1 . 已知三棱锥
的四个顶点均在同一球面上,
,
,且三棱锥
体积的最大值为
,则该球的表面积为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41e5db1d2fd912f77923e4c120a7dc19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e45b0e1c3f6f5bc4cc81290bf263d90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d70dc2c20619a4fc12a0cfda59af5b69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41e5db1d2fd912f77923e4c120a7dc19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860884c0017c8bceb5b0edff796c144f.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
2 . 如图所示,在直三棱柱
中,若
,
,则下列说法中正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080db3af81b29ed10144a1c2e2a4fb8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6c02338caa8ea2a27a5a37226a8a472.png)
A.三棱锥![]() ![]() |
B.点![]() ![]() ![]() ![]() |
C.![]() ![]() ![]() ![]() ![]() ![]() ![]() |
D.点![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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|
3卷引用:辽宁省大连市部分学校2024届高三下学期联合模拟考试数学试题
名校
3 . 如图,平行四边形
中,
,
.现将
沿
起,使二面角
大小为120°,则折起后得到的三棱锥
外接球的表面积为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1cc201327a8ee3fd646948d3f0c5d9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05d35d8d8bb0dc17f2f86fe5b230a2b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/661ff55b5ebbadfb600989af3cfce2fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/282d4a8c3476b2b81e3fd73898e64539.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3931333820859378ea6723ff3075189.png)
A.![]() | B.![]() | C.![]() | D.![]() |
您最近一年使用:0次
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843次组卷
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5卷引用:辽宁省大连市部分学校2024届高三下学期联合模拟考试数学试题
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解题方法
4 . 如图,矩形
中,
,
是边
的中点,将
沿直线
翻折成
(点
不落在底面
内),连接
.若
为线段
的中点,则在
的翻折过程中,以下结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f80f51c31583fea58fde645474d60b8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3aa2a83fed9bf4cb09d84a980452e346.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fa7bbd7831e9ff4f8cffc8889d34f05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebd60c083fb876623162f6291c73ae34.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ee8456443402a25b1e25d35ff7e1c98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
A.![]() ![]() | B.不存在某个位置,使![]() |
C.线段![]() ![]() | D.![]() |
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解题方法
5 . 柯西不等式在数学的众多分支中有精彩应用,柯西不等式的n元形式为:设
,
,
不全为0,
不全为0,则
,当且仅当存在一个数k,使得
时,等号成立.
(1)请你写出柯西不等式的二元形式;
(2)设P是棱长为
的正四面体ABCD内的任意一点,点P到四个面的距离分别为
,
,
,
,求
的最小值;
(3)已知无穷正数数列
满足:
①存在
,使得
;
②对任意正整数i、
,均有
.
求证:对任意
,
,恒有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50a272adba0f1120109824440f0e252c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba031aac09bdee5b36549bb6e68bdb5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50a272adba0f1120109824440f0e252c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1602c6064af12eed3fd1291f8272d93c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/944ab11422d7221e45aa4cc6d868828b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34039940c47c92f3660e9dc7c27e5961.png)
(1)请你写出柯西不等式的二元形式;
(2)设P是棱长为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5edf900c810371fb21297c15f86d8743.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b31ac1def558351e2e3ed1235c570530.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/342d0252c1b2f7d2a84b5c985d19d547.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d31659f106fba3c9750661eb0e3c3eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dde93376f5d29f8f7d501122759b0ab.png)
(3)已知无穷正数数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
①存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c24ecf9e59082e563372b12981d03fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1b5cbf6a7e19a347e95de7f119094fb.png)
②对任意正整数i、
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8598147874a35becc05e7bf4d90ce096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a33ac34aa03dc7f0a5faad6dc664ec6.png)
求证:对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5818ede14d21f6df9ef9c2bfe09286c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c229aec38946b710076588b7710381c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d191d6de821fbb06a51b5a20112db6de.png)
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6 . 在空间直角坐标系中,已知
,则几何体
的体积为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfae9b085798f2e25e25d877c32bdd23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
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7 . 三棱锥P﹣ABC所有棱长都等于2,动点M在三棱锥P﹣ABC的外接球上,且
的最大值为s,最小值为t,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9ebcb8b266d2ac748308d22a89649ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca1c938b970b3a6e21436fda8c641f8e.png)
A.2 | B.![]() | C.![]() | D.3 |
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8 . 已知球O的半径为4,平面
,
与球面分别相交,得圆C与圆D,AB为圆C与圆D的公共弦,若
,
,则点O到直线AB的距离为______ ,四面体ABCD的体积为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16e315d525af44daca685149d7d1a51e.png)
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解题方法
9 . 已知四棱柱
中,
平面
,在底面四边形
中,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/810ee7bc82b6f452afb3fc18691abc3b.png)
,点
是
的中点.
平面
,求三棱锥
的体积;
(2)设
且
,若直线
与平面
所成角等于
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5845ccc0d735dc14c92a8926d9b1def6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/141bdb7ecc7677ecc56e139ac01c5078.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/810ee7bc82b6f452afb3fc18691abc3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a53a4b47eaf893d0d1b2c9595e6b126f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f0db5b8d1bf3bee0237d7c50c9cda64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b367f040bb205eabcf9e79c0248c4d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffedd2a59e23b39074afabd1a5a3bb26.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19b6269558518a9e0446c9f0f4c6d2bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544530e1133b2924ccfbe691141a5641.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69bcb3226e013650b7d8827c31dd41d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d5bca00fa20e6e80480b9d06d2e52ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
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10 . 正四棱柱
中,
,动点
满足
,且
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f21c7c194c5bc2986a21fd441c81495.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a871156caa5d7222c7944afecf5f9ea1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba1c15263cbb460891a4af6a9c693060.png)
A.当![]() ![]() ![]() |
B.当![]() ![]() ![]() |
C.若直线![]() ![]() ![]() ![]() |
D.当![]() ![]() ![]() |
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