名校
1 . 如图,在三棱锥
中,平面
平面
,
是以
为斜边的等腰直角三角形,
,
,
为
中点,
为
内的动点(含边界).
![](https://img.xkw.com/dksih/QBM/2023/10/14/3346119356293120/3348038980730880/STEM/fa000488120d40448813e4729225b425.png?resizew=207)
(1)求证:
平面
;
(2)求平面
与平面
夹角的余弦值;
(3)若
平面
,求直线
与平面
所成角的正弦值的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d077f6da8b2c00b152d4679aa2ed7f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4494a85de0be0b97a69348115aef8513.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fcc2aba06dbc28f39d111a10233ff12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
![](https://img.xkw.com/dksih/QBM/2023/10/14/3346119356293120/3348038980730880/STEM/fa000488120d40448813e4729225b425.png?resizew=207)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3e126c16032892966489053f44b9048.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a0a858194b17ae1e609ed341d75194.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35d58f9019097bd05037aefd5c322916.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
您最近一年使用:0次
2023-10-17更新
|
304次组卷
|
2卷引用:北京市朝阳区北京工业大学附属中学2023-2024学年高二上学期10月月考数学试题
解题方法
2 . 如图,在四棱锥
中,平面
平面
为
的中点,
,
.
(1)求证:平面
平面
;
(2)求平面
与平面
的余弦值;
(3)在线段
上是否存在点
,使得
平面
?若存在,求出点
的位置;若不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93edc7bb513f40a89173121c8570cd65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23d11e19c84255eb0431415c2dec553d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8f9dab3914e54230b717252736be326.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83d2d775c03b3ea1674d5b861d6fb0fe.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/17/4b62daca-038b-4477-b354-be2de38bf9e5.png?resizew=164)
(1)求证:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93edc7bb513f40a89173121c8570cd65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/955e030d649a3c7885071b4bf849993c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64eb31601464364be2baf4aa87404bcd.png)
(3)在线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc5adb5eb60ae4435a12d93854066298.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0457394ce4f2dc8d940c565c94dcf557.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
您最近一年使用:0次
名校
3 . 如图,在四棱锥
中,
平面ABCD,
,
,E为CD的中点,M在AB上,且
,
(1)求证:
平面PAD;
(2)求平面PAD与平面PBC所成锐二面角的余弦值;
(3)点F是线段PD上异于两端点的任意一点,若满足异面直线EF与AC所成角为
,求AF的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3299fc3474a4b67ffc38e5397c9b98d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b684dd5c86b7568976bf92dc02ce729.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d74b1d0480790400a9223e4437afdba.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/26/e5d3c671-7b37-404d-a398-7c67966640a0.png?resizew=162)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd0285afe567ca0b32f0ccafc30167cc.png)
(2)求平面PAD与平面PBC所成锐二面角的余弦值;
(3)点F是线段PD上异于两端点的任意一点,若满足异面直线EF与AC所成角为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79a97bb4dcfab4ec7539bc783d563c49.png)
您最近一年使用:0次
2023-07-25更新
|
676次组卷
|
13卷引用:北京市中国人民大学附属中学2021-2022学年高二10月统练数学试题(一)
北京市中国人民大学附属中学2021-2022学年高二10月统练数学试题(一)北京市顺义区第一中学2023-2024学年高二上学期10月考试数学试题北京市第八十中学2023-2024学年高二上学期10月阶段测评数学试题天津市耀华中学2020-2021学年高三上学期第二次月考数学试题天津市南开大学附属中学2023届高三下学期2月统练(一)数学试题天津市北师大静海附属学校2024届高三上学期第三次月考数学试题天津市九校联考2022届高三下学期一模数学试题天津市滨海新区塘沽第一中学2023届高三上学期线上统练摸底考试数学试题天津市九十六中学2022-2023学年高三上学期期末数学试题天津市滨海新区塘沽第一中学2023届高三下学期十二校联考(一)数学模拟试题(已下线)第07讲 空间向量的应用 (1)(已下线)第07讲 拓展一:异面直线所成角(传统法与向量法,5类热点题型讲练)-【帮课堂】2023-2024学年高二数学同步学与练(人教A版2019选择性必修第一册)(已下线)通关练03 用空间向量解决距离、夹角问题10考点精练(58题) - 【考点通关】2023-2024学年高二数学高频考点与解题策略(人教A版2019选择性必修第一册)
名校
4 . 在三棱柱
中,侧面
为矩形,
平面
, D,E分别是棱
的中点.
(1)求证:
平面
;
(2)若
,求直线
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e168672b47d7e64dc1b404f8882c7dcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e56fdf217165748fafe938b64fa08179.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e168672b47d7e64dc1b404f8882c7dcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c95c0160e73beb94a4a1cbc0168e9a5.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/11/68a4621c-c81b-4a05-bdfa-c2b9b738db1d.png?resizew=123)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6ae72f5e5891249caa10c43224da89c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dddfef906818cc8ddd00f867b77f227.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb717228e1762d335814a3adc90eae45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dddfef906818cc8ddd00f867b77f227.png)
您最近一年使用:0次
名校
解题方法
5 . 如图,四棱锥
中,平面
平面
,
,
,
,
,
,
为
上一点,
.
(1)求证:
平面
;
(2)求证:
平面
;
(3)求平面
与平面
的夹角的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4aa9084b8fe0fe05c4388d1f835587b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd8e727e4efc22b49649f71ae9c9d84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9536a2be7b84612f45cc875a00c5a5d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/905584add4587cfc006afdce3e7ef91c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7873eeda444826cf6a15f86f25f6e0b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f29c3e772e56008790298824122792.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e3ad276e6b32bd203fdacb42b1fe6d7.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/25/242a61c5-a7bb-4b1d-84e9-65015b03d5b2.png?resizew=157)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f4c3f9dd5d0343597a7f58a1989b537.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8c2b786c64e6a9ed2ec5670cde74f86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34be4e71cabf458f17a6cd7f24bc70af.png)
(3)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bee4a6b8ef3e79b4482388c3391d8b18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93f76b6ac1b8875af7156f3239dae6f7.png)
您最近一年使用:0次
名校
解题方法
6 . 在如图所示的几何体ABCDFE中,面ABCD是边长为2的正方形,AE⊥面ABCD,DF∥AE,且DF
AE=1,N为BE的中点.M为CD的中点,
(1)求证:FN∥平面ABCD;
(2)求二面角N﹣MF﹣D的余弦值;
(3)求点A到平面MNF的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a50a39604477d1d9326eb455cda2e838.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/26/0a941b6a-de00-4126-8dba-fad8a237b7a1.png?resizew=161)
(1)求证:FN∥平面ABCD;
(2)求二面角N﹣MF﹣D的余弦值;
(3)求点A到平面MNF的距离.
您最近一年使用:0次
2023-05-25更新
|
1687次组卷
|
10卷引用:北京市清华大学附属中学2021-2022学年高二下学期统练一数学试题
北京市清华大学附属中学2021-2022学年高二下学期统练一数学试题重庆市重庆十八中两江实验中学校2023届高三上学期第一次适应性强化训练数学试题江苏省南京师范大学苏州实验学校2022-2023学年高二上学期9月月考数学试题(已下线)第09讲 空间向量的应用 -【暑假自学课】2022年新高二数学暑假精品课(人教版2019必修第二册+选择性必修第一册)(已下线)专题24 空间向量及其应用(讲义)-2023年高考数学一轮复习精讲精练宝典(新高考专用)(已下线)7.6 空间向量求空间距离(精练)辽宁省辽南协作校2022-2023学年高二上学期期末考试数学试题(已下线)专题1.9 空间向量的应用-重难点题型精讲-2022-2023学年高二数学举一反三系列(人教A版2019选择性必修第一册)辽宁省营口市大石桥市第三高级中学等2校2022-2023学年高二上学期期末数学试题(已下线)第11讲 用空间向量研究距离、夹角问题11种常见考法归类-【暑假自学课】2023年新高二数学暑假精品课(人教A版2019选择性必修第一册)
7 . 如图,在四棱锥
中,底面
是边长为
的正方形,侧面
为等腰直角三角形,且
,点
为棱
上的点,平面
与棱
交于点
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/20/09bdcc80-e0b1-4e87-904b-91e9d73ddd4b.png?resizew=152)
(1)求证:
;
(2)从条件①、条件②、条件③这三个条件中选择两个作为已知,求平面
与平面
所成锐二面角的大小.
条件①:
;
条件②:平面
平面
;
条件③:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852aabd89edffc1b94344ff3f1f31ccd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d96c0afa541ea653e6fa345ba93b287c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fa3254460ecbacecb3e57c5dce227f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/20/09bdcc80-e0b1-4e87-904b-91e9d73ddd4b.png?resizew=152)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd6020b78ff385667b30088ecadeadd3.png)
(2)从条件①、条件②、条件③这三个条件中选择两个作为已知,求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e08c14e87a2bcf7090eab2fea73667d2.png)
条件①:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/338c6c83ab4abc895ac36ab888a55be6.png)
条件②:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93edc7bb513f40a89173121c8570cd65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
条件③:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65355f6a872f7148e4efd9e3bf877860.png)
您最近一年使用:0次
2023-05-12更新
|
988次组卷
|
4卷引用:北京市海淀区北京交大附中2024届高三上学期12月诊断练习数学试题
名校
8 . 如图,在四棱锥
中,底面
为正方形,
平面
,
,
分别为棱
,
的中点,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/19/badd85f7-5e1e-47a5-8eeb-dfeb67a7413c.png?resizew=156)
(1)求证:
平面
;
(2)求直线
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.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/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f83a04565a8ebaa111894b724b0ba266.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/19/badd85f7-5e1e-47a5-8eeb-dfeb67a7413c.png?resizew=156)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7592c4f01c8e06c7ee90df5b9413a9f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7609a1407f1e965fc9f1235552dcf9e.png)
您最近一年使用:0次
2023-04-17更新
|
1154次组卷
|
9卷引用:北京市第三十五中2021-2022学年高二12月月考数学试题
名校
9 . 在四棱锥P-ABCD中,底面ABCD为直角梯形,
,
,
,E为线段AD的中点.PE⊥底面ABCD,点F是棱PC的中点,平面BEF与棱PD相交于点G.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/29/7970cfeb-0c9d-4ce5-82be-2be538a59247.png?resizew=128)
(1)求证:
;
(2)若PC与AB所成的角为
,求直线PB与平面BEF所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d730ae4307db56b47849c3a19dedfb3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4795ee1f96b430529934e2231b38885d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/639bec6242a4b3f7bfb4b7033a67328c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/29/7970cfeb-0c9d-4ce5-82be-2be538a59247.png?resizew=128)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16b8b871bc2a1c85da6a27451dbbf522.png)
(2)若PC与AB所成的角为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15615de1a6df206dbd081251f676578e.png)
您最近一年使用:0次
2023-04-28更新
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1357次组卷
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6卷引用:北京市第五中学2021届高三上学期10月月考数学试题
名校
解题方法
10 . 如图,在四棱锥
中,
平面
,
,
,
,
,点
为
的中点.
(1)求证:平面PBC⊥平面PAC;
(2)求二面角E﹣CD﹣A的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e52411c8437d0640c5b3d87cf5ebebb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1134c8e3440abb6cd385af2c169037fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52a923784f083b7f4777891afe06b44e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/16/a6557ddf-6220-4bb4-80c1-d56caebf70aa.png?resizew=179)
(1)求证:平面PBC⊥平面PAC;
(2)求二面角E﹣CD﹣A的余弦值.
您最近一年使用:0次
2023-06-14更新
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714次组卷
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10卷引用:北京师范大学附属实验中学2021-2022学年高二年级12月月考数学试题
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