1 . 金刚石是天然存在的最硬的物质,如图1所示是组成金刚石的碳原子在空间中排列的结构示意图,组成金刚石的每个碳原子,都与其相邻的4个碳原子以完全相同的方式连接.从立体几何的角度来看,可以认为4个碳原子分布在一个正四面体的四个顶点处,而中间的那个碳原子处于与这4个碳原子距离都相等的位置,如图2所示.这就是说,图2中有
,若正四面体
的棱长为
,则( )
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/1/b596b17b-dd04-4470-b69c-a3a7ee9e678e.png?resizew=400)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ea3b99368bbae2422f7709e573c176.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/1/b596b17b-dd04-4470-b69c-a3a7ee9e678e.png?resizew=400)
A.![]() | B.![]() |
C.![]() | D.![]() |
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3卷引用:山东省威海市2021-2022学年高二上学期期末数学试题
山东省威海市2021-2022学年高二上学期期末数学试题山东省济南市章丘区章丘区第四中学2022-2023学年高二上学期期末数学试题(已下线)模块一 专题1 空间向量的基本运算 A基础卷 期末终极研习室(2023-2024学年第一学期)高二人教A版
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解题方法
2 . 在圆锥
中,
是母线
上靠近点
的三等分点,
,底面圆的半径为
,圆锥
的侧面积为
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18e5ef91fb27dd684a27ae7f1993cfba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a6e2867f32d3f1c3cd36cd3a11a8580.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ddaa2c8b030ac1dd8029a2413b166c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18e5ef91fb27dd684a27ae7f1993cfba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14c055a02fba0827ffcaa92f73ce7720.png)
A.当![]() ![]() ![]() ![]() |
B.当![]() ![]() ![]() |
C.当![]() ![]() ![]() |
D.当![]() ![]() ![]() |
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5卷引用:湖南省永州市2021-2022学年高三上学期第二次适应性考试数学试题
湖南省永州市2021-2022学年高三上学期第二次适应性考试数学试题(已下线)NO.3 练悟专区——客观题满分练(一)-2022年高考数学二轮复习讲练测(新教材·新高考地区专用)(已下线)押新高考第12题 立体几何-备战2022年高考数学临考题号押题(新高考专用)辽宁省大连市第一〇三中学2022届高三下学期第八次模拟考试数学试题浙江省杭州第十四中学2021-2022学年高二下学期期末模拟数学试题
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3 . 如图所示,圆锥的高
,底面圆
的半径为
,延长直径
到点
,使得
,分别过点
、
作底面圆
的切线,两切线相交于点
,点
是切线
与圆
的切点.
![](https://img.xkw.com/dksih/QBM/2022/1/21/2899052386713600/2900924662915072/STEM/c17c3011-0e63-4f82-8113-5bb43bb7b463.png?resizew=161)
(1)证明:
平面
;
(2)若平面
与平面
所成锐二面角的余弦值为
,求该圆锥的体积
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0819cd060cdfb72896f379db29a4724.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c76a0cbea833ae927c2f05602a965ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://img.xkw.com/dksih/QBM/2022/1/21/2899052386713600/2900924662915072/STEM/c17c3011-0e63-4f82-8113-5bb43bb7b463.png?resizew=161)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44b190c8d3d7d7d0e6e959e8a52eae90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c15b63176f43bc7a0654d0f6f45e7429.png)
(2)若平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b37793a3a810e823e10c340986f55ddd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83303d3784492506fc44f2b4d6b07bc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
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3卷引用:山东省济宁市2021-2022学年高二上学期期末数学试题
名校
解题方法
4 . 正四面体是一种柏拉图多面体,正四面体与自身对偶;正四面体的重心,四条高的交点,外接球、内切球球心共点.4个半径为1的小球装入一个正四面体内,下列四个结论中错误的是( )
A.四面体最小体积![]() |
B.四面体最小表面积![]() |
C.四面体最短棱长![]() |
D.四面体最小高![]() |
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3卷引用:辽宁省葫芦岛市2021-2022学年高三上学期期末数学试题
2022高三·全国·专题练习
5 . 如图所示,点
为正方体
的中心,点
为棱
的中点,若
,则下面说法正确的是( )
![](https://img.xkw.com/dksih/QBM/2022/1/9/2890852478820352/2893807157952512/STEM/ff797cc1e55a4d91b96ed2ed69b6ee6a.png?resizew=263)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3da8c338342e38c9aa3f274c053fd5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20c431cd12f858f0bc8dabb1d8c0b8e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ced06b71073e1bb777f326f06016ce17.png)
![](https://img.xkw.com/dksih/QBM/2022/1/9/2890852478820352/2893807157952512/STEM/ff797cc1e55a4d91b96ed2ed69b6ee6a.png?resizew=263)
A.直线![]() ![]() ![]() |
B.点![]() ![]() ![]() |
C.四面体![]() ![]() ![]() |
D.过点![]() ![]() ![]() ![]() |
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6 . 下列说法正确的是( )
A.若G是四面体OABC的底面三角形ABC的重心,则![]() |
B.在四面体OABC中,若![]() |
C.已知平行六面体![]() ![]() ![]() ![]() |
D.若向量![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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4卷引用:浙江省台州市2021-2022学年高二上学期期末数学试题
浙江省台州市2021-2022学年高二上学期期末数学试题山东省菏泽市2022-2023学年高二上学期期末数学试题(已下线)第03讲 空间向量及其运算的坐标表示(7大考点)-2022-2023学年高二数学考试满分全攻略(人教A版2019选择性必修第一册)福建省永安市第九中学2023-2024学年高二上学期期中考试数学试题
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7 . 三棱锥
中,
,
,
,直线
与平面
所成的角为
,点
在线段
上.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/28/1401ce1c-58c3-4119-8bb6-4cd452cd97c2.png?resizew=160)
(1)求证:
;
(2)若点
在
上,满足
,点
满足
,求实数
使得二面角
的余弦值为
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/463c7753d6f7614f90b19245bb3e439e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1affed1ad8e53a73308c85849a72444.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cbb05b8b630052ff544249ebd72d95d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d88591679796c52024d11c4de641bdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/28/1401ce1c-58c3-4119-8bb6-4cd452cd97c2.png?resizew=160)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70734a8e672376bb0bd1522e229f86a2.png)
(2)若点
![](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/d4bb04187b181054c7ddc7f0e35e3e5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c49e8906f0de208b36a18e448f7ecc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/445b51117626fbd3373e32acc514c64b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d33adb74906403b0b00fcbd9fa691d8b.png)
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4卷引用:重庆市巴蜀中学2021-2022学年高二上学期期末数学试题
解题方法
8 . 如图所示,圆柱OO1内有一个棱长为2的正方体ABCD-A1B1C1D1,正方体的顶点都在圆柱上下底面的圆周上,E为BD上的动点,则下面选项正确的是( )
![](https://img.xkw.com/dksih/QBM/2022/1/19/2897931869822976/2899251800326144/STEM/ffbbe4ee-9edf-4ef7-b2ba-09b0df4f4ee2.png?resizew=189)
![](https://img.xkw.com/dksih/QBM/2022/1/19/2897931869822976/2899251800326144/STEM/ffbbe4ee-9edf-4ef7-b2ba-09b0df4f4ee2.png?resizew=189)
A.△![]() ![]() |
B.圆柱OO1的侧面积为![]() |
C.异面直线AD1与C1D所成的角为![]() |
D.四面体A1BC1D的外接球的表面积为![]() |
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4卷引用:广东省茂名市2022届高三一模数学试题
名校
解题方法
9 . 有一个圆锥形铅垂,其底面直径为10cm,母线长为15cm.P是铅垂底面圆周上一点,则关于下列命题:①铅垂的侧面积为150cm2;②一只蚂蚁从P点出发沿铅垂侧面爬行一周、最终又回到P点的最短路径的长度为
cm.其中正确的判断是( )
![](https://img.xkw.com/dksih/QBM/2022/1/19/2897872790519808/2899176186249216/STEM/05f550a3-8e0f-476c-bc68-976af6082273.png?resizew=132)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14fc6d812e0f2a7ff2b7bb4f4c92369d.png)
![](https://img.xkw.com/dksih/QBM/2022/1/19/2897872790519808/2899176186249216/STEM/05f550a3-8e0f-476c-bc68-976af6082273.png?resizew=132)
A.①②都正确 | B.①正确、②错误 | C.①错误、②正确 |
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解题方法
10 . 如图,四面体ABCD的表面积为S,体积为V,E、F、G、H分别是AB、BC、CD、DA上的点,且
平面EFGH,
平面EFGH,设
,则下列结论正确的是( )
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/28/d133caf6-1033-4fdc-ac46-1a32fc37ce3e.png?resizew=179)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/429551ecb5930b2f033019e4d5b37ad7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e3c0430146b7b8d40ebb721a4d0de19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9aae7639b35ae15b9e78a1e1314b94a.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/28/d133caf6-1033-4fdc-ac46-1a32fc37ce3e.png?resizew=179)
A.四边形EFGH是正方形 |
B.AE和AH与平面EFGH所成的角相等 |
C.若![]() ![]() ![]() |
D.若![]() ![]() ![]() |
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