名校
解题方法
1 . 如图,在四棱锥
中,
,且
,底面
是边长为
的菱形,
.
(1)证明:面![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0628681907ac8d7fdb94d8bc1b15feb9.png)
面
;
(2)若直线
与平面
所成角的正弦值为
,点
为棱
上的动点,求平面
与平面
夹角的正弦值的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/545e18836bc7fee22f8f813a6f525d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32d0710321d97361e5782124bbf7f0c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38387ba1cadfd3dfc4dea4ca9f613cea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bbaccd578a43b2397c8bdd50592fa07.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/15/3781a6dd-fd38-4594-b49f-2802f180b55e.png?resizew=172)
(1)证明:面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0628681907ac8d7fdb94d8bc1b15feb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1633988fd62a652de726ee92a917b52d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63a253c7fdf589ee3dece13d5b5b5732.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dee14db57f0c762aad845cf5b4a243c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64484f96568410926e0c50898eba6e32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
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2023-10-13更新
|
983次组卷
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3卷引用:广东省深圳实验学校高中部2023-2024学年高二上学期第一阶段数学试题
广东省深圳实验学校高中部2023-2024学年高二上学期第一阶段数学试题广东省广州市协和中学2023-2024学年高二上学期期中数学试题(已下线)难关必刷01 空间向量的综合应用-【满分全攻略】2023-2024学年高二数学同步讲义全优学案(人教A版2019选择性必修第一册)
名校
2 . 如图所示,已知四棱锥
,满足
为
中点
,
,
.
(1)求证
平面![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)若
与
夹角的余弦值为
,且
,求
与平面
夹角的正弦值
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23b6e08fde74010412a6f14ad4dfbcc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cde654b8bf0fb3f49acb19a8f4a1e48f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc38fef49dcb25f204076ba365827169.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/16/c8406212-99c0-432d-a97b-d2c85fe5033d.png?resizew=134)
(1)求证
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db807b09cc550f476b3f8fa0c6a14425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/821309f088a175c00dc0f4828334503d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b28971fe3e3f9d3849400e98b46fc3ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852aabd89edffc1b94344ff3f1f31ccd.png)
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2023-09-29更新
|
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4卷引用:浙江省名校协作体2022-2023学年高二下学期联考数学试题
浙江省名校协作体2022-2023学年高二下学期联考数学试题辽宁省大连市第八中学2023-2024学年高二上学期期中考试数学试题(已下线)难关必刷01 空间向量的综合应用-【满分全攻略】2023-2024学年高二数学同步讲义全优学案(人教A版2019选择性必修第一册)(已下线)2023-2024学年高二上学期数学期末预测能力卷(人教A版2019)
解题方法
3 . 圆柱
中,四边形
为过轴
的截面,
,
,
为底面圆
的内接正三角形,
.
(1)证明:
平面
;
(2)求三棱锥
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e65ac334119ccd6204402a7aba29a55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2257da1e2425f2ea9ac7440f52659ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e65ac334119ccd6204402a7aba29a55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41b30ab3e9dda0c794ce649cc959a5d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d37160545bf07e848d23fca6a7b1da9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd81adb13f5a7550b0f94f770900a613.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/14/6a956016-4309-4dcd-b08c-a5c838e768b2.png?resizew=119)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d3267664e1d0a09def7c38743f0193f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a34fdf9e6d2d87d01ad0bbb6a73ee05.png)
(2)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e1c9c7f004f2712bb6ac2b727acd899.png)
您最近一年使用:0次
名校
4 . 已知如图平面四边形
,
,
,
,
,现将
沿
边折起,使得平面
平面
,点
为线段
的中点.
平面
;
(2)若
为
的中点,求
与平面
所成角的正弦值;
(3)在(2)的条件下,求二面角
的平面角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cb3f9a5da641be35117fd35ba07a6aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27db558e8db4c957654c8e5cecd2d2dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f945a69cf7e8213e50622125cde652f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcfac9ab1dc776c9ec076ab2a132fcd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff9c7cbcc38b28d45c8539710e5b260a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcf6dc837ae85207789b94d109c5c2eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca67a5b8f69507c8b80379e86f90a8ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2b4e753ef119608188c46a50ec597e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4eb7e9ad5486cf1c5e506b20c5469e8.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/438f34bc8b04e8c494b91306ac6fe352.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aeb5255e2159617505e0c87d01437a57.png)
(3)在(2)的条件下,求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f04e376d75882fa61c533dbf33ea6f17.png)
您最近一年使用:0次
2023-09-14更新
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8卷引用:江西省宜春市宜丰县宜丰中学2023-2024学年高二上学期开学考试数学试题
江西省宜春市宜丰县宜丰中学2023-2024学年高二上学期开学考试数学试题(已下线)【一题多解】立体几何 新旧呼应(已下线)第二章 立体几何中的计算 专题一 空间角 微点10 二面角大小的计算综合训练【培优版】(已下线)第八章 立体几何初步(压轴题专练)-单元速记·巧练(人教A版2019必修第二册)(已下线)高一下学期期中复习解答题压轴题十八大题型专练(2)-举一反三系列(人教A版2019必修第二册)(已下线)第8章 立体几何初步 单元综合检测(重点)-《重难点题型·高分突破》(人教A版2019必修第二册)(已下线)专题20 空间直线、平面的垂直-《重难点题型·高分突破》(人教A版2019必修第二册)(已下线)专题20 平面与平面的位置关系-《重难点题型·高分突破》(苏教版2019必修第二册)
名校
5 . 已知平行四边形ABCD如图甲,
,沿AC将
折起,使点D到达点P位置,且
,连接PB得三棱锥
如图乙.
平面ABC;
(2)在线段PC上是否存在点M,使二面角
的余弦值为
,若存在,求出
的值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686853e0629f88f347f1436711cece87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7ac5396c5ea442e0364b50c1db3d2da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/307d38cc7012c328f1f22aa793fe76d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
(2)在线段PC上是否存在点M,使二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2e9ac46aabe38e5ea1a8cb0febc98af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5ec70bc9d4f8f5df312e2f09ee3bcb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/450aa785fe5021ea99abb8f496293f5a.png)
您最近一年使用:0次
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6 . 如图,在三棱柱
中,底面是边长为2的等边三角形,
,D,E分别是线段
的中点,
在平面
内的射影为D.
平面
;
(2)若点F为棱
的中点,求三棱锥
的体积;
(3)在线段
上是否存在点G,使二面角
的大小为
,若存在,请求出
的长度,若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/462b1c65b1b233ab98a90c164c0968c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d8485573874c071b1d56e0c1d06d5c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d26d8a9d64ad3c8cba28840b41ed7837.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34be4e71cabf458f17a6cd7f24bc70af.png)
(2)若点F为棱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f7ba05c54b3de1f4378f7c8eb58328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/820a55066be63da11d346175942b09aa.png)
(3)在线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f7ba05c54b3de1f4378f7c8eb58328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6460959e79869f0c8cf8613cd211688.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15615de1a6df206dbd081251f676578e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fec5bd77cfc1313bc200480cc66c766.png)
您最近一年使用:0次
2023-09-01更新
|
1442次组卷
|
10卷引用:黑龙江省哈尔滨市第三中学校2023-2024学年高二上学期开学测试数学试题
黑龙江省哈尔滨市第三中学校2023-2024学年高二上学期开学测试数学试题黑龙江省哈尔滨第一中学校2023-2024学年高二上学期开学测试数学试题(已下线)专题8.13 立体几何初步全章综合测试卷(提高篇)-举一反三系列(已下线)第八章 立体几何初步(压轴题专练)-单元速记·巧练(人教A版2019必修第二册)(已下线)专题8.9 空间角与空间距离大题专项训练-举一反三系列(已下线)高一下学期期中复习解答题压轴题十八大题型专练(2)-举一反三系列(人教A版2019必修第二册)(已下线)专题20 空间直线、平面的垂直-《重难点题型·高分突破》(人教A版2019必修第二册)(已下线)第8章 立体几何初步 单元综合检测(难点)-《重难点题型·高分突破》(人教A版2019必修第二册)(已下线)高一下学期期末复习解答题压轴题二十四大题型专练(2)-举一反三系列(人教A版2019必修第二册)(已下线)重难点专题14 利用传统方法解决二面角问题-【帮课堂】(苏教版2019必修第二册)
名校
7 . 如图,在四棱柱
中,四边形
是平行四边形,
,
,
,
,
为
的中点,且
.
(1)求证:
平面
;
(2)若平面
与平面
的夹角的余弦值为
,求三棱锥
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ced06b71073e1bb777f326f06016ce17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65a3e478bb87d094e3a0af30dd10ae8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e075468e7fb0bf30229aec01a7205977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c078a10649b3a953631690021aa59879.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59fc4baf66de980a95f267051e6b190d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/31/f1a7d905-8650-42ac-910a-75b64a3ee109.png?resizew=155)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af6ab0386dde0643de8caf33f946072f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)若平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bf9628142422a4884bd59538da6d312.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ebb05874eb3353d754af24c9974273e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/924d32b574fe69e43724304cf39513e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a35ac94f0ea5d79ae4530f5a3116f670.png)
您最近一年使用:0次
2023-10-13更新
|
897次组卷
|
3卷引用:山西省孝义市2023-2024学年高二上学期10月月考数学试题
名校
8 . 如图,四棱锥
的侧面
是边长为2的正三角形,底面
为正方形,且平面
平面
,
,
分别为
,
的中点.
;
(2)在线段
上是否存在一点
使得
平面
,存在指出位置,不存在请说明理由.
(3)求二面角
的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852aabd89edffc1b94344ff3f1f31ccd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93edc7bb513f40a89173121c8570cd65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/014c4c0d6c8e50e5c6c83e857f9ecac7.png)
(2)在线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8014e499e7852b587b3b36af14b7816.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d62de810f5160223afa54fd882acb9b.png)
(3)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c80c2240af31b07857eaac003b3d8132.png)
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2023-07-27更新
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2008次组卷
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8卷引用:福建省福州高级中学2022-2023学年高一下学期第四学段(期末)考试数学试题
福建省福州高级中学2022-2023学年高一下学期第四学段(期末)考试数学试题江西省吉安市第三中学2023-2024学年高二上学期开学考试(艺术类)数学试题(已下线)【一题多解】立体几何 新旧呼应(已下线)第二章 立体几何中的计算 专题一 空间角 微点8 二面角大小的计算(三)【培优版】(已下线)高一下学期期中复习解答题压轴题十八大题型专练(2)-举一反三系列(人教A版2019必修第二册)云南省大理白族自治州祥云县祥云祥华中学2023-2024学年高一下学期4月二调数学试题福建省宁德市博雅培文学校2023-2024学年高一下学期5月月考数学试题(已下线)专题02 高一下期末真题精选(2)-期末考点大串讲(人教A版2019必修第二册)
名校
9 . 如图,在四棱锥
中,底面
是正方形,侧面QAD是正三角形,侧面
底面
,M是QD的中点.
平面
;
(2)求侧面QBC与底面
所成二面角的余弦值;
(3)在棱QC上是否存在点N使平面
平面AMC成立?如果存在,求出
,如果不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55c6caa0455442437177ab9b995df37b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb11df029afb11e4233989b1338cb3a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d0edb1508fc95765f3bb316bcb5252d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c550269f3199038726f55cbd281c13a.png)
(2)求侧面QBC与底面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(3)在棱QC上是否存在点N使平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79a2e866037fb17d7fb74b462ef2f34d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6525116388ec2bf0e2828bdc3cc5d3b9.png)
您最近一年使用:0次
2023-07-31更新
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1619次组卷
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10卷引用:吉林省长春市实验中学2022-2023学年高一下学期期末数学试题
吉林省长春市实验中学2022-2023学年高一下学期期末数学试题(已下线)第10章 空间直线与平面(压轴题专练)-2023-2024学年高二数学单元速记·巧练(沪教版2020必修第三册)(已下线)第八章 立体几何初步(提升卷)-重难点突破及混淆易错规避(人教A版2019必修第二册)(已下线)6.5.2平面与平面垂直-【帮课堂】(北师大版2019必修第二册)(已下线)高一下学期期末复习解答题压轴题二十四大题型专练(2)-举一反三系列(人教A版2019必修第二册)(已下线)高一下学期期末数学试卷(巩固篇)-举一反三系列(人教A版2019必修第二册)江苏省宿迁市泗阳县实验高级中学2023-2024学年高一下学期第二次调研测试(5月)数学试题安徽省安庆市第一中学2023-2024学年高一下学期5月同步测试数学试卷河南省信阳市浉河区信阳高级中学2023-2024学年高一下学期6月月考数学试题西安市交大附中2023—2024学年高一下学期第二次月考数学试题
名校
解题方法
10 . 在菱形
中,已知
,
.
是对角线
上一点,沿
把菱形折成二面角
,将折成二面角后的
点记作
,设
,点
在平面
上的射影记为
.
(1)当
是
的中点时,如图1,求证
平面
;
(2)当
落在菱形的边
上时,如图2,求二面角
的取值范围;
(3)设折痕
与菱形的边
交于点
,求四棱锥
体积的最大值(说明:可以用到必修一探究实践活动中得到的不等式![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d936ea1443a8c881633d5e04fdd3434.png)
).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e918b70b02a73685e3c536c7f380e2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8efa92fbb689ce6f9ab3384918f48774.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3160d33de228937b8a691519fced22e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca67a5b8f69507c8b80379e86f90a8ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/15/926446e4-566c-43a8-8e7c-da6d5f318095.png?resizew=342)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/662698361c6b3ddaf0c28a3c87be53e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bf12905647aeeded72bbca21a63f319.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8efa92fbb689ce6f9ab3384918f48774.png)
(3)设折痕
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee724bdc07aa2bacb42d4779585a2a12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d936ea1443a8c881633d5e04fdd3434.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d3ac744d01bdb607d6c2ffd7ef64a24.png)
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