名校
1 . 如图,在四棱锥
中,
为直角梯形,
,
,平面
平面
.
是以
为斜边的等腰直角三角形,
为
上一点,且
.
![](https://img.xkw.com/dksih/QBM/2023/10/20/3350341524086784/3351791761915904/STEM/8ed66266088b4f0db9f9771c407736b3.png?resizew=98)
(1)证明:直线
平面
;
(2)求二面角
的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faeb97acf19bd3b2c6c77c2814df4d2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f571396be1aa4a8914a66f7d7abd6381.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bd6a2b112facda441f4e34bf5c145fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/235d1553f6806c1eee3b17b94d23f0f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c009f663ad2b0c3ba521daf4b86b066f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d1e4c39b516f9b56b5a131de6fb9902.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b5e290c6b2c5508a3bf6117afbf7e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa8246b4ac03ec7cf2cec56887cc981.png)
![](https://img.xkw.com/dksih/QBM/2023/10/20/3350341524086784/3351791761915904/STEM/8ed66266088b4f0db9f9771c407736b3.png?resizew=98)
(1)证明:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca6d50356a01ae13936f1bd8efa94c98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c66d99a6a8415ddad22bbed33b64cfb.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01200dd11d32f00d4a74f2ef3ca79118.png)
您最近一年使用:0次
名校
解题方法
2 . 四边形
是边长为1的正方形,
与
交于
点,
平面
,且二面角
的大小为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/9/662676a7-4f5f-4e81-ae57-e62a7c32acdc.png?resizew=160)
(1)求点
到平面
的距离;
(2)求直线
与平面
所成的角.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.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/b796bbaeb8450404c2d146283562006e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79a97bb4dcfab4ec7539bc783d563c49.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/9/662676a7-4f5f-4e81-ae57-e62a7c32acdc.png?resizew=160)
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f571a1aac46c6d0cf440c0ec2846bf9.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
您最近一年使用:0次
2023-04-06更新
|
713次组卷
|
4卷引用:上海市杨浦区2023届高三二模数学试题
3 . 如图,四棱锥
的底面是矩形,
平面
,
为
的中点,且
,
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/9/d1dc5688-5fbd-4ef2-9bb8-7728c9ee88a9.png?resizew=567)
(1)求点
到平面
的距离;
(2)求二面角
的大小;
(3)已知
为
的中点,若一只蚂蚁从
点出发,沿着四棱锥的表面爬行,求这只蚂蚁爬到点
的最短距离(结果精确到0.01).
![](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/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc11331a7b2d2619b40ee6d34c3bd620.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/491c3a4f72b84ebadd28b90711435adc.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/9/d1dc5688-5fbd-4ef2-9bb8-7728c9ee88a9.png?resizew=567)
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b35708245a5da381178284f5ac7ce9c6.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6ea5de1a95497e2818198d0c2a57669.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
您最近一年使用:0次
2023-01-05更新
|
206次组卷
|
2卷引用:上海市上海财经大学附属中学2022-2023学年高二上学期期末数学试题
解题方法
4 . 如图,在正三棱柱
中,底面
的面积为
,侧面积为60,
是
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/9/ada03c52-4b09-40db-9440-ad18660afe66.png?resizew=129)
(1)求异面直线
与
所成的角的大小;
(2)求直线
与平面
所成的角的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41322821ce31416fdac8dd6e0aa41c71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/9/ada03c52-4b09-40db-9440-ad18660afe66.png?resizew=129)
(1)求异面直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93ecad355286188fd317939fa50f9555.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93ecad355286188fd317939fa50f9555.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d9a8181f7a7fe7f3fac872ce9534f15.png)
您最近一年使用:0次
名校
解题方法
5 . 如图所示,在棱长为2的正方体
中,
分别为线段
的中点.
与
所成的角;
(2)求三棱锥
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad056c25c0fdcbcc765eb5cbc6093f2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89b866a756d422faec0f7eb229dfaabf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
(2)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1be192377dae2ccbf1e0348b5bdd8bd7.png)
您最近一年使用:0次
2023-01-03更新
|
356次组卷
|
7卷引用:上海市复旦大学附中2018届高三上学期10月月考数学试题
6 . 在四棱锥P﹣ABCD中,底面是边长为2的菱形,∠DAB=60°,对角线AC与BD相交于点O,PO⊥平面ABCD,PB与平面ABCD所成的角为60°.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/9/75820310-db78-455d-9a78-afec674717ab.png?resizew=193)
(1)求四棱锥P﹣ABCD的体积;
(2)若E是PB的中点,求异面直线DE与PA所成角的大小(结果用反三角函数值表示).
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/9/75820310-db78-455d-9a78-afec674717ab.png?resizew=193)
(1)求四棱锥P﹣ABCD的体积;
(2)若E是PB的中点,求异面直线DE与PA所成角的大小(结果用反三角函数值表示).
您最近一年使用:0次
2022-11-08更新
|
399次组卷
|
5卷引用:上海市复旦大学附属中学2020-2021学年高二下学期期末数学试题
名校
解题方法
7 . 如图,在长方体ABCD﹣A1B1C1D1中,AA1=1,AB=AD=2,E、F分别是AB、BC的中点.
![](https://img.xkw.com/dksih/QBM/2022/11/5/3103321081421824/3103391535513600/STEM/4a1ee999540a404280d256d1314b94cf.png?resizew=155)
(1)证明:A1、C1、F、E四点共面;
(2)求直线CD1与平面A1C1FE所成的角的大小.
![](https://img.xkw.com/dksih/QBM/2022/11/5/3103321081421824/3103391535513600/STEM/4a1ee999540a404280d256d1314b94cf.png?resizew=155)
(1)证明:A1、C1、F、E四点共面;
(2)求直线CD1与平面A1C1FE所成的角的大小.
您最近一年使用:0次
2022-11-06更新
|
388次组卷
|
7卷引用:上海市控江中学2022-2023学年高二上学期期中数学试题
上海市控江中学2022-2023学年高二上学期期中数学试题沪教版(2020) 选修第一册 新课改一课一练 第3章 3.4.3 求角的大小上海市进才中学2022届高三下学期期中数学试题沪教版(2020) 选修第一册 新课改一课一练 期末测试B(已下线)专题16 空间向量及其应用(练习)-2(已下线)第20讲 空间向量与立体几何-3(已下线)专题11空间向量与立体几何必考题型分类训练-2
名校
8 . 如图,直三棱柱
内接于高为
的圆柱中,已知
,
,
,
为
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/14/08a10c66-ec46-4866-ab5c-dd8b88b8779f.png?resizew=136)
(1)求圆柱的表面积;
(2)求二面角
的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71f2185273bf04c11118c7954f7ec822.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ed8f7d3d7043d4b1eb98fc5c4e2fcd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a67d8576417f761dd5f583ad3a1555a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1799f2f26ed09738aa08fdf64ca86242.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/14/08a10c66-ec46-4866-ab5c-dd8b88b8779f.png?resizew=136)
(1)求圆柱的表面积;
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aeb2af3f9181e6fcce86c71aee45c9e1.png)
您最近一年使用:0次
2022-10-11更新
|
1348次组卷
|
8卷引用:上海市杨浦区同济大学第一附属中学2024届高三上学期期中数学试题
上海市杨浦区同济大学第一附属中学2024届高三上学期期中数学试题上海市洋泾中学2023届高三上学期10月月考数学试题(已下线)第20讲 空间向量与立体几何-2上海市奉贤区2023届高三上学期期中数学试题(已下线)3.4 空间向量在立体几何中的应用(作业)(夯实基础+能力提升)-【教材配套课件+作业】2022-2023学年高二数学精品教学课件(沪教版2020选修第一册)(已下线)重难点01 空间角度和距离五种解题方法-【满分全攻略】2022-2023学年高二数学下学期核心考点+重难点讲练与测试(沪教版2020选修一+选修二)上海市敬业中学2024届高三上学期10月月考数学试题吉林省白城市通榆县毓才高级中学有限责任公司2023-2024学年高二上学期10月期中数学试题
19-20高二下·上海浦东新·阶段练习
名校
解题方法
9 . 如图,在四棱锥
中,底面正方形ABCD的边长为2,
底面ABCD,E为BC的中点,PC与平面PAD所成的角为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/7/82dc9d8b-cf9e-4be5-b89e-08d2315dc386.png?resizew=143)
(1)求PA的长度;
(2)求异面直线AE与PD所成角的大小.
![](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/717d026dd0029aab76fd410d88f67bd8.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/7/82dc9d8b-cf9e-4be5-b89e-08d2315dc386.png?resizew=143)
(1)求PA的长度;
(2)求异面直线AE与PD所成角的大小.
您最近一年使用:0次
2022-09-07更新
|
311次组卷
|
7卷引用:上海市杨浦高级中学2022-2023学年高二下学期开学考试数学试题
上海市杨浦高级中学2022-2023学年高二下学期开学考试数学试题(已下线)上海市华东师范大学第二附属中学2019-2020学年高二下学期(4月)月考数学试题上海市交通大学附属中学闵行分校2021-2022学年高二上学期10月月考数学试题沪教版(2020) 选修第一册 同步跟踪练习 第3章 3.4 3 求角的大小 第1课时 求线线角与线面角的大小(已下线)第3章 空间向量及其应用(基础、常考、易错、压轴)分类专项训练(原卷版)上海市七宝中学2023-2024学年高二上学期10月月考数学试题上海市七宝中学2023-2024学年高二上学期9月月考数学试题
名校
解题方法
10 . 如图所示,在四棱锥
中,
平面
,
平面
,
,
,又
,
,
为
中点.
平面
;
(2)求平面
与平面
所成的锐二面角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faeb97acf19bd3b2c6c77c2814df4d2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca5dd496ee0c1170ef6dcc48266ee444.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ef796b46e68fe77b117ff0483d2370c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2ffc6952e988d04f22f0fb2f7f0ab7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ef796b46e68fe77b117ff0483d2370c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84e0b7d845cbceccd3e76ca461fcc534.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa7aeb2a8d1437eeb4482c3b6ad9f315.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c76d296e1cf0e421b3969c70064f6fcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdb64061e933aea7669294640c331bcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/defa5b53043ae802bb1af7d14374406d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1925035dc7e4d98cd72f96fbb60ec2d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc9c9cfa597b444b5c9dbae7a825a695.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d923a338dd2d2e29336b42574d38448.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc9c9cfa597b444b5c9dbae7a825a695.png)
您最近一年使用:0次
2022-08-30更新
|
1123次组卷
|
7卷引用:上海市杨浦区同济大学第一附属中学2024届高三上学期开学考试数学试题