名校
解题方法
1 . 如图,一块三角形铁片ABC,已知
,
,
,现在这块铁片中间发现一个小洞,记为点D,
,
.如果过点D作一条直线分别交AB,AC于点E,F,并沿直线EF裁掉
,则剩下的四边形EFCB面积的最大值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33c99fc553ecb3e2afdb7058201bc642.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26b566395b41fc587143d533f9756024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f4aca5534bce25acaeb7379deed8f8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20d54e182a943f960a1885b8556d07a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c105d6ba18fbb0581fb982175e2eac9.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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名校
解题方法
2 . 设集合
,(
,
)且A中任意两数之和不能被5整除,则n的最大值为____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6bd3eec1b1e189c92c2db22ea773cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd4613271f782a90ab580131d09d03d1.png)
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名校
3 . 已知函数
和
的定义域分别为
和
,若对任意
,恰好存在
个不同的实数
,使得
(其中
),则称
为
的“
重覆盖函数”.
(1)试判断
是否为
的“2重覆盖函数”?请说明理由;
(2)若
,为
,的“2重覆盖函数”,求实数
的取值范围;
(3)函数
表示不超过
的最大整数,如
.若
为
的“2024重覆盖函数”,求正实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6795cae2df43a722e1355e9562d93c09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c296e45b84cf67a98939aa7334e7d478.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3eddf991be37d25d033f78bd3511809.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2d5df7922a4e98e8e07bf418dd48a7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad4cc356abee7ec3437ea301dbfbb6a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/234c48b3478c6912aa97d8e20ca82188.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)试判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d09a854c282bb3a196a91eb25ca01e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/885f35f354492e6c09f1a91d45d3221c.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87b66d7174f5f4ccee109340d93e5311.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1261e4befbd80b30fdf656321b8537e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d82bab9f2808b11904d680eae089356.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4987dca9120f6a58139fd3e412ed77c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a899e901b141a0a6d56e3387ecf9f047.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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名校
4 . 已知函数
,若关于
的方程
有4个不同的实根
、
,且
,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0c407c501c66aa304839fe61ff531d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9c0d827ef8598ba6b70b34b2bdcd1e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2961ff859ef31f3a71c69841dc64176f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3604274ad6707a906eba371a9e884144.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f176de330ed3f93fee6f5a398de57c3.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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2024-05-02更新
|
535次组卷
|
4卷引用:湖南省长沙市明德中学2023-2024学年高一下学期期中考试数学试卷
湖南省长沙市明德中学2023-2024学年高一下学期期中考试数学试卷江西省宜春市宜丰中学2023-2024学年高一下学期6月月考数学试题浙江省东阳市外国语学校2023-2024学年高二下学期5月月考数学试题(已下线)模型12 对数函数绝对值 “积定法”的零点模型(高中数学大模型)
名校
5 . 设函数
,若对于任意实数
,函数
在区间
上至少有2个零点,至多有3个零点,则
的取值范围是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/545e70674ab9a9d9afa707bb32b921ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6581916f5a65edfea257c804efee007e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bccd6a6e85bdf500218a3e75b31f3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/074c228ffc7b1e306f8410afe7bc4b5c.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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2024-04-11更新
|
931次组卷
|
2卷引用:湖南省长沙市第一中学2023-2024学年高一下学期开学自主检测数学试卷
6 . (1)对实系数的一元二次方程可以用求根公式求复数范围内的解,在复数范围解方程
;
(2)对一般的实系数一元三次方程
(
),由于总可以通过代换
消去其二次项,就可以变为方程
.在一些数学工具书中,我们可以找到方程
的求根公式,这一公式被称为卡尔丹公式,它是以16世纪意大利数学家卡尔丹(J. Cardan)的名字命名的.卡尔丹公式的获得过程如下:三次方程
可以变形为
,把未知数
写成两数之和
,再把等式
的右边展开,就得到
,即
.将上式与
相对照,得到
,把此方程组中的第一个方程两边同时作三次方,
,并把
与
看成未知数,解得
于是,方程
一个根可以写成
.
阅读以上材料,求解方程
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed344791b8b035ca04d4b5af7364cae5.png)
(2)对一般的实系数一元三次方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48ad9d68d15b5d5121fcf99ebddaa986.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20849c00c47cbdc43f18d53341b6c4e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae0f3c81f415857813838d4b9b714d56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ea05ab19c339e26f8268fbc7b6e918d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5dd275a6062b21f9c3e9155c7e0ba62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5dd275a6062b21f9c3e9155c7e0ba62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ead1b77b69e6b51d6d483331fd01d41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0bed1a02239821a616bc173181e7ed2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c26aacdd3362aa65b2966045cbfcddf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f02c3aa1326c9b1e069b6997cd29bfa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11792ad247341c0dbc80663dd0fa6f77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ead1b77b69e6b51d6d483331fd01d41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1e8aa11c220ffef18a553784e1ecc16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/491db400b0e81be11e3fd8729fe61a41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36accab23dbd172687769aea43e5781c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411a315870ed3e6d0e8ea885f1a04bcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a9930c09269f4f03794e38c17f6da67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5dd275a6062b21f9c3e9155c7e0ba62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49d63387694fd1caafce80adfb43c86b.png)
阅读以上材料,求解方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93c3d494147195cf4f5e1fa3f6f5a0b9.png)
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名校
解题方法
7 . 已知
为常数,函数
.
(1)当
时,求关于
的不等式
的解集;
(2)当
时,若函数
在
上存在零点,求实数
的取值范围;
(3)对于给定的
,且
,证明:关于
的方程
在区间
内有一个实数根.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11fe72001cc55e5c4c5d96f641aabb42.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d0700dce7edc5ec1981b0483eef1b6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e9c599e8d420006448905acec2b8234.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7eca46642891f6b8e3e30edd9b37dd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29f48e1c656aace41360467f254e359d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
(3)对于给定的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e5223ece2f8f76850c49e2505304532.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a78288f56f67c4f126209f9d2ee76a3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a95496c9d918148f5d86a6d48a136b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f803a468e5d66004e57372a5bf2c5e1b.png)
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名校
解题方法
8 . 我国人脸识别技术处于世界领先地位.所谓人脸识别,就是利用计算机检测样本之间的相似度,余弦距离是检测相似度的常用方法.假设二维空间中有两个点
,
,O为坐标原点,余弦相似度为向量
,
夹角的余弦值,记作
,余弦距离为
.已知
,
,
,若P,Q的余弦距离为
,Q,R的余弦距离为
,且
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06a2b8dae45cd30a752aedb1ca66c30a.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a3efb79f35db8448f3391252ab7d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df332f01628130c084fd46aaca0a4b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4f605ec0729ce6d72237ad662a06862.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fc9656d8286c4d6fa309d6ae347c89e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1df320d266e857a428ff0bac1eb680bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef2376fa0549e6fae46bae54d6ace942.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/926778fe15991308849cdba1822595a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1b00c13a8bb6db096dd866a2c37dee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b20bb38fbd8ff8e5f8e985a0babf498d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e28a96ee1e62b7fbc1fb0257ad0deeed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06a2b8dae45cd30a752aedb1ca66c30a.png)
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2024-04-04更新
|
533次组卷
|
3卷引用:湖南省长沙市实验中学2023-2024学年高一下学期第一次月考数学试题
名校
解题方法
9 . 若函数
对定义域内的每一个值
,在其定义域内都存在唯一的
,使
成立,则称该函数为“依赖函数”.
(1)判断函数
是否为“依赖函数”,并说明理由;
(2)已知函数
在定义域
上为“依赖函数”,若存在实数
,使得对任意的
,不等式
都成立,求实数
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.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/c13ca0f27aa97d8d1bec1f6879f460d6.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4351bd617a7516709fbfdf31dc993c7.png)
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5b3cc08473fd879e63795139f628efa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bdae0482d51063c22282f2e49332526.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fc53c366cc45062f75b446f5e0420d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bf039c46a25e331446c6ee1e9af3c82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b91b18127b51a93a54db0e96390bbf3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5873c01192b7d33b7483f444f90b5b0.png)
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名校
解题方法
10 . 设连续函数
的定义域为
,如果对于
内任意两数
,都有
,则称
为
上的凹函数;若
,则称
为凸函数.若
是区间
上的凹函数,则对任意的
,有琴生不等式
恒成立(当且仅当
时等号成立).
(1)证明:
在
上为凹函数;
(2)设
,且
,求
的最小值;
(3)设
为大于或等于1的实数,证明:
.(提示:可设
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a7a1783349936cc7254a4a8694c6812.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1bedaf3854b48806b82b3b804451cf8.png)
![](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/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0fec4d10407498ec4692b33ebe1bb7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1784a3a9dd90c51dab965445d65f08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2697ef67790838c84cc238a0334c5d47.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/008ab9b6200370bd8d534a6317cb88e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7160d93f92089ef36f3dab809d3114b8.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6da13af19b32430759c9c1d1aea894e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b16ad49f62d7362441e3b92efe7f87d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b47ade684a2e49ef6139afe6ab59a29.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1d4c274b53adfbffc4b19e7adbc39d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6694499b581256296277c515f6dcdc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10cda4049695561dab3e0803c3a287fe.png)
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