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1 . 已知有穷正项数列
,若将每个项依次围成一圈,满足每一项的平方等于相邻两项平方的乘积,则称该数列可围成一个“HL-Circle”.例如:数列
都可围成“HL-Circle”.
(1)设
,当
时,是否存在
使该数列可围成“HL-Circle”,并说明理由:
(2)若
的各项不全相等,且可围成“HL-Circle”.
(i)求
的取值集合;
(ii)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a2723483435ce9c39acae4032d12e9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ec293ea5d248edf3b66a54f81b5e5f0.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7fab51121848ce166035ceab6f4e00b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c7a1d739890a8951586e23b78b035bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(i)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(ii)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9989990eebe2cc2bdddafd5e1d6c0a85.png)
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解题方法
2 . 在圆台
中,圆
的半径是2,母线
,圆
是
的外接圆,
,
,则三棱锥体积最大值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e65ac334119ccd6204402a7aba29a55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2899e607479d8d1c47d954ae9ebb7144.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c4f6f74444b2b7947fc6e35c8d62322.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84d81d732204a3c2384a27606f858677.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68b40d0d2f3cdd8981bb792ad87efb42.png)
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3 . 对于正整数n,
是小于或等于n的正整数中与n互质的数的数目.函数
以其首名研究者欧拉命名,称为欧拉函数,例如
(
与
互质),则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce7cc0ad7521b5771950aea983f0c1c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce7cc0ad7521b5771950aea983f0c1c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4c9e69c7d5a3d7a5633a373a8a39544.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/786c6406780167f9744d0f9e9682e471.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8d02ea8c4988c5c28ab93f0d70fb55a.png)
A.若n为质数,则![]() | B.数列![]() |
C.数列![]() | D.数列![]() |
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3卷引用:湖北省宜荆荆2024届高三下学期五月高考适应性考试数学试题
湖北省宜荆荆2024届高三下学期五月高考适应性考试数学试题 吉林省通化市梅河口市第五中学2024届高三三模数学试题(已下线)高二数学期末模拟试卷02【好题汇编】-备战2023-2024学年高二数学下学期期末真题分类汇编(北师大版2019选择性必修第二册)
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4 . 奔驰定理是一个关于三角形的几何定理,它的图形形状和奔驰轿车logo相似,因此得名.如图,P是
内的任意一点,角A,B,C所对的边分别为a,b,c,总有优美等式:
.
的内心,
,延长AP交BC于点D,求
;
(2)若P是锐角
的外心,
,
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6304b350f56f7ee6a0c51d9ece5ee7db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64edb981d2bd90d1dc58e9d140d3903d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/219679f2c28a0418f62d9861b7aec02f.png)
(2)若P是锐角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a2264c134952d41fb9bcb90e6c72c83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/568c6a1484da196de32bd04994d9d3d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b88584cf1df43e28d03592c7998b1653.png)
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5 . 对于平面向量
,定义“
变换”:
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8560ca9023cf64637ce1467f338556bd.png)
(1)若向量
,
,求
;
(2)已知
,
,且
与
不平行,
,
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d128ae3e21294e2eac5bcc775ccfb03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a18fd5445fb8a04b925a2745a56f613.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fddd9dc1110e60973b7b9e43bb1f9d15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8560ca9023cf64637ce1467f338556bd.png)
(1)若向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ea462b0382581d99c8bba51d9b79f09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3e186ebc624ebacde9a03b96289f1ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2e900404ba71110c5861ced9634646f.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22601439d36b6a93453d738c2b803eb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc499d2e731df31957eeaa355bfbac4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4f605ec0729ce6d72237ad662a06862.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fc9656d8286c4d6fa309d6ae347c89e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a63cf7e5f25165ccf0e24d32add179ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a176f300a2462e4f1ffef99d30c21e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39e719e667f2783febbec38dea080b98.png)
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解题方法
6 . 已知数列
是斐波那契数列,其数值为:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a4fb1a2d1cb1152ef78d7332d45b681.png)
.这一数列以如下递推的方法定义:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/642c7410a3134bed37df637e8d382c88.png)
.数列
对于确定的正整数
,若存在正整数
使得
成立,则称数列
为“
阶可分拆数列”.
(1)已知数列
满足![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c9e8b08ba803f851cf12404e742775.png)
.判断是否对
,总存在确定的正整数
,使得数列
为“
阶可分拆数列”,并说明理由.
(2)设数列
的前
项和为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75541174a021adfd2e3356ca2ad56f7b.png)
,
(i)若数列
为“
阶可分拆数列”,求出符合条件的实数
的值;
(ii)在(i)问的前提下,若数列
满足
,
,其前
项和为
.证明:当
且
时,
成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a4fb1a2d1cb1152ef78d7332d45b681.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc50612eece655796b752da6b4bc3f3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/642c7410a3134bed37df637e8d382c88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d74fa7fa6330976d7eb8e523a62cd09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cdfd9c3f8933cddb63d87dbe2812994.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(1)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ab0309e2cd35585ea9fb2cc3017abf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c9e8b08ba803f851cf12404e742775.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/753358ca020523f27725f5187bb8e988.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/136a003907c455bfd58875c96c138772.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ab0309e2cd35585ea9fb2cc3017abf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d813f3ca8db41a4db6c18eac30fef98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75541174a021adfd2e3356ca2ad56f7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79282bbe9f6408297d6378878c423bec.png)
(i)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d813f3ca8db41a4db6c18eac30fef98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(ii)在(i)问的前提下,若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca5f894d605847c6df0c4df24cf8e1fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69a340cb0e3c456ec64ffdf89d7cd6ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09881de0dc186bbcd1e60eb00159ee97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09881de0dc186bbcd1e60eb00159ee97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e48268fd6d3f92032eb54fbf65c01405.png)
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7 . 设无穷数列
的前
项和为
,且
,若存在
,使
成立,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.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/661eb98b215405edbdc6434ce55b89cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/699dfd96d64e59252e384847629c7a75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4208fc3a5068fad9bcd233cca865c2fe.png)
A.![]() |
B.![]() |
C.不等式![]() ![]() |
D.对任意给定的实数![]() ![]() ![]() ![]() |
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8 . 在不大于
的正整数中,所有既不能被2整除也不能被3整除的个数记为
.
(1)求
,
的值;
(2)对于
,
,是否存在m,n,p,使得
?若存在,求出m,n,p的值;若不存在,请说明理由;
(3)记
表示不超过
的最大整数,且
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bc1e9444e6cbbcccfb19bef934fda45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c581f06adc031bd163f98c461300d862.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f0f3595c506dd94a3399da87f0b33ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/985ea7ad3004613e28dd691829437c11.png)
(2)对于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5510ef06b326f131933224473550d41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acf45fc1d20ec9adb3b25794ac938855.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b80b43936d042aae836465212e716964.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19bbe68c798af91a4f5fbf939c4ed315.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a3651b3fedba1f0e9998fa88acefd08.png)
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2024-06-07更新
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469次组卷
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3卷引用:安徽省A10联盟2024届高三4月质量检测考试数学试题
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9 . 某人从银行贷款100万,贷款月利率为
年还清,约定采用等额本息按月还款(即每个月还相同数额的款,240个月还清贷款的利息与本金),则每月大约需还款( )(参考数据:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47b06df422eec83ae3ce4a5ccb2960a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/341d0337c8eebc8172d78fc6dc8b8cb9.png)
A.7265元 | B.7165元 | C.7365元 | D.7285元 |
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|
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2卷引用:陕西省西安市第一中学2024届高三下学期模拟考试数学(文科)试题
10 . 在直角坐标平面内有线段
,已知点
是线段
上靠近
的三等分点,点
是线段
上靠近
的三等分点,……,点
是线段
(
,
)上靠近
的三等分点,设点
的横坐标为
.
(1)求证:数列
为等比数列;
(2)若
,
,求
的通项公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/473913c0887bb64d386f4c02f1853452.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04b56e44e4f0424a2b7a45567120a2e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/473913c0887bb64d386f4c02f1853452.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e47cd514b2920609e3781c87df6ab70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11603c89c66f064b263af841dae023f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04b56e44e4f0424a2b7a45567120a2e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039af66355ed85ff4c204931b882b694.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f614bf2bc35c4aaf1123d829f7fa82dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d82c65a855b1eed9c43e6829f6c3bffb.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1928c254cfada1f75a5cd1e34db5a63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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