解题方法
1 . 假设在某种细菌培养过程中,正常细菌每小时分裂1次(1个正常细菌分裂成2个正常细菌和1个非正常细菌),非正常细菌每小时分裂1次(1个非正常细菌分裂成2个非正常细菌).若1个正常细菌经过14小时的培养,则可分裂成的细菌的个数为______ .
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4卷引用:内蒙古名校联盟2024届高三下学期联合质量检测文科数学试题
2 . 北京天安门广场中心屹立着一座中国最大的纪念碑——人民英雄纪念碑,它专门为缅怀近现代英雄而建,它不仅仅是一个简单的建筑,更是民族精神的象征.某学生为测量该纪念碑的高度
,选取与碑基
在同一水平面内的两个测量点
.现测得
米,在点
处测得碑顶
的仰角为
,则纪念碑高
为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/885db9da8b713067ff7afe37b6e7f074.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/8ac09dc1ca2cdd7aef28c218763d3e4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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3 . 蚊香具有悠久的历史,我国蚊香的发明与古人端午节的习俗有关,如图为某校数学社团用数学软件制作的“蚊香”.画法如下:在水平直线上收长度为1的线段
,作一个等边三角形
,然后以点
为圆心,
为半径逆时针画圆弧交线段
的延长线于点
(第一段圆弧),再以点
为圆心,
为半径逆时针画圆弧交线段
的延长线于点
,再以点
为圆心,
为半径逆时针画圆弧……以此类推,当得到的“蚊香”恰好有15段圆弧时,“蚊香”的长度为_____________ .
![](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/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
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4 . 下表中的数阵为“森德拉姆筛”,其特点是每行每列都成等差数列
表中对角线上的一列数2,5.10,17,26,37,…构成数列
,则
( )
2 | 3 | 4 | 5 | 6 | 7 | … |
3 | 5 | 7 | 9 | 11 | 13 | … |
4 | 7 | 10 | 18 | 16 | 19 | … |
5 | 9 | 13 | 17 | 21 | 25 | … |
6 | 11 | 16 | 21 | 26 | 31 | … |
7 | 13 | 19 | 25 | 31 | 37 | … |
… | … | … | … | … | …… |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bbc12ecb6d1d18f4a7ae777bde43d27.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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5 . “杨辉三角”是二项式系数在三角形中的一种几何排列.从第1层开始,第
层从左到右的数字之和记为
,如
,
,…,则
的前9项和![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc4e70b360f988fdbd92300ab22c4613.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e7df0430db8db9fc354ffdd038fb432.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c996a43ff8843aec0be0a9d0ac0e9ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc4e70b360f988fdbd92300ab22c4613.png)
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名校
解题方法
6 . “费马点”是由十七世纪法国数学家费马提出并征解的一个问题.该问题是:“在一个三角形内求作一点,使其与此三角形的三个顶点的距离之和最小.”意大利数学家托里拆利给出了解答,当
的三个内角均小于
时,使得
的点
即为费马点;当
有一个内角大于或等于
时,最大内角的顶点为费马点.
在
中,内角
,
,
的对边分别为
,
,
.
(1)若
.
①求
;
②若
的面积为
,设点
为
的费马点,求
的取值范围;
(2)若
内一点
满足
,且
平分
,试问是否存在常实数
,使得
,若存在,求出常数
;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1eab88a16df610f20dd46a44ba098d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.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)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b7f7180b86108862c7aa44c950f872a.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab7aaa871ceb78e5b80b531a7cf4f1c9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec15e5cb6d4dc2cf6ba0bedd87514448.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39b8d91afc34e4a9b0fdbb6bafb9087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca347a0ea5e4d813a81407796be5fea7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
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7 . 敬亭山,位于安徽省宣城市北郊,是中国历史文化名山,原名昭亭山,晋初为避帝讳,易名敬亭山.李白在《独坐敬亭山》中写道:众鸟高飞尽,孤云独去闲.相看两不厌,只有敬亭山.相传该诗题写于太白独坐楼(如图(1)).为了测量该楼的高度
(如图(2)),一研究小组选取了与该楼底部
在同一水平面内的两个测量基点
与
,现测得
,
,在
点处测得该楼顶端
的仰角为
则该楼的高度
为_______ m.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c20dce79366a9249f852cdd6b45d836.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/278183c10de47de6e0a8e7f249d2a035.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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8 . 对于等差数列和等比数列,我国古代很早就有研究成果,北宋大科学家沈括在《梦溪笔谈》中首创的“隙积术”,就是关于高阶等差级数求和的问题.现有一货物堆,从上向下查,第一层有2个货物,第二层比第一层多3个,第三层比第二层多4个,以此类推,记第
层货物的个数为
,则数列
的前10项和![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5af1f1b7e9e20d799ee3c06b89a0611c.png)
_________________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/241515dbec4be59ea1099bb33e3aa26f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5af1f1b7e9e20d799ee3c06b89a0611c.png)
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9 . 我国汉代数学家赵爽为了证明勾股定理,创造了一幅“勾股圆方图”,后人称其为“赵爽弦图”.类比赵爽弦图,用3个全等的小三角形拼成了如图所示的等边
,若
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a71a9d21f77e9535de152bb33f802bb.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/221a091e823526ce02a78be01068c01d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c0ba1776a7c0bac5141407836e12153.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a71a9d21f77e9535de152bb33f802bb.png)
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2卷引用:四川成华区某校2023-2024学年高一下学期期中考试数学试题
10 . 古希腊数学家托勒密对三角学的发展做出了重要贡献,他的《天文学大成》包含一张弦表(即不同圆心角的弦长表),这张表本质上相当于正弦三角函数表.托勒密把圆的半径60等分,用圆的半径长的
作为单位来度量弦长.将圆心角
所对的弦长记为
.如图,在圆
中,
的圆心角所对的弦长恰好等于圆
的半径,因此
的圆心角所对的弦长为60个单位,即![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af62753905ae48459d5543255f1d951c.png)
.若
为圆心角,
,则
.______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae3e21b4e394a3a01fb51a67fafaccb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a3504172c63a1ae2481c130c426e30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af62753905ae48459d5543255f1d951c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8699ac473ac35962c9756a7891bcfd3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bbe344a129538a507ab3047227239a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d4bf37b295de4fc2cc16c00027f8245.png)
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