1 . 如图,一个棱长为6的透明的正方体容器(记为正方体
)放置在水平面
的上方,点
恰在平面
内,点
到平面
的距离为2,若容器中装有水,静止时水面与表面
的交线与
的夹角为0,记水面到平面
的距离为
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](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/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7253ffd3fc633d861810ee2e872188b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10d8eb4a9f462ca0c1d49c3fe91e720d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
A.平面![]() ![]() |
B.点![]() ![]() |
C.当![]() |
D.当![]() ![]() |
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2 . 泉州花灯技艺源于唐朝中期从形式上有人物灯、宫物灯、宫灯,绣房灯、走马灯、拉提灯、锡雕元宵灯等多种款式.在2024年元宵节,小明制做了一个半正多面体形状的花灯,他将正方体沿交于一顶点的三条棱的中点截去一个三棱锥,共截去八个三棱锥,得到一个有十四个面的半正多面体,如图所示.已知该半正多面体的体积为
,M为
的中心,过M截该半正多面体的外接球的截面面积为S,则S的最大值与最小值之比( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3d754c4ad51e4482e12a615d20a13fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
A.![]() | B.![]() | C.3 | D.9 |
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3 . 如图所示,点
为正方体形木料
上底面的动点,则下列结论正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
A.三棱锥![]() |
B.存在点![]() ![]() ![]() |
C.不存在点![]() ![]() ![]() |
D.经过点![]() ![]() ![]() ![]() ![]() ![]() |
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真题
4 . 定义一个集合
,集合中的元素是空间内的点集,任取
,存在不全为0的实数
,使得
.已知
,则
的充分条件是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0047f659c182291c84c224df6b5e993f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5b35f6fb648951da2a58b8f7f0b1873.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/096820d860ddf8f3155bf36f5fded491.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef3082f30bdeb19124814eacce2ac58f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c70219b67f6352be3d0a0400bde89d41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51f0e65d0421529cf42d98794b027e82.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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5 . 设一个简单几何体的表面积为
,体积为
,定义系数
,已知球体对应的系数为
,定义
为一个几何体的“球形比例系数”.
(1)计算正方体和正四面体的“球形比例系数”;
(2)求圆柱体的“球形比例系数”范围;
(3)是否存在“球形比例系数”为0.75的简单几何体?若存在,请描述该几何体的基本特征;若不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2e32d2d3759b3edd79ef82c1f61d3f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb75fdaefb95f5061a0b33c2559f446b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d66edadaad08a904172a9c162529b57.png)
(1)计算正方体和正四面体的“球形比例系数”;
(2)求圆柱体的“球形比例系数”范围;
(3)是否存在“球形比例系数”为0.75的简单几何体?若存在,请描述该几何体的基本特征;若不存在,说明理由.
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6 . 已知正方体
和点
,有两个命题:
命题甲:存在
条过点
的直线
,满足
与正方体的每条棱所成角都相等;
命题乙:存在
个过点
的平面
,满足
与正方体的每个面所成锐二面角都相等;
则下列判断正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
命题甲:存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
命题乙:存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
则下列判断正确的是( )
A.![]() | B.![]() |
C.![]() | D.![]() ![]() |
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7 . 某雕刻师在切割玉料时,切割出一块如图所示的三棱锥型边料,测得在此三棱锥
中,侧面
底面
,且
,该雕刻师计划将其打磨成一颗球形玉珠,则磨成的球形玉珠的直径的最大值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/891579e7c231584a8e16b8eeff79888e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3d7090639341730951c1bc3c9b6164e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca67a5b8f69507c8b80379e86f90a8ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe3faa82da218bcd6aaf987a4743954c.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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8 . 在圆台
中,圆
的半径是2,母线
,圆
是
的外接圆,
,
,则三棱锥体积最大值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e65ac334119ccd6204402a7aba29a55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2899e607479d8d1c47d954ae9ebb7144.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c4f6f74444b2b7947fc6e35c8d62322.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84d81d732204a3c2384a27606f858677.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68b40d0d2f3cdd8981bb792ad87efb42.png)
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9 . 如图所示,四面体
的各棱长均为
分别为棱
的中点,
为棱
上异于顶点的点,则以下结论正确的为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41e5db1d2fd912f77923e4c120a7dc19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2612d8c87010adfe87728a591b63e3a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/374fe9986ebbc986fc422e514ab93a51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c2bc5e50b8dfa02601c70822252854a.png)
A.![]() |
B.直线![]() ![]() ![]() |
C.四面体![]() ![]() |
D.平面![]() |
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10 . 空间内一点P可用三个有次序的数
来确定,其中r为原点O与点P间的距离;
为有向线段
与z轴正向的夹角;
为从正z轴来看自x轴按逆时针方向转到
所转过的角,这里M为点P在
面上的投影,这样的三个数
叫做点P的球面坐标,其中
,
,
,如图所示. 球面距离是指球面上两点之间的最短路径长度,这条路径是通过这两点的大圆上的劣弧(大圆是过球心的平面与球面相交形成的圆).
,
,求A,B间的球面距离;
(2)若
,
,记P,Q间的球面距离为d,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cef479716723efbb3e7fdc71e1a7904c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27ed74dbeba7d418a559f9c97c1df414.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf3369e0ea90e8d5cf4b6b3c45c0fd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b870a01c388175a446747d5fdaa0bf4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/363136f32811f5f8424775d6fb5a4897.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daac76dc6806917c5d76429d503aaed2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f80a89e5af8bee9f1815f52cb1db3022.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3be4358b49a194e363f77a604bc5dff8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cdca5d42af7a42337f5559a7d0babc1.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ed3ae064cd66c85f3f4a21fba7a81c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dda2a523239e2bfd6cd958533ac087ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1bfcbb2c0f8bf457a33aeba31d95c8f.png)
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