解题方法
1 . 对集合
,定义其特征函数
,考虑集合
和正实数
,定义
为
和式函数.设
,则
为闭区间列;如果集合
对任意
,有
,则称
是无交集合列,设集合
.
(1)证明:L和式函数的值域为有限集合;
(2)设
为闭区间列,
是定义在
上的函数.已知存在唯一的正整数
,各项不同的非零实数
,和无交集合列
使得
,并且
,称
为
和式函数
的典范形式.设
为
的典范数.
(i)设
,证明:
;
(ii)给定正整数
,任取正实数
和闭区间列
,判断
的典范数
最大值的存在性.如果存在,给出最大值;如果不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1304eb00ab95d664dc84385f602a8f09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f69939291758b5eaa19146f76709e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aee6c8ae5004f2ffe7f8392b4d3c39b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/238908949859936af0e109ef684599b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f69939291758b5eaa19146f76709e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f69939291758b5eaa19146f76709e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/937c09d82c480e4d67f8a48d3f66c5f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a02da5d46478a54d279755a295d548f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1b56da93ba7a2dec958070eb2666240.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05386869739fb11a190c637ba8a93174.png)
(1)证明:L和式函数的值域为有限集合;
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f69939291758b5eaa19146f76709e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20b4010030e10725398b64d4dcc09429.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab0fa51de98f090eda3e3f60a26475db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecfcda4333678bafacc4c676c2836977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee06844034f61cab7d421d55179ee367.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/359a16305129aeea0953efd9100f4b9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7b4e32041b54703ade8e8c2cee01f13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ed82555c7d6fc6b449fbdb1f68fef1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20b4010030e10725398b64d4dcc09429.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20b4010030e10725398b64d4dcc09429.png)
(i)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1462612f3654548c39489985987cb67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7870c36161f465fc992534b5fc3777f3.png)
(ii)给定正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f69939291758b5eaa19146f76709e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20b4010030e10725398b64d4dcc09429.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
2 . 设
是正实数数列.
(1)若
收敛,求证:存在严格递增的无界正实数数列
满足
收敛.
(2)若
收敛,是否一定存在严格递增的正整数数列
,满足
收敛,且
?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc7376941fa463c63b1d4d4ea866b78c.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccecde965d7557d5ee35dea8ae7164a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60c988a3683540149b687486af0ed3a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/120dcd9c3adc5b08ab9d84f228cc4b90.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97ad99ac2f9cbe69281dcdc7d4195d8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fba58d775c69de6d132c58581d614792.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/246e5563a2f86de45879b21393d814f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c69eef9b8e90f6a153b87738f759bcf.png)
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3 . 设
,满足
.
(1)证明:若
,则当
时,
.
(2)若存在
满足
,证明
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e761714f6940c2c06c5750e2ed80cc4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fbd27b6b4143c730ab9d36393a5fe14.png)
(1)证明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33c61cfbfd3bf888856b7dc9b2a84c4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c7b69e93488fcd2a195cb9793e94fc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac247d375e0da7fddafad1aa8186aa51.png)
(2)若存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c7b69e93488fcd2a195cb9793e94fc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e4439c7de7291f79def06d548603de7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffa3205b1df826d63914dcb55bb3ab43.png)
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4 . 对三次函数
,如果其存在三个实根
,则有
.称为三次方程根与系数关系.
(1)对三次函数
,设
,存在
,满足
.证明:存在
,使得
;
(2)称
是
上的广义正弦函数当且仅当
存在极值点
,使得
.在平面直角坐标系
中,
是第一象限上一点,设
.已知
在
上有两根
.
(i)证明:
在
上存在两个极值点的充要条件是
;
(ii)求点
组成的点集,满足
是
上的广义正弦函数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3037bb4ec2e6dfb182b22df30899cab7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05b8ec9d4206ea66a02de5c4a1e1e911.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c2dfbd59c0d4efc09e09ad82e83e431.png)
(1)对三次函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d2c831570d29c0fcbe5da38473ee828.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55d6fa911e3396b34fb470c10b063fde.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7121bf913ba5f136cb6d35db030ed70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c287a0f6a3521b83db37422a1aa309bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/944ede342597c070831052dc06bca45d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fe4d0e11cd9b9421c4d18121ffd181a.png)
(2)称
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7041eb865c44a89770acd4fd71024bac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d9919ff015350c4e25aa0c05c09c329.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b56f913087e3bbf8cd9dd7c9bba7dc21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c65d3d6119b18fd2427497cbd413c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2232cbe8d56d936da2ea9c3a78d87f14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce43981e5251e382690797f24907de2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03d97b51756740950b8a9304755b4224.png)
(i)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cd50020c0e3198d4a6b2d26a413b1b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42027a6a90b0a513981ebd5ed4431460.png)
(ii)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0fa6d6bca6428b15c6e95504904e944.png)
您最近一年使用:0次
解题方法
5 . 设M是由复数组成的集合,对M的一个子集A,若存在复平面上的一个圆,使得A的所有数在复平面上对应的点都在圆内或圆周上,且
中的数对应的点都在圆外,则称A是一个M的“可分离子集”.
(1)判断
是否是
的“可分离子集”,并说明理由;
(2)设复数z满足
,其中
分别表示z的实部和虚部.证明:
是
的“可分离子集”当且仅当
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33a2a51a8d747c5a61f259a3ddf3bd0e.png)
(1)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6f12a019ea4cab2a4143b39043157ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e6670f3947ae0329e5d9788b96c50f8.png)
(2)设复数z满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c86a4bfb6dd4bafcbe3c5c1aaead277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aff32d9320e0d72844f155f5c2acedb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/739598c5b7f2c8a97353a987b7392536.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f77809bc2f616691dd7417b3d31df5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fae53a4b5ae5f0288d4d1ed6b41a7b11.png)
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6 . 17世纪德国天文学家约翰内斯·开普勒提出描述行星运动的三大基本定律:
(a)行星绕太阳运动的轨道为椭圆(圆可视为特殊的椭圆),太阳位于椭圆的一个焦点上,所有行星的轨道可近似看成在同一平面内;
(b)行星在其椭圆轨道上的相等时间内,与太阳连线所扫过的面积相等.
(c)行星的公转周期的平方与它们的椭圆轨道长轴的立方成正比.
开普勒三定律为我们理解行星运动提供了重要的基础,并且被广泛应用于天体力学和行星轨道计算中.设a,b,
,地球、太阳、火星均可视为点,太阳位于
,地球的公转轨道可近似看成圆
,火星的公转轨道可近似看成圆
,且火星的公转周期约为地球公转周期的1.882倍.霍曼转移轨道E是以太阳所在位置为其中一个焦点,并且与
均相切的椭圆.2020年,我国自主研制的火星探测器天问一号从地球发射,经霍曼转移轨道到达火星,如下图所示.
(1)计算霍曼转移轨道E的离心率.(参考数据:
,计算结果保留两位小数)
(2)设天问一号位于E上的一点P,当P不在
上时,
上存在依赖于P的两点A,B,使得
为观测地球的最大视角(即地球不可能位于该角的外部),问:轨道平面内是否存在定圆
,使得直线AB恒与
相切?证明你的结论.
(a)行星绕太阳运动的轨道为椭圆(圆可视为特殊的椭圆),太阳位于椭圆的一个焦点上,所有行星的轨道可近似看成在同一平面内;
(b)行星在其椭圆轨道上的相等时间内,与太阳连线所扫过的面积相等.
(c)行星的公转周期的平方与它们的椭圆轨道长轴的立方成正比.
开普勒三定律为我们理解行星运动提供了重要的基础,并且被广泛应用于天体力学和行星轨道计算中.设a,b,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b644521da261e452421307913a47dacf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92476f5898293a343fe2c3895c12a249.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d43cb1f811bcd47ae65285be9854a55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea324a7d90c1c12472d2ab412c29e0e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2aaa30d92dfea3fa999ffa88aaf89153.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/24/56c5d108-58bb-4d12-a973-26b3b768ae13.png?resizew=300)
(1)计算霍曼转移轨道E的离心率.(参考数据:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08faef2ef9706bc0f8343a3b89462e25.png)
(2)设天问一号位于E上的一点P,当P不在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f10392437ab60e58109787b9b0952f2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f10392437ab60e58109787b9b0952f2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb686e4f5e3938575bc547e849d5513f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb2f4c73bee61643cfcd522cc70a3bca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb2f4c73bee61643cfcd522cc70a3bca.png)
您最近一年使用:0次
解题方法
7 . 设
为坐标原点,
为抛物线
上异于
的一点,
,
.
(1)求
的最小值;
(2)求
的取值范围;
(3)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/745de5ef1fd897d16e37464172d5e8c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e9f2b482e8a8e0e1b5c720a3574af70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e24f172a287592897ea4378a2ad29013.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f4dfec890cdfdda355e19463f3be813.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fb8a80473da8d3f571def3f3f34086d.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7e66ea801d8df6d13f924cae67fc1db.png)
您最近一年使用:0次
名校
解题方法
8 . 在数学中,双曲函数是与三角函数类似的函数,最基本的双曲函数是双曲正弦函数与双曲余弦函数,其中双曲正弦函数:
,双曲余弦函数:
.(e是自然对数的底数,
).
(1)计算
的值;
(2)类比两角和的余弦公式,写出两角和的双曲余弦公式:
______,并加以证明;
(3)若对任意
,关于
的方程
有解,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3321510a9eb73909a36c084a8630e89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0099b9b80ed478824fa95677ebe9d5b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11204e2fb6e560bf7a4ca26eaebfc526.png)
(1)计算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e694af0c9f990ecb8b54b1c08bcc578e.png)
(2)类比两角和的余弦公式,写出两角和的双曲余弦公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d92c32edc0e000405b7a6b9c48549959.png)
(3)若对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f78f05631a2ecb8bc3d379ca6c81f93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eed807cc52eca7b462a3850b5e5e02b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
2023-06-21更新
|
1008次组卷
|
8卷引用:山东省济南市山东师大附中2022-2023学年高一下学期数学竞赛选拔(初赛)试题
山东省济南市山东师大附中2022-2023学年高一下学期数学竞赛选拔(初赛)试题上海市宝山区2022-2023学年高一下学期期末数学试题(已下线)模块六 专题5 全真拔高模拟1(已下线)专题14 三角函数的图象与性质压轴题-【常考压轴题】(已下线)第10章 三角恒等变换单元综合能力测试卷-【帮课堂】(苏教版2019必修第二册)上海市闵行(文琦)中学2023-2024学年高一下学期3月月考数学试卷(已下线)专题06 期末解答压轴题-《期末真题分类汇编》(上海专用)上海市市西中学2023-2024学年高一下学期期末复习数学试卷
9 . 设
为一个质数,且
也是一个质数,证明:
的小数表示形式中包含0至9的所有数码.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ca4c3cdbb63bf2e3c1c3d67d46d3acc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cbadc8f735cb509c99741c41f32d097.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1eccdf91bcfb339dee1e546ebcf17bc.png)
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10 . 设
,记:
,其中求和是对1,2,…,n的所有
个k元组合
进行的,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0610af5abf3cc278643f5696f58b625d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12b645e448faf42a5eec1a1e02b85325.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c015bd3b114c45ca0881c8468b4b35d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/800cc73c3cf1e852b3cb2eb9dc1ff3f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/663211001b3f08bcbe2cf4a9c9ee0505.png)
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