1 . 如图,在正方体
中,棱长为4,
分别为
的中点,
分别为
上的一点,且满足
,
,设正方体
的体积为
,几何体
的体积为
,则下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3da8c338342e38c9aa3f274c053fd5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/133c979108b412083d6c73ba0760a3d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad056c25c0fdcbcc765eb5cbc6093f2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8423aab028326469e6c07c0fe50700ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28981d8133f48dbe6a2eae89e75529b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65d009bbb173545f1930a36fc36c8888.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3da8c338342e38c9aa3f274c053fd5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4764374bd2fb78e59cd0b283637baeb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ba758e83b255c00600ab7de43223395.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63055a5d6916f99d07fede49120753f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/4/8de54558-021f-4ba3-9d45-534931be2bf0.png?resizew=154)
A.![]() | B.点![]() ![]() |
C.当![]() ![]() | D.当![]() ![]() |
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解题方法
2 . 远看曲靖一中文昌校区紫光楼主楼,一顶巨大的“博士帽”屹立在爨园之中.其基础主体结构可以看做是一个倒扣的正四棱台
.如图所示,过
作底面
的垂线,垂足为G.记
,
,
,面
与面
所成角为
,面
与面
所成角为x,
,
,
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3da8c338342e38c9aa3f274c053fd5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3953cec61ac602ce5eb59b7912352179.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87d7c9001717a5a04c4eee2c98bed4f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70d70d5a39044f808980ca8587766e27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50d55ce97d97240622bee2dd066809ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0bc51695e51aa8cd2f97d220c8f5340.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0bc51695e51aa8cd2f97d220c8f5340.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e06bc9ebc09748382b644bac7bcb3215.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2361a524115b309de21149fc57188eca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4a88b719166fcc1431f876bc8c5656c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ee13374ece581c58330639a820b07f2.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/5/d5233f67-e141-49ab-93c1-56d659930a30.png?resizew=373)
A.正四棱台![]() ![]() |
B.![]() |
C.![]() |
D.![]() |
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2卷引用:广东省广州市广东实验中学2024届高三上学期大湾区数学冲刺卷(三)
解题方法
3 . 为体现市民参与城市建设、共建共享公园城市的热情,同时搭建城市共建共享平台,彰显城市的发展温度,某市在中心公园开放长椅赠送点位,接受市民赠送的休闲长椅.其中观景草坪上一架长椅因其造型简单别致,颇受人们喜欢(如图1).已知
和
是圆
的两条互相垂直的直径,将平面
沿
翻折至平面
,使得平面
平面
(如图2)此时直线
与平面
所成角的正弦值为( )
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/4/be3bb93e-c205-41a5-88f9-c69964bfa20b.png?resizew=329)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52856669e6adf246c92923b4bb120d91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6dad72203e297fff4933dddec465b13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7abd284f76d9f5769bc189508ce2572b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80280197cb6326f703eb3de1c0bc211f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/4/be3bb93e-c205-41a5-88f9-c69964bfa20b.png?resizew=329)
A.![]() | B.![]() | C.![]() | D.![]() |
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2024·全国·模拟预测
名校
解题方法
4 . 已知
中,
,在线段
上取一点
,连接
,如图①所示.将
沿直线
折起,使得点
到达
的位置,此时
内部存在一点
,使得
平面
,如图②所示,则
的值可能为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4187969ad00f4ddf09c832937d2ee7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dddb32a7a5c157fdf8aa049b2d665b4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f955e5cc9f108de6f3ca01e5eb84c52e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c92179e42115a1ee77e20f7ddd48ec9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ebcd36cffd9845d233f6d7e105c0d3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e42198b8e646ccfbbec323fd364c4d90.png)
A.![]() | B.![]() | C.![]() | D.1 |
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6卷引用:2024年全国高考名校名师联席命制数学(理)信息卷(十一)
(已下线)2024年全国高考名校名师联席命制数学(理)信息卷(十一)湖南省衡阳市第八中学2024届高三上学期第一次模拟测试数学试题广东省广州市执信中学2024届高三第二次调研数学试题(已下线)考点14 余弦定理及应用 --2024届高考数学考点总动员【讲】(已下线)专题02 解三角形(2)-【常考压轴题】(已下线)专题19 直线与平面的位置关系-《重难点题型·高分突破》(苏教版2019必修第二册)
5 . 如图,在正方体
中,
均为棱的中点,现有下列4个结论:
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/3/e7adc206-fdc0-43d4-8fad-f52f29460469.png?resizew=180)
①平面
平面
;
②梯形
内存在一点
,使得
平面
;
③过
可作一个平面,使得
到这个平面的距离相等;
④梯形
的面积是
面积的3倍.
其中正确的个数为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b7fa1c1d6321b28a2a2b7ffd0b27253.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/3/e7adc206-fdc0-43d4-8fad-f52f29460469.png?resizew=180)
①平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d46f725fb1c57d0855a0a6cc26bf562a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ac66de8543430fd51e7c18042e626dd.png)
②梯形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ac66de8543430fd51e7c18042e626dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3834d7ec7531f3c3c0ce9b286f7a49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2803a2949df8ff5494549cf41207f963.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1885efcff0b903e314057dd153578600.png)
③过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6795cae2df43a722e1355e9562d93c09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b7f4d18c6a2e405bbd7ea1e814649c6.png)
④梯形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ac66de8543430fd51e7c18042e626dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98013a5042685a1db94249e70c62c09a.png)
其中正确的个数为( )
A.4 | B.3 | C.2 | D.1 |
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4卷引用:陕西省西安市长安区教育片区2024届高三上学期模拟考试数学(理)试题
陕西省西安市长安区教育片区2024届高三上学期模拟考试数学(理)试题陕西省西安市长安区教育片区2024届高三上学期模拟考试数学(文)试题陕西省西安市部分学校2024届高三上学期12月联考数学(文)试题(已下线)热点6-1 线线、线面、面面的平行与垂直(6题型+满分技巧+限时检测)
名校
解题方法
6 . 如图,在圆台
中,截面
分别交圆台的上下底面于点
,
,
,
四点.点
为劣弧
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/27/b1a293cc-0531-4fb5-a1dc-c34eaf1d8ca5.png?resizew=149)
(1)求过点
作平面
垂直于截面
,请说明作法,并说明理由;
(2)若圆台上底面的半径为1,下底面的半径为3,母线长为3,
,求平面
与平面
所成夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abbe2aba242716238b79c46bb1f40e88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22d1a8ed65b138016acff8c465165337.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/27/b1a293cc-0531-4fb5-a1dc-c34eaf1d8ca5.png?resizew=149)
(1)求过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22d1a8ed65b138016acff8c465165337.png)
(2)若圆台上底面的半径为1,下底面的半径为3,母线长为3,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73926ba72492b6a6a935dae64ed59e2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/105f1100148bbf6d789b9048281755a1.png)
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解题方法
7 . 已知四面体
.分别对于下列三个条件:
①
;②
;③
,
是
的充要条件的共有几个( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcf0148aeba9b23f1dc95d753790b60.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b5f215a42c4b7078d8d65923eb9980e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7de966c316db1013defc56372fcf814e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e8f950a001b6108071e8d630c058b82.png)
是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a77e3c1c236141d6118429fade0a9b9d.png)
A.0 | B.1 | C.2 | D.3 |
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8 . 已知三棱锥
,则下列论述正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41e5db1d2fd912f77923e4c120a7dc19.png)
A.若点S在平面![]() ![]() ![]() |
B.若点S在平面![]() ![]() ![]() ![]() |
C.若![]() ![]() ![]() ![]() ![]() ![]() |
D.若![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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3卷引用:2024届河北省部分高中高考一模数学试题
9 . 如图,在圆台
中,上底面的半径为1,下底面的半径为3,母线长为3.在截面
与截面
中,
,
.
(1)求证:截面
截面
;
(2)求四棱台
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abbe2aba242716238b79c46bb1f40e88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ecc1cb55a57dde481f8dd07ab150676.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ab3181632564c50284bfa4853343b75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c72e543ab8584eee527a13ce394be7d6.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/29/372fb7a5-3bf6-436f-9f93-f97bed6cdc08.png?resizew=161)
(1)求证:截面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eebbdc18cc3cdeac3de285024ca9310.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c86d9511b26493ac1eaf8739c32c57f.png)
(2)求四棱台
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab8ad09b0cb888934e7532fdf8956930.png)
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解题方法
10 . 我国古代数学著作《九章算术》中研究过一种叫“鳖(biē)臑(nào)”的几何体,它指的是由四个直角三角形围成的四面体,那么在一个长方体的八个顶点中任取四个,所组成的四面体中“鳖臑”的个数是
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4卷引用:上海市杨浦区2024届高三上学期模拟质量调研数学试题
上海市杨浦区2024届高三上学期模拟质量调研数学试题重庆缙云教育联盟2024届高三高考第一次诊断性检测数学试卷(已下线)专题01 两个计数原理与排列组合(7类压轴题型)-【常考压轴题】2023-2024学年高二数学压轴题攻略(人教A版2019选择性必修第三册)上海市宝山中学2023-2024学年高二下学期3月考数学试卷