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1 . 给出集合
对任意
,都有
成立
.
(1)若
,求证:函数
;
(2)由于(1)中函数
既是周期函数又是偶函数,于是张同学猜想了两个结论:
命题甲:集合
中的元素都是周期为6的函数;
命题乙:集合
中的元素都是偶函数;
请对两个命题给出判断,如果正确,请证明;如果不正确,请举反例
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/305c81b6a05c983ef0dd04962d546bd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4166972dec0aa3e8694a44eeb941a08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33b005e1e4b8e41c0028cd464835c464.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4ca8bdc812627d925f00ed7c145d696.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71d6c8ce1327c39675b26deeb0cfa49c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5314a9d2205a2beba0dcffb8fd943b18.png)
(2)由于(1)中函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71d6c8ce1327c39675b26deeb0cfa49c.png)
命题甲:集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
命题乙:集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
请对两个命题给出判断,如果正确,请证明;如果不正确,请举反例
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2 . 证明:
(1)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd8fffe92548da698866a9888e57d472.png)
(2)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87c85bf2963df20faabb0e5b2f512e55.png)
(3)已知
,
,求证
.
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd8fffe92548da698866a9888e57d472.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87c85bf2963df20faabb0e5b2f512e55.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c0975b825228b2a91799eee6894987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57802c9ef29020d70c31d10d8c19125f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca26c7f4cf6a970f1f7c400ac2fc53ac.png)
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解题方法
3 . 如果![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e75cb7615ca33b128496114742a1f2dd.png)
(1)求证:
;
(2)若
为三角形的三个内角,判断
与
的大小关系,并予以证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e75cb7615ca33b128496114742a1f2dd.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3e10e76a8cf6e3eb92e57ee971a218a.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e263f0291e3df8b6fb866abaf3f4576.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbeb449ec91ac7e5798e3b347fd0d107.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89a25e473dce119ddd92f32fef1dc576.png)
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4 . (1)在用“五点法”作出函数
的大致图象的过程中,第一步需要将五个关键点列表,请完成下表:
(2)设实数
且
,求证:
;(可以使用公式:
)
(3)证明:等式
对任意实数
恒成立的充要条件是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/467a953b54798b6e2dcd6d76f8817938.png)
0 | |||||
0 | |||||
1 |
![](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/6d95727eed094e7ceb6663ee9d39bda3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/141ba74bc522b95958aea59cdc8c93d0.png)
(3)证明:等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c83576aaf57c7ebdcf56110fdbb0c12a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1d8ae1706a9ea5df3eca17eaa5c8b71.png)
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解题方法
5 . 已知数列
中
,关于
的函数
有唯一零点,记
.
(1)判断函数
的奇偶性并证明;
(2)求
;
(3)求证:
;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72f4d4fa1b049045d58a9571a0709004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/072010cccaa77474c07b66816ce4ae92.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72f4d4fa1b049045d58a9571a0709004.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f8c399c162dbd37d2aa304a4a3a1fd.png)
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解题方法
6 . (1)证明:若
,求证:
;
(2)已知
,
均为锐角,且满足
,
,求
值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d677e6f94d57c506bd007617c50a19e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99db5e19a19614908bee34c4ae100286.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c2c1ac87d07d6c7a1a62eb333d112d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8628325b9e41358aec8f97a50da7f27a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbfea1e888676a13ad69c72fba0405ea.png)
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7 . 已知函数
的定义域为D,若对任意的实数
,都有
成立(等号当且仅当
时成立),则称函数
是D上的凸函数,并且凸函数具有以下性质:对任意的实数
,都有
(
,
)成立(等号当且仅当
时成立).
(1)判断函数
、
是否为凸函数,并证明你的结论;
(2)若函数
是定义域为R的奇函数,证明:
不是R上的凸函数;
(3)求证:函数
是
上的凸函数,并求
的最大值(其中A、B、C是
的三个内角).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbc1bc250c8a6523a1be394ff48d4a51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39243f2c10a8291d75d65694b2dec94a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f333263260646c494225db8a7476c00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bd2179ba09ac27fce32baf170528ea6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddec2919fb0760a9b54440e581d6f7a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac69e6db1df13ed64756b4f391ae9fac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e167b43045b3297248e334c41c621b8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73223617c8855826298d435673787a94.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c57e815c01a412466a6aa12d3e883a77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/128d3060b444b2ee0ef61f2420c5109b.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
(3)求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2b9643da0c0fea4f099f9a9133d6076.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71163f419555f2ed76075c8ff659fbfc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a8080fef9bdfa92ae70f3e314eef3e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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8 . 三角比内容丰富,公式很多.若仔细观察、大胆猜想、科学求证,你也能发现其中的一些奥秘.现有如下两个恒等式:
(1)
;(2)
.
根据以上恒等式,请你猜想出一个一般性的结论并证明.
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cda84160c36ea80bab0cf253b271b16a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52105165d9a478ed469397902da1e3d9.png)
根据以上恒等式,请你猜想出一个一般性的结论并证明.
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解题方法
9 . 如图一:球面上的任意两个与球心不在同一条直线上的点和球心确定一个平面,该平面与球相交的图形称为球的大圆,任意两点都可以用大圆上的劣弧进行连接.过球面一点的两个大圆弧,分别在弧所在的两个半圆内作公共直径的垂线,两条垂线的夹角称为这两个弧的夹角.如图二:现给出球面上三个点,其任意两个不与球心共线,将它们两两用大圆上的劣弧连起来的封闭图形称为球面三角形.两点间的弧长定义为球面三角形的边长,两个弧的夹角定义为球面三角形的角.现设图二球面三角形
的三边长为
,
,
,三个角大小为
,
,
,球的半径为
.![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf538440bd45e5881f2b22994560ba7a.png)
(2)①求球面三角形
的面积
(用
,
,
,
表示).
②证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.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/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f435efcc7869eec21bdba1ed81dc3f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf538440bd45e5881f2b22994560ba7a.png)
(2)①求球面三角形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f435efcc7869eec21bdba1ed81dc3f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
②证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f314e3f1d6311f0476623d4e55484a3e.png)
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2023-04-21更新
|
386次组卷
|
4卷引用:浙江省A9协作体2022-2023学年高一下学期期中联考数学试题
浙江省A9协作体2022-2023学年高一下学期期中联考数学试题(已下线)13.3 空间图形的表面积和体积(分层练习)江苏省徐州市第一中学2022-2023学年高一下学期期中数学试题(已下线)11.1.5 旋转体-【帮课堂】(人教B版2019必修第四册)
解题方法
10 . 三角求值、证明
(1)已知
,
,求
的值.
(2)已知
,求
的值.
(3)求证:
.
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9a3fd3a067dce1354bc941df061508a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9a7de5b70003502e40b95b3b7d3d933.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7aee715ac87a76f7a00996af77481ed.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e5a67ff28e81a57b17fa65f1636916.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5918316afd3e8247dd1109237e992701.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bcadb740e82e3bfcf26107039756fe1.png)
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