名校
1 . 固定项链的两端,在重力的作用下项链所形成的曲线是悬链线,1691年,莱布尼茨等得出“悬链线”方程
,其中
为参数.当
时,就是双曲余弦函数
,类似的我们可以定义双曲正弦函数
.它们与正、余弦函数有许多类似的性质.
(1)求证:
;
(2)对
,不等式
恒成立,求实数
的取值范围;
(3)若
,试比较
与
的大小关系,并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08144630f70f5bba0c73252569d97841.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4580cc037c0c760c728cdbb74a8154c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d848439a448faa1d4cd9fa20ca206215.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe0a24e9c7616bf8afac5a0ffb0aa1fb.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bdd3fb4f930b309f261929ba7a1f055.png)
(2)对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fe9f3099ed9429dc5b4e38a350e524a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24464f0ea26038d85cc22a1786257605.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a83c8948a168ff2c567aee048cabff9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a2cebaab3423dfb2f2c944dfc43df8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb966b7b2dd6581640bcee2d97dacf77.png)
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2 . 设
是定义在区间
上的函数,如果对任意的
,有
,则称
为区间
上的下凸函数;如果有
,则称
为区间
上的上凸函数.
(1)已知函数
,求证:
(ⅰ)
;
(ⅱ)函数
为下凸函数;
(2)已知函数
,其中实数
,且函数
在区间
内为上凸函数,求实数
的取值范围.
![](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/dbc1bc250c8a6523a1be394ff48d4a51.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/8455657dde27aabe6adb7b188e031c11.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/8455657dde27aabe6adb7b188e031c11.png)
(1)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7128f99cbbab0279aa548f03d400f20d.png)
(ⅰ)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9042d7f774a2d79b2fc4f410ced2b10.png)
(ⅱ)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7128f99cbbab0279aa548f03d400f20d.png)
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bff01d42e61c8adeac0615b4b33db5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7160d93f92089ef36f3dab809d3114b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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解题方法
3 . (1)求值:
.
(2)在非直角
中,求证:
;
(3)高斯是德国著名的数学家,近代数学的奠基人之一,享有数学“王子”的称号,他和阿基米德、牛顿并列为世界的三大数学家,用其名字命名的“高斯函数”为:设
,符号
表示不大于x的最大整数,则
称为“高斯函数”,例如
,
,
.在非直角
中,角A、B、C满足
,若
,试求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/127c94c6a31959c2271cd7f716076961.png)
(2)在非直角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e35270d268704ef49b5e206d7df8d61f.png)
(3)高斯是德国著名的数学家,近代数学的奠基人之一,享有数学“王子”的称号,他和阿基米德、牛顿并列为世界的三大数学家,用其名字命名的“高斯函数”为:设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7179c645736d68c90023f83d7f11ed01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/797715acd30d07aabbed52bd10b234e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/447edcfb531a10755c19709915f0376e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1656bbf55c56dfccabcc5d025fa28ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bbc49013b6496bac591b07c6336cb98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7dc63dac12b3dc8fea7623e82d7eb50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10e8fbc147d6555a34240af94cc0a1ee.png)
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解题方法
4 . 在四棱锥
中,已知![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/638537c0a30676c73fea76c80e0f8bd0.png)
,
是线段
上的点.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/9/b2b11e44-dc79-423f-b312-152974c5961e.png?resizew=158)
(1)求证:
底面
;
(2)是否存在点
使得三棱锥
的体积为
?若存在,求出
的值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/638537c0a30676c73fea76c80e0f8bd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd7a66e971ec041fbb0b09318df77f30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd3d71dcafa623cc5a69ae60a4735286.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/9/b2b11e44-dc79-423f-b312-152974c5961e.png?resizew=158)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5f1897a7e856b42f8cee0f286ad913d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)是否存在点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45d492a2248463e0c0199a25d0f76d23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d86ab7c97cd8a0b15ba5efc1be94230.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fb4402e082c123111c12fc6cc3acbc9.png)
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解题方法
5 . 已知函数
.
(1)当
时,直接写出
的单调区间(不要求证明),并求出
的值域;
(2)设函数
,若对任意
,总有
,使得
,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14b2d3738f56987d159a343dc160f384.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cbeede118c407a800b05757b9a1393e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fdabdbbbde9b3ee68df66171b0145785.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3d5a5e70f64f0933ae1e4ddec5fa2c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61761abb364ece2281af24d9b1f008de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e63bbadc6250f7139836ede33205550.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
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2024-03-07更新
|
508次组卷
|
11卷引用:四川省德阳市德阳中学校2023-2024学年高一下学期入学考试数学试卷
四川省德阳市德阳中学校2023-2024学年高一下学期入学考试数学试卷(已下线)专题17 三角值域问题安徽省合肥市一中、六中、八中三校2020-2021学年高一上学期期末数学试题安徽省合肥一中、六中、八中2020-2021学年高一上学期期末联考数学试题安徽省淮南市寿县第一中学2020-2021学年高一下学期入学考试数学试题安徽省淮北市树人高级中学2020-2021学年高一下学期开学考试数学试题(已下线)大题好拿分期中考前必做30题(压轴版)-2020-2021学年高一数学下册期中期末考试高分直通车(沪教版2020必修第二册)(已下线)第7章 三角函数 单元测试(单元综合检测)(难点)(单元培优)-2021-2022学年高一数学课后培优练(苏教版2019必修第一册)(已下线)7.3 三角函数的图像和性质(难点)(课堂培优)-2021-2022学年高一数学课后培优练(苏教版2019必修第一册)山东省淄博市美达菲双语高级中学2022-2023学年高一下学期3月月考数学试题湖南省株洲市第二中学2022届高三下学期期中数学试题
6 . 如图,在四棱台
中,
,
,
.
平面
;
(2)若
,四棱台
的体积为
,求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34e0a957a55460c72673c0f2ee90dbb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/feefd792abfb990702d3ef1c8baec6c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c77cb2f11c66a269bbd9d63e4bb6d1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd14966183389b10618cbe33fd777407.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/901567f610a4e89005799f11e347166e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7e5332f18038cc811f7fff449c2d99d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82b724168afaee2ecddf97257180be18.png)
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7日内更新
|
53次组卷
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2卷引用:四川省绵阳市东辰学校2024届高三下学期模拟押题卷理科数学试题(一)
7 . 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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7日内更新
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266次组卷
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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
名校
解题方法
8 . 在锐角
中,内角A,B,C所对的边分别为a,b,c,满足
.
(1)求证:
;
(2)若
,求a边的范围;
(3)求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c8e5ce6c55a720a332a08c07f1a89a1.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a2264c134952d41fb9bcb90e6c72c83.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a3c442579603164f3fc19458677d307.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9563e5c29f03707996eb761fba29ce21.png)
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解题方法
9 . 已知函数
.
(1)求函数
的单调递增区间;
(2)在
中,a,b,c分别是角A、B、C所对的边,记
的面积为S,从下面①②③中选取两个作为条件,证明另外一个成立.
①
;②
;③
.
注:若选择不同的组合分别解答,则按第一个解答计分.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b40af7c8a04c3318d5fc0ad7d06a9a6.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d3c28e2fc237e76e757b8a82c619802.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96da42826c1216bd581e822a807d39af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b19988d792b5252222f0a7acf7ef5fd.png)
注:若选择不同的组合分别解答,则按第一个解答计分.
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10 . 已知函数
.
(1)求函数
的单调递增区间;
(2)在
中,a,b,c分别是角A、B、C所对的边,记
的面积为S,若
,
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23ec6a00564735fcf0642c001c14d600.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)在
![](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/8a0d5cf8c22d0cf93274939d92963665.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96da42826c1216bd581e822a807d39af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b19988d792b5252222f0a7acf7ef5fd.png)
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