1 . 已知
是抛物线
上位于第一象限的一点,且
到
的焦点的距离为5.
(1)求抛物线
的方程;
(2)设
为坐标原点,
为
的焦点,
,
为
上异于
的两点,且直线
与
斜率乘积为
.
(i)证明:直线
过定点;
(ii)求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b218bde519e649de7e9948fb6f5339a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e6c830bfa9a1b979a1a9665166424bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3edbd40e04e2a943051fa83d6e511add.png)
(i)证明:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
(ii)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a7dbc5f85292795579155dfd3baff0e.png)
您最近一年使用:0次
名校
解题方法
2 . 已知抛物线
的焦点为
,点
在
上,且
的最小值为1.
(1)求
的方程;
(2)过点
的直线与
相交于
,
两点,过点
的直线与
相交于
,
两点,且
,
不重合,判断直线
是否过定点.若是,求出该定点;若不是,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e63d3f6fe96b1f26cf87eba3165ee1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56c9f195bc306a5ff74ad29bbd3a6ed1.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fdb8f8f39e1d3eea420b0de792d20f38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](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/f1ddda95c7c302aa246077dc7f0283e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
您最近一年使用:0次
2023-11-23更新
|
599次组卷
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6卷引用:辽宁省葫芦岛市协作校2023-2024学年高二上学期第二次考试数学试题
3 . 已知直线
的方向向量与直线
的方向向量共线且过点
;
(1)求
的方程;
(2)若
与抛物线
交于点
为坐标原点,设直线
,直线
的斜率分别是
;求
及
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/089243ad31b799d65b9e86ec2a116188.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63f739a29d4fbb88ba1337e8456ef1f8.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/745de5ef1fd897d16e37464172d5e8c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/876856cdb3900b117f280615c185d347.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/4ad608ff2250d3f93f509ef52fff7611.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b881044b5c73db6fcce110525741b02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acb8956ebcde7cf0f1011a2fc1888e15.png)
您最近一年使用:0次
2023-11-19更新
|
731次组卷
|
3卷引用:辽宁省高级中学2023-2024学年高二上学期期中数学试题
名校
解题方法
4 . 已知抛物线
的焦点为
,过
作直线
交抛物线于
两点,点
,若直线
的斜率分别为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49a53876badc982691e887414c6b86db.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7089148c36cb3c39af71de653756396a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52087fb2a64abe13e70a208089c5c70d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03df57efff473b3cfeb8503796b7d6b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90963760acac7bfad3ae03088c6c80b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49a53876badc982691e887414c6b86db.png)
您最近一年使用:0次
2022-08-31更新
|
377次组卷
|
3卷引用:辽宁省协作校2022-2023学年高二上学期期中考试数学试题
名校
解题方法
5 . 已知抛物线
上一点
到其焦点的距离为5.
![](https://img.xkw.com/dksih/QBM/2022/1/21/2899466200055808/2903016204541952/STEM/56a65b2f05d64e879ce862b54d7a75de.png?resizew=346)
(1)求
与
的值;
(2)过点
作斜率存在的直线
与拋物线交于
两点(异于原点
),
为
在
轴上的投影,连接
与
分别交抛物线于
,问:直线
是否过定点,若存在,求出该定点,若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3764ba3aa0a241787f4661026bb14053.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/715ddd908bb767484dfc2b7458b8ce64.png)
![](https://img.xkw.com/dksih/QBM/2022/1/21/2899466200055808/2903016204541952/STEM/56a65b2f05d64e879ce862b54d7a75de.png?resizew=346)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa118c0fb33ab21dfc494cb868fa983d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.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/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7785afeeaf274892253d04b4f693b367.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
您最近一年使用:0次
2022-01-26更新
|
707次组卷
|
2卷引用:辽宁省铁岭市昌图县第一高级中学2022-2023学年高二上学期期中数学试题
19-20高一·浙江杭州·期末
6 . 如图,点
为椭圆
的左顶点,过
的直线
交抛物线
于
,
两点,点
是
的中点.
![](https://img.xkw.com/dksih/QBM/2020/11/13/2591931775950848/2591979700248576/STEM/ded9309600ca4232a5ebe6fd4963480a.png?resizew=232)
(Ⅰ)若点
在抛物线
的准线上,求抛物线
的标准方程:
(Ⅱ)若直线
过点
,且倾斜角和直线
的倾斜角互补,交椭圆
于
,
两点,
(i)证明:点
的横坐标是定值,并求出该定值:
(ii)当
的面积最大时,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f6d926590ae5d35e8a6123074413e3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79c9c87eba774f6bc072663d32d11fc4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/2020/11/13/2591931775950848/2591979700248576/STEM/ded9309600ca4232a5ebe6fd4963480a.png?resizew=232)
(Ⅰ)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(Ⅱ)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
(i)证明:点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(ii)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38d0469336f71edd52dc9148c67db052.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
您最近一年使用:0次
2020-11-13更新
|
1056次组卷
|
7卷引用:辽宁省沈阳市第一二〇中学2021-2022学年高二上学期期中考试数学试题
辽宁省沈阳市第一二〇中学2021-2022学年高二上学期期中考试数学试题浙江省杭州地区(含周边)重点中学2020-2021学年高三上学期期中数学试题(已下线)【新东方】高中数学20210323-005【高二下】福建省厦门市第一中学2020-2021学年高二上学期数学市质检模拟卷试题(已下线)【新东方】杭州新东方高中数学试卷356湖南省长沙市雅礼中学2020-2021学年高三上学期第5次月考数学试题湖南省邵阳市第二中学2022-2023学年高三上学期第二次月考数学试题
名校
解题方法
7 . 已知抛物线
,
为
上一点且纵坐标为4,
轴于点
,且
,其中点
为抛物线的焦点.
(1)求抛物线
的方程;
(2)已知点
,
,
是抛物线
上不同的两点,且满
,证明直线
恒过定点,并求出定点的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37ab7408ffcefcb8e5e1ad4a9c58f1b1.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/dfba66e25f01d30e190e1458375da661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0adb5ea2e5533717253ff00a6ecdf562.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1599d6f3e744e56f029921cfe4b60f89.png)
![](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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da006b1390397b520a0e8767cd49e7f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
您最近一年使用:0次
2020-09-17更新
|
1198次组卷
|
10卷引用:辽宁省沈阳市第一二〇中学2021-2022学年高二上学期期中考试数学试题
辽宁省沈阳市第一二〇中学2021-2022学年高二上学期期中考试数学试题黑龙江省双鸭山市第一中学2020-2021学年高二上学期期中考试数学(理)试题四川省绵阳市涪城区东辰国际学校2020-2021学年高二上学期期中数学理科试题四川省泸州市泸县第五中学2021-2022学年高二上学期期中考试数学(理)试题(已下线)专题3.3 抛物线-2020-2021学年高二数学同步课堂帮帮帮(人教A版2019选择性必修第一册)(已下线)专题八 抛物线-2021-2022学年高二数学同步单元AB卷(人教A版2019选择性必修第一册)重庆市实验中学2021-2022学年高二上学期第二次阶段性测试数学试题山西省太原市实验中学2021-2022学年高二上学期12月月考数学试题重庆市巴蜀中学2021届高三上学期高考适应性月考(一)数学试题(已下线)专题13 抛物线及其性质——2020年高考数学母题题源解密(山东、海南专版)
名校
解题方法
8 . 已知函数
,函数图象上有两动点
、
.
(1)用
表示在点
处的切线方程;
(2)若动直线
在
轴上的截距恒等于
,函数在
、
两点处的切线交于点
,求证:点
的纵坐标为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/108c18cb76d7d34b05c991a644c8b136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a3efb79f35db8448f3391252ab7d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df332f01628130c084fd46aaca0a4b7.png)
(1)用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)若动直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
您最近一年使用:0次
2020-05-16更新
|
323次组卷
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2卷引用:辽宁省实验中学2019-2020学年高二下学期期中考试数学(理)试题
9 . 已知抛物线C:y2=2px(p>0)的焦点为F,过F且斜率为
的直线l与抛物线C交于A,B两点,B在x轴的上方,且点B的横坐标为4.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/6/b682b646-6bac-487e-b8b0-1292ca7b59c0.png?resizew=157)
(1)求抛物线C的标准方程;
(2)设点P为抛物线C上异于A,B的点,直线PA与PB分别交抛物线C的准线于E,G两点,x轴与准线的交点为H,求证:HG•HE为定值,并求出定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb0b6452a3d526d760136e5c8936ba8f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/6/b682b646-6bac-487e-b8b0-1292ca7b59c0.png?resizew=157)
(1)求抛物线C的标准方程;
(2)设点P为抛物线C上异于A,B的点,直线PA与PB分别交抛物线C的准线于E,G两点,x轴与准线的交点为H,求证:HG•HE为定值,并求出定值.
您最近一年使用:0次
11-12高二下·辽宁大连·期中
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解题方法
10 . 如图所示,直线
与抛物线
交于
两点,与
轴交于点
,且
,
(1)求证:点
的坐标为
;
(2)求证:
;
(3)求
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e8953ded144195804384dcb494d5e2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4daf40bad1cc89311930cce356672354.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77d681a7a9830fe11598654e141aa28e.png)
(1)求证:点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53a948d2f7732d7f03e986c63712089b.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3825ccc273ef9a672a606432d165b866.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
![](https://img.xkw.com/dksih/QBM/2018/1/3/1852660358438912/1853172702126080/STEM/35dca44a68d945a9b7154d43765efc44.png?resizew=278)
您最近一年使用:0次
2018-01-04更新
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551次组卷
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4卷引用:2011-2012学年辽宁省庄河六中高二下学期期中考试文科数学试卷
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