名校
1 . 已知函数
.
(1)当
时,求曲线
在
处的切线方程;
(2)当
时,证明:
为单调递增函数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6687279e5f0000eb9d36582b8e1a1e63.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bea9227dd0104da58e0c40952cc87ed.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbe45993e6bd636a4f34886bb3d72f42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
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2 . 已知函数
,
.
(1)讨论
的单调性;
(2)若
有两个零点,求实数
的取值范围;
(3)若
对任意的
恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5eaa6d55cdb24cff59f22f8a09b27160.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/026d50aeb347823e800aa11442b80331.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/780876acd6f251de9b8510f4def91b5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2fb40a36a293471742ce75f6b9635b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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3卷引用:辽宁省辽阳市辽阳县辽阳石油化纤公司高级中学2024届高三下学期模拟考试数学试题
3 . 如图,已知菱形ABCD和菱形ADEF的边长均为2,
,
,M,N分别为AE、BD上的动点,且
.
平面EDC;
(2)当MN的长度最小时,求AF与平面MND所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b31285c747ac70d3ea0beac9b658d027.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/793e03b60fc5ac445d6d44c508e7e73e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b138770c405307ea4fc828624645e2c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edcf19a7f0dd0cdf59516ae585025110.png)
(2)当MN的长度最小时,求AF与平面MND所成角的正弦值.
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解题方法
4 . 如图,在四棱锥
中,底面
是直角梯形,
,
,
,
,M是
的中点
平面
;
(2)若
,求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f79863ffcfa63117ca6741b20a48e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4795ee1f96b430529934e2231b38885d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc5fbc998343c146358c29aa219afcb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0530f462e5ec1e58c46e1f7644d0cc21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4aa9084b8fe0fe05c4388d1f835587b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f35614aff055b98b76ca262f64e629d.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/364d6c88726d8c3bb8ed297057332bac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f35614aff055b98b76ca262f64e629d.png)
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解题方法
5 . 已知数列
中,
,设
为
前
项和,
,已知数列
,设
的前
项和
.
(1)求
;
(2)若
对任意
恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73af653d11c3d6c2673300a6622a5279.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7d3d55a85012933f91c5d8d27d8801d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f4f2c7992392f3e89a243767109c674.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/930bc56406e69b785b37a83d48e36724.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
6 . 已知数列
是等差数列,
是等比数列,且
,
,
,
.
(1)求
的通项公式:
(2)设
的前
项和为
,证明:
;
(3)设
,求数列
的前
项和.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/627e48c5ab76f5d1874c57a40d32d89e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6abbd310bfce5fc6df04add486e95070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89fe0f4e8a80a2840c0f6929a8a6351b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/144958d8f106c2a5384404a612c35a31.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0441e285d990afd18061376145503267.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d0b808acfe2c8d5dbc280b5b83efc28.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac07953530e3c248b3438fb200fb1661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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解题方法
7 . (1)已知数列
,其中
,且数列
为等比数列,求常数p;
(2)设
,
是公比不相等的两个等比数列,
,证明:数列
不是等比数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/196a7987e250ec273e4ec1614f53aebc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92d9d2a1b8240835f63bba14a00d6647.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d44ddab6e0c60119be69985ae7fa65b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
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8 . 意大利画家列奥纳多·达·芬奇曾提出:固定项链的两端,使其在重力的作用下自然下垂,项链所形成的曲线是什么?这就是著名的“悬链线问题”,后人给出了悬链线的函数表达式
,其中
为悬链线系数,
称为双曲余弦函数,其函数表达式
,相反地,双曲正弦函数的函数表达式为
.
(1)证明:①
;
②
.
(2)求不等式:
的解集.
(3)已知函数
存在三个零点,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65623d246ccde18e941c9bda7011ef65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71ff88c570435584c4df32454224c442.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0639494fc8cc7a048c7621f972eae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6a59c8dc71935b342d42cb4a54eed27.png)
(1)证明:①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fec3182982e6dcf905ea35d6b5be5f48.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe43cb3653c29dd797074b27780695a9.png)
(2)求不等式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/baf091e70e33483f99554568eb54a10a.png)
(3)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f307ed8ec3f398d3d3e445266396acdf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
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解题方法
9 . 已知函数
,其中
.
(1)直接写出
的单调区间;
(2)若当
时,
恒成立,求
的取值范围;
(3)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3724946c15f9ac4d229dc1c650d5eda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5f7f23e7f20dd8bc65a4967cd306782.png)
(1)直接写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(2)若当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afdf7c8f771c4605d81e519e13c6dc87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b9d149be7bce4fad26fb372bbd025ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/395eb4e94fb286dc737631260ecde98a.png)
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10 . 古希腊数学家托勒密对凸四边形(凸四边形是指没有角度大于180°的四边形)进行研究,终于有重大发现:任意一凸四边形,两组对边的乘积之和不小于两条对角线的乘积,当且仅当四点共圆时等号成立.且若给定凸四边形的四条边长,四点共圆时四边形的面积最大.根据上述材料 ,解决以下问题,如图,在凸四边形
中,
,
,
,
(图1),求线段
长度的最大值;
(2)若
,
,
(图2),求四边形
面积取得最大值时角
的大小,并求出四边形
面积的最大值;
(3)在满足(2)条件下,若点
是
外接圆上异于
的点,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21ea52361458ce2e49ed0fe99d8e6c02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa7aeb2a8d1437eeb4482c3b6ad9f315.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/422f54faa21cdabc65b912b0e76eb68e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/212bfbd5575772ca36d6fc3e7b246e49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07160f14b3b453bebb64cb2bf96dc85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89c41757ae282475fb29ec1e8e02045d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(3)在满足(2)条件下,若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbdb21011ea821b91d539cb763aac649.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b94fd6403a7f18702993f80e29bfe1.png)
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3卷引用:辽宁省协作校2023-2024学年高一下学期5月期中考试数学试题
辽宁省协作校2023-2024学年高一下学期5月期中考试数学试题安徽省阜阳市第三中学2023-2024学年高一下学期第二次调研(期中)数学试题(已下线)第9题 解三角形在几何图形中的应用(高一期末每日一题)