名校
1 . 已知空间向量
,
,则向量
在向量
上的投影是____________ .(用坐标表示)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e14e6ce38d3aa1cf65e5ae2ec9eb157e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/380198c5e99b2e86c1b78d5344bc3ead.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
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2 . 已知甲组数据
,
,…,
的茎叶图如图所示,其中数据的整数部分为茎,数据的小数部分(仅一位小数)为叶,例如第一数据为5.3.
、方差
、中位数M;
(2)乙组数据为
,
,…,
,且甲、乙两组数据合并后的30个数据的平均值
,方差
,求乙组数据的平均值
和方差
,写出必要的计算过程和步骤.
![](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/c5bbae9353140c2235610852d04a9a3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/848dac3d6db7c23c58f399970b3f9b18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfbc29b47b83fdc5368770b7b1acb439.png)
(2)乙组数据为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b715e7842b95f654f16056a7c7f2abe9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13423c094861baf4b759b7f3d8c3c226.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a9f28d2342d61e9fae0744b0a8a1cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b382035a4e07f9a8679d024bf1f7b7fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5cf8479865093acc003149c1f00fd73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c0da8a9d862e0005c44a8cb8fd262bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab1295cbd36fdc55a55b549aa2dd5887.png)
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3 . 平面
上三点的坐标分别是
.小林同学在点
处休息,进而小猫沿着
所在直线来回跑动,小猫离小林同学最近时的坐标为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad939769ee081dd5ecf86533c36f5eb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
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2024-05-27更新
|
147次组卷
|
2卷引用:上海市进才中学2023-2024学年高一下学期期中数学试题
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解题方法
4 . 已知点
,
,若点
满足
,则点
的坐标为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f1e95fb519f59c46f40e4ab44660073.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/444c5d393b85df77fc5c8ca43374cc5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4079b9c1702107192ed48da27fd117a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
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2024-05-27更新
|
355次组卷
|
2卷引用:上海市进才中学2023-2024学年高一下学期期中数学试题
名校
5 . 某读书会有5名成员,假期他们每个人阅读的本数分别如下:3,5,4,2,1,则这组数据的60%分位数为____________ .
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6 . 如图所示的空间直角坐标系A-xyz中,所有坐标均为整数的点称为整点;已知正方体
的棱长为a,点P满足
,其中
,
;
,且直线
与平面
所成角大小为
,求点P的轨迹长度;
(2)若
,
,求正方体经过点
的截面面积S的取值范围;
(3)若
,求三棱锥
内(不包括表面边界)整点的个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f9dd1458186cfbdf0701ba835572721.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79296cd4046a71e163a8f3e647a176ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/994fe2e2ccff825adfe4567a84e0fd79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6655c413f509068d30b165f9d92bdba0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c52091eb745de866044477641a7c55f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15615de1a6df206dbd081251f676578e.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/500d68f2678989a5ce7431cfd51b019d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2d03c3b051baf2d868a53b9d2dba307.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e15cbd7c42d7b15d7ba8d2b28ab8df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb650f48c879ea25127662b47d16feec.png)
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7 . 下表是“膜法世家”形象代言人选举得票情况统计,其中周柯宇的票数被污损了无法看清,那么应该当选的人是( )
姓 名 | 张元英 | 林正英 | 米 卡 | 周柯宇 | 林 墨 | 合 计 |
票 数 | 250 | 200 | 380 | 320 | 1550 |
A.米卡 | B.周柯宇 | C.无法确定 | D.合计 |
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8 . 高斯是德国著名的数学家,近代数学奠基者之一,享有“数学王子”的称号,他和阿基米德、牛顿并列为世界三大数学家,为了纪念他,人们把函数
(
)称为高斯函数,其中
表示不超过x的最大整数. 已知:
,(
,
)
,
,
,求
的值;
(2)若
,
,
,求证:
;
(3)设
,求S除以2023的余数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5b7f26fe1977bda9de200debe99f020.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/851eae00e3369068e33a7e6420483883.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f161c2a3717f1b6c62d0d7dae0b606.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2382ddc29a4fec13b4d0a11deb2cfd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d290a4927c50661098e2fbea58d77b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e167b43045b3297248e334c41c621b8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0956ed974bee14fffcf9ada9cafe49d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/964a54ca7774d6fc04aaf8b85fda5fd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06ee581ca3282490206a8bc11dfb5ccf.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81a45b5fe1fea8e19026d491596392fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c59e7c7a84a4bdb959e95536d0404ceb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f54e945ef81cf9c9d985aa6f2f72dc5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1972a6ab2655f4cb706f7dd379902a87.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a03304e01039bd5039d463207d81ce65.png)
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9 . 《九章算术》中的“商功”篇主要讲述了以立体几何为主的各种形体体积的计算,其中堑堵是指底面为直角三角形的直棱柱.如图,在堑堵
中,
分别是
,
的中点,
是
的中点,若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee477d164d38ff067fc4ce342cf6d380.png)
____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f1f229274a6e17977cc047814212589.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7d81947c03e5677e0d8828fe7fcd7d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee477d164d38ff067fc4ce342cf6d380.png)
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10 . 某同学用“五点法”画函数
在某一个周期内的图像时,列表并填入了部分数据,如下表:
(1)请直接写出表中
的值,并求出函数
的解析式和最小正周期;
(2)若关于
的方程
在
上有两个不同的实数解,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c0bfe1c23b5467382e6c2df68106e60.png)
0 | |||||
![]() | |||||
0 | 1 | 0 |
(1)请直接写出表中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
(2)若关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae615b3edf7b6d7208c02acd74ced3bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4da008a98340a4940ca753902937a688.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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