判断
是否对“
”封闭,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cffa35373ec4e4684107b42adb7a5161.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8af42f2ec410ad1b28be69d3415ed10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a9f1ff137e76626b7b608cef7c36349.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2468403b3eba9e40bfa36f464e927738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cffa35373ec4e4684107b42adb7a5161.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2468403b3eba9e40bfa36f464e927738.png)
(1)已知集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50dae68b646f975a8a0c6bed67d028ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e57ed82958abb00776e75987aa62d723.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)在(1)的条件下,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9ecef225b27b5857d2d76fbe5b34982.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caf22d7d1a965bda25168a233fb6290c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bff58fdfe14af9fb8aa0e0cf0d60853a.png)
(3)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2468403b3eba9e40bfa36f464e927738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81a3375139541ed61929fa658f16bb7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2468403b3eba9e40bfa36f464e927738.png)
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8 . 对于给定集合
,若集合
中任意两个不同元素之和仍是集合
中的元素,则称集合
是“封闭集合”.设
为实常数且
,集合
,证明:集合
为“封闭集合”的充要条件是:存在整数
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cd5371a6f0f82c65dd22f75f8b807c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a3ac83c571110d41a396d12d8eea1c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01addf1c0dae299be04495dec2a3c733.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70a141495f9abd68126822a2ae920aed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efd44d872abf0e0480c139e86d9bb5c1.png)
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9 . 已知数集
具有性质
:对任意的
与
两数中至少有一个属于
.
(1)分别判断数集
与
是否具有性质![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fc5c19ac440746be97d8b46af5d288a.png)
(2)证明:
,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecf37bb0ce613aa440869a7db3abc26d.png)
(3)当
时,若
,若数集
具有性质
,求数集
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f69edbd87b8d3d77f6e2ad2a3b460323.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fc5c19ac440746be97d8b46af5d288a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a76fd7d70aaa306d18a76e238365cfa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30692e554e0aa2bcfabd25c2ca4b391a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0914b68f106a912420705b2f3928ca42.png)
(1)分别判断数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8564068ee1f073615fab9694b210e1b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4acdcfc22edc7cbc41fb7145e9976d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fc5c19ac440746be97d8b46af5d288a.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/853eace02560e7f1490694276c29a856.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecf37bb0ce613aa440869a7db3abc26d.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2da4f52abad1deccf4bd6e100b9d36fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4332a0bc93d1044dc972ce0fbf755ae4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0914b68f106a912420705b2f3928ca42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fc5c19ac440746be97d8b46af5d288a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0914b68f106a912420705b2f3928ca42.png)
您最近一年使用:0次
2022-10-18更新
|
319次组卷
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5卷引用:上海市七宝中学2015-2016学年高一上学期第一次月考数学试题
上海市七宝中学2015-2016学年高一上学期第一次月考数学试题上海市大同中学2022-2023学年高一上学期10月月考数学试题上海市复兴高级中学2022-2023学年高一上学期10月月考数学试题(已下线)上海市华东师范大学第二附属中学2023-2024学年高一上学期10月月考数学试题(已下线)单元高难问题01集合中的新定义问题-【倍速学习法】(沪教版2020必修第一册)
名校
10 . 已知集合
具有性质
:对任意
,
(
),
与
至少一个属于
.
(1)分别判断集合
,与
是否具有性质
,并说明理由;
(2)证明:
;
(3)
具有性质
,当
时,求集合
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d45a296e38b585f04206530b9e53d36f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7600d2cfbdc6146db96cc545706004f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18db8b768e5060b3471415e4b55ac30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/059a6c5a965c335b8da05e697da2c7c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13ee542834ccbb57fcc55b1680ca9db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(1)分别判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5b42882dd156f60b1bbcc394155ee88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d9bf9b7e8523d5cdca10de9ae70770e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2faf3937abcb6a59071c17bc6bb10f6.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d020cd453031ae9eede7961ec78f21a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18d8e8f821111de8075e5c3dfb22a5d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
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2022-11-08更新
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3卷引用:上海市光明中学2022-2023学年高一上学期10月月考数学试题