名校
1 . 用正方形按如图所示的规律拼图案,其中第①个图案中有5个正方形,第②个图案中有9个正方形,第③个图案中有13个正方形,第④个图案中有17个正方形,此规律排列下去,则第⑨个图案中正方形的个数为( )
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/21/69f672da-bbeb-4d7d-92fe-7ee9aeff3878.png?resizew=582)
A.32 | B.34 | C.37 | D.41 |
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2 . 两位数
和两位数
,它们各个数位上的数字都不为0,将数
和数
的个位数字与十位数字交叉相乘再求和所得的结果记为
.例如:
.又如:
.则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3dd8061aa6b0803d371f9d418b6f0dd.png)
____________ ;若一个两位数
,两位数
(
,
,且
,
都取整数),交换
的十位数字和个位数字得到新两位数
,当
与
的个位数字的5倍的和能被11整除时,称这样的两个数
和
为“快乐数对”,则所有“快乐数对”
的最大值为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4b40e34f3f8e31cd6882fbceed3f72f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27f365c6a314b26500b860f145810e47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ec4f99213b50854169318c1002993c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3dd8061aa6b0803d371f9d418b6f0dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8676003f043a1220213c8fa169a7e71c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c25fa0a6b9c3be012cb5b6908b155eb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/200f9560f8f90d4addcf5fb6fdcc7956.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df9807e2182a8ef6ded07aef5ca0d6d6.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/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7093538ecfb10a639b23863e7331a66d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7093538ecfb10a639b23863e7331a66d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4b40e34f3f8e31cd6882fbceed3f72f.png)
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2023-09-14更新
|
45次组卷
|
2卷引用:重庆市七校2023-2024学年高一上学期开学联考数学试题
3 . 如图都是由三角形按一定规律组成的,其中第①个图形共有3个顶点,第②个图形共有6个顶点,第③图形共有10个顶点,…,按此规律排列下去,第⑥个图形顶点的个数为( )
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/12/74b406b9-7dd0-490e-a5cd-22485f332573.png?resizew=227)
A.21 | B.28 | C.36 | D.45 |
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4 . 如图,将一个边长为
的正三角形的每条边三等分,以中间一段为边向外作正三角形,并擦去中间这一段,如此继续下去得到的曲线称为科克雪花曲线.将下面的图形依次记作![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b4af5e2031dfa48ec4b9d0fcdc7a98f.png)
(1)求
的周长;
(2)求
所围成的面积;
(3)当
时,计算周长和面积的极限,说明科克雪花曲线所围成的图形是“边长”无限增大而面积却有极限的图形.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b4af5e2031dfa48ec4b9d0fcdc7a98f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/14/0fa60d42-f0cd-4eba-8dfd-6a3181ea0799.png?resizew=313)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a107eb946e0fe41629c644b7628d5cba.png)
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23-24高二上·上海·课后作业
5 . 是否存在常数
,使等式
对任意正整数
都成立?证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e74f7d13fcf37d55852f588ef4389be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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23-24高二上·上海·课后作业
6 . (1)依次计算下列各式的值:
,
,
,
.
(2)根据第(1)题的计算结果,猜想
(
为正整数)的表达式,并用数学归纳法证明相应的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5296c0056db0e2b5331c9b9a6d45962.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6898c8129d4cdb8ae988854f553723c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61c1db1861753db975385da6ec85f743.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9af7498b656097fe51a3d5ff044dd48.png)
(2)根据第(1)题的计算结果,猜想
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e30da1db1ed26dec3eacb75d5471f66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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7 . 如图,由
开始,作一系列的相似三角形,OA的长度是
.
(2)设
,
,
,如此类推,证明:
.
(3)用这个方法作更多的直角三角形,直至最后一个三角形的斜边OM与OA重合为止,求OM.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5b352fa3e781df195ceccca90c3932a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96a4fe52baabb3071d55134f157a6079.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea32ddf9fa4087e121d209f0792d46ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e0a959cec22d164b15827e6a6c2ad31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a746fe7421122a76f5ff42ecd3d4127e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1a72688efac22042640c0a96d4e74aa.png)
(3)用这个方法作更多的直角三角形,直至最后一个三角形的斜边OM与OA重合为止,求OM.
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8 . 如图(1),四边形
是一边长为14cm的正方形.
,
,
,
依次将
,
,
,
分成
的两部分,得到正方形
.依循相同的规律,
,
,
,
依次将
,
,
,
分成
的两部分,得到正方形
.不断重复这个步骤,得到正方形
,…,
,….
.
(2)求
.
(3)一蚂蚁从
出发,沿路径
爬行,如图(2)所示,证明:该蚂蚁所爬行的总距离不能大于21cm.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632f2bf1cd0435041fa04b01901d1c8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43a71fc9c0068109dad1382354570665.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c296e45b84cf67a98939aa7334e7d478.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f7ba05c54b3de1f4378f7c8eb58328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f66fb71b75b63594ebeeeebd1963eed5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94ce22f30a8de2af135de3c89403aff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c60fe5130254a1d38bb4fd0015630f6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5bf350a619ef25d8d9b988f3db804e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04b56e44e4f0424a2b7a45567120a2e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2857fac4963b129d99e79dcb3e13d295.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab94459e87c666facddbe1a23ae1899d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de8ce8c07e34224e2d25130ed27c9a98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc34876d748f30fa4fc2eb6a686b5ff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97286f42173154565d6a99b55e1cb84c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1c94a6f24e6fae9888155f8461f878d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f54e9e692cbee655e280d075b2dc19fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c60fe5130254a1d38bb4fd0015630f6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c266cba73c7b8ad86e52aa34892e2239.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/941e96b41fa7eb82df5892eca05fb9c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36134f01da0f13b340e82e8835324f25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc34876d748f30fa4fc2eb6a686b5ff5.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af4439b8724e3907474b2a3798a5be52.png)
(3)一蚂蚁从
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70bce947f6fbe8ba99956a8929672887.png)
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9 . 莱布尼茨三角是与杨辉三角数阵相似的一种几何排列,但与杨辉三角不同的是,莱布尼茨三角每个三角形数组顶端的数等于底边两数之和.记第2行的第2个数字为
,第3行的第2个数字为
,…,第
行的第2个数字为
,则
( ).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4f27f84764f1cca89ce3d93fc1cf603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7230de53663c75658c58bbf206a0085.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9013bf1819f272929b9cadba31520e6.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/7/cc41dcfb-d4b9-4cf4-b3bd-cc6790dfda61.png?resizew=200)
A.![]() | B.![]() | C.![]() | D.![]() |
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10 . 分形几何是一门新兴学科,图1是长度为1的线段,将其三等分,以中间线段为边作无底边正三角形得到图2,称为一次分形;同样把图2的每一条线段重复上述操作得到图3,称为二次分形;……,则第5次分形后图形长度为( )
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/6/7561f618-96ee-4dfb-84cb-4211ef194bc0.png?resizew=315)
A.![]() | B.![]() | C.![]() | D.![]() |
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