名校
1 . 对于正整数集合
,如果去掉其中任意一个元素
之后,剩余的所有元素组成的集合都能分为两个交集为空集的集合,且这两个集合的所有元素之和相等,就称集合
为“平衡集”.
(1)判断集合
是否是“平衡集”并说明理由;
(2)求证:若集合
是“平衡集”,则集合
中元素的奇偶性都相同;
(3)证明:四元集合
,其中
不可能是“平衡集”.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffbfa3e226e067ec597ebf0bbc2e87d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62c4392f75c09edaec2e70c9eccb2b85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(1)判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e5f6cb6141a374d04b6a14a1b27e282.png)
(2)求证:若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(3)证明:四元集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a784e0ba1c17aba6990123fe39b89114.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59a3f266cb6beee27f3d831c1169d3d2.png)
您最近一年使用:0次
2 . 对任意正整数n,记集合
,
.
,
,若对任意
都有
,则记
.
(1)写出集合
和
;
(2)证明:对任意
,存在
,使得
;
(3)设集合
.求证:
中的元素个数是完全平方数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a39352d44787ecda055946f530893f97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5359f5f086b89cc656cdd4f79a3b7baa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5df0913f90131b298c8f6f57437f69b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20c1352dca7dc3caf67c1cb937d52795.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d35755ad07f05e7bfe00176d6334389f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75a28908216e6879a09b372d957be1e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90df641ab645927ee577e79faf18dcdd.png)
(1)写出集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43a71fc9c0068109dad1382354570665.png)
(2)证明:对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e840ba7606959ccea36793f3ef0775d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b576952835af1f3492f0f3e6d00093e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90df641ab645927ee577e79faf18dcdd.png)
(3)设集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/112d102cbe7ce2bebe0c76e87e89a00c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
您最近一年使用:0次
2023-11-15更新
|
161次组卷
|
4卷引用:北京市第三十五中学2023-2024学年高二上学期期中考试数学试题
3 . 已知函数
,
,且满足
.
(1)求实数a的取值范围;
(2)求证函数
存在唯一零点;
(3)设
,证明
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fb0c7b952731190aea730a9fb18a603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66692ec49a458f9e48c7315d03dfc37b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a02872c8c4d0f941ad55b2f88fa58ea.png)
(1)求实数a的取值范围;
(2)求证函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c419949314258c61e4436e16477fa42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67a51414243ca45bcca00d14a9865f93.png)
您最近一年使用:0次
名校
4 . 设A为非空集合,令
,则
的任意子集R都叫做从A到A的一个关系(Relation),简称A上的关系.例如
时,
{0,2},![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76986e6f96cbb9d7d6d0fbcf0bf2321a.png)
,
,
{(0,0),(2,1)}等都是A上的关系.设R为非空集合A上的关系.给出如下定义:
①(自反性)若
,有
,则称R在A上是自反的;
②(对称性)若
,有
,则称R在A上是对称的;
③(传递性)若
,有
,则称R在A上是传递的;
如果R同时满足这3条性质,则称R为A上的等价关系.
(1)已知
,按要求填空:
①用列举法写出
______________________;
②A上的关系有____________个(用数值做答);
③用列举法写出A上的所有等价关系:{(0,0),(1,1),(2,2)},{(0,0),(1,1),(2,2),(0,1),(1,0)},{(0,0),(1,1),(2,2),(0,2),(2,0)},_______________,_______________,共5个.
(2)设
和
是某个非空集合A上的关系,证明:
①若
,
是自反的和对称的,则
也是自反的和对称的;
②若
,
是传递的,则
也是传递的.
(3)若给定的集合A有n个元素(
),
,
,...,
为A的非空子集,满足
且两两交集为空集.求证:
为A上的等价关系.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff994543fe18b563c7127c8b2a874358.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f95c06ed271ff0a6407a3bf5deec5871.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5689e50a9353ba69ff5b71e7b6a3c795.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8225d0531fba46cbb4a3af4dd2d6751f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76986e6f96cbb9d7d6d0fbcf0bf2321a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f95c06ed271ff0a6407a3bf5deec5871.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/934909fce1b90557163c6f43d4f0790d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8225d0531fba46cbb4a3af4dd2d6751f.png)
①(自反性)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd170c506a8ce70f550f5751ae016ca6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e96fb327d44b08d715e86db04cc9785.png)
②(对称性)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/328ae22ec119ce8f0faac8dc554a2c10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c950781f08495bc2a4c20454c26c48d8.png)
③(传递性)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/227539cbcd96eb67cbcf7c94de56598d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78dd7c7d34bcfcae1f423a684aae9542.png)
如果R同时满足这3条性质,则称R为A上的等价关系.
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5689e50a9353ba69ff5b71e7b6a3c795.png)
①用列举法写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39b0949c177d28fe5b6ec4a0de58c80a.png)
②A上的关系有____________个(用数值做答);
③用列举法写出A上的所有等价关系:{(0,0),(1,1),(2,2)},{(0,0),(1,1),(2,2),(0,1),(1,0)},{(0,0),(1,1),(2,2),(0,2),(2,0)},_______________,_______________,共5个.
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efc18a5bb2e53586331b2a58538a48b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f20f21a9d50b61dac519a3ddab539d.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efc18a5bb2e53586331b2a58538a48b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f20f21a9d50b61dac519a3ddab539d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05725d20ff805152beff52c7a5e8d735.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efc18a5bb2e53586331b2a58538a48b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f20f21a9d50b61dac519a3ddab539d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7077a5e7dce0e2f0e678b1147deae46.png)
(3)若给定的集合A有n个元素(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5818ede14d21f6df9ef9c2bfe09286c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f4bae4bf0e8cf84b9e1c6c7258b06d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79ff32c9e80fd90fcdb360f9a5a21c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/591eaeea196d5720d0762ced03e8ce3b.png)
您最近一年使用:0次
20-21高二下·上海浦东新·期末
名校
5 . 已知定义在R上的函数
与
.
(1)对于任意满足
的实数p,q,r均有
并判断函数
的奇偶性,并说明理由
(2)函数
与
(均为奇函数,
在
上是增函数,
在
上是增函数,试判断函数
与
在R上是否是增函数?如果是请证明,如果不是请说明理由.
(3)函数
与
均为单调递增的一次函数,
为整数当且仅当
为整数.求证:对一切
,
为整数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
(1)对于任意满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9534ea8db35f625f10fdd3271417b46a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ace78ab406e053a72c7f7bdb3a7ec8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
(2)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8938db94f49dcbe0c383fba0241bb0da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bdfed8d6862125dc1fecfce0322a750.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
(3)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4166972dec0aa3e8694a44eeb941a08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2728a4ef67b88090a84c1e5746c7f6b8.png)
您最近一年使用:0次
6 . 阅读下面题目及其解答过程,并补全解答过程.
以上解答过程中,设置了①~⑤五个空格,如下的表格中为每个空格给出了两个选项,其中只有一个正确,请选出你认为正确的,并填写在答题卡的指定位置.
已知函数![]() (Ⅰ)当 ![]() ![]() (Ⅱ)求证:函数 ![]() ![]() 解答:(Ⅰ)当 ![]() ![]() 因为 ![]() 所以当 ![]() ![]() 因为函数 ![]() ![]() 所以 ![]() ![]() 所以 ![]() 所以 ![]() 所以函数 ![]() (Ⅱ)证明:任取 ![]() ![]() 因为 ![]() 所以 ![]() 所以⑤. 所以 ![]() 所以函数 ![]() ![]() |
空格序号 | 选项 | |
① | A.![]() | B.![]() |
② | A.![]() | B.![]() |
③ | A.![]() | B.![]() |
④ | A.![]() | B.![]() |
⑤ | A.![]() | B. ![]() |
您最近一年使用:0次
名校
7 . 设
,函数
为常数,
.
(1)若
,求证:函数
为奇函数;
(2)若
.
①判断并证明函数
的单调性;
②若存在
,
,使得
成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1e69392d21261afd8e5e5f096634669.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6559f6c5bcd240cf567c7e472b12a1a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fc679a2fdf60535af5af9b4b517a585.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e10e1c43b86a8cd4360ca9b57232164.png)
①判断并证明函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
②若存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38e96e9a314387fa1c76e86179ee0121.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45340678c2ec1bc8cd68c0a3a2ab8902.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/551ba93905ba57cee861f59f2c883603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
2020-11-06更新
|
678次组卷
|
8卷引用:浙江省金华市东阳市横店高级中学2021-2022学年高二下学期4月月考数学试题
解题方法
8 . 已知函数
(其中
),
(1)试判断并证明函数
的单调性;
(2)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92cf6f1cf39478c7b037aa45b5d89468.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd8dc2c7954a61c17ce444232f965ab5.png)
(1)试判断并证明函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a34e7e4fb503c94e0008d1fc7a561f6.png)
您最近一年使用:0次
9 . 现定义:设
是非零实常数,若对于任意的
,都有
,则称函数
为“关于的
偶型函数”
(1)请以三角函数为例,写出一个“关于2的偶型函数”的解析式,并给予证明
(2)设定义域为的“关于的
偶型函数”在区间
上单调递增,求证在区间
上单调递减
(3)设定义域为
的“关于
的偶型函数”
是奇函数,若
,请猜测
的值,并用数学归纳法证明你的结论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e02cab1add26335b3cb43d5b54c7c853.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55550151ed0b0264fce45814acfc725a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(1)请以三角函数为例,写出一个“关于2的偶型函数”的解析式,并给予证明
(2)设定义域为的“关于的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1f0b8dcc8ea36ef8093122d4efbedc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a8f80511de15d3dfb871ca2f400424.png)
(3)设定义域为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af5cf9c12181dd8683944b2b30bf8e08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38fcec7af3520884b173b29bda6c657a.png)
您最近一年使用:0次
2019-12-31更新
|
333次组卷
|
5卷引用:第四章++数列1(基础过关)-2020-2021学年高二数学单元测试定心卷(人教A版2019选择性必修第二册)
(已下线)第四章++数列1(基础过关)-2020-2021学年高二数学单元测试定心卷(人教A版2019选择性必修第二册)(已下线)第二章 推理与证明(基础过关)-2020-2021学年高二数学单元测试定心卷(人教版选修2-2)上海市静安区2019-2020学年高三上学期期末数学试题2020届上海市静安区高三一模(期末)数学试题(已下线)热点02 函数及其性质-2021年高考数学【热点·重点·难点】专练(上海专用)
10 . 已知函数
是定义在
上的不恒为零的函数,对于任意非零实数
满足
,且当
时,有
.
(Ⅰ)判断并证明
的奇偶性;
(Ⅱ)求证:函数
在
上为增函数,并求不等式
的解集.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c8bd00a1b1c012681aab8513b755cbc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea38ff7b3050c464f0270c4a146d2350.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fde64f4d3c38e43fbdee24eadc4b0dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/018857ec6e498113b3b12a730d9313da.png)
(Ⅰ)判断并证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
(Ⅱ)求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8938db94f49dcbe0c383fba0241bb0da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/984c0d730cbd5d1a09e4dda6d93ce729.png)
您最近一年使用:0次