名校
解题方法
1 . 柯西是一位伟大的法国数学家,许多数学定理和结论都以他的名字命名,柯西不等式就是其中之一,它在数学的众多分支中有精彩应用,柯西不等式的一般形式为:设
,则
当且仅当
或存在一个数
,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
内的任意一点,点
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
,均有
.求证:对任意
,
,恒有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81a8a1b208f491296432e9e6bf0e91c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0653d6a0e8778ad47b06d5f6b88cffa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/419c991c4022ef12d4801e119018b587.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f31a068fb311eff550b3088a212fb2f0.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/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.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)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c24ecf9e59082e563372b12981d03fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ee33826e02eda7aa6221649355a5709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9db6b0bf3d360830fff618193c595b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a33ac34aa03dc7f0a5faad6dc664ec6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5818ede14d21f6df9ef9c2bfe09286c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cca1d86c9f078347773f700fee49d1d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d191d6de821fbb06a51b5a20112db6de.png)
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2024-06-04更新
|
342次组卷
|
2卷引用:河北省邯郸市2024届高三下学期高考保温数学试题
解题方法
2 . 如图,三棱台
的底面
为锐角三角形,点D,H,E分别为棱
,
,
的中点,且
,
;侧面
为垂直于底面的等腰梯形,若该三棱台的体积最大值为
,则下列说法可能但不一定正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cabc3303519ac16fc998913ad9f349c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5825e3891ce507d4af2e0d9d1a0b74b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e72c5599e326e06217772e409413fd5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e168672b47d7e64dc1b404f8882c7dcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01ce827a56e0203a293d774d134f2adf.png)
A.该三棱台的体积最小值为![]() | B.![]() |
C.![]() | D.![]() |
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解题方法
3 . 已知三棱锥
的各顶点均在半径为2的球
表面上,
,
,则三棱锥
的内切球半径为__________ ;若
,则三棱锥
体积的最大值为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/418b19e808e7ac6acb2065b005595693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a621bfa70a014bdcdb58697f099b597.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4278c0911e7df78965e78cff69cac5f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fbddb854a1a634484936c64ab4a9102.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5402d4268ec1028e5307063539c10d7.png)
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名校
4 . 直四棱柱的各顶点都在半径为2的球O的球面上,下列说法正确的是( )
A.若![]() ![]() |
B.若![]() ![]() |
C.若![]() ![]() |
D.若![]() ![]() |
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2024高三·全国·专题练习
5 . 如图,已知
的直角边
,点
是
从左到右的四等分点(非中点).已知椭圆
所在的平面⊥平面
,且其左右顶点为
,左右焦点为
,点
在
上.
(1)求三棱锥
体积的最大值;
(2)证明:二面角
不大于60°.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32b4da8d254b7d76b7615d7f7cd824a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40f98254a6193566587a70c7d95fdabf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0520e9675bc1b416e8b8f01eb69fd13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40f98254a6193566587a70c7d95fdabf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/21/5195191a-2f2b-4a8d-8747-94b7508e5dff.png?resizew=168)
(1)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f26d79ba498b2f1dbaaa690fb38eac.png)
(2)证明:二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c29eadbcfaf2fb50b07d0f5fa165a0b.png)
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6 . “十字贯穿体”是由两个完全相同的正四棱柱“垂直贯穿”构成的多面体,其中一个四棱柱的每一条侧棱分别垂直于另一个四棱柱的每一条侧棱,两个四棱柱分别有两条相对的侧棱交于两点,另外两条相对的侧棱交于一点(该点为所在棱的中点)若某“十字贯穿体”由两个底面边长为2,高为
的正四棱柱构成,则下列说法正确的是( )
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/25/21a440cd-9ec9-4605-bb75-6d4d6b251da2.png?resizew=151)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8af2fdf1944afebb51cb6a5e6c74aadd.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/25/21a440cd-9ec9-4605-bb75-6d4d6b251da2.png?resizew=151)
A.一个正四棱柱的某个侧面与另一个正四棱柱的两个侧面的交线互相垂直 |
B.该“十字贯穿体”的表面积是![]() |
C.该“十字贯穿体”的体积是![]() |
D.一只蚂蚁从该“十字贯穿体”的顶点A出发,沿表面到达顶点B的最短路线长为![]() |
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解题方法
7 . 已知四点均在半径为
(
为常数)的球
的球面上运动,且
,若四面体
的体积的最大值为
,则球
的表面积为( )
A.![]() | B.![]() | C.![]() | D.![]() |
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2024高三·江苏·专题练习
8 . 在三棱锥
中,
平面
,
,
,
,
,点M在该三棱锥的外接球O的球面上运动,且满足
,则三棱锥
的体积最大值为__________
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5f1897a7e856b42f8cee0f286ad913d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/236f52e04ddbd7253f44b97c4756ef9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c4cd2370ee4086a8d8e8e16e069d3f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66e26d8a5ed926ba85eb734239f12b1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e621c984c64e8008d43c867bb22768ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1b9ddeedeff87f7722aedd6c1109d57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c70e9a28e85ccb37507499c56eff84d2.png)
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2024·四川成都·模拟预测
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9 . 已知三棱锥中,
,
,
,二面角
的余弦值是
.则当三棱锥
的体积最大时,其外接球的表面积是
您最近一年使用:0次
名校
解题方法
10 . 如图,在直四棱柱
中,底面
为菱形,
为
的中点,点
满足
,则下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c41258baa6ddd40fe74d9a0f0f2a398.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866bf34d3d5e15836ecf6888d62a6e43.png)
A.若![]() ![]() |
B.若![]() ![]() ![]() |
C.若![]() ![]() ![]() |
D.若![]() ![]() ![]() ![]() ![]() |
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2024-02-20更新
|
1210次组卷
|
3卷引用:湖南省长沙市雅礼中学2024届高三月考试卷数学(六)