名校
解题方法
1 . 如图,已知圆台
的下底面半径为2,上底面半径为1,母线与底面所成的角为
为母线,平面
平面
为
的中点.
![](https://img.xkw.com/dksih/QBM/2022/4/26/2966603934867456/2971240273485824/STEM/4ec3219b-03e8-43b5-89a8-1065b806c921.png?resizew=256)
(1)证明:平面
平面
;
(2)当点
为线段
的中点时,求直线
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae0a4f38420bb9215dbc9c875b755838.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2aad0fd1f6be95656b17ad937cb51c6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/550d549789c10796087d258cf1d1bde7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4ab35b3793bbd542738c481937772fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
![](https://img.xkw.com/dksih/QBM/2022/4/26/2966603934867456/2971240273485824/STEM/4ec3219b-03e8-43b5-89a8-1065b806c921.png?resizew=256)
(1)证明:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4080387bf118a3b382ede20ce8d60e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/945d27bb4d47e78d472186cb02314a8b.png)
(2)当点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bd9c299d7ed6e96cd09cae67d7bba41.png)
您最近一年使用:0次
2022-05-03更新
|
738次组卷
|
3卷引用:江苏省郑梁梅高级中学2022-2023学年高二下学期4月月考数学试题
江苏省郑梁梅高级中学2022-2023学年高二下学期4月月考数学试题湖南省衡阳市2022届高三下学期二模数学试题(已下线)考点18 空间中的角度和距离问题-1-(核心考点讲与练)-2023年高考数学一轮复习核心考点讲与练(新高考专用)
名校
2 . 如图,正方形
和直角梯形
所在平面互相垂直,
,
,且
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/7/5196057a-6554-4001-8019-85ac67b33f8b.png?resizew=123)
(1)证明:
平面
;
(2)求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a6480f384476190883f06c0289c7519.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2ea9d3df7c2bcdf135dedd1554fb82b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cce38d8a8a7043586aad206f8153d0bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a77f26a7be722e00baa984f769ec8d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ce84f6062f12bf6ef42d7b733cd2248.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/7/5196057a-6554-4001-8019-85ac67b33f8b.png?resizew=123)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc521258fcaeaf7acffc5ae98c3af6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/646084b7f3902efa4c462ed67599265a.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77a34e44c5d7e1d22521fb293994f5b0.png)
您最近一年使用:0次
2022-09-06更新
|
1014次组卷
|
6卷引用:江苏省淮安市盱眙县马坝高级中学2023-2024学年高三上学期期中数学试卷
名校
3 . 如图,直三棱柱
中,
,
、
、
分别是
、
、
的中点.
![](https://img.xkw.com/dksih/QBM/2022/1/2/2886003315892224/2918442117898240/STEM/cde6e544-ff60-4cd9-a43b-e18b13d4ac09.png?resizew=152)
(1)求证:
;
(2)若
,求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39e8a50eb8b05d9b6950911b93625c3a.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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e26d9636ad77369535852c6e4493446a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
![](https://img.xkw.com/dksih/QBM/2022/1/2/2886003315892224/2918442117898240/STEM/cde6e544-ff60-4cd9-a43b-e18b13d4ac09.png?resizew=152)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f5c3348ca18ee9788c1bcfc7827f4be.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18cdf258eb8bca2ee1ead36140c64539.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e98457d387bb6318d7953175e9f9d92.png)
您最近一年使用:0次
2022-02-17更新
|
250次组卷
|
2卷引用:江苏省淮安市淮阴中学2021-2022学年高三上学期12月月考数学试题
名校
4 . 如图,
且
且
且
平面
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/7/94b7fc63-da4b-4947-9cb0-b8f632c4ef3d.png?resizew=175)
(1)若
为
的中点,
为
的中点,求证:
平面
;
(2)求二面角
的正弦值;
(3)求直线
到平面
的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34e0a957a55460c72673c0f2ee90dbb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db2bc58f6c66b96a3624cbaf06689847.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fa14ce2ff04d7d29a6296792279c64c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d156737daa15bf9c634e9eac1687ecd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/615dea62b4775453e2f0330c4d3e5719.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/7/94b7fc63-da4b-4947-9cb0-b8f632c4ef3d.png?resizew=175)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e55e398e8520d8a36fb5a625a085b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edcf19a7f0dd0cdf59516ae585025110.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10fc7991ea17d54ff5f4445ac5699463.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a8d99c75180422fecf6d3f3d2910b34.png)
(3)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8257b6bd25104e07b9ad935c0a3aac4.png)
您最近一年使用:0次
2021-08-12更新
|
726次组卷
|
6卷引用:江苏省淮安市马坝高级中学2021-2022学年高二下学期期中数学试题
名校
5 . 如图,四棱锥
的侧面
是正三角形,底面
是直角梯形,
,
,
为
的中点.
![](https://img.xkw.com/dksih/QBM/2021/1/30/2647183122931712/2650614553337856/STEM/654e91d9-13cf-41b0-a52d-ba0bf967da00.png)
(1)求证:
;
(2)若
,求线
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/55a675310c8ba418e5a59beb7317e21e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c709ccf950ed4d37ad9e9234dd2446b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bacde79b2d53b9a47b73c4376b1032e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://img.xkw.com/dksih/QBM/2021/1/30/2647183122931712/2650614553337856/STEM/654e91d9-13cf-41b0-a52d-ba0bf967da00.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a40b744f59a78172b82d3ea6a3216103.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74107ed86d62c4a2ca1630d626dff115.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/892909e49156f7dcc0650fcd65243877.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
您最近一年使用:0次
2021-02-04更新
|
872次组卷
|
4卷引用:江苏省淮安市淮安区2021-2022学年高二下学期期中数学试题
6 . 如图,在四棱锥
中,底面
是菱形,G、P是线段
、
的中点.
![](https://img.xkw.com/dksih/QBM/2020/12/18/2616987940626432/2619498007617536/STEM/dbb6316d-7f27-48d3-b0c5-845ce79f4b90.png)
(1)证明:
平面
;
(2)若
,
,
,求平面
与平面
所成锐二面角的余弦值.
![](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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/defa5b53043ae802bb1af7d14374406d.png)
![](https://img.xkw.com/dksih/QBM/2020/12/18/2616987940626432/2619498007617536/STEM/dbb6316d-7f27-48d3-b0c5-845ce79f4b90.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1651b87ec052cefebae47d1e589c5146.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21665d21bbfb04410c78345de1fd15ae.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb8fb552b9e21dbaba74d11aa747790.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c90adb28e2332fb7cb1e02cf00fac44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa540507612bcb666413642e858eaded.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21665d21bbfb04410c78345de1fd15ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67a1822040c893d9c4961acc810afe02.png)
您最近一年使用:0次
2020-12-22更新
|
115次组卷
|
2卷引用:江苏省淮安市淮阴中学2020-2021年高三上学期12月阶段检测数学试题
名校
解题方法
7 . 如图①,在菱形
中,
且
,
为
的中点,将
沿
折起使
,得到如图②所示的四棱锥
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/4/3d120806-7143-43c1-813e-ad31c1ab32c8.png?resizew=340)
(1)求证:平面
平面
;
(2)若
为
的中点,求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5b8b98b2f83279a49e94d9f48c5e6f2.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/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e742966e3711cfa53dce04022acf4bcc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a88c44f558705de3bcefcfc0ece96b8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5164a3cc47e266446d49127e2ef10c37.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/4/3d120806-7143-43c1-813e-ad31c1ab32c8.png?resizew=340)
(1)求证:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed5f0cfc1049f84a04c81bd213afb8d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02a7ba7cd0c654714c967a900513ba16.png)
您最近一年使用:0次
2020-08-14更新
|
1637次组卷
|
12卷引用:江苏省淮安市盱眙中学2021-2022学年高三上学期期中数学试题
江苏省淮安市盱眙中学2021-2022学年高三上学期期中数学试题四川省成都市2021届高三毕业班摸底测试数学理科试题四川省武胜烈面中学校2020-2021学年高三9月月考数学(理)试题甘肃省天水市第一中学2021届高三十模数学(理)试题安徽省合肥市第八中学2021届高三下学期高考模拟最后一卷理科数学试题重庆复旦中学2020-2021学年高二下学期期中数学试题福建省上杭一中、永定一中2022届高三上学期第一次联考数学试题广东省深圳市福田区外国语高级中学2022届高三上学期第二次调研考试数学试题湖南省长沙市长郡中学2021-2022学年高三上学期月考(三)数学试题湖北省黄石市有色一中2021-2022学年高二上学期10月月考数学试题湖北省鄂州市鄂城区秋林高级中学2022-2023学年高二上学期10月月考数学试题重庆市永川北山中学校2023届高三下学期入学考试数学试题
名校
解题方法
8 . 如图,四棱锥
的底面是正方形,每条侧棱的长都是底面边长的
倍,P为侧棱SD的中点,试用向量法解决下面的问题.
![](https://img.xkw.com/dksih/QBM/2020/10/10/2568133224710144/2569596584730624/STEM/ffc8d4c1-121f-4625-ba19-265acee96a6d.png?resizew=294)
(1)求证:
;
(2)若
,求线段BP的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faeb97acf19bd3b2c6c77c2814df4d2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://img.xkw.com/dksih/QBM/2020/10/10/2568133224710144/2569596584730624/STEM/ffc8d4c1-121f-4625-ba19-265acee96a6d.png?resizew=294)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c177e06cc3f703e8ca7be7c491fa2942.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef0402dd5ae3db10281f9f1e11738bcb.png)
您最近一年使用:0次
2020-10-12更新
|
461次组卷
|
5卷引用:江苏省淮安市高中校协作体2022-2023学年高二下学期期中数学试题
9 . 如图,在四棱锥P-ABCD中,底面ABCD是矩形,侧面PAD⊥底面ABCD,E为PA的中点,过C,D,E三点的平面与PB交于点F,且PA=PD=AB=2.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/26/f61746a9-5bc6-42b0-a4f5-aac3ff460a4b.png?resizew=226)
(1)证明:
;
(2)若四棱锥
的体积为
,则在线段
上是否存在点G,使得二面角
的余弦值为
?若存在,求
的值;若不存在,请说明理由.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/26/f61746a9-5bc6-42b0-a4f5-aac3ff460a4b.png?resizew=226)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab1a1fd2fc33e89f357cef772ff6cd0e.png)
(2)若四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a391005600bdd69c96750589f9adb048.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a155d285d8d487adf9fac93a48bb0700.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9868f77d5ab5073b6145f1c6d272122e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6729fb0c5e5e9549035590144b73144.png)
您最近一年使用:0次
2020-08-17更新
|
496次组卷
|
4卷引用:江苏省淮安市涟水县第一中学2024届高三上学期12月考试数学试题
江苏省淮安市涟水县第一中学2024届高三上学期12月考试数学试题湖南省常德市第二中学2020届高三下学期临考冲刺数学(理)试题(已下线)专题20 立体几何综合-2020年高考数学(理)母题题源解密(全国Ⅱ专版)(已下线)广东省江门市棠下中学2022-2023学年高三上学期数学试题变式题17-22
名校
10 . 如图,在平行四边形ABCD中,
,四边形ACEF为正方形,且平面
平面ACEF.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/30/dbae6f82-443c-46fa-9876-b870458f380b.png?resizew=185)
(1)证明:
;
(2)求平面BEF与平面BCF所成锐二面角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a64e473873047e3df9b53a2493b6cd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cf9a6db3571fa57bfa2d5e4d44c51b3.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/30/dbae6f82-443c-46fa-9876-b870458f380b.png?resizew=185)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9f76925ed99b7172956319974258a9b.png)
(2)求平面BEF与平面BCF所成锐二面角的余弦值.
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2020-02-10更新
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405次组卷
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2卷引用:江苏省淮安市涟水中学2020-2021学年高三上学期阶段检测(三)数学试题