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解题方法
1 . “费马点”是由十七世纪法国数学家费马提出并征解的一个问题.该问题是:“在一个三角形内求作一点,使其与此三角形的三个顶点的距离之和最小.”意大利数学家托里拆利给出了解答,当
的三个内角均小于
时,使得
的点
即为费马点;当
有一个内角大于或等于
时,最大内角的顶点为费马点.
在
中,内角
,
,
的对边分别为
,
,
.
(1)若
.
①求
;
②若
的面积为
,设点
为
的费马点,求
的取值范围;
(2)若
内一点
满足
,且
平分
,试问是否存在常实数
,使得
,若存在,求出常数
;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1eab88a16df610f20dd46a44ba098d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b7f7180b86108862c7aa44c950f872a.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab7aaa871ceb78e5b80b531a7cf4f1c9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec15e5cb6d4dc2cf6ba0bedd87514448.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39b8d91afc34e4a9b0fdbb6bafb9087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca347a0ea5e4d813a81407796be5fea7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
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2 . 如图,在四棱锥
中,底面
是菱形,平面
平面
,
是边长为2的正三角形,
,
是
中点,过点
,
,
的平面与
交于点
.
;
(2)求证:
;
(3)求二面角
的正切值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93edc7bb513f40a89173121c8570cd65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55a675310c8ba418e5a59beb7317e21e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1720da6d65e7fa854d98322d3864240.png)
![](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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/666734423f1818d76a74f171b7420b68.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ad6f8846d4294ae3789a6ddd17af5b6.png)
(3)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc6600c93130ba16f270a449e261d15f.png)
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3 . 已知向量
,
.
(1)若
,求
;
(2)若
,
①求
;
②已知
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a67711170219688d03144200ae396e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d5de571daa349850dbc7fc98b7b990a.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/076559f08d17fb25e82886e791719e52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cc9750c313ee972124cb62c4a6fb7ea.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44726b1d833cfef78c52b027817a69ce.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/032f8f657f9163ebc64db95d214f4091.png)
②已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d394016b9fecac73f38cbc4ff18dee2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6f90c4754e6b6fc862d72943fb35569.png)
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解题方法
4 . 在棱长为2的正方体
中,
分别是
的中点,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1797b6b52b9ef3d5a39cfc583bafac80.png)
A.直线![]() ![]() |
B.过点![]() ![]() |
C.三棱锥![]() ![]() |
D.点![]() ![]() ![]() |
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5 . 已知复数
,
满足
,
则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adcc78b50e4eb2ef6076b0ef4fab732d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/467e944474d73e57a5afc1934d51daee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6db0e748196e89b9d821e0289c751d9.png)
A.![]() | B.![]() |
C.在复平面内![]() ![]() | D.![]() ![]() |
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6 . 在棱长均为2的正三棱柱
中, E为
的中点.过AE的截面与棱
分别交于点F, G.
(1)若F为
的中点,求三棱柱被截面AGEF分成上下两部分的体积比 ![](https://staticzujuan.xkw.com/quesimg/Upload/formula/301709b97852b4c2b951cfa2b2cfa2d2.png)
(2)若四棱锥
的体积为
求截面 AGEF 与底面ABC所成二面角的正弦值;
(3)设截面AFEG的面积为
面积为S₁,△AEF面积为
当点F在棱
上变动时,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/770abdd10660689c605577f9cb6d9db1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa717be2a43918b9eb13655f86042a93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b828011b4a0be31af4f30d829d7e30b1.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/6/18/c1fc03ae-c978-409e-8fa2-9e23147b9838.png?resizew=123)
(1)若F为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a5fb47a5caaa26007a7606acf05c52c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/301709b97852b4c2b951cfa2b2cfa2d2.png)
(2)若四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04eb4a437f9aee581cb2d0fff0d9d588.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c181e33261a6c8df1e5291b0a119dc9.png)
(3)设截面AFEG的面积为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/431bceebfd7dbb3dcbd0348e486a97fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71c9821bd2db0d34acbd7acd0fb2bb8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a5fb47a5caaa26007a7606acf05c52c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d17646ce1c1499a928584904d416a2a2.png)
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7 . 已知向量
,则
在
方向上的投影向量为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a17870c32f1beadcfec4b22fddd945d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
8 . 如图,已知在矩形
中,
,点
是边
的中点,
与
相交于点
,现将
沿
折起,点
的位置记为
,此时
,则二面角
的余弦值为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cd4fb83908d0cabf3cc17ea6ce4a3c5.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/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac451db3443cabb204f96c31fd4a02e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5b3bd5e6bc2a0a277d279bb01af9584.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d30374a2dad85e336ceaa462be7e00e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d7f66b146fccec50efa1c316aa1afe0.png)
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9 . 为获得某中学高一学生的身高(单位:
)信息,采用随机抽样方法抽取了样本量为50的样本,其中男女生样本量均为25,计算得到男生样本的均值为176,标准差为10,女生样本的均值为166,标准差为20.则总样本的方差为__________
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efa9fbcfb9595e2f031aa691db4564b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88ce13774b09ff2edddaf21a072cf60a.png)
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解题方法
10 . 若
、
是两个不重合的平面:①若
内的两条相交直线分别平行于
内的两条直线,则
;②设
、
相交于直线
,若
内有一条直线垂直于
,则
;③若
外一条直线
与
内的一条直线平行,则
.以上说法中成立的有( )个
![](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/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/202af51f5ebe87ec0017f439a6ad7fbf.png)
![](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/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5986f2991d45fbf3578f08f27d9fd7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f236f93b96ed780adb8a79e473efefd.png)
A.0 | B.1 | C.2 | D.3 |
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