名校
1 . 如图,在圆台
中,
分别为上、下底面直径,且
,
,
为异于
的一条母线.
为
的中点,证明:
平面
;
(2)若
,求二面角
的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/192f4f9446c954a291f779d963f90257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/601050d23e9d0b81ee6c5eda991dbdf9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86605a29fe8fff454e0db6b86047a8fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/439cf259dd6137aa31bb99244a04ddfc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c95c0160e73beb94a4a1cbc0168e9a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afabf56cc68ea438a890f9fea04b708e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab3e0dba5705e1d749cfb21ebbb2ed93.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebc9e0457471047bc750ecd31989414a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6647d7d03d64dc6eac2c9651badd9376.png)
您最近一年使用:0次
2023-03-29更新
|
5586次组卷
|
14卷引用:江苏省八市(南通、泰州、扬州、徐州、淮安、连云港、宿迁、盐城)2023届高三二模数学试题
江苏省八市(南通、泰州、扬州、徐州、淮安、连云港、宿迁、盐城)2023届高三二模数学试题重庆市缙云教育联盟2023届高三二模数学试题(已下线)专题07立体几何的向量方法(已下线)押新高考第20题 立体几何(已下线)江苏省八市2023届高三二模数学试题变式题17-22专题16空间向量与立体几何(解答题)江苏省部分四星级高中2023-2024学年高三上学期期初调研数学试题(已下线)江苏省南通市如皋市2023-2024学年高三上学期期初调研数学试题广东省湛江市第一中学2023-2024学年高二上学期第一次大考数学试题江苏省八市2023届高三下学期第二次调研测试数学试题江苏省镇江市扬中市第二高级中学2023-2024学年高三上学期期末模拟数学试题3(已下线)空间向量与立体几何江苏省南京外国语学校2023-2024学年高三上学期期中模拟数学试题2024届安徽省阜阳市皖江名校联盟高三模拟预测数学试题
名校
解题方法
2 . 在
中,
,过点
作
,交线段
于点
(如图1),沿
将
折起,使
(如图2),点
分别为棱
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/29/af1034bc-5ab5-4b98-9116-da4bc36f5d26.png?resizew=378)
(1)求证:
;
(2)在①图1中
,②图1中
,③图2中三棱锥
的体积最大.
这三个条件中任选一个,补充在下面问题中,再解答问题.
问题:已知__________,试在棱
上确定一点
,使得
,并求平面
与平面
的夹角的余弦值.
注:如果选择多个条件分别解答,按第一个解答计分.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16770045e02c32c6b246f1e88c580647.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b5f215a42c4b7078d8d65923eb9980e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1347b1707478d309af4287a00e852b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5889e1f093f2c35273d3132ef8434e4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cca04b2a2b61d62a809776670a60c09.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/29/af1034bc-5ab5-4b98-9116-da4bc36f5d26.png?resizew=378)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73038c8fab9ef31d42b3ee0631b3dd1c.png)
(2)在①图1中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23e84ed4d1ef85e452a30c6b8f7981b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1371c97ec3d0ea7b3ef979f5538d330.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/891579e7c231584a8e16b8eeff79888e.png)
这三个条件中任选一个,补充在下面问题中,再解答问题.
问题:已知__________,试在棱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5448218bd8c5b4f4a3714e0b0292d2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/212a67f115d1cbe69f100b489babe5f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f5ce42fe8ea626c297e3b2a2ab95149.png)
注:如果选择多个条件分别解答,按第一个解答计分.
您最近一年使用:0次
2023-03-28更新
|
1239次组卷
|
6卷引用:湖南省岳阳市2023届高三下学期二模数学试题
湖南省岳阳市2023届高三下学期二模数学试题四川省内江市2023届高三第三次模拟考试数学(理科)试题(已下线)专题07立体几何的向量方法专题16空间向量与立体几何(解答题)宁夏银川一中2022-2023学年高二下学期期中考试数学(理)试题(已下线)专题06 立体几何 第一讲 立体几何中的证明问题(分层练)
名校
解题方法
3 . 在苏州博物馆有一类典型建筑八角亭,既美观又利于采光,其中一角如图所示,为多面体
,
,
,
,
底面
,四边形
是边长为2的正方形且平行于底面,
,
,
的中点分别为
,
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2407d4b99e51c6a8d33cc32972549f9.png)
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/28/1b33c98e-854e-4684-9ba8-f1a7ce79dff8.png?resizew=453)
(1)证明:
平面
;
(2)求平面
与平面
夹角的余弦值;
(3)一束光从玻璃窗面
上点
射入恰经过点
(假设此时光经过玻璃为直射),求这束光在玻璃窗
上的入射角的正切值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afb94145069d895e289f871c9deb403a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7bd02e0adeae92ba9526261b1baf797.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/542b5bc10c7341c04c22244f3ec16e6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03733d1465d041a6d6da32bf91a7cff8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5845ccc0d735dc14c92a8926d9b1def6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9142a8490de14a87eda628ffa7e28982.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632f2bf1cd0435041fa04b01901d1c8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8f3392a792c219bf3f365281ad9bb70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15dc61d5de97b5a40be925b278ae494c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1859959fdb4c5edd8056893f94a10a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2407d4b99e51c6a8d33cc32972549f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8930099c42933f19d18446c471738a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad1a56baf43ffdf67bc8460856e31fec.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/28/1b33c98e-854e-4684-9ba8-f1a7ce79dff8.png?resizew=453)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c536d18163bd4bc3d7573e206a8d538.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e31228c7fd89c98d6235ad993d51d413.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e31228c7fd89c98d6235ad993d51d413.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9edc50f7febbc2d5d8dcdc23a3630a7.png)
(3)一束光从玻璃窗面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e31228c7fd89c98d6235ad993d51d413.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e31228c7fd89c98d6235ad993d51d413.png)
您最近一年使用:0次
2023-03-28更新
|
980次组卷
|
3卷引用:天津市河东区2023届高三一模数学试题
名校
解题方法
4 . 圆柱
中,四边形
为过轴
的截面,
,
,
为底面圆
的内接正三角形,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/29/026f0a8f-416d-493d-a7f4-421edb8b6065.png?resizew=189)
(1)证明:
平面
;
(2)求平面
与平面
所成角的正弦值.
![](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/2023/3/29/026f0a8f-416d-493d-a7f4-421edb8b6065.png?resizew=189)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d3267664e1d0a09def7c38743f0193f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a34fdf9e6d2d87d01ad0bbb6a73ee05.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7495688c046142f688c822209c0e968e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a34fdf9e6d2d87d01ad0bbb6a73ee05.png)
您最近一年使用:0次
2023-03-26更新
|
352次组卷
|
2卷引用:四川省宜宾市2023届高三下学期第二次诊断性测试理科数学试题
名校
解题方法
5 . 如图1,四边形ABCD是等腰梯形,E,F分别是AD,BC的中点,
.将四边形ABFE沿着EF折起到四边形
处,使得
,如图2,G在
上,且
.
平面DFG;
(2)求平面DFG与平面
夹角的余弦值
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a4f0b5ec9e40e70c00eaae68d1d3888.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52bf6ff245d22c6dbebbb36bb780d3ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11197fb5a297ccd643d34ecdbd04f794.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce1b066f8869d0ff4513f7a99745125.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98ade35115ffaa4d6d6f1c2e136bd5f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07391ef575d28f09bc5cda0ff8130a54.png)
(2)求平面DFG与平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74bca84ad86c648d3bb20c8909c8da3f.png)
您最近一年使用:0次
2023-03-17更新
|
1647次组卷
|
3卷引用:河北省邯郸市2023届高三一模数学试题
名校
解题方法
6 . 如图所示的几何体为一个正四棱柱被两个平面
与
所截后剩余部分,且满足
,
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/13/1477b758-6170-42d8-b055-c1aacdcbecac.png?resizew=185)
(1)当
多长时,
,证明你的结论;
(2)当
时,求平面
与平面
所成角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72406478fda1c6e3b8052467385a3bc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09219dbd440c70d66bf2bf8b4c2bfe2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46264ad39c95ef05658e3fa15373c6d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d37cb657446616b7d679dfd9d2bbef5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2743a47b0c3e422512b4c76cc7112232.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/13/1477b758-6170-42d8-b055-c1aacdcbecac.png?resizew=185)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e79216c6a32bb699aeb36144da020490.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51bb339ba41929e8f693b3618d5ee4b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72406478fda1c6e3b8052467385a3bc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09219dbd440c70d66bf2bf8b4c2bfe2f.png)
您最近一年使用:0次
2023-03-10更新
|
922次组卷
|
4卷引用:辽宁省名校联盟2023届高三下学期3月份联合考试数学试题
解题方法
7 . 如图在三棱柱
中,
为
的中点,
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/10/cf0761e8-879f-471a-8959-7cdc975ed67e.png?resizew=201)
(1)证明:
;
(2)若
,且满足:______,______(待选条件 ).
从下面给出的①②③中选择两个 填入待选条件 ,求二面角
的正弦值.
①三棱柱
的体积为
;
②直线
与平面
所成的角的正弦值为
;
③二面角
的大小为60°;
注:若选择不同的组合分别解答,则按第一个解答计分.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f121eabff3c62c1a196d9ca5f6f83f0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d34755c55d43c21b0f05897ef81c1b1f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/10/cf0761e8-879f-471a-8959-7cdc975ed67e.png?resizew=201)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a27bad0636a087e38bb1d253d66a231d.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/623f49f1a30f13a3f6706142ed0f92f4.png)
从下面给出的①②③中选择
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7de0f1e677c4a1fd058cd6d13dc2dc90.png)
①三棱柱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adbd3e8cf8325999cde03adf845d3dd0.png)
②直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b470c4e195cf7a07b7a331ce4b436e03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e168672b47d7e64dc1b404f8882c7dcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e108d5c61e85e0741ec2c484fc5768.png)
③二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45cbb74984939d59964559c3560ef7ba.png)
注:若选择不同的组合分别解答,则按第一个解答计分.
您最近一年使用:0次
8 . 如图所示的几何体是一个半圆柱,点P是半圆弧
上一动点(点P与点A,D不重合),
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/1/17e2fac0-5a16-4149-8200-7847dede756f.png?resizew=214)
(1)证明:
;
(2)若点P在平面ABCD的射影为点H,设
的中点为E点,当点P运动到某个位置时,平面
与平面
的夹角为
,求此时DH的长度.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6f9353ca110c8b81561455b232dbc15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5441d73845911db1993bf903c4d8700f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/1/17e2fac0-5a16-4149-8200-7847dede756f.png?resizew=214)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6ab33b35c2c84b097ced930fa180fb1.png)
(2)若点P在平面ABCD的射影为点H,设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6f9353ca110c8b81561455b232dbc15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f571a1aac46c6d0cf440c0ec2846bf9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10fc7991ea17d54ff5f4445ac5699463.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79a97bb4dcfab4ec7539bc783d563c49.png)
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名校
解题方法
9 . 如图,四边形ABCD是圆柱底面的内接四边形,是圆柱的底面直径,
是圆柱的母线,E是AC与BD的交点,
,
.
(1)记圆柱的体积为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4764374bd2fb78e59cd0b283637baeb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63055a5d6916f99d07fede49120753f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f737b04ce09bc7e1ed86dc9b3c85203b.png)
(2)设点F在线段AP上,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0724089d732523d6f5d0f0fbc6f64984.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5744f53b3376ffbe7a6bc5044c861273.png)
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2023-02-23更新
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6964次组卷
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15卷引用:2023届安徽省、云南省、吉林省、黑龙江省高三下学期2月适应性测试数学试题
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2023·河北·模拟预测
名校
解题方法
10 . 如图,用一垂直于某条母线的平面截一顶角正弦值为
的圆锥,截口曲线是椭圆,顶点A到平面的距离为3.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/7/030c760c-c753-4169-b610-21ab5b11bd12.png?resizew=181)
(1)求椭圆的离心率;
(2)已知P在椭圆上运动且不与长轴两端点重合,椭圆的两焦点为
,
,证明:二面角
的大小小于
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7294f5ae2a24ff42e84cd9773b2a7287.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/7/030c760c-c753-4169-b610-21ab5b11bd12.png?resizew=181)
(1)求椭圆的离心率;
(2)已知P在椭圆上运动且不与长轴两端点重合,椭圆的两焦点为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c29eadbcfaf2fb50b07d0f5fa165a0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
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