名校
1 . 已知角
的终边上一点
,
.
(1)请用定义证明:
;
(2)已知函数
在区间
的最大值
,求实数
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aee82283f06cedef32eb15b87964f5d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b0527f3801a1f5fae326d9411555b7d.png)
(1)请用定义证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa1ec2d7289bc848c59d03ef876073d6.png)
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21c48512814068f0781df94dabd78a7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ea7e406afac9609ca4015d25066af1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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2 . 定义:
是无穷数列,若存在正整数k使得对任意
,均有
则称
是近似递增(减)数列,其中k叫近似递增(减)数列
的间隔数
(1)若
,
是不是近似递增数列,并说明理由
(2)已知数列
的通项公式为
,其前n项的和为
,若2是近似递增数列
的间隔数,求a的取值范围:
(3)已知
,证明
是近似递减数列,并且4是它的最小间隔数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36b98ef143f8159f3a7dafa1fd2f2370.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5868622de607b54d53fc6c481dc6302d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebd6e7277f682a7f7adf2243ac5c9e2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97b2c4b8c1ebc9a3622f7d09de41496f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/846fa57d92d6ad44d6a0cafad1e71ed4.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73841553d9289a6463664c8ea4647127.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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2020-05-19更新
|
398次组卷
|
4卷引用:上海市文建中学2022-2023学年高一上学期期中数学试题
上海市文建中学2022-2023学年高一上学期期中数学试题2020届上海市宝山区高三下学期二模数学试题(已下线)上海市华东师范大学第二附属中学2019-2020学年高一下学期期末数学试题上海市七宝中学2022届高三上学期十月月考数学试题
解题方法
3 . 在推导很多三角恒等变换公式时,我们可以利用平面向量的有关知识来研究,在一定程度上可以简化推理过程.如我们就可以利用平面向量来推导两角差的余弦公式:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e6276ff5468f5aa9c6eaff479c26cc7.png)
具体过程如下:
如图,在平面直角坐标系
内作单位圆O,以
为始边作角
.它们的终边与单位圆O的交点分别为A,B.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/29/3378e1b0-11ac-4e21-89d7-e7bef545c1e9.png?resizew=334)
则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a98717138350884b83b2bc3335ac3262.png)
由向量数量积的坐标表示,有:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/437ebce60a1d755209353f0d94462154.png)
设
的夹角为θ,则
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/665d77a90728ca9eb4d63b07dbe89e80.png)
另一方面,由图3.1—3(1)可知,
;由图可知,
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/29/8e003e58-f755-4f57-ba40-42e3c44c2f0e.png?resizew=348)
.于是
.
所以
,也有
,
所以,对于任意角
有:
(
)
此公式给出了任意角
的正弦、余弦值与其差角
的余弦值之间的关系,称为差角的余弦公式,简记作
.
有了公式
以后,我们只要知道
的值,就可以求得
的值了.
阅读以上材料,利用下图单位圆及相关数据(图中M是AB的中点),采取类似方法(用其他方法解答正确同等给分)解决下列问题:
(1)判断
是否正确?(不需要证明)
(2)证明:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/889623d5e61054f38a35aedd644c9ff5.png)
(3)利用以上结论求函数
的单调区间.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e6276ff5468f5aa9c6eaff479c26cc7.png)
具体过程如下:
如图,在平面直角坐标系
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3e5af20b2f8c1fba4470f9650989e51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfa404d3ff313b0a28a76a48d7d87234.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/29/3378e1b0-11ac-4e21-89d7-e7bef545c1e9.png?resizew=334)
则
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a98717138350884b83b2bc3335ac3262.png)
由向量数量积的坐标表示,有:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/437ebce60a1d755209353f0d94462154.png)
设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/538844ce819df320039e394ba92356f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/665d77a90728ca9eb4d63b07dbe89e80.png)
另一方面,由图3.1—3(1)可知,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/655ee7e11f540619722504916419e009.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/29/8e003e58-f755-4f57-ba40-42e3c44c2f0e.png?resizew=348)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18eedcc65589e7529da85a578bd0ecb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e366809cf946d825277ad151abb374a2.png)
所以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a689c643b92f5fafe77fb2c754b0184.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e6276ff5468f5aa9c6eaff479c26cc7.png)
所以,对于任意角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e288596fa3811dd2c17bded60e82e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e6276ff5468f5aa9c6eaff479c26cc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9e74ca761ffa2566a9851c5ce9ccaaf.png)
此公式给出了任意角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e288596fa3811dd2c17bded60e82e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd927b4b5a7875528c1b54aa4bb8b2dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9e74ca761ffa2566a9851c5ce9ccaaf.png)
有了公式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9e74ca761ffa2566a9851c5ce9ccaaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1455db71a4123b3317dcfce3e2005e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22d521f8d021b20757d7a68107fcef1d.png)
阅读以上材料,利用下图单位圆及相关数据(图中M是AB的中点),采取类似方法(用其他方法解答正确同等给分)解决下列问题:
(1)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90f93aa4ff886e380c9b7c05dbafd08d.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/889623d5e61054f38a35aedd644c9ff5.png)
(3)利用以上结论求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1414c4eb3a476aac49f6a35d62b1f7ac.png)
您最近一年使用:0次
2020-05-22更新
|
713次组卷
|
3卷引用:大题好拿分期中考前必做30题(压轴版)-2020-2021学年高一数学下册期中期末考试高分直通车(沪教版2020必修第二册)
(已下线)大题好拿分期中考前必做30题(压轴版)-2020-2021学年高一数学下册期中期末考试高分直通车(沪教版2020必修第二册)贵阳市普通高中2018-2019学年度高一上学期数学期末质量监测试题贵州省贵阳市2018-2019学年高一(上)期末数学试题
名校
4 . 已知函数
的部分图象如下图所示.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/22/2a8c5b30-20f5-4f39-b178-cb0fac4bc648.png?resizew=170)
(1)求函数
的解析式;
(2)已知关于x的方程
在
内恰有两个不同的解
,
.
①求实数
的取值范围.
②证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7bd53c018e24caee7d661de800a1573.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/22/2a8c5b30-20f5-4f39-b178-cb0fac4bc648.png?resizew=170)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(2)已知关于x的方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca9a696ab843d9dfd2f718cec4760823.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0e6ecdea5cd5a1a31b61bbd2671937.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
①求实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
②证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0932b838c884d09a225396da0eefe591.png)
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2020-03-16更新
|
897次组卷
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2卷引用:江西省景德镇一中2020-2021学年高一上学期期中考试数学(2班)试题
5 . 已知函数
,其中
为自然对数的底数.
(1)证明:
在
上单调递增.
(2)设
,函数
,如果总存在
,对任意
,
都成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b493a1557ab271024d0026d2203fef84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8938db94f49dcbe0c383fba0241bb0da.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cef58eb649b6d20935789175977c77bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b8af73bbdedee43e2a99d06ee9c67b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86ba8542fbe02e78cf3948c9abea9855.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c8a6ab0f521c14a67580b934ce6b41d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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2020-02-23更新
|
1130次组卷
|
4卷引用:大题好拿分期中考前必做30题(压轴版)-2020-2021学年高一数学下册期中期末考试高分直通车(沪教版2020必修第二册)
(已下线)大题好拿分期中考前必做30题(压轴版)-2020-2021学年高一数学下册期中期末考试高分直通车(沪教版2020必修第二册)广东省2019-2020学年高一上学期期末数学试题广东省云浮市2019-2020学年高一上学期期末数学试题(已下线)上海高一上学期期中【压轴42题专练】(2)
6 . 已知平面向量
,设函数
(
为常数且满足
),若函数
图象的一条对称轴是直线
.
(1)求
的值;
(2)求函数
在
上的最大值和最小值:
(3)证明:直线
与函数
的图象不相切.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcb7484264801ab0b062cf825c02619f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27a56b1f745a78c71ddfbb44837231c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6581916f5a65edfea257c804efee007e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ceea6fed130d78179f6edba061360174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f9f520533f7a4e500860d8ed1ae9e80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e60b49815c5c25b48e6c74d7077851a.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6581916f5a65edfea257c804efee007e.png)
(2)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f9f520533f7a4e500860d8ed1ae9e80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18c9aeed3c8c5a04e48d011c607f9142.png)
(3)证明:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1d412ff61c13ff59052ced69a46e26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f9f520533f7a4e500860d8ed1ae9e80.png)
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2019-12-09更新
|
240次组卷
|
2卷引用:浙江省宁波市慈溪市2019-2020学年高三上学期期中数学试题
真题
名校
7 . 已知函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11882a302c0a5e9f47833eb2416d0725.png)
(Ⅰ)求
的最小正周期和最小值;
(Ⅱ)已知
,
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11882a302c0a5e9f47833eb2416d0725.png)
(Ⅰ)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(Ⅱ)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efbbc96fd62a9c2557c2f683c91eb3ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/126cc244e8b5a9cb557789613ba9d725.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4e7d4aa8b7a6719c0c1e2898930641.png)
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2019-01-30更新
|
1916次组卷
|
6卷引用:2013-2014学年山西省吕梁学院附中高一下学期期中考试数学试卷
名校
8 . 如图,已知矩形
,
,
,点
为矩形内一点,且
,设
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/30/abedc139-0dd3-4138-99bd-6a77170deee8.png?resizew=146)
(1)当
时,求证:
;
(2)求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d783fe7f3ce673d5d21281174e7a7968.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e09e0ce09a4711bb308fccef46faf4f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5c9d96d2dc0082bb375c3b0e7214bdf.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/30/abedc139-0dd3-4138-99bd-6a77170deee8.png?resizew=146)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05e2c8d466ab8eb5ecd38060b53bbe8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/642cec60c6719f5e18a7e1227040e481.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ef9d59074422189c31b540dcbdc680b.png)
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2019-07-11更新
|
1627次组卷
|
8卷引用:辽宁省协作校2019-2020学年高一下学期期中考试数学试题
辽宁省协作校2019-2020学年高一下学期期中考试数学试题广东省佛山市2017-2018学年高一上学期期末教学质量检测数学试题【全国百强校】甘肃省天水市一中2017-2018学年高一下学期第三学段(期末)考试数学试题江苏省无锡市2018-2019学年高二下学期期末质量数学(文)试题(已下线)第06讲 第五章 平面向量、数系的扩充与复数的引入(单元测试)(测)-《2020年高考一轮复习讲练测》(浙江版)吉林省长春市第二实验中学2020-2021学年高一下学期4月月考数学试题(已下线)专题06 平面向量的坐标表示(1)-《重难点题型·高分突破》(人教A版2019必修第二册)(已下线)专题06 平面向量的坐标表示(1)-《重难点题型·高分突破》(苏教版2019必修第二册)
名校
9 . 在平面直角坐标系中,
为坐标原点,
三点满足
.
(1)求证:
三点共线;
(2)已知
,
的最小值为
,求实数
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7453d507d0c0c7c10d5f73dca6dceb5d.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/311995bf3e561f255d947b8a75ca0d38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f18c358e515979c5eea9a6617ebc151.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
2018-04-25更新
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791次组卷
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7卷引用:福建省厦门市双十中学2016-2017学年高一下学期期中考试数学试题
名校
解题方法
10 . 定义行列式运算:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783b15b916672ddca749ed64109bd01.png)
,若函数
(
,
)的最小正周期是
,将其图象向右平移
个单位后得到的图象关于原点对称.
(1)求函数
的单调增区间;
(2)数列
的前
项和
,且
,求证:数列
的前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2803b474b5f8f96c7e6e9e741e20d073.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783b15b916672ddca749ed64109bd01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34f1955111f224144de4fb2aa8c2eec4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f1726b4c39cc78d2f03cf51c4216937.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9b078ff54940a718915c2d1425d2031.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4456675a5dbe545462a22cef9aca8fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa6af3e2115ce0aaf5b99ac70c4441d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86ebba6ed1add0fe647c0226614b9290.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d88591679796c52024d11c4de641bdb.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7109f604b724dfab348530752a0891ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7df78e3319e7592af36eacedf746b0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf893b061515c5b9e7979e12b2af5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c02e80983b88cdf6b540502816c87d13.png)
您最近一年使用:0次
2017-11-27更新
|
792次组卷
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2卷引用:四川省成都市郫都区2019-2020学年高一下学期期中考试数学(理)试题