名校
1 . 已知在任意一个三角形的三条边上分别向外做出三个等边三角形,则这三个等边三角形的中心也构成一个等边三角形;我们称由这三个等边三角形中心构成的三角形为其外拿破仑三角形.在锐角
中,角
所对的边分别为
且
,以
的边
分别向外作的三个等边三角形的中心分别记为
,且
的面积为
,记
为
的外接圆半径.
,求
;
(2)若
,求
面积的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38335830b93ac4d99c28a8e209eecb3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9f054a8f158340bc788136124632325.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39900a2d790732077ffb571427a134fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b372e638a9dbed3bbf92454d816d212.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2a5a1f585c19f582edbb8cf73c58c6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4310db23fc79936c7182361e652bab1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c0f2a532c86e3f4a1cd508839363b05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d46be1fb12ca82fdef75149d22bc1c94.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8843442efe0889ebbd393484ff8596cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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2 . 由于四边形不具有稳定性,所以求四边形面积公式需要有限制条件.我们将四个点在圆上的四边形称为圆内接四边形,圆内接四边形具有对角互补的性质.印度数学家婆罗摩笈多发现了圆内接四边形的面积公式为
,其中
、
、
、
分别为圆内接四边形的4条边,
,与海伦公式有类似之处.已知在圆内接四边形
中,
,
,
,
,则四边形
的面积为___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26f9a99467e1c9715852266155be6a9d.png)
![](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/575008e0b065f0d535251a041203f99f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efc6e4b936d7a800e839a30c3839574d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65a3e478bb87d094e3a0af30dd10ae8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b515965c22d2950b592c096c6e3bdfd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bcffaa7a79cedadb925149e28e39a43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
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3 . 成都天府绿道专为骑行而建,以绿道为线,串联上百个生态公园,一路上树木成荫、鸟语花香,目前已然成为成都新的城市名片.成都市政府为升级绿道沿途风景,计划在某段全长200米的直线绿道
一侧规划一个三角形区域
做绿化,如图,已知
,为提升美观度,设计师拟将绿化区设计为一个锐角三角形.
米,求
的长;
(2)求绿化区域
面积的取值范围;
(3)绿化完成后,某游客在绿道
的另一侧空地上寻找最佳拍照打卡点,该游客从A到
,再从
到B,然后从
到
,最终返回
点拍照.已知
,求游客所走路程的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b4f5cdf18c053283a4122005414b825.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03f4b1ad684fc71ca2641f68f960361a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
(2)求绿化区域
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(3)绿化完成后,某游客在绿道
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.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/7f9e8449aad35c5d840a3395ea86df6d.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/7d31f25ee405fc6c411115b0f40ff9e1.png)
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4 . 已知数列
的通项公式为
,
,在
中依次选取若干项(至少3项)
,
,
,
,
,
,使
成为一个等比数列,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f48873417a7667f628e5c7052c96f416.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae8ebcd5b8f8ae04513834a70575816f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e3aaabba90eefb861068bf014f9d0b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d745f422af19d28303b7fbc3a08a9ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37e5531913e2f170465d8df01795cd51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b625d02d9054af978b5fb6ec9bee787.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37e5531913e2f170465d8df01795cd51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ec07a126ada2c921c5b4337f77854cf.png)
A.若取![]() ![]() ![]() |
B.满足题意的![]() |
C.在![]() ![]() |
D.如果把![]() ![]() |
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解题方法
5 .
年
月
日,雅万高铁正式开通运营,标志着印度尼西亚迈入高铁时代,中国印度尼西亚共建“一带一路”取得重大标志性成果.中国高铁正在成为共建“一带一路”和国际产能合作的重要项目.国内某车辆厂决定从传统型、智能型两种型号的高铁列车车厢中选择一种进行投资生产.已知投资生产这两种型号车厢的有关数据如下表(单位:百万元)
已知
,每销售
节智能型车厢时,需上交
百万元用于当地基础建设.假设生产的车厢当年都能销售完.
(1)设
、
分别为该厂投资传统型和智能型两种型号车厢的年利润,分别求出
、
与年产量
之间的函数关系式;
(2)①分别求出生产两种型号车厢的平均利润的最大值;
②要使生产两种型号车厢的平均利润最大,该厂应该选择生产哪种型号车厢?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e946baf1316ac1f219398ecedadf6cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d07ae0b4264da6a8812454ffd2f20d94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/feda926749de04fa585f73f84c568f0c.png)
年固定成本 | 每节车厢成本 | 每节车厢价格 | 每年最多生产的节数 | |
传统型 | ![]() | ![]() | ![]() | ![]() |
智能型 | ![]() | ![]() | ![]() | ![]() |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc3e8da4c8513ae60f5116baa70f35f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a7a1d0792285a953c9341734564a1c8.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54015ff5b49e3283901da1291b6b921d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46f6872ffb1934339c53c2c2282d5889.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54015ff5b49e3283901da1291b6b921d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46f6872ffb1934339c53c2c2282d5889.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)①分别求出生产两种型号车厢的平均利润的最大值;
②要使生产两种型号车厢的平均利润最大,该厂应该选择生产哪种型号车厢?
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2024-03-07更新
|
372次组卷
|
2卷引用:湖北省武汉市华中师范大学第一附属中学2023-2024学年高一上学期1月期末检测数学试题
解题方法
6 . 某种生命体M在生长一天后会分裂成2个生命体M和1个生命体N,1个生命体N生长一天后可以分裂成2个生命体N和1个生命体M,每个新生命体都可以持续生长并发生分裂.假设从某个生命体M的生长开始计算,记
表示第n天生命体M的个数,
表示第n天生命体N的个数,则
,
,则下列结论中正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87ea014220aa658c8baa6e1f43e686a2.png)
A.![]() | B.数列![]() |
C.![]() | D.若![]() ![]() |
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7 . 如图,以等腰直角三角形
的直角边
为斜边,在
外侧作等腰直角三角形
,以边
的中点
为圆心,作一个圆心角是
的圆弧
;再以等腰直角三角形
的直角边
为斜边,在
外侧作等腰直角三角形
,以边
的中点
为圆心,作一个圆心角是
的圆弧
;
;按此规律操作,直至得到的直角三角形
的直角顶点
首次落到线段
上,作出相应的圆弧后结束.若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51b1e185d6a0ab350cdc947beeb82040.png)
__________ ,所有圆弧的总长度为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27d507b1aa980ec0a7bfcaecf267a2e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b7d857811cbd619f868d951aa7a0ab8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a42f1bd1ae451df237cbced30e0407b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5973afa9369da0f72e137bd85c8774be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/115ffa9d5103e55ae94fd1bd374fb588.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2ff7caec4fdd8fb54a3ffbff9692414.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7926d00c6bb116891abd4de1dff2e7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5973afa9369da0f72e137bd85c8774be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57831b59ea828a088ed9daec8babb774.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73314082aec86935318f5efc466b1dce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66bfd7725d6dfe43da8bb92cded1d92c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b7d857811cbd619f868d951aa7a0ab8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c4f6f74444b2b7947fc6e35c8d62322.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2ff7caec4fdd8fb54a3ffbff9692414.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/473913c0887bb64d386f4c02f1853452.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c97922b30f361bae90bf20d7f4c6628.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e819b87f90651d89fcd258c276294e43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/115ffa9d5103e55ae94fd1bd374fb588.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a4edfbe9729909b23dc7fde2135a606.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51b1e185d6a0ab350cdc947beeb82040.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/22/99770be8-f330-4962-ad55-f5de34737927.png?resizew=246)
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8 . 某企业2023年9~11月份生产的产品产量
(单位:千件)与收益
(单位:万元)的统计数据如下表:
(1)根据上表数据,从下列三个函数模型①
,②
,③
(
且
)中选取一个恰当的函数模型描述该企业2023年9~11月份生产的产品产量
(单位:千件)与收益
(单位:万元)之间的关系,并写出这个函数关系式;
(2)问该企业12月份生产的产品产量
应控制在什么范围内,才能使该企业12月份的收益在4950万元以上(含4950万元)?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/032ed803966d385c48539f9a61465f75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
月份 | 9月 | 10月 | 11月 |
产品产母![]() | 30 | 40 | 80 |
收益![]() | 4200 | 4800 | 3200 |
(1)根据上表数据,从下列三个函数模型①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89c1972246a0a3d1c987d25205dbdd99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fab8c2e00bef63a0feb061ea85f8f88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/109de1a139e125d3a54ada44c4300efd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c400a615a16a1662de98dfb4e49d58d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
(2)问该企业12月份生产的产品产量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/032ed803966d385c48539f9a61465f75.png)
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解题方法
9 . “天眼”探空、神舟飞天、高铁奔驰、北斗组网等,我国创造了一个又一个科技工程奇迹.为了顺应我国科技发展战略,某高科技公司决定启动一项高科技项目,启动资金为2000亿元,为保持每年可获利20%,每年年底需从利润中取出200亿元作为研发经费.设经过n年之后,该项目资金为
亿元.
(1)写出
的值,并求出数列
的通项公式.
(2)求至少要经过多少年,该项目的资金才可以达到或超过翻一番(即为原来的2倍)的目标.(取
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(1)写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(2)求至少要经过多少年,该项目的资金才可以达到或超过翻一番(即为原来的2倍)的目标.(取
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f96ad2a5315cc4dd120bb806f0981bc0.png)
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10 . 如图所示,线段
为半圆的直径,
为圆心,
为半圆弧上不与
重合的点,
.作
于
于
,设
,则下列不等式中可以直接表示
的是( )
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/21/4ee89e17-218f-4431-a68e-9e65cb874101.png?resizew=172)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea9aa3f533894ca87a394f7a1dfd0ea3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ebef5bab02280cdc99cc7f689135cd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b757f0c42ae5c9a2d6a4b19e5877b27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcd8be8efc6841d26ad2df41a69111e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cd2ff4dbbafc93088d7f58af9b01a23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45b71e97f78a3d1383908276f857b850.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/21/4ee89e17-218f-4431-a68e-9e65cb874101.png?resizew=172)
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