1 . 已知
的其中两个顶点为
,点
为
的重心,边
,
上的两条中线的长度之和为
,记点
的轨迹为曲线
.
(1)求
的方程;
(2)过点
作斜率存在且不为0的直线
与
相交于
两点,过原点
且与直线
垂直的直线
与
相交于
两点,记四边形
的面积为S,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33d776753746914c2410a3946c357f35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5439f5ff9bd5deec0f0ef35c6f605b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33d776753746914c2410a3946c357f35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ac86e1c253297a377e14fb9a1689be8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0739793f234f8e86adc6177801ae7295.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8af2fdf1944afebb51cb6a5e6c74aadd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae3e029070ad0d2ce680d5336ed7150a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58d3ae70ecef31f90e511eba69f99b0c.png)
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5卷引用:山东省泰安市2024届高三下学期高考模拟((三模))数学试题
名校
解题方法
2 . 某灯具配件厂生产了一种塑胶配件,该厂质检人员某日随机抽取了100个该配件的质量指标值(单位:分)作为一个样本,得到如下所示的频率分布直方图,则(同一组中的数据用该组区间的中点值作代表)( )
A.![]() |
B.样本质量指标值的平均数为75 |
C.样本质量指标值的众数小于其平均数 |
D.样本质量指标值的第75百分位数为85 |
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2094次组卷
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8卷引用:山东省泰安市2024届高三下学期高考模拟((三模))数学试题
山东省泰安市2024届高三下学期高考模拟((三模))数学试题河南省三门峡部分名校2024届高三下学期高考模拟考试(一)数学试题河北省保定市曲阳县第一高级中学2023-2024学年高一下学期5月月考数学试卷山东省菏泽市2024届高三下学期模拟预测数学试题(三)(已下线)专题14.1统计(2))-重难点突破及混淆易错规避(苏教版2019必修第二册)福建省宁德市博雅培文学校2023-2024学年高一下学期5月月考数学试题(已下线)专题08 统计图表与用样本估计总体必考点-《期末真题分类汇编》(江苏专用)(已下线)专题03 高一下期末考前必刷卷01(基础卷)-期末考点大串讲(人教A版2019必修第二册)
名校
3 . 甲、乙、丙、丁各自研究两个随机变量的数据,若甲、乙、丙、丁计算得到各自研究的两个随机变量的线性相关系数分别为
,
,
,
,则这四人中,______ 研究的两个随机变量的线性相关程度最高.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7134e1107b26fbae9be177bcd71255fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d01e6150645e12a6a36b72b17d8edb24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7df389c5b47b9bbb35f0b7f292ca982d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5f9db69e3a499d7e5b98dd79f97caed.png)
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5卷引用:山东省泰安市新泰市第一中学东校2023-2024学年高二下学期第二次质量检测数学试题
名校
解题方法
4 . 设
是单位圆上不同的两个定点,点
为圆心,点
是单位圆上的动点,点
满足
(
为锐角)线段
交
于点
(不包括
),点
在射线
上运动且在圆外,过
作圆的两条切线
.
(1)求
的范围
(2)求
的最小值,
(3)若
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7818812e33052be4de712cbbbb21e2e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ec8858389f4c3156a946ba8bf0d8a7b.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b8d5cf36f04941f4ad49fe4c5e26133.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d8eb37a4dd75318dcbd836395e575bd.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e438bc5acc5cc10b3e7138279949a2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f63a42e22f8bc63465f595caf10e5842.png)
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4卷引用:山东省泰安市宁阳县第一中学2023-2024学年高一下学期4月月考数学试题
名校
解题方法
5 . 若
,都存在唯一的实数
,使得
,则称函数
存在“源数列”
.已知
.
(1)证明:
存在源数列;
(2)(ⅰ)若
恒成立,求
的取值范围;
(ⅱ)记
的源数列为
,证明:
前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72620c113a6fe83273803a9ac24baa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c59e7c7a84a4bdb959e95536d0404ceb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a038de5f1ce88d3baa95c2fd30abf7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9e8b81696639769354c282560245f0b.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)(ⅰ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38d5aa1a74419f1557aae998dbdadf87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(ⅱ)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/773bccec5a6fe68146daa59088db27d8.png)
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|
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5卷引用:山东省泰安市第一中学2023-2024学年高二下学期3月月考数学试题
山东省泰安市第一中学2023-2024学年高二下学期3月月考数学试题福建省厦门市2024届高三下学期第二次质量检测数学试题(已下线)湖南省长沙市四县区2024届高三下学期3月调研考试数学试题变式题16-19辽宁省沈阳市第二中学2023-2024学年下学期期中考试数学试卷 江苏省南通市2024届高三高考考前押题卷(最后一卷)数学试题
6 . 已知各项均不为0的递增数列
的前
项和为
,且
(
,且
).
(1)求数列
的前
项和
;
(2)定义首项为2且公比大于1的等比数列为“
-数列”.证明:
①对任意
且
,存在“
-数列”
,使得
成立;
②当
且
时,不存在“
-数列”
,使得
对任意正整数
成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.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/2810eaf7cba4fb3420b7124c2702b26c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8050391385b496e9c059201e4f12600a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
(2)定义首项为2且公比大于1的等比数列为“
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
①对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acb931a4b3eaa34e2c6f7dab5650f8af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cad52924df9291d5d191d18e09374ee1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95702d51d453347207cf73c6d5472717.png)
②当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1eae62173655a75678b9514af29d56f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7399fcd570d1de4057f2059759d18cc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abb534fa10cb56e77d73ecbfe64f555f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9abb6c2bc0ec36972b7f0a8d09552e8b.png)
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7 . 下列说法中正确的是( )
A.一组数据![]() |
B.某中学有高中生3500人,初中生1500人,为了解学生学习情况.用分层抽样的方法从该校学生中抽取一个容量为100的样本,则抽取的高中生人数为70 |
C.若样本数据![]() ![]() |
D.随机变量![]() ![]() ![]() ![]() |
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5卷引用:山东省泰安市2024届高三下学期一轮检测数学试题
8 . 动圆
与圆
和圆
都内切,记动圆圆心
的轨迹为
.
(1)求
的方程;
(2)已知圆锥曲线具有如下性质:若圆锥曲线的方程为
,则曲线上一点
处的切线方程为:
,试运用该性质解决以下问题:点
为直线
上一点(
不在
轴上),过点
作
的两条切线
,切点分别为
.
(i)证明:直线
过定点;
(ii)点
关于
轴的对称点为
,连接
交
轴于点
,设
的面积分别为
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2eae17f0752a563676c3de835849e782.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89a8532bf1f0cdb5c1262167fa3c00fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)已知圆锥曲线具有如下性质:若圆锥曲线的方程为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/896f353fb463bb5bbfad2d94e8b04971.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23b4f86e48e2b0d63c1865c60ed1e4d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1684f875065fa386c8254eac2e0a3875.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/224d30ca84f1aeeeda7a718e751a4925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/790ef3382b1c731f2885eecfd92c2a86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
(i)证明:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
(ii)点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e663220a66eff19da6a71e46b397db2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d153ab3afbbc7f88d04eb8e5ba5c411.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3637753af5ce86be9c23a9beb6b5067.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c59c4295f918205f5598ecc9a96d8867.png)
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4卷引用:山东省泰安市新泰第一中学2024届高三下学期高考模拟测试(一)数学试题
9 . 球面被平面所截得的一部分叫做球冠,截得的圆叫做球冠的底,垂直于截面的直径被截得的一段叫做球冠的高.球被平面截下的一部分叫做球缺,截面叫做球缺的底面,垂直于截面的直径被截下的线段长叫做球缺的高,球缺是旋转体,可以看做是球冠和其底所在的圆面所围成的几何体.如图1,一个球面的半径为
,球冠的高是
,球冠的表面积公式是
,与之对应的球缺的体积公式是
.如图2,已知
是以
为直径的圆上的两点,
,则扇形
绕直线
旋转一周形成的几何体的表面积为__________ ,体积为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eabd5f3a86afe49dcd70571e2b96cfd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aef5e6ee09edc8b6e274cd45a2e7af02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d24f6e152f122a7b0c7b72cd40429fda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f784040120239d7e75c591029d56c31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bf126cfed85fa9b7720ec6f7b0008dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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山东省泰安市新泰第一中学2024届高三下学期高考模拟测试(一)数学试题山东省临沂市2024届高三下学期一模考试数学试题(已下线)第六章 突破立体几何创新问题 专题二 融合科技、社会热点 微点3 融合科技、社会热点等现代文化的立体几何和问题综合训练【培优版】(已下线)高一 模块3 专题1 小题进阶提升练
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10 . 下列结论正确的是( )
A.一组样本数据的散点图中,若所有样本点![]() ![]() ![]() |
B.已知随机变量![]() ![]() ![]() |
C.在![]() ![]() ![]() ![]() ![]() |
D.分别抛掷2枚质地均匀的骰子,若事件![]() ![]() ![]() |
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