名校
1 . 设整数
,对于
任一排列
,记
,求
的值,并计算取到最小值时排列
的数目.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94ea993dd2879ecfefc8d2f312825662.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c9e9321f74373775e8148da90dfe698.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0227c93e5e723d3a5358cffe4121960.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b01d24c5d4d3d7b6d78aa396bc18af8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c0ad7e7853a069537387b5192f73844.png)
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2 . 求所有正整数
,满足正
边形能内接于平面直角坐标系
中椭圆
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8715a3f984d2627afd7c40c61347b7cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48da128547c4cf9745e8e4b99988a3db.png)
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解题方法
3 . 如果
是离散型随机变量,则
在
事件下的期望满足
其中
是
所有可能取值的集合.已知某独立重复试验的成功概率为
,进行
次试验,求第
次试验恰好是第二次成功的条件下,第一次成功的试验次数
的数学期望是__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa0010cb466163db1349fc1040f6b439.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90a71f2ff30791e8b210727912600096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97ad71223cc853bc21bf203e7a5321f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/385f040f4037e9934620d6971da08131.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
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解题方法
4 . 双五棱锥是由两个侧面均为边长为1的正三角形的五棱锥上下拼接而成的,如图所示.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/17/7b4e1fa0-9c3c-43ca-b5b3-188c8bc7d378.jpg?resizew=133)
(1)求双五棱锥的内切球半径;
(2)求分别位于拼接面(正五边形)两侧的相邻的两个正三角形构成的二面角的余弦值.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/17/7b4e1fa0-9c3c-43ca-b5b3-188c8bc7d378.jpg?resizew=133)
(1)求双五棱锥的内切球半径;
(2)求分别位于拼接面(正五边形)两侧的相邻的两个正三角形构成的二面角的余弦值.
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5 . 整数列
,
,
,对
有
,
为固定正整数,求使
成立的
的个数______
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82f241a73bf197764cebade47dcd9a55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/401b7a6ae9616f1f3a27ff11c8abf2f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ed352cca4f2ffd027a1f203bc4e51bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75cc773ef279d9960a90052c5b710e8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90f6a09c6d8b804e7214da49256f0317.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b25b0878c79813d60afc929c6319434.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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6 . A,B,C为
内角,x,y,z为实数,求以下三式中恒成立的个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ad95998658d2ab5aad3714e2276bb6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5870e3149fc4cec4c7066ea95db8e5c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ad95998658d2ab5aad3714e2276bb6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5870e3149fc4cec4c7066ea95db8e5c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7df217a60bbc39645dd98a10ebc809e5.png)
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7 . 设有两个集合
,如果对任意
,存在唯一的
,满足
,那么称
是一个
的函数.设
是
的函数,
是
的函数,那么
是
的函数,称为
和
的复合,记为
.如果两个
的函数
对任意
,都有
,则称
.
(1)对
,分别求一个
,使得
对全体
恒成立;
(2)设集合
和
的函数
以及
的函数
.
(i)对
,构造
的函数
以及
的函数
,满足
;
(ii)对
,构造
的函数
以及
的函数
,满足
,并且说明如果存在其它的集合
满足存在
的函数
以及
的函数
,满足
,则存在唯一的
的函数
满足
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cc020b0997a2f37b214718112b79d8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2317480af8719266bb92ee59d6b2c3b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf13fc6f6469b51cba724df6701f2016.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42acae4bf2a6bead9d904b70d0480fc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f484115a9df1b6060d6b14df85c6f38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42acae4bf2a6bead9d904b70d0480fc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4b9b582f8426f1269b54ee910b97f5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5609d6ec9ef33f11e359f35d27289623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e50933c5e999ca64d3046fe1fa84dbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/169ccf0e6e7ada48126feac1e67de0c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/276509f01529d982ab21e479a4619268.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83c775f6d86ae37c1d2251ee935f17c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42acae4bf2a6bead9d904b70d0480fc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/674c6027fe6ac3a582bced0c2cde8c8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cc020b0997a2f37b214718112b79d8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a719e840660f5c08ec081ee661d7ac77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e9af5ad8e107db8512bc54e627d565d.png)
(1)对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/585be45e95d45f86456790525ef6e24d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/136c6a4d3de85560224cc18a46c60231.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a4e2f595418c82b94ef66a6f526e63d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/636289ad84b4a3a51095dd32ca201f94.png)
(2)设集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/169ccf0e6e7ada48126feac1e67de0c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5609d6ec9ef33f11e359f35d27289623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
(i)对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2836fa0f9aca03548ec72d1bd49d9ff0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07005ca97e96fafcef55ec126ba84f12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0603e63b4cc9944c237e4a776f0688b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e52c39d6ffd048bfa7bc97d849029ea.png)
(ii)对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2836fa0f9aca03548ec72d1bd49d9ff0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07005ca97e96fafcef55ec126ba84f12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0603e63b4cc9944c237e4a776f0688b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e52c39d6ffd048bfa7bc97d849029ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3b1a5427d8ff23df0f3ec194756c84c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686c919499ba2092f18a1555c5e1bce2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a886d45a46bdde67115c5911cb85ea6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5f2e87a902efe698955b1b7d805690b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/735dd0765a19f97a03c9db4babb64ba0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d249478e181957c41657b2c61c1ad50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/916a915b809c8d0028cbe245e0149e83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47f5817ab7b88e9d8a83dd086ffdb3c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3909505a98004589b98f9d14e1bd41d.png)
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解题方法
8 . 已知定义域为
的函数
,其中
代表不超过
的最大整数.设数列
满足:
是
在
上最大值,数列
满足:
且
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ed2f490aac02631c2ed9e6b76354a49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38618cc6b1ffc6c8a84c906260337b3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4737369d70b1a3144eedcb0c924df616.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e976c0663fa749ca749f99842d21ca03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b5cc159f3dfa0534358359ea245338b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adb8851954f243423781831867aedbdf.png)
A.![]() ![]() |
B.![]() ![]() ![]() |
C.![]() |
D.![]() |
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9 . 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)
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10 . 记集合
正n边形可用尺规作出},熟知3,4,5
,7,9,11
,则以下角中能被尺规作出的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db4858547b6b1e37d3122536d5ef66fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfd506a569d791a6d1796b8656ba55cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6e1da46078e9930685d44653cadf878.png)
A.21° | B.25° | C.48° | D.62° |
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