解题方法
1 . 如图所示是两个半径不同的同心圆,半径分别为1和
是小圆
上的一个动点,
是大圆
上的三个不同的动点,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7231c8dd7928b46f06a4286903c47363.png)
![](https://img.xkw.com/dksih/QBM/2021/5/6/2715469668352000/2799679180554240/STEM/60ad64d120bf444cb1732c9242531b77.png?resizew=165)
(1)求证![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3bbe8a224c56e320004764101b9871f.png)
(2)求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2353c7e8ad00d9d25c71535bd45d7568.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28855a0db693584aabac1df99dfade3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7231c8dd7928b46f06a4286903c47363.png)
![](https://img.xkw.com/dksih/QBM/2021/5/6/2715469668352000/2799679180554240/STEM/60ad64d120bf444cb1732c9242531b77.png?resizew=165)
(1)求证
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3bbe8a224c56e320004764101b9871f.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/219924df5ba319fe45e8356621730fe5.png)
您最近一年使用:0次
21-22高二上·北京西城·期中
名校
2 . 对于给定的正整数n,记集合
,其中元素
称为一个n维向量.特别地,
称为零向量.设
,
,
,定义加法和数乘:
,
.对一组向量
,
,…,
(
,
),若存在一组不全为零 的实数
,
,…,
,使得
,则称这组向量线性相关.否则,称为线性无关.
(1)对
,判断下列各组向量是线性相关还是线性无关,并说明理由.
①
,
;②
,
,
;③
,
,
,
.
(2)已知向量
,
,
线性无关,判断向量
,
,
是线性相关还是线性无关,并说明理由.
(3)已知
个向量
,
,…,
线性相关,但其中任意
个都线性无关,证明下列结论:
①如果存在等式
(
,
),则这些系数
,
,…,
或者全为零,或者全不为零;
②如果两个等式
,
(
,
,
)同时成立,其中
,则
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d916961d15be6e499999e2fbe51c456.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac852f327effde190b9ebf3dd08e037c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58a13d67dfa8d6f81c9aa06ca42c9e68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aff8d9b6533ff319420cdc5e8740b04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2668c56f2e7bb75917c8b7d7b09c8926.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b18d4591126e3801e10b93dadf2df3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14dfee189ba5596b1fe73fef044b4687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4146baa4c775fef17a3eb033b2c1691.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a40932863dcc8cb8356bbd487eaf196.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d43fd777be9b2e4fdfe3fbe9a7cdde79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e15e74e0eba2b8dbf3d09f6e6382f4c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37c567e4139b2be7a0ac01f3430d2d54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1445d8e79cc2309e20001952f2b742d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d2dc187e5bae6ab191f2442948763ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5259301dfd6e050de40be969068229b4.png)
(1)对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74044239e1bc20b87b6be3f40b282675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be93e52fa914240b43743270b76a8fb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74044239e1bc20b87b6be3f40b282675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be93e52fa914240b43743270b76a8fb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d159653ffeca0f732a520f9d52eaf5fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cb112690d7eb559b09a4f7666ff10ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7308ce4466feaf0c214ac02503c83029.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/250be63dcc2692aceeed802b008e767b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca2e34d347c50b4de2ac8a306ead0e9c.png)
(2)已知向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac852f327effde190b9ebf3dd08e037c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e257e6c38a24ced4f9423f7f51b0e92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f168ab643bde113e92134b3fa8139665.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85f7e488464e41e1a1e1eed427154aa6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81c769ca2d340eb5ad4d16cd54852990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e6faf15e812aa5e0f9272c8dbaa3591.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b2ea5126ca30d68cc8b70f03860ba2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a40932863dcc8cb8356bbd487eaf196.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d43fd777be9b2e4fdfe3fbe9a7cdde79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c0c36e75691a88bd6e38e6269e69fef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f1e94f660b7d05de4be4b5fbd9041f4.png)
①如果存在等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ad741efadd0b14577a80df4c243e3b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7368220470ebd17a0fd2bcf1f9fd495.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/299d6864de87fd34448379b795776334.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27628b047da341c79074ea4aa938ddc8.png)
②如果两个等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ad741efadd0b14577a80df4c243e3b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/898ac87221725132b69725b4042bb441.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7368220470ebd17a0fd2bcf1f9fd495.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d66379e8d6cb9d00a3287d434fa3c9f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/299d6864de87fd34448379b795776334.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f165fa05c199307b89a91027dd6d696.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecd9175a6d8d1d75238a4b60db254846.png)
您最近一年使用:0次
2021-11-19更新
|
2662次组卷
|
13卷引用:北京市第四中学2021-2022学年高二上学期期中考试数学试题
(已下线)北京市第四中学2021-2022学年高二上学期期中考试数学试题北京市昌平区第二中学2022-2023学年高二上学期10月月考数学试题江苏省南通市如皋市2024届高三上学期1月诊断测试数学试题湖北省孝感市高级中学2024届高三上学期期末数学试题广东省深圳市南山区华侨城中学2024届高三下学期一模适应性考试数学试题重庆市九龙坡区育才中学2023-2024学年高一下学期寒假检测定时训练数学试题(已下线)最新模拟重组精华卷1-模块一 各地期末考试精选汇编(已下线)新题型01 新高考新结构二十一大考点汇总-3(已下线)专题22 新高考新题型第19题新定义压轴解答题归纳(9大核心考点)(讲义)(已下线)黄金卷03(2024新题型)福建省福州外国语学校2023-2024学年高一下学期3月月考数学试卷辽宁省东北育才学校双语校区2023-2024学年高一下学期期中考试数学试题湖南省衡阳市第八中学2024届高三下学期适应考试数学试题
解题方法
3 . 已知
是两个单位向量,
,
,
,
.
(1)若
,求
;
(2)若
,求
的最大值及相应的
值;
(3)若
,
,求证:
.
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eaccfa28fcd8b60194990aca32418470.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b307e7885a99d20f3f5cdeb22b5321b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/300ac50f6c88b25c6c083309913eecbd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6cfbefa60582db308925afcd5a8a42e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a62ed270b6d736b9fcfbb6f8425699f.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07bd57dd68879c0d78a2c42b9e724584.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fec223312cbc1aa48586f94fd03232d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8abd7755b18fc0d7bbb014b1c5936f1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2490175a5a77617d7e22087a7f11ead9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e287bb061bde44afaab70747ffe18769.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/152d4811e2f1341399c1cfa77f6e8d23.png)
.
您最近一年使用:0次
4 . 对
,定义
.
(1)求
的最小值;
(2)
,有
恒成立,求A的最大值;
(3)求证:不存在
,且m>n,使得
为恒定常数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e97769855336d73371930df1f187875e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f166b8917034ebc7522d1a160707f6a.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a7de87e4f04e15189c927b34b2e5afb.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bde2576b383ae3c851529435805b3adc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b206982b923b94befb9985e51f6499cb.png)
(3)求证:不存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c61a450b5c1c412aca3294e9eb4e9874.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c29ca729804f56f23d760ab66b79f68.png)
您最近一年使用:0次
2021-07-19更新
|
532次组卷
|
3卷引用:北京市一零一中学2020-2021学年高一下学期期末数学试题
5 . 向量是解决数学问题的一种重要工具,我们可以应用向量的数量积来解决不等式等问题.
(1)(ⅰ)若
,
,比较
与
的大小;
(ⅱ)若
,
,比较
与
的大小;
(2)
,
为非零向量,
,
,证明:
;
(3)设
为正数,
,
,
,求
的值.
(1)(ⅰ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fecf76d71aed3b37bd48550bf48c1086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4f683269ae8936d010ba111e9c9be5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3a8f20036b1e7cfb0800f141d843718.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f01e75d42bcb00df9c20734d9f3c547.png)
(ⅱ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fecf76d71aed3b37bd48550bf48c1086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/012b990e9c0d9f8da823df2ef36b26dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3a8f20036b1e7cfb0800f141d843718.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f01e75d42bcb00df9c20734d9f3c547.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ae98586d80f892771c90ab39eaced90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee437e6ff470c2f67b8429f57b90ae37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0118005d052d96ec2490facb71145b05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b04e531f59cead9c6f1017dbf1c953f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79e891ae2a63b7c20e00cb05e9acb71.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a93b98eef83e6b1b364a4cd6c55148ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e37739c262e686df999f5b89595c264.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc786b1cdc9c0bf814f43abdb1d2ad67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb5b9c5de247a0aef2e56f58a88a8698.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7f4495f96d35d8cb294a872223b923a.png)
您最近一年使用:0次
名校
6 . 如图,在四边形
中,
为对角线
与
中点连线
的中点,
为平面上任意给定的一点.
![](https://img.xkw.com/dksih/QBM/2021/6/18/2745403371003904/2768546705989632/STEM/46cde7beb3864628b0df3a3de5e77c2e.png?resizew=93)
(1)求证:
;
(2)若
,
,
,
,点
在直线
上运动,当
在什么位置时,
取到最小值?
(3)在(2)的条件下,过
的直线分别交线段
、
于点
、
(不含端点),若
,
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://img.xkw.com/dksih/QBM/2021/6/18/2745403371003904/2768546705989632/STEM/46cde7beb3864628b0df3a3de5e77c2e.png?resizew=93)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5632cbd5252f07934ecdb6e6951246f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4a0816f8db1e1cad53f05b8dc1837ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ed8128329f973dff60d13e4039957b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7652bbae10722db2cf0458d9da4a54c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f82432e6351f381c0008e5bc6b545f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbc4a9a1594591b27b24998ea3f5a2e5.png)
(3)在(2)的条件下,过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3834d7ec7531f3c3c0ce9b286f7a49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da25790e5a2cc740b8a5ef8809dab3a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/660018ccbfd4abf386c30cad0f9ea8eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d27e62ba44cbfec6823ebc1f0c7457fb.png)
您最近一年使用:0次
2021-07-20更新
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432次组卷
|
3卷引用:上海市复兴高级中学2020-2021学年高一下学期期末数学试题
7 . (1)对于平面向量
,
,求证:
,并说明等号成立的条件;
(2)我们知道求
的最大值可化为求
的最大值,也可以利用向量的知识,将
构造为两个向量的数量积形式,即:令
,
,则转化为
,求出最大值.利用以上向量的知识,完成下列问题:
①对于任意的
,求证:
;
②求
的最值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ff409cd3886c767afb13c9a869c5f23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6570cd7c2f81c9fcffd2c64664f1564.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73b1d21f757e06a46d40f9b8c4f525aa.png)
(2)我们知道求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/144e44ad8402f5ee368f64a87ab8c4e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9e0b006e08261158ac9b1cd631da051.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd288d4152caf5fc8187a1a901c8949f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/070a34585f3a61222c25cebdd532184f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d68ce4ff798e7cdcc8f4256c2fd6570.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bb9fa34fafe6cd3a7db5c79cda3e0c3.png)
①对于任意的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32ac12138178cb539a9e1c8f77587038.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/920a38dd1573498365963519c3bd2daa.png)
②求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61c5e59b2552eb5f033aea9e034e87ba.png)
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