1 . 如图,在矩形ABCD中,DE平分∠ADC时,将△ABE沿AE折叠至△AFE,点F恰好落在DE上.
(1)求证:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2b34233772c4c26d6669499d9b1f15a.png)
(2)如图,延长CF交AE于点G,交AB于点H.
①求证:
;
②求
的值.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/1/459fe26e-22b8-421b-af21-cb1bfefaf4fc.png?resizew=378)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2b34233772c4c26d6669499d9b1f15a.png)
(2)如图,延长CF交AE于点G,交AB于点H.
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/245f8f7bc4a3e71d38fca7a19a7ab8d3.png)
②求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/624cf36a6102fe01057b25d72aa5462b.png)
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2 . 勾股定理,是几何学中一颗光彩夺目的明珠,被称为“几何学的基石”. 中国是发现和研究勾股定理最古老的国家之一. 据记载,在公元前1120年,商高答周公曰“故折矩,以为勾广三,股修四,径隅五,既方之,外半其一矩,环而共盘,得成三四五,两矩共长二十有五,是谓积矩. ”因此,勾股定理在中国又称“商高定理”. 数百年后,希腊数学家毕达哥拉斯发现并证明了这个定理,因此“勾股定理”在西方被称为“毕达哥拉斯定理”. 三国时期,吴国的数学家赵爽创制了一幅“勾股圆方图”,用数形结合的方法给出了勾股定理的详细证明. 如图所示的勾股圆方图中,四个全等的直角三角形与中间的小正方形拼成一个大正方形. 若中间小正方形面积(阴影部分)是大正方形面积一半,则直角三角形中较小的锐角
的大小为_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
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3 . 已知实数
,满足
,
.当
时,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7d9712c3b25f3030e166e136d3a4686.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52e1e0dd06a2d9d19da5383e257b4df0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/947f5c378ddee70618ad72e8ee4d1170.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d87401acf8df1dd6bf6780a0f3f0cb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f20407be10b42cf632ddef38b09cf6.png)
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4 . 若定义在
上的函数
满足:
的单调区间与
的单调区间完全相同,则称
为“二阶和谐函数”.
(1)求证:
是“二阶和谐函数”;
(2)若
是“二阶和谐函数”,求实数a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16e56e3224d58e0cfd256493f3fc63df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a903745cd2cb536443d07579b606ece5.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5491a30b301f7e0ce29db9e2c864b9bc.png)
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5 . 如图1,在正方形ABCD中,点E是CD上一动点,将正方形沿着BE折叠,点C落在点F处,连接BE,CF,延长CF交AD于点G
(1)求证:
;
(2)如图2,在已知条件下,延长BF交AD于点H.若
,
,求线段DE的长;
(3)将正方形改成矩形,点E是CD上一动点,同样沿着BE折叠,连接CF,延长CF,BF交直线AD于G,H两点,若
,
,求
的值(用含k的代数式表示).
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/8/95706dd2-78af-4151-b667-b9d67c953e80.png?resizew=488)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f3093ffb7f78608f593447157f13d0e.png)
(2)如图2,在已知条件下,延长BF交AD于点H.若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7166924a6936059fb904c039eba339b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/002767b8780fc8bf7aafc7beb2bae1e1.png)
(3)将正方形改成矩形,点E是CD上一动点,同样沿着BE折叠,连接CF,延长CF,BF交直线AD于G,H两点,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45ae51f42830225591fa7b9f26052b6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7166924a6936059fb904c039eba339b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a5731d679887ca3bbf45d476c5d16b4.png)
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6 . 如图①,在
中,
,两条直角边边长分别为
,b,斜边长为c,如图②,现将与
全等的四个直角三角形拼成一个正方形
.
(1)利用图②证明勾股定理即在
中证明:
;
(2)若
的两直角边之比为
,现随机向图②内掷一枚小针,则针尖落在四个直角三角形区域的概率是多少?
(3)如图③所示,过点D作DF的垂线交小正方形对角线EF的延长线于点G,连接CG,延长BE交CF于点M,交CG于点H,若
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1a3b94566843e92ce775ff6bdf386fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72a2dae4056db39071c7144f64e68f32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1a3b94566843e92ce775ff6bdf386fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/14/b5e0c17b-c03e-48f0-9c15-a2d436f5e8b9.png?resizew=488)
(1)利用图②证明勾股定理即在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1a3b94566843e92ce775ff6bdf386fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3133ef62aad6bdd6637140620f068fad.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1a3b94566843e92ce775ff6bdf386fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9583f85fd629aec99507236814acaf45.png)
(3)如图③所示,过点D作DF的垂线交小正方形对角线EF的延长线于点G,连接CG,延长BE交CF于点M,交CG于点H,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6ff22cc8e398040e9fae2aee87572ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f68c7b6efb2fec162dbc8425758dc03.png)
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解题方法
7 . 如图,圆锥的顶点为
,底面圆心为
为两条互相垂直的直径,
是底面圆周上的动点(异于
),且
在直径
的两侧.已知
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/21/f29a2a51-826c-42a6-bf5d-b8e480b904a5.png?resizew=212)
(1)若
,求证:
;
(2)若在线段
上存在点
(异于
),使得
平面
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fc5c19ac440746be97d8b46af5d288a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e58eed25e261cb5698f6471f084624cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2eacd9c1ce5e65fec29c32f40d86b73d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16e4e4c7a79d9d3cdb9ac5949d53e33e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0662d6db42144cbfbb137cd9301ca535.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db00726c76522557a59a557495f5a597.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/21/f29a2a51-826c-42a6-bf5d-b8e480b904a5.png?resizew=212)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efe856e706e882d49be9f47875967f75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f575d2b9f151e73a5c00c0f91b2d3239.png)
(2)若在线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67300207553ae70b997bde84ca730cf8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05b09e2d46f94b9ca3caf3f8283619c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13dc118b643fa00a546f0647f4ee3ea9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0628681907ac8d7fdb94d8bc1b15feb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/316ef5ac26be0e388d1f0c80519947de.png)
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2022-10-17更新
|
433次组卷
|
3卷引用:安徽省示范高中培优联盟2022-2023学年高二上学期秋季联赛数学试题
名校
解题方法
8 . 用文具盒中的两块直角三角板(
直角三角形和
直角三角形)绕着公共斜边翻折成
的二面角,如图
和
,
,
,
,
,将
翻折到
,使二面角
成
,
为边
上的点,且
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/9/fa55fa66-60f8-40e4-9104-56ad505dd9fa.png?resizew=351)
(1)证明:
;
(2)求直线
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79a97bb4dcfab4ec7539bc783d563c49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6b86c22b670a8e9f3896f9e8883fbbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6b86c22b670a8e9f3896f9e8883fbbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42798067d911f96e5784cb138319c907.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/035bb7882c582c2de36cae3d772ec63a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97fc2087ec10f3de2f253044992eac52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f89deb952f57f4b3fa4887b098b7b91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8cfd06965af6014208127f2880b476b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42798067d911f96e5784cb138319c907.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/719103f93166bab4828257608e641a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bd93fd4788cbe4ae8ea1c783633f127.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6b86c22b670a8e9f3896f9e8883fbbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/297f713ddbcc4578e73c8afe3a52abfa.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/9/fa55fa66-60f8-40e4-9104-56ad505dd9fa.png?resizew=351)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bf447e844152545c47a9f67fc3248c0.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a21897349d3d7c94419692106887153.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/719103f93166bab4828257608e641a9a.png)
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2022-09-07更新
|
635次组卷
|
2卷引用:安徽省淮南市第二中学2022-2023学年高二上学期10月月考数学试题
名校
9 . 观察下列关于自然数的等式:
(1)
①
(2)
②
(3)
③
根据上述规律解决下列问题:
(1)写出第4个等式:__________
__________;
(2)写出你猜想的第
个等式(用含
的式子表示),并证明其正确性.
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/676ea1b90c909d2cce4624bc05d991d5.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b093206b2c520d106b178b4fdcd0db13.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/397afc86d7b0034c4d040c5ffad35969.png)
根据上述规律解决下列问题:
(1)写出第4个等式:__________
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6706fe00b4e231e62d9ecbec567d526b.png)
(2)写出你猜想的第
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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解题方法
10 . 已知数列
和
,
且
,函数
,其中
.
(1)求函数
的单调区间;
(2)若数列
各项均为正整数,且对任意的
都有
.求证:
(ⅰ)
;
(ⅱ)
,其中
为自然对数的底数.
![](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/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ee03c109e4f64f3539de74ef30f06fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc65a38fceb3231eada88b96f0c63d14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b140e221ddf537b8964fff8557cca0.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e97769855336d73371930df1f187875e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0b0b716fda9b1efd9e47e2d80543f2d.png)
(ⅰ)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf779c918958c14824cd7d952a4bb4bc.png)
(ⅱ)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/462a81927ad910cd66ae9a5fd5813502.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/797bbd18359c9a29842b39109b3a0aac.png)
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2022-04-07更新
|
921次组卷
|
3卷引用:安徽省滁州市2022届高三下学期第二次教学质量检测理科数学试题