1 . 已知函数
.
(1)求函数
的单调递增区间;
(2)若函数
有且仅有三个零点,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a944f056d09785e9e3e859601bee6ae.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6af861abcb2667803bb5985c0ae55a7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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2 . 已知函数
.
(1)求
的最小值;
(2)设函数
,讨论
零点的个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ffb296c54f3ac746d3a0c503f888126.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cffa662f0273f0921c1fa4727f632395.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.png)
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2卷引用:2024届湖南省长沙市第一中学高考最后一卷数学试题
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3 . 若平面直角坐标系内
两点满足: (1)点
都在
的图象上; (2)点
关于原点对称,则称点对
是函数
的一个“姊妹点对”,且点对
与
记为一个“姊妹点对”. 已知函数
,则
的“姊妹点对”有__________ 个.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41be4f4fbf21555f325caf280c392c00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41be4f4fbf21555f325caf280c392c00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7113de81db260e6666292f39b447b848.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/319365a080c254009c27fc59d690bc7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
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4 . 已知函数
.
(1)判断并证明
的零点个数
(2)记
在
上的零点为
,求证;
(i)
是一个递减数列
(ii)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58ae99833cb675e5e36c58f345eb03e7.png)
(1)判断并证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d64af919a56a107e0fc0a417e481648.png)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d64af919a56a107e0fc0a417e481648.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/870ebc2f7aabb028024894568d749934.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
(i)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1fd18a909cecbaee7115d6b15631d83.png)
(ii)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/167f4f8fd4d3de714b87f05e57a3ba3b.png)
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5 . 已知函数
.
(1)求
的单调区间;
(2)证明:
;
(3)若函数
有三个不同的零点,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75b791a36d153e5023d1a8bbaa7e625b.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dee430ad7c01ddaa606363af678ae70.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74ad33aefccaa62d263f45ea411c699a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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6 . 若函数
恰好有四个零点,则实数
的取值范围是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5c19c25d49326bcdf412233a526318a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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7 . 已知
的三个角
的对边分别为
且
,点
在边
上,
是
的角平分线,设
(其中
为正实数).
(1)求实数
的取值范围;
(2)设函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/736d82fce28c323d02a4183610777845.png)
①当
时,求函数
的极小值;
②设
是
的最大零点,试比较
与1的大小.
![](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/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe29c42302504e7fd8577dbc7d130ac7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a9ecf347b1b8b1edd8f354a0fc1f152.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(1)求实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(2)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/736d82fce28c323d02a4183610777845.png)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f627a70fa1006b30c2db5b1fcfaae82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
②设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
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|
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湖南省岳阳市2024届高三教学质量监测(三)数学试题(已下线)模块5 三模重组卷 第1套 全真模拟卷(已下线)压轴题07三角函数与正余弦定理压轴题9题型汇总-2浙江省东阳市外国语学校2023-2024学年高二下学期5月月考数学试题
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8 . 已知函数
.
(1)是否存在实数
,使得
和
在
上的单调区间相同?若存在,求出
的取值范围;若不存在,请说明理由.
(2)已知
是
的零点,
是
的零点.
①证明:
,
②证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1397a4cb36ba5e0176b45213b6083314.png)
(1)是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.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/d562dc22dfb3b81d0c3f88b54d063c2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16b5f36cbdb64b34f98763993dc0e972.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
①证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e45981620389be345fa37839336b33b7.png)
②证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a6d676125daa80de10a38c4825aee9e.png)
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2024·全国·模拟预测
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9 . 已知函数
,
.
(1)若
存在零点,求a的取值范围;
(2)若
,
为
的零点,且
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e7009deb27743a0e51075f457881d5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26d8dafc71b106f39f4e15442220897b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/185472116a4292fdf2dddcf6402ee2ca.png)
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10 . 已知函数
.
(1)求曲线
在
处的切线方程;
(2)若
,讨论曲线
与曲线
的交点个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ae2ff77b0364d1d446008e48ab51402.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68c6b6a11760d0724b0b60e55970e229.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7e6ace7ba273101018ae81be08c9024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/004a9c3c84ff65384fa0b75b9ec2a2cd.png)
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