1 . 已知①设函数
的值域是
,对于
中的每个
,若函数
在每一处
都等于它对应的
,这样的函数
叫做函数
的反函数,记作
,我们习惯记自变量为
,因此
可改成
即为原函数的反函数.易知
与
互为反函数,且
.如
的反函数是
可改写成
即为
的反函数,
与
互为反函数.②
是定义在
且取值于
的一个函数,定义![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db6c5d65e0ccd27748fcbc420c6a2e22.png)
,则称
是函数
在
上的
次迭代.例如
,则
.对于一些相对复杂的函数,为求出其
次迭代函数,我们引入如下一种关系:对于给定的函数
和
,若函数
的反函数
存在,且有
,称
与
关于
相似,记作
,其中
称为桥函数,桥函数满足以下性质:
(i)若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e0e8a92152787274ed6e06a21ef1661.png)
(ii)若
为
的一个不动点,即
,则
为
的一个不动点.
(1)若函数
,求
(写出结果即可)
(2)证明:若
,则
.
(3)若函数
,求
(桥函数可选取
),若
,试选取恰当桥函数,计算
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e66c24e657d998beb013ad1fb311d33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f93e3581e920716e710e22b31006bdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f93e3581e920716e710e22b31006bdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63ced31d098cfb0cf14d906e97e6353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e66c24e657d998beb013ad1fb311d33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/955f3c2b80eebc3f88c804112e5f41f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/955f3c2b80eebc3f88c804112e5f41f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb87ce86e2c9a1a8188b03b74438fdd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb87ce86e2c9a1a8188b03b74438fdd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e66c24e657d998beb013ad1fb311d33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25ef38135b0e7906687d8a4918a4cb67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b161347f6a2fcfd9bf0acf1e8a03fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43737e3ca063dfc210d0c72924a4930.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/808bd2fc4b344e7669fca65b4fa122df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b161347f6a2fcfd9bf0acf1e8a03fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/808bd2fc4b344e7669fca65b4fa122df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b161347f6a2fcfd9bf0acf1e8a03fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db6c5d65e0ccd27748fcbc420c6a2e22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c110a1293773729278a214c7fe8d544e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a33cfe27fd2276a7c542f062c17b4d85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/052ddf3664af9ab2990f3ea622997e00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/910ca4e4f009554b599eab90e1d94c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](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/544f91d4fb22c571db9f8481b72a0419.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71567deb76e48f8a2424b06536cbe465.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a66b033ee7a03c7b3508583481465275.png)
![](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/544f91d4fb22c571db9f8481b72a0419.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1249f186df944244da02e1b8c754005.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544f91d4fb22c571db9f8481b72a0419.png)
(i)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1249f186df944244da02e1b8c754005.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e0e8a92152787274ed6e06a21ef1661.png)
(ii)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66f66a2b3d90f0d935d6c8ebaf675349.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4f199ad3fad8657afa38f370b319a75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/192bb45bd15b200f40b34377bc58905b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a33cfe27fd2276a7c542f062c17b4d85.png)
(2)证明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1249f186df944244da02e1b8c754005.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99315f5b2ae9bea18e06401b41d3780c.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/becd598a11b876d858728161a7a09705.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a33cfe27fd2276a7c542f062c17b4d85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04ab7717944da2b6cc305b6a65f91408.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c85f9d9be0ba965ff7beb0e011267f29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bee7f1ccd52c7d526b6d466b970e769.png)
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2 . 对于数列A:
,经过变换T:交换A中某相邻两段的位置(数列A中的一项或连续的几项称为一段),得到数列
.例如,数列A:
经交换M,N两段位置,变换为数列
:
.设
是有穷数列,令
.
(1)如果数列
为3,2,1,且
为1,2,3.写出数列
;(写出一个即可)
(2)如果数列
为9,8,7,6,5,4,3,2,1,
为5,4,9,8,7,6,3,2,1,
为5,6,3,4,9,8,7,2,1,
为1,2,3,4,5,6,7,8,9.写出数列
;(写出一组即可)
(3)如果数列
为等差数列:2015,2014,…,1,
为等差数列:1,2,…,2015,求n的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adc250de2317c83a904f0ebce5fc2989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bbf1fc7753df7951558e56b71b78c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adc250de2317c83a904f0ebce5fc2989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a403f562ca2b40d07069aace053e8af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79a60554299801b8c94f80e04398e589.png)
(1)如果数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
(2)如果数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.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/5002f030017f6f0b34a61b2e15c5a9cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bfefd19874aeff8340735b3df468333.png)
(3)如果数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
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名校
3 . 对称变换在对称数学中具有重要的研究意义.若一个平面图形K在m(旋转变换或反射变换)的作用下仍然与原图形重合,就称K具有对称性,并记m为K的一个对称变换.例如,正三角形R在
(绕中心O作120°的旋转)的作用下仍然与R重合(如图1图2所示),所以
是R的一个对称变换,考虑到变换前后R的三个顶点间的对应关系,记
;又如,R在
(关于对称轴
所在直线的反射)的作用下仍然与R重合(如图1图3所示),所以
也是R的一个对称变换,类似地,记
.记正三角形R的所有对称变换构成集合S.一个非空集合G对于给定的代数运算.来说作成一个群,假如同时满足:
I.
,
;
II.
,
;
Ⅲ.
,
,
;
Ⅳ.
,
,
.
对于一个群G,称Ⅲ中的e为群G的单位元,称Ⅳ中的
为a在群G中的逆元.一个群G的一个非空子集H叫做G的一个子群,假如H对于G的代数运算
来说作成一个群.
(2)同一个对称变换的符号语言表达形式不唯一,如
.对于集合S中的元素,定义一种新运算*,规则如下:
,
.
①证明集合S对于给定的代数运算*来说作成一个群;
②已知H是群G的一个子群,e,
分别是G,H的单位元,
,
,
分别是a在群G,群H中的逆元.猜想e,
之间的关系以及
,
之间的关系,并给出证明;
③写出群S的所有子群.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77ab1256702aef4e9f1a5eb6c12ecc96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77ab1256702aef4e9f1a5eb6c12ecc96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f8278c090ec35994a2300a2f6e03cd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2858005b9ae89ae080d83dcc13cf8e81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b9a0da1382342078b9b0bc326a0b58e.png)
I.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8362f15e544684164f38ff9ad7c38ac7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68f73696ca1660407be38423825ac579.png)
II.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/509a09a7391de2cc86e5e44ccccc981b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47512437070ec582249e3fe8a9422516.png)
Ⅲ.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27321be7cc5aec6555c61775f6638cea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebf00e8864c86c3ce8118ea76bf69773.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4a34726666c0499373270f6ca37136f.png)
Ⅳ.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebf00e8864c86c3ce8118ea76bf69773.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e78818e18abc456ae7a86110636386ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b2db6609d50b3b58c4c98ee07396606.png)
对于一个群G,称Ⅲ中的e为群G的单位元,称Ⅳ中的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/856b4ab24ff3b7d9e0b4d1c945232aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/655c66701407d942ef38d482e6b3ffd7.png)
(2)同一个对称变换的符号语言表达形式不唯一,如
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/317369bcdd0bc35e2ca45ff7ee37ec09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7703f78bf42acd363d895107b6edae18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54ec72c22e432256b92c8c87f31f4bd2.png)
①证明集合S对于给定的代数运算*来说作成一个群;
②已知H是群G的一个子群,e,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3377b3f59d9c7ac048d59262ecbaf389.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c15c2fe2621766b6e71a4e61686f3bea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/856b4ab24ff3b7d9e0b4d1c945232aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e90425090dfd36313d564a97289b3b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3377b3f59d9c7ac048d59262ecbaf389.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/856b4ab24ff3b7d9e0b4d1c945232aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e90425090dfd36313d564a97289b3b1.png)
③写出群S的所有子群.
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5卷引用:安徽省芜湖市安徽师范大学附属中学2024届高三第二次模拟考试数学试题
安徽省芜湖市安徽师范大学附属中学2024届高三第二次模拟考试数学试题安徽省天域全国名校协作体2024届高三下学期联考(二模)数学试题山东省菏泽市单县第一中学2024届高三下学期3月月考数学试题(已下线)安徽省天域全国名校协作体2024届高三下学期联考(二模)数学试题变式题16-19(已下线)压轴题01集合新定义、函数与导数13题型汇总-2
4 . 设A,B是两个非空集合,如果对于集合A中的任意一个元素x,按照某种确定的对应关系
,在集合B中都有唯一确定的元素y和它对应,并且不同的x对应不同的y;同时B中的每一个元素y,都有一个A中的元素x与它对应,则称
:
为从集合A到集合B的一一对应,并称集合A与B等势,记作
.若集合A与B之间不存在一一对应关系,则称A与B不等势,记作
.
例如:对于集合
,
,存在一一对应关系
,因此
.
(1)已知集合
,
,试判断
是否成立?请说明理由;
(2)证明:①
;
②
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42acae4bf2a6bead9d904b70d0480fc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0915685a3eae67d5c6bc3bd722030876.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79aedd00413c6ff9b2696a63a854867.png)
例如:对于集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1aac2c0e4c6fc7ae8950a38098cb062f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8794b3ea2ca1d6d2b70dcec2a991dd3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/210402b31fd895e4fd6921cb25c1ee88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0915685a3eae67d5c6bc3bd722030876.png)
(1)已知集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cf4f47caab35fc473167ca17c7b5f4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eae2c499889a4619a5102a4b2e6b8129.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e386b0005c8f091434060361a07955d8.png)
(2)证明:①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b06ec5553f5aeef37ec8ca6f0d9caba8.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/229c5c40da18cb86a81e709d802d4c1e.png)
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4卷引用:浙江省台州市2024届高三下学期第二次教学质量评估数学试题
浙江省台州市2024届高三下学期第二次教学质量评估数学试题(已下线)压轴题01集合新定义、函数与导数13题型汇总 -1河北省名校联盟2024届高三下学期4月第二次联考数学试题 (已下线)情境10 存在性探索命题
5 . 对于给定的一个
位自然数
(其中
,
),称集合
为自然数
的子列集合,定义如下:
{
且
,使得
},比如:当
时,
.
(1)当
时,写出集合
;
(2)有限集合
的元素个数称为集合
的基数,一般用符号
来表示.
(ⅰ)已知
,试比较
大小关系;
(ⅱ)记函数
(其中
为
这
个数的一种顺序变换),并将能使
取到最小值的
记为
.当
时,求
的最小值,并写出所有满足条件的
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6be3d8ff4885ce8cf21ed3b7e4c9059.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff4ab89a62749697c6a67e4fe8e6f3c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12b0e3b00fe47801afb53ec56706c21a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5123cae73867329882792f626287b970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d688a7cacda715fc5c2fad9a2adaddee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab691edda624f588e85d493423b3e398.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76fb6d4810762396e3fbe728687a0794.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba7a9e06bedb3aca590121cc47e64e52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e534ff8ca5451dce6629223e002d5878.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d407ed5fd8a5fd413426fc1fc118422.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eeaa1b4ec60977b69d48d3d023f5d731.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5123cae73867329882792f626287b970.png)
(2)有限集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc8279d9dd0b7750953cb9e2098b3b90.png)
(ⅰ)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b43ea4161df6e6178c26c524935af465.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b53fe2a5d83d2e3e97f3a49d1f845370.png)
(ⅱ)记函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a2a7d5abc0e14bf1da403fba5b27644.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c558c7204d256c96b74b9c991c0e5c1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/086eb439f6a1578fdba904825340772d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aad1bf0868f56ad3bda73d4ca5851cb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbf91726683a3963e941231877c8c6ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e731e11a03c0f5d2768e87a3442634d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c326b4a68d5148e8e5a5ebc15d3b132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aad1bf0868f56ad3bda73d4ca5851cb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e731e11a03c0f5d2768e87a3442634d.png)
您最近一年使用:0次
2024·全国·模拟预测
6 . 矩阵可以理解为一个二维数列,在数列的研究中有重要的应用.记一个m行l列的矩阵A为
,A中第i行第j列的元素为
(
,2,3,…,m,
,2,3,…,l),记一个l行n列的矩阵B为
,我们定义一个双目运算符“
”使得矩阵
,那么有以下规则:
a.A的列数必须与B的行数相等.
b.C是一个m行n列的矩阵.
c.C中的元素
.
d.若有n个相同的矩阵A相
后得到一个新矩阵,可将其记作
.
e.
运算满足结合律,不满足交换律.
(1)求
.
(2)数列
满足:
,其中
.存在唯一的矩阵D使得
(
,
).
①求矩阵D,并用矩阵相
的形式表示出矩阵
(
,
);
②用矩阵相
的形式表示出矩阵
(
,
).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83047b671b93bcf1d8b3de1360dac981.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4ba1bbe411bc71bca016d3fd82352f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c45176df950dfe48b8ca7eac08ee349.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2aa4dd10e848cc71144873a428946dc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b73757f14449bd45efd1fab48851416a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e16415b61722f9961e412386e6819f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1133585a2bffb160518d7d25c1b7ff96.png)
a.A的列数必须与B的行数相等.
b.C是一个m行n列的矩阵.
c.C中的元素
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2316502be5a59c336036f5c197e5b481.png)
d.若有n个相同的矩阵A相
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e16415b61722f9961e412386e6819f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/685ef7cc8f28e8e4456d7bde3be8f672.png)
e.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e16415b61722f9961e412386e6819f.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7469755f4d88a52733f0e667bc4d3a9.png)
(2)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/763b41172fa5f9f9ef85ab59df78bc39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/253c5ef19dc5ac506e52293d1587a344.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4173753db5fd43cbf7908f56ba39e6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4db0fb7f1f1a2a9ec694af3da44436d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
①求矩阵D,并用矩阵相
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e16415b61722f9961e412386e6819f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/accb0659581a88737e1b9f9af850440d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
②用矩阵相
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e16415b61722f9961e412386e6819f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b60bb82a91915332c067f0cefc676681.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
您最近一年使用:0次
名校
7 . 已知无穷数列
满足
,其中
表示x,y中最大的数,
表示x,y中最小的数.
(1)当
,
时,写出
的所有可能值;
(2)若数列
中的项存在最大值,证明:0为数列
中的项;
(3)若
,是否存在正实数M,使得对任意的正整数n,都有
?如果存在,写出一个满足条件的M;如果不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25ba6d5fdf4c491c1332483be3cfab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d37f161c1dd788025cef9910858df7a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c03a27be8ae82e24b86cc52a92204c28.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c9b6e51986fe5d7a7265e0e93adcb4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daf464629fa321a6ff7401ab79f07083.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2a65d8762e567f485f39f81564b593a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5612ce06759d0f77ca029d10083f7d1e.png)
您最近一年使用:0次
2023-05-05更新
|
3798次组卷
|
19卷引用:北京市朝阳区2023届高三二模数学试题
北京市朝阳区2023届高三二模数学试题北京卷专题18数列(解答题)北京一零一中学2024届高三上学期统考一数学试题北京市景山学校2024届高三上学期10月月考数学试题(已下线)北京市第四中学2024届高三上学期10月月考数学试题(已下线)北京市第四中学2024届高三上学期10月月考数学试题变式题16-21北京市东城区东直门中学2024届高三上学期期中数学试题(已下线)专题01 条件开放型【练】【北京版】2024年全国普通高中九省联考仿真模拟数学试题(二)(已下线)高三数学开学摸底考02(新考法,新高考七省地区专用)(已下线)【一题多变】取大取小 分类讨论广东省2024届高三数学新改革适应性训练二(九省联考题型)(已下线)数列新定义北京市第二中学2023-2024学年高三下学期开学考试数学试卷(已下线)(新高考新结构)2024年高考数学模拟卷(二)上海市杨浦区复旦大学附属中学2024届高三下学期3月月考数学试题北京市顺义区第九中学2023-2024学年高三下学期3月月考数学试题广东省云浮市云安区云安中学2024届高三下学期3月模拟考试数学试题北京市海淀实验中学2024届高三上学期10月月考数学试题
名校
解题方法
8 . 球面几何学是在球表面上的几何学,也是非欧几何的一个例子.对于半径为R的球
,过球面上一点
作两条大圆的弧
,
,它们构成的图形叫做球面角,记作
(或
),其值为二面角
的大小,点
称为球面角的顶点,大圆弧
称为球面角的边.不在同一大圆上的三点
,可以得到经过这三点中任意两点的大圆的劣弧
,这三条劣弧组成的图形称为球面
,这三条劣弧称为球面
的边,
三点称为球面
的顶点;三个球面角
称为球面
的三个内角.
的单位球面上有不同在一个大圆上的三点
.
(1)球面
的三条边长相等(称为等边球面三角形),若
,求球面
的内角和;
(2)类比二面角,我们称从点
出发的三条射线
组成的图形为三面角,记为
.
其中点
称为三面角的顶点,
称为它的棱,
称为它的面角. 若三面角
的三个面角的余弦值分别为
.
(ⅰ)求球面
的三个内角的余弦值;
(ⅱ)求球面
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16d65cecaf8a3dc2953f4109c75a981e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/667349d99185bb045030b733352ff7fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdcdeb8eb2d5a8a7f1c81071ae349504.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f2dcc2105ebb1c89bfb1572a7e076e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a19c1bcb8431ae315ecd29c6478d3eff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ef528373d472534670a8fd7fb301492.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a1b98b1478ed9480a9e1a62ec3b82da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](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/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e037d52e5d75070cd02df4727b5922d.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/24e0c10fb103930eabd5fa18e8f9bb06.png)
(1)球面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814dc3914cdf4d5af2f4cfadf41c260.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(2)类比二面角,我们称从点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f673831a6738e1c317fede2920436d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5de938433cfaf25cb38dd5c9d915bb2b.png)
其中点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f673831a6738e1c317fede2920436d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c93fa5e252ef36adfbffa39410f2b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4278c0911e7df78965e78cff69cac5f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27029c4cc0fe55c8f4dbdb33beca9980.png)
(ⅰ)求球面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(ⅱ)求球面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
您最近一年使用:0次
名校
9 . 学习几何体结构素描是学习素描的重要一步.如图所示,这是一个用来练习几何体结构素描的石膏几何体,它是由一个圆柱
和一个正三棱锥
穿插而成的对称组合体.棱
和面
与圆柱侧而相切,点
是棱
与圆柱侧而的切点.直线
分别与面
,面
交于点
,圆柱
在面
,面
上分别截得椭圆
.在平面
和平面
中,椭圆
上分别有两组不重合的两点
和
(图中未画出).且满足关系
.已知三棱锥
的外接球表面积为
,圆柱的底面直径为
,请问平面
,平面
上是否分别存在点
,使得对于满足
的直线
分别恒过定点
.若存在,试求
和
夹角的余弦值:若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/270ddac9587bf1ea553914cb69595ab2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0628681907ac8d7fdb94d8bc1b15feb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/270ddac9587bf1ea553914cb69595ab2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d096cd7bd8a5a2219fd7dd166bbb8460.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/270ddac9587bf1ea553914cb69595ab2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13d726b14a313e000a81526162f35748.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0628681907ac8d7fdb94d8bc1b15feb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13d726b14a313e000a81526162f35748.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c121f8f04b49c7036888316b5f35e93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c121f8f04b49c7036888316b5f35e93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8461e08e60b84222ef056f49e71fa20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad044057e3a5b2f033d9ad0006b6f50e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b46bf815a5a193e9f5fc73078fb39d38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/357e0872d9e98d662a780e7686de86ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af67b85b1c91d1b489332503f6aaa8ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c37d0645be8436fa5c7a6ee8ba7de527.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/357e0872d9e98d662a780e7686de86ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cefbad5a2e9e1e812b54c5e972cf98d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1d1e25f3c8094bf9da6eabef95e8214.png)
![](https://img.xkw.com/dksih/QBM/2024/1/13/3410267677220864/3410816027746304/STEM/273256e47234495eaf4d1e8d41ba7a4a.png?resizew=183)
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10 . 高铁的建设为一个地区的经济发展提供了强大的推进力,也给人们的生活带来极大便捷.以下是2022年开工的雄商高铁线路上某个路段的示意图,其中线段
、
代表山坡,线段
为一段平地.设图中
坡的倾角满足
,
长
长
长
.假设该路段的高铁轨道是水平的(与
平行),且端点
分别与
在同一铅垂线上,每隔
需要建造一个桥墩(不考虑端点
建造桥墩)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/11/c74f6394-8bbb-41cf-9c85-b0607eb123fd.png?resizew=200)
(1)求需要建造的桥墩的个数;
(2)已知高铁轨道的高度为
,设计过程中每
放置一个桥墩,设桥墩高度为
(单位:
),单个桥墩的建造成本为
(单位:万元),求所有桥墩建造成本总和的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94e731b605eba332df44bea683c4aee5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d27db0d05bd138ad752b4cddc2e3bed6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8392f26db20adb474c5e7673036fe3d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d4ef601f9b9db4f7fb5f0b0dd2e0a61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/617b624dd6e80853ea8dda1b5591cdd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bed8f425894336757c6feb514a884fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1ae536809b1161fd4e83fdc7f42be96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b4ee159aea403220848b14b8e81dc20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1611d15c7e1e4ab4a4a61537b3989d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/11/c74f6394-8bbb-41cf-9c85-b0607eb123fd.png?resizew=200)
(1)求需要建造的桥墩的个数;
(2)已知高铁轨道的高度为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49ec72d07385ed62ec9540083958be2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1611d15c7e1e4ab4a4a61537b3989d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eabd5f3a86afe49dcd70571e2b96cfd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e15e00f40396e914d1d9955bd7785f1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1369d3503abf9bfcbdab0d6a4717017c.png)
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2023-03-06更新
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1403次组卷
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3卷引用:上海市2023届高三模拟数学试题