名校
解题方法
1 . 如图,正三棱台
的上下底面边长分别为3和6,侧棱长为3,则下列结论中正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
A.过AC的平面截该三棱台所得截面三角形周长的最小值为![]() |
B.棱长为![]() |
C.直径为![]() |
D.该三棱台可以整体放入直径为![]() |
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2 . 厦门一中为提升学校食堂的服务水平,组织全校师生对学校食堂满意度进行评分,按照分层抽样方法,抽取200位师生的评分(满分100分)作为样本,在这200个样本中,所有学生评分样本的平均数为
,方差为
,所有教师评分样本的半均数为
,方差为
,总样本的平均数为
,方差为
,若
,抽取的学生样本多于教师样本,则总样本中学生样本的个数至少为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfbe7f95b5d89f9409ec24536da9e826.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a63cadbf6b0d54955a3c3d1b7a62b14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb525270c748eddaaecc4a549cca250e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9289410bd35c9d57326b93cc7f4c4767.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d41042207515dd2e8349c805e6aee400.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/671f43c79d612c93a6d160335e86e177.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d954d1e6b433661e694eddc231be784.png)
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3 . 正多面体是指各个面都是全等的正多边形,并且各个多面角都是全等的多面角,又称为柏拉图多面体,因为柏拉图及其追随者对它们所作的研究而得名.自然界中有许多的柏拉图多面体,如甲烷、金刚石分子结构模型都是正四面体,氯化钠的分子结构模型是正六面体,萤石的结晶体有时是正八面体,硫化体的结晶体有时会接近正十二面体的形状……柏拉图多面体满足性质:
(其中V,F和E分别表示多面体的顶点数,面数和棱数).
(2)如图所示的正方体
中,点
为正方体六个面的中心,假设几何体
的体积为
,正方体
的体积为
,求
的值;
(3)判断柏拉图多面体有多少种?并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a098e3851f80b3d3c273d34416c4778e.png)
(2)如图所示的正方体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/455cfa98d3b692be03f4e927d6a10b92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4d86e8d1bef7032ab58f3c85d47c28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4764374bd2fb78e59cd0b283637baeb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63055a5d6916f99d07fede49120753f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f737b04ce09bc7e1ed86dc9b3c85203b.png)
(3)判断柏拉图多面体有多少种?并说明理由.
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4 . 中国古代建筑中重要的构件之一——柱(俗称“柱子”
多数为木造,属于大木作范围,其中,瓜棱柱是古建筑木柱的一种做法,即木柱非整根原木,而是多块用榫卯拼合而成.宁波保国寺大殿的瓜棱柱,一部分用到了“包镶式瓜棱柱”形式,即在一根木柱周围,根据需要再用若干根一定厚度的木料包镶而成的柱子,图1为“包镶式瓜棱柱”,图2为此瓜棱柱的横截面图,中间大圆木的直径为
,外部八根小圆木的直径均为
,所有圆木的高度均为
,且粗细均匀,则中间大圆木与一根外部小圆木的体积之比为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04582116cd765fcc5a52f44279ad6c94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e7f30167e1135806fe9de641870b1e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e86b1cfe63800f6fc02f999e64dd24b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eabd5f3a86afe49dcd70571e2b96cfd.png)
A.![]() | B.![]() |
C.3 | D.![]() |
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5 . 下列四个选项中正确的是( )
A.有两个面平行,其余各面都是平行四边形的多面体是棱柱 |
B.圆台![]() ![]() ![]() |
C.正四棱柱的底面边长为2,侧棱长为4,且它的所有顶点在球![]() ![]() |
D.某圆柱下底面圆直径为![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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2卷引用:福建省泉州市泉州一中、泉港一中、厦外石狮分校三校联盟2023-2024学年高一下学期5月期中联考数学试题
名校
解题方法
6 . 定义空间中既有大小又有方向的量为空间向量.起点为
,终点为
的空间向量记作
,其大小称为
的模,记作
等于
两点间的距离.模为零的向量称为零向量,记作
.空间向量的加法、减法以及数乘运算的定义与性质和平面向量一致,如:对任意空间向量
,均有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53bcfdf754e71318fb8329b8e7c09264.png)
,
,
;对任意实数
和空间向量
,均有
;对任意三点
,均有
等.已知体积为
的三棱锥
的底面均为
,在
中,
是
内一点,
.记
.
(1)若
到平面
的距离均为1,求
;
(2)若
是
的重心,且对任意
,均有
.
(i)求
的最大值;
(ii)当
最大时,5个分别由24个实数组成的24元数组
满足对任意
,均有
,且对任意
均有
求证:
不可能对任意
及
均成立.
(参考公式:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c082be7f93f355e1ca70588a4a89aead.png)
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28fde0a8b4ec1e2fff42cee3fc54c0f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28fde0a8b4ec1e2fff42cee3fc54c0f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49da589810153e2ec39ed656a2b61f4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00442d96d695db2c58bf1fb7165fca94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b12de8a4f788ff23d36e74c811354779.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e12e95f703ad30ab9a3d38376830989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53bcfdf754e71318fb8329b8e7c09264.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea3ff5e2f25dfebafaf8db07712ff706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff47a4801df7bc7bce1cb52327a7b174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8e0a953946d9e878aa017c7f24ffb40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0714b48d55f6b0854fb90a4255bc49c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6fa157b4f65f3a9aa1f7f82de02e99e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e1e19465c82977a26ca6900622ee1bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/718ba76bf48024ca425948e470e60042.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c761455094dc4913de76122017a243dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e48ac6b0dda0647d7dad3287ce4ad258.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d131fd570dc36b912396dc2dd06405c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4aeda1e642ce85f1c0394bc419bda8e.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f49f84442a1b38f27ac977214cd4b688.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/902a402a179a09f74f2391fb5cb4ae6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/247daad150250fc13a230d5375adda93.png)
(i)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
(ii)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab39849dc21c8c68cd5cde0911d5db23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a61e6011a0717ef57516821d0407a656.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aae3155971b2bb3c9d68b43e14b7186f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6ffe9f4e3243bd760835af03fa7ffe1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8c053ebe33366203ad0eca474760118.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d05f59bfd6b1f55920e73653bf87a46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6ffe9f4e3243bd760835af03fa7ffe1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db09e9844b90e46a6f2f5a710b6a3451.png)
(参考公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c082be7f93f355e1ca70588a4a89aead.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2343b61be295955a2b9baea86202f32.png)
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2024-06-13更新
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2卷引用:福建省厦门第一中学2023-2024学年高一下学期6月适应性练习数学试卷
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7 . 已知函数
,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bda318adf65c6b6fd45f651365f52346.png)
(1)求
的最大值
(2)写出
与
的大小关系,并给出证明
(3)试问
能否作为
三边长?若能,给出证明,并探究
的外接圆的半径是否为定值?若不能,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1044dcf4fba551e1b7fbfeb895ea08c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bda318adf65c6b6fd45f651365f52346.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b692262286e03cc0536598789fab8699.png)
(2)写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88fd5c1ef0fc722337a4984834829c84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e7bb46b41cd3f1f9b5621c20bf7fe07.png)
(3)试问
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b799aaa36edd0d10fc38925ce2e55045.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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2卷引用:福建省泉州市安溪第八中学2023-2024学年高一下学期6月份质量检测数学试题
8 . 高中教材必修第二册选学内容中指出:设复数
对应复平面内的点
,设
,
,则任何一个复数
都可以表示成:
的形式,这种形式叫做复数三角形式,其中
是复数
的模,
称为复数
的辐角,若
,则
称为复数
的辐角主值,记为
.复数有以下三角形式的运算法则:若
,则:
,特别地,如果
,那么
,这个结论叫做棣莫弗定理.请运用上述知识和结论解答下面的问题:
(1)求复数
,
的模
和辐角主值
(用
表示);
(2)设
,
,若存在
满足
,那么这样的
有多少个?
(3)求和:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b40b6895776e0807c2baecbc8f33a8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8b9ad2fcfff3dd546c5fdbedfe6238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1116c1a2be36c2952f3f621854433824.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/983fa8f4d178a0a909226523a33d521c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b40b6895776e0807c2baecbc8f33a8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/437f03842c607c5554d86177ce090def.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8eec3e684af41f9ed4db5b931b9ccfb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f56cfb41ee7cb758fee138ab09e0d13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30cac4804764e9ffa2a2c9c37e450713.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6481f56ecdb2488e91835028d3cc7e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b604ddba45cd6dbf1b937f9db82906d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77476f0974841f574785fc9940b2f8ca.png)
(1)求复数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/042b282f488b75517fb269e8b2512125.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e1d604600d084879cf3199cd0282345.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce48af55c99256efdc68fac0767d944c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f56cfb41ee7cb758fee138ab09e0d13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f3b1a317184018ea9efc8154a878658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65a40142c84be68ee2918c3a8303388c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/388d3d213a231cccf854a29eef611d01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dffae22ae38d7238130e81a9e554d94b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(3)求和:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f152097ab61600de85e8181d056dab9b.png)
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2卷引用:福建省安溪铭选中学2023-2024学年高一下学期6月份质量检测数学试题
名校
解题方法
9 . 如图(1),正三棱柱
,将其上底面ABC绕
的中心逆时针旋转
,
,分别连接
得到如图(2)的八面体
,依次连接该八面体侧棱
的中点分别为M,N,P,Q,R,S,
(ⅰ)求证:
共面;
(ⅱ)求多边形
的面积;
(2)求该八面体体积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a3a008a5ce2f3e0d93bf1b31f1e941d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1b73c7e51c2fbe79faa78e5287d2ccc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ff5cc57686ee7429fee0907651083c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78a40d2cf43fce0c99dff3470d554eb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ff5cc57686ee7429fee0907651083c4.png)
(ⅰ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8ae231960760617a585b8478185d8ac.png)
(ⅱ)求多边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac3662c929bd88085eb96dd4797482de.png)
(2)求该八面体体积的最大值.
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解题方法
10 . 球冠是指一个球面被平面所截得的曲面,截得的圆面是底,垂直于圆面的直径被截得的部分是高.如图,已知球的半径为20cm,球冠的高为10cm,现有3根长度相等的支柱
,
,
用于支撑球冠,立于水平的桌面上.若
,为使稳固支撑球冠,则应满足![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38dd8aad9cfb5e1fe2a45d80923f1571.png)
___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/602d0265aa2b89076b0cff90853f7cfa.png)
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