1 .
个有次序的实数
所组成的有序数组
称为一个n维向量,其中
称为该向量的第
个分量.特别地,对一个n维向量
,若
,
,称
为n维信号向量.设
,则
和
的内积定义为
,且
.
(1)写出所有3维信号向量;
(2)直接写出4个两两垂直的4维信号向量;
(3)证明:不存在14个两两垂直的14维信号向量;
(4)已知
个两两垂直的2024维信号向量
满足它们的前
个分量都是相同的,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f2b043b989216035c6fd985f1dd6a3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97de4e0337716e1d89eb1a6cfd7b8335.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6e51ca089ee13a138e985e20f1b7b3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c43d0d6f87afa8b4fd5f6cf81f2bdcdc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da796531c7b6c590a22b811df1fcef53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/293e6a784d135c77e3bded6f48f6eec9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b6be373930634c9aa53fec30bec8896.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea2978e42bc0f5abe31fe2536969afa9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19c7c807358869b70becd16ca80e1714.png)
(1)写出所有3维信号向量;
(2)直接写出4个两两垂直的4维信号向量;
(3)证明:不存在14个两两垂直的14维信号向量;
(4)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb9cae65660b220cc622b87ed9eea092.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf2182d0dad848ccc76944d976befbf2.png)
您最近一年使用:0次
2 . 设非零向量
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c367ebf81da8ce860b8d4db598ce3b0.png)
,并定义![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8de1a1abedbffcec3416ebfbba00c22b.png)
(1)若
,求
;
(2)写出![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9ee7554e993fa6d1035ea7da1621b6f.png)
之间的等量关系,并证明;
(3)若
,求证:集合
是有限集.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48b45cac4b26830e829a80640bf01673.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c367ebf81da8ce860b8d4db598ce3b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f4776b8be0546414c6a82e0f7c21315.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8de1a1abedbffcec3416ebfbba00c22b.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b69cf5eb74f6f3b69186a665b06696d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9abc628cb2ec8b1250ac0e86a034611.png)
(2)写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9ee7554e993fa6d1035ea7da1621b6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f4776b8be0546414c6a82e0f7c21315.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fffedfb01c0a6802e19c44067252fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac41950e0db22f2216407b7e3999b51d.png)
您最近一年使用:0次
2023-07-25更新
|
497次组卷
|
3卷引用:北京市丰台区2022-2023学年高一下学期期末考试数学试卷
北京市丰台区2022-2023学年高一下学期期末考试数学试卷(已下线)专题07 向量应用-《重难点题型·高分突破》(苏教版2019必修第二册)【北京专用】专题06平面向量(第二部分)-高一下学期名校期末好题汇编
3 . 给出集合
{
对任意
,都有
成立}.
(1)若
,求证:函数
;
(2)由于(1)中函数
既是周期函数又是偶函数,于是张同学猜想了两个结论:命题甲:集合M中的元素都是周期为6的函数:命题乙:集合M中的元素都是偶函数;请对两个命题给出判断,如果正确,请证明;如果不正确,请举反例:
(3)设p为常数,且
,求满足
成立的常数p的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9163ebe812708ee5337d62298c2e3363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acfc163d35aa285e48bcc21d6e392b0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33b005e1e4b8e41c0028cd464835c464.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c2d58d309affa76c49e442468c013dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5314a9d2205a2beba0dcffb8fd943b18.png)
(2)由于(1)中函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c2d58d309affa76c49e442468c013dd.png)
(3)设p为常数,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c7f87b7ddd102e6a5e9cfb282bb47ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e06a648a9da06fd9b05bb13cb8a5c047.png)
您最近一年使用:0次
名校
解题方法
4 . 设函数
定义在区间
上,若对任意的
、
、
、![](https://staticzujuan.xkw.com/quesimg/Upload/formula/177e54e8deea5da9dc6bc82eb3de0c2c.png)
,当
,且
时,不等式
成立,就称函数
具有M性质.
(1)判断函数
,
是否具有M性质,并说明理由;
(2)已知函数
在区间
上恒正,且函数
,
具有M性质,求证:对任意的
、![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
,且
,有
;
(3)①已知函数
,
具有M性质,证明:对任意的
、
、![](https://staticzujuan.xkw.com/quesimg/Upload/formula/291c25fc6a69d6d0ccfb8d839b9b4462.png)
,有
,其中等号当且仅当
时成立;
②已知函数
,
具有M性质,若
、
、
为三角形
的内角,求
的最大值.
(可参考:对于任意给定实数
、
,有
,且等号当且仅当
时成立.)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea9c587f6257331045c362ef25677c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/770cf3716f1e9dc8023a898df7f33783.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/177e54e8deea5da9dc6bc82eb3de0c2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d589f18d16b1a6bbd5108409c53fd05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a49c641617f38855f6abc7baf36af8e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f05279fb93940ea0741b64227cc58c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a70644524df044d4a24b998a81d44c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bee6881a170f6ef9ed5c133b95c2f448.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/475a20b276768b190ac15c9aa5c352ef.png)
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea9c587f6257331045c362ef25677c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80fcd5a1ca4f9abf76c88db3a3542b38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb5348b540c0b2e012191ae95351aaac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d589f18d16b1a6bbd5108409c53fd05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33bd24e647a626899a243a3f3984f90a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/450fb41cf5543a06035606ff29a9e934.png)
(3)①已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb5348b540c0b2e012191ae95351aaac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/291c25fc6a69d6d0ccfb8d839b9b4462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d589f18d16b1a6bbd5108409c53fd05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f183be2a65b185fd240990dffdec3ba7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b62e63003be4ad8c4c51e36e71df2ac3.png)
②已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b923078510697d5f7f9ea392eb76dd9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/089e6e44271b4c08be46dda1e7403741.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/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a8080fef9bdfa92ae70f3e314eef3e3.png)
(可参考:对于任意给定实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/205ca5a7d5bede14db0175445bb6d508.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f6b79d363c080275b93b8cc4b279653.png)
您最近一年使用:0次
2021-12-27更新
|
701次组卷
|
5卷引用:上海市黄浦区2022届高三一模数学试题
上海市黄浦区2022届高三一模数学试题(已下线)上海市黄浦区2022届高三上学期一模数学试题(已下线)第04讲 函数最值与性质-3上海市文来高中2023届高三上学期期中数学试题(已下线)专题06 期末解答压轴题-《期末真题分类汇编》(上海专用)
名校
5 . 已知
为
维向量,若
,则称
为可聚向量.对于可聚向量
实施变换
:把
的某两个坐标
删除后,添加
作为最后一个坐标,得到一个
维新向量
,如果
为可聚向量,可继续实施变换
,得到新向量
,……,如此经过
次变换后得到的向量记为
.特别的,二维可聚向量变换后得到一个实数.若向量
经过若干次变换后结果为实数,则称该实数为向量
的聚数.
(1)设
,直接写出
的所有可能结果;
(2)求证:对于任意一个
维可聚向量
,变换
总可以进行
次;
(3)设
,求
的聚数的所有可能结果.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46a84dcda19adf4e49271709ba3cfe48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c8d69c214ec6ae9233c7c2f21685177.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4754574a3b4f416cae9e3251347907c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/214d9a4f6648f9b37bb3bb9ee078c0b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4453ce4c6d461f1370048150f671497e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aadf9ab510510120699c5eee39ab18b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/201195e69e7d74cf49a9ba32c8b2c3ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/201195e69e7d74cf49a9ba32c8b2c3ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7db499662471ba1e6879d326104637d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b65d545d67d024b7d19575df07bd4ef3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ceee7288022589e3ee1c46f843321b27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/201195e69e7d74cf49a9ba32c8b2c3ab.png)
(2)求证:对于任意一个
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ac19e2a797cd0a408316988a63b3755.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aadf9ab510510120699c5eee39ab18b.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5ac76ff59a7ce5a47c563659ef4e80a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
您最近一年使用:0次
名校
解题方法
6 . 在数学中,双曲函数是与三角函数类似的函数,最基本的双曲函数是双曲正弦函数与双曲余弦函数,其中双曲正弦函数:
,双曲余弦函数:
.(e是自然对数的底数,
).
(1)计算
的值;
(2)类比两角和的余弦公式,写出两角和的双曲余弦公式:
______,并加以证明;
(3)若对任意
,关于
的方程
有解,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3321510a9eb73909a36c084a8630e89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0099b9b80ed478824fa95677ebe9d5b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11204e2fb6e560bf7a4ca26eaebfc526.png)
(1)计算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e694af0c9f990ecb8b54b1c08bcc578e.png)
(2)类比两角和的余弦公式,写出两角和的双曲余弦公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d92c32edc0e000405b7a6b9c48549959.png)
(3)若对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f78f05631a2ecb8bc3d379ca6c81f93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eed807cc52eca7b462a3850b5e5e02b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
2023-06-21更新
|
1016次组卷
|
8卷引用:上海市宝山区2022-2023学年高一下学期期末数学试题
上海市宝山区2022-2023学年高一下学期期末数学试题(已下线)模块六 专题5 全真拔高模拟1(已下线)专题14 三角函数的图象与性质压轴题-【常考压轴题】山东省济南市山东师大附中2022-2023学年高一下学期数学竞赛选拔(初赛)试题(已下线)第10章 三角恒等变换单元综合能力测试卷-【帮课堂】(苏教版2019必修第二册)上海市闵行(文琦)中学2023-2024学年高一下学期3月月考数学试卷(已下线)专题06 期末解答压轴题-《期末真题分类汇编》(上海专用)上海市市西中学2023-2024学年高一下学期期末复习数学试卷
名校
7 . 对于三维向量
,定义“
变换”:
,其中,
.记
,
.
(1)若
,求
及
;
(2)证明:对于任意
,经过若干次
变换后,必存在
,使
;
(3)已知
,将
再经过
次
变换后,
最小,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/384c75b6d80b247b341e4d19f231a7dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41bd66e602e9c043218806708e943c2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed50f0b03a7cc5f809e222d283dfc2f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09b05756fbd0f41a4fb35e7379e6b6f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e68d20604666dd9b1be3a5756aa1e06a.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9f42fda276fc8add9ffded503884a0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a5c19921380da55f5f1a00809a34503.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35234a3829d238ea479fef9cec166468.png)
(2)证明:对于任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f389ec068eb1d1aa586b79097d70a7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daac610026ebae0358e9c56d7bf91ff8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03385c625de63ac95bff151de1e2ebe2.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab5d893313655986257eec42d3fcf7ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d344174267f996c7cefecfd6985d380.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6308724fa5b677baf09b81469bf042b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
2023-07-11更新
|
1416次组卷
|
6卷引用:北京市东城区2022-2023学年高一下学期期末统一检测数学试题
北京市东城区2022-2023学年高一下学期期末统一检测数学试题北京市第十一中学2023-2024学年高二上学期期中练习数学试题广东省东莞市石竹实验学校2023-2024学年高一下学期3月月考数学试卷(已下线)专题02 高一下期末真题精选(1)-期末考点大串讲(人教A版2019必修第二册)【北京专用】专题07平面向量(第三部分)-高一下学期名校期末好题汇编(已下线)专题08 期末必刷解答题专题训练的7种常考题型归类-期末真题分类汇编(北师大版2019必修第二册)
解题方法
8 . 已知向量
与向量
的对应关系可用
表示.
(1)证明:对于任意向量
,
及常数m,n,恒有
成立;
(2)设
,
,求向量
及
的坐标;
(3)求使
成立的向量
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ccc04dc7b6641bf952daad9ef0997d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8acb51ec302e0df826ec67a5aeedd4a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e0e8b77133bc519162f519e42b1f463.png)
(1)证明:对于任意向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d99a1dabace6b00a0a70e09c7af43f0.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78b3515de9a26c0397039e077f272b1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d3ce134a42f342277ff7b53f6b1324f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eadcaa2e51643c637a490aef7bff19d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bba365f31076d7e75f149f284ae081b4.png)
(3)求使
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9412addd95de4764030cfdbe77e78e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73a0b19e69be46452425916a0fcb49c9.png)
您最近一年使用:0次
2023-04-13更新
|
105次组卷
|
3卷引用:4.2平面向量及运算的坐标表示 课后巩固提升习题2020-2021学年高一下学期数学北师大版(2019)必修第二册
4.2平面向量及运算的坐标表示 课后巩固提升习题2020-2021学年高一下学期数学北师大版(2019)必修第二册第二章 4.2平面向量及运算的坐标表示-北师大版(2019)高中数学必修第二册(已下线)6.3.2+6.3.3+6.3.4平面向量的正交分解及坐标表示【第三练】“上好三节课,做好三套题“高中数学素养晋级之路
解题方法
9 . 对于函数
,若存在非零常数M,使得对任意的
,都有
成立,我们称函数
为“M函数”;对于函数
,若存在非零常数M,使得对任意的
,都有
成立,我们称函数
为“严格M函数”.
(1)求证:
,是“M函数”;
(2)若函数
,是“
函数”,求k的取值范围;
(3)对于定义域为R的函数
对任意的正实数M,
均是“严格M函数”,若
,求实数a的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b232cd355157f66f1f0c6b02a03c5e86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bb6324279df94decba955e04ccfa9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ea657922fae2c5875761f5c3ce4b6ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b232cd355157f66f1f0c6b02a03c5e86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bb6324279df94decba955e04ccfa9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0741d41839ae1ee0914daad3c00f9243.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e85d70b23039da0296f97e25fc99791.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/951d13b1ddae2726049144b5b21c4b0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d49f8a63ddbca52039fa9ab44cda6b29.png)
(3)对于定义域为R的函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bb676cb3d49edadeaf419b3038591c4.png)
您最近一年使用:0次
2023-04-30更新
|
383次组卷
|
2卷引用:山东省青岛市西海岸新区2022-2023学年高一下学期期中数学试题
名校
解题方法
10 . 阅读材料:三角形的重心、垂心、内心和外心是与三角形有关的四个特殊点,它们与三角形的顶点或边都具有一些特殊的性质.
(一)三角形的“四心”
1.三角形的重心:三角形三条中线的交点叫做三角形的重心,重心到顶点的距离与重心到对边中点的距离之比为2:1.
2.三角形的垂心:三角形三边上的高的交点叫做三角形的垂心,垂心和顶点的连线与对边垂直.
3.三角形的内心:三角形三条内角平分线的交点叫做三角形的内心,也就是内切圆的圆心,三角形的内心到三边的距离相等,都等于内切圆半径r.
4三角形的外心:三角形三条边的垂直平分线的交点叫做三角形的外心,也就是三角形外接圆的圆心,它到三角形三个顶点的距离相等.
(二)三角形“四心”的向量表示
在
中,角
所对的边分别为
.
1.三角形的重心:
是
的重心.
2.三角形的垂心:
是
的垂心.
3.三角形的内心:
是
的内心.
4.三角形的外心:
是
的外心.
研究三角形“四心”的向量表示,我们就可以把与三角形“四心”有关的问题转化为向量问题,充分利用平面向量的相关知识解决三角形的问题,这在一定程度上发挥了平面向量的工具作用,也很好地体现了数形结合的数学思想.
结合阅读材料回答下面的问题:
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/29/ade3b84f-f648-4ea3-9d1f-1f223bf926d7.png?resizew=182)
(1)在
中,若
,求
的重心
的坐标;
(2)如图所示,在非等腰的锐角
中,已知点
是
的垂心,点
是
的外心.若
是
的中点,求证:
.
(一)三角形的“四心”
1.三角形的重心:三角形三条中线的交点叫做三角形的重心,重心到顶点的距离与重心到对边中点的距离之比为2:1.
2.三角形的垂心:三角形三边上的高的交点叫做三角形的垂心,垂心和顶点的连线与对边垂直.
3.三角形的内心:三角形三条内角平分线的交点叫做三角形的内心,也就是内切圆的圆心,三角形的内心到三边的距离相等,都等于内切圆半径r.
4三角形的外心:三角形三条边的垂直平分线的交点叫做三角形的外心,也就是三角形外接圆的圆心,它到三角形三个顶点的距离相等.
(二)三角形“四心”的向量表示
在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
1.三角形的重心:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b93c2e06509eb7087d76b21ab73701b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
2.三角形的垂心:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afb403a2aeaf2bff58aaab2eee17910f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
3.三角形的内心:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/794748a2cd3415724caca156e359abbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
4.三角形的外心:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b175450140c6866cf1a807f71b06013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
研究三角形“四心”的向量表示,我们就可以把与三角形“四心”有关的问题转化为向量问题,充分利用平面向量的相关知识解决三角形的问题,这在一定程度上发挥了平面向量的工具作用,也很好地体现了数形结合的数学思想.
结合阅读材料回答下面的问题:
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/29/ade3b84f-f648-4ea3-9d1f-1f223bf926d7.png?resizew=182)
(1)在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e260e695638c2651ce4b9b85b16b325b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
(2)如图所示,在非等腰的锐角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60cde0e1f12e802bc1b490a4c70f4f41.png)
您最近一年使用:0次
2022-07-16更新
|
1362次组卷
|
2卷引用:贵州省贵阳市2021-2022学年高一下学期期末监测考试数学试题