21-22高一·湖南·课后作业
解题方法
1 . 证明不等式:
(1)若
,
,
,
都是正数,求证:
;
(2)若
,
,
是非负实数,则
;
(3)若
,
是非负实数,则
;
(4)若
,
,则
.
(1)若
![](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)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2e7387a3fbab6508695365955f55258.png)
(2)若
![](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)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2d92a6b95fdfdedb405447340293bdc.png)
(3)若
![](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/727ff3ac24b506706045956c16336f94.png)
(4)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19339e3904e9541ff26b30ae5f1242b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4f58b9bc974b789928f6490acb43fb3.png)
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21-22高二·湖南·课后作业
2 . 阅读“多知道一点:平面方程”,并解答下列问题:
(1)建立空间直角坐标系,已知
,
,
三点,而
是空间任意一点,求A,B,C,P四点共面的充要条件.
(2)试求过点
,
,
的平面ABC的方程,其中a,b,c都不等于0.
(3)已知平面
有法向量
,并且经过点
,求平面
的方程.
(4)已知平面
的方程为
,证明:
是平面
的法向量.
(5)①求点
到平面
的距离;
②求证:点
到平面
的距离
,并将这个公式与“平面解析几何初步”中介绍的点到直线的距离公式进行比较.
(1)建立空间直角坐标系,已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33b5103a4c35ab0c395c68690a300023.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f6f9d8550d619061ab0ba1105ec6a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adf322f683d50ecd3c7d4d5996122726.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1b82ad92798b264062c062f4a9a1a5c.png)
(2)试求过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd3ea554707fa3fc12fc9de51c94e4fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5622d4be6bba8c7a6851dc082ef34fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d1f4b53c90e4c31dd35b4bb548c5193.png)
(3)已知平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b163c34a920cb649829c376e7870007a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33b5103a4c35ab0c395c68690a300023.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
(4)已知平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a0cbd6b024b3fdff2f5fb5602da1a3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27e0ae1c14104ee9985e3ba31c604531.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
(5)①求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae715c996c1a6b5e35a3807c671bd6e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd24c686fbaaa68705d654b880481ffe.png)
②求证:点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e874a5821372c21a768cd1f5e20536d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a0cbd6b024b3fdff2f5fb5602da1a3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c828e62664e7373ed1f6dde8aa9653c.png)
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23-24高二上·全国·课后作业
解题方法
3 . 如图,
是抛物线
对称轴上一点,过点M作抛物线的弦AB,交抛物线于A,B.
,求弦AB中点的轨迹方程;
(2)过点M作抛物线的另一条弦CD,若AD与y轴交于点E,连接ME,BC,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bbfc5253a678d786c9a8091fff43729.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/745de5ef1fd897d16e37464172d5e8c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e258ab9e600435b37465092243d99f6.png)
(2)过点M作抛物线的另一条弦CD,若AD与y轴交于点E,连接ME,BC,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/490111e9e5aefb170410f0501134d22e.png)
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解题方法
4 . 如图,点A,B分别位于异面直线a,b上,过AB中点O的平面
与a,b都平行,M,N分别是a,b上异于A,B的另外两点,MN与
交于点P.求证:P是MN的中点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
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5 . 图中四边形ABCD,DCEF,FEGH都是正方形,用复数方法证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd6e7b446e3329dfe2894751f93cf0c1.png)
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23-24高二上·全国·课后作业
6 . 设圆O的弦
的中点为M,过点M任作两弦
,弦
与
分别交
于点E,F.
的中点;
(2)如果将圆分别变为椭圆、双曲线或抛物线,你能得到类似的结论吗?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5c4cd264c97c1f261229925cc5a6761.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
(2)如果将圆分别变为椭圆、双曲线或抛物线,你能得到类似的结论吗?
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21-22高二·江苏·课后作业
解题方法
7 . 已知数列
和
是两个无穷等差数列,公差分别为
和
,求证:数列
是等差数列,并求它的公差.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.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/5344eadd4711db34e3f935aedd5fb270.png)
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21-22高二·江苏·课后作业
8 . 已知
,点
,
,直线
.求证:点P到直线l的距离等于
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a449474e7cf366add572ca32014a23e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7775aa57ca0e62216f3039ed88dceed0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f6d06b9a4d3a891ea7c986b1ab4e925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c75f0d9463771a3aba1865a9d9d398aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ac654a052f98d1ccb7fede1f122cec3.png)
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21-22高一·湖南·课后作业
解题方法
9 . 用向量运算刻画三角形的重心.
(1)已知
,求一点G满足
.
(2)求证:满足条件
的点G是
的重心.
(提示:说明点G同时在
的三条中线上.)
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c41af21f3c86faeb4b18e6ba7abf67.png)
(2)求证:满足条件
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c41af21f3c86faeb4b18e6ba7abf67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(提示:说明点G同时在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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2022-02-22更新
|
835次组卷
|
8卷引用:专题02 平面向量的运算-2021-2022学年高一《新题速递·数学》(人教A版2019)
(已下线)专题02 平面向量的运算-2021-2022学年高一《新题速递·数学》(人教A版2019)(已下线)6.2.3向量的数乘运算(分层作业)-【上好课】2022-2023学年高一数学同步备课系列(人教A版2019必修第二册)(已下线)6.2.3向量的数乘运算(精练)-【精讲精练】2022-2023学年高一数学下学期同步精讲精练(人教A版2019必修第二册)(已下线)专题03 向量的数乘(2)-《重难点题型·高分突破》(苏教版2019必修第二册)(已下线)【高一模块四】 回归1 平面向量的课本典型例题和习题(已下线)1.3 向量的数乘湘教版(2019)必修第二册课本习题 习题1.3(已下线)第02讲 平面向量的运算-【寒假预科讲义】(人教A版2019必修第一册)
21-22高二·全国·课后作业
10 . 在数轴上,对坐标分别为
和
的两点A和B,用绝对值定义两点间的距离,表示为
.
(1)在数轴上任意取三点A,B,C,证明
.
(2)设A和B两点的坐标分别为
和2,分别找出(1)中不等式等号成立和等号不成立时点C的范围.
![](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/f7a1a35fb8112b8b0661c9e5281839dd.png)
(1)在数轴上任意取三点A,B,C,证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88bcaabd563b35f69c5059c8d4e71a98.png)
(2)设A和B两点的坐标分别为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81fb134b2b48acc99213fff6ccfee65f.png)
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