1 . 我们知道,在平面内取定单位正交基底建立坐标系后,任意一个平面向量,都可以用二元有序实数对
表示.平面向量又称为二维向量.一般地,n元有序实数组
称为n维向量,它是二维向量的推广.类似二维向量,对于n维向量,也可定义两个向量的数量积、向量的长度(模)等:设
,
,则
;
.已知向量
满足
,向量
满足
.
(1)求
的值;
(2)若
,其中
,当
且
时,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39c2af42141367e6e9ff0296c31daa7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62b3b354facacd72bc68da6ac07be453.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a48d974578eb15ca117e0cb1b59788d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99aa60676891adca75eac086182a15c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2581496116ddfba6dd03722fd771d5a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5babafd9f4e5c3c222ba25a3de66794.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a48d974578eb15ca117e0cb1b59788d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7cb2f5c0569962cd7c1026f388cb661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99aa60676891adca75eac086182a15c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4492fb816272cd60cf3456c6a064020e.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efa3e5481ce1f11ea4cb1d1ddc71413.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/301fa5679316c282923735aff9285559.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95ac252e9126ab540c0102b941f0ee42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36b98ef143f8159f3a7dafa1fd2f2370.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74cac554f22f3655ef6691b2ef821eac.png)
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2 . 如图,在平面直角坐标系中,
和
是
轴上关于原点对称的两个点,过点
倾斜角为
的直线
与抛物线
交于
两点,且
.
为
的焦点,求证:
;
(2)过点
作
轴的垂线,垂足为
,若
,求直线
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.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/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bb4dd4670828f75bc573b52cdd02e1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e08136e6bc876f29a13d1204d9d621db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fea0b6471289760543596f5f45aa43ae.png)
(2)过点
![](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/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14cc1572ca21da2e3271484f127a5094.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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2024-06-03更新
|
515次组卷
|
3卷引用:第一套 艺体生新高考全真模拟 (三模重组卷)
名校
3 . 对于求解方程
的正整数解
(
,
,
)的问题,循环构造是一种常用且有效地构造方法.例如已知
是方程
的一组正整数解,则
,将
代入等式右边,得
,变形得:
,于是构造出方程
的另一组解
,重复上述过程,可以得到其他正整数解.进一步地,若取初始解时满足
最小,则依次重复上述过程 可以得到方程
的所有正整数解 .已知双曲线
(
,
)的离心率为
,实轴长为2.
(1)求双曲线
的标准方程;
(2)方程
的所有正整数解为
,且数列
单调递增.
①求证:
始终是4的整数倍;
②将
看作点,试问
的面积是否为定值?若是,请求出该定值;若不是,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c09e7065aa112872161285c5f3bfc022.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3460cd2f27a53941986606734a9b479a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a79b9eaa5e7ab7a1e5c512b571914dc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145b6046bc80d0ffecc61ac67c87ca1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38ca4e60aab76ce3be3b5ffb9137f163.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ef282595213e6ac1c04b09c8703e176.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e62b8fca47d65828c45fc8e38fe8beb4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99c804f8356fa6120aa13b2d11bfea10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73f33dbb2fc073ae2d62891732e52dc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e31df193bb9a9b93b02f2daa2fb747c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e1e81a0995ee5492c4281539c65bf00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ca5fd57c2c2fcc3c7a574fdd1467d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18483c9c195ecd922772527fa85c0fcb.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f3fa0b40fb0d9b8c62e37316ab3b04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adb4c637bd4364a8d3b8d13889befd4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e976c0663fa749ca749f99842d21ca03.png)
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba78250f4e67347c7e80c543078d02e6.png)
②将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adb4c637bd4364a8d3b8d13889befd4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52b66d595bfea3722fbc56e2fdccd548.png)
您最近一年使用:0次
名校
4 . 贵州省“美丽乡村”篮球联赛在比赛间隙进行芦笙舞、侗族大歌等非物质文化遗产展演,这项活动将体育运动与当地民族民俗文化相触合,创造出独特的文体公共产品.为了打造更具吸引力的赛事,某平台发起了群众观赛意见反馈调查,共收回了200份调查问卷.
(1)通过进一步分析关注赛事群众的调查问卷得知,关注表演的女性用户有24名,现从关注赛事的群众中抽取一人,设“抽取的一人为男性”为事件A,“抽取的一人关注表演”为事件B,若
,则以此次调查的数据为依据,估计从平台用户中任意抽取一名用户,该用户关注表演的概率为多少;
(2)是否有
的把握认为是否关注赛事与性别有关?
附:
,其中
.
性别 | 关注赛事 | 不关注赛事 |
男 | 84 | 36 |
女 | 40 | 40 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9ae197ece5a94e48fe6f05dea69e936.png)
(2)是否有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b253ec362b46012c30d1eb2d3030ee2a.png)
附:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e3821f70c08c5180e9b3086d3c9610f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/356b05e46b10ee51c3e43546d73ec96c.png)
0.050 | 0.010 | 0.005 | 0.001 | |
k | 3.841 | 6.635 | 7.879 | 10.828 |
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名校
5 . 如图,等边
内有一动点
是等边三角形(点
在直线
两侧),直线
与直线
交于点
.
的大小是否为定值?若是定值,求出其大小;若不是定值,请说明理由.
(2)若
,求线段
长的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c856f329068123c34ad5c5c2c3dd5ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9521effc99744da7d1445af9681a3ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ed24d7acba0b3a7ebf4d520296424a3.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e72613c053b42aff06d307f2bf662ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
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6 . 如图,点
是正方形
的边
上一动点(异于
),连
,以
为对角线作正方形
与
交于点
,连
.
三点共线;
(2)若
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/526768fab505a5637e515731c9940380.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9c15e609c856cb7856d6101aa198fef.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5ab52edccf02d432e0aaa4542a147bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/956c4a03127f33e7fb7acc3476f717ce.png)
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2024·全国·模拟预测
解题方法
7 . 我国中学生的近视率一直是社会关注的焦点.某市疾控中心为调查该市高中生的视力状况,从某高中3000名学生中随机抽取了100名学生用五分记录法统计了其裸眼视力,得到如图1所示的频率分布直方图:
![](https://img.xkw.com/dksih/QBM/editorImg/2024/5/7/b4485223-5bfb-48c0-8ea1-f0c8048befa2.jpg?resizew=261)
为改善学生的视力状况,该校积极落实学生近视防控工作,建立视力监测制度,几年后,再次抽取100名学生,用五分记录法统计其裸眼视力,得到如下频数分布表:
(1)若裸眼视力位于
为轻度近视,用样本估计总体,用频率估计概率,估计近视防控工作开展前全校患轻度近视的学生人数;
(2)在图2中作出近视防控工作开展后100名学生裸眼视力的频率分布直方图;
![](https://img.xkw.com/dksih/QBM/editorImg/2024/5/7/c4c40aa3-e4be-4d90-bcd9-7be450d56561.jpg?resizew=293)
(3)估计近视防控工作开展后该校学生裸眼视力比开展前学生裸眼视力的平均值提高了多少(同一组中的数据用该组区间的中点值为代表).
![](https://img.xkw.com/dksih/QBM/editorImg/2024/5/7/b4485223-5bfb-48c0-8ea1-f0c8048befa2.jpg?resizew=261)
为改善学生的视力状况,该校积极落实学生近视防控工作,建立视力监测制度,几年后,再次抽取100名学生,用五分记录法统计其裸眼视力,得到如下频数分布表:
裸眼视力 | ![]() | ![]() | ![]() | ![]() |
人数 | 5 | 20 | 60 | 15 |
(1)若裸眼视力位于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/412eb4f4db252fe8ee0abbf0ad9228ba.png)
(2)在图2中作出近视防控工作开展后100名学生裸眼视力的频率分布直方图;
![](https://img.xkw.com/dksih/QBM/editorImg/2024/5/7/c4c40aa3-e4be-4d90-bcd9-7be450d56561.jpg?resizew=293)
(3)估计近视防控工作开展后该校学生裸眼视力比开展前学生裸眼视力的平均值提高了多少(同一组中的数据用该组区间的中点值为代表).
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2024高一下·全国·专题练习
8 . 某校高一某班的某次数学测试成绩(满分100分,单位:分)如下:56,58,62,63,63,65,66,68,69,71,72,72,73,74,75,76,77,78,79, ,90,95,由于保存不利,其中
内的成绩被墨水覆盖.根据该数据绘制的频率分布直方图(如图)也被墨水覆盖了部分区域.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/486c7705dbd7b7b9ec5dd17b4891088b.png)
(2)求成绩在区间
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d90faf85d1548242098a6fe3accd84f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d90faf85d1548242098a6fe3accd84f.png)
您最近一年使用:0次
解题方法
9 . 法国数学家弗朗索瓦·韦达发现了一元二次方程的根与系数之间的关系,将其推广到高次方程,并在其著作《论方程的识别与订正》中正式发表,后来人们把这个关系称为韦达定理,即如果
是关于x的实系数一元n次方程
在复数集C内的n个根,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/988acbe8533ef50c899650a057717cf5.png)
试运用韦达定理解决下列问题:
(1)已知
,
,
,求
的最小值;
(2)已知
,关于x的方程
有三个实数根,其中至少有一个实效根在区间
内,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44258e924e42ec263b5236499252d4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f44bf708f38a916de0572d8ef1cf45a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/988acbe8533ef50c899650a057717cf5.png)
试运用韦达定理解决下列问题:
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3eb9b6fe8959ae9e71e857b6d6fed49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/751e274e9107d780c39ba9c49d6daefb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/460b68eaa42bc8929edf81e21ad0bca0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bc2818de1c0d7d347718672b0bcec32.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73254f32b6da29ecc32df2e9f87a4c97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a070d41a05c5193153ae18e0235a492.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4a90cfdbfa05577b6ec0b22739e7c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/754826457671db8939098215943e656a.png)
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