解题方法
1 . 汉诺塔(Tower of Hanoi),是一个源于印度古老传说的益智玩具. 如图所示,有三根相邻的标号分别为A、B、C的柱子, A柱子从下到上按金字塔状叠放着
个不同大小的圆盘,要把所有盘子一个一个移动到柱子B上,并且每次移动时,同一根柱子上都不能出现大盘子在小盘子的上方,请问至少需要移动多少次?记至少移动次数为
,例如:
,
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd23d0f62b8a9a65f548a987709ebf41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6948a20e6292daa6dec7e1b1f81df75d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e13f0b574e0464b11e8febe22f1cf73.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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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
|
511次组卷
|
3卷引用:浙江省绍兴市第一中学2024届高三下学期5月模拟数学试题
名校
解题方法
3 . 欧拉函数
表示不大于正整数
且与
互素(互素:公约数只有1)的正整数的个数.已知
,其中
,
,…,
是
的所有不重复的质因数(质因数:因数中的质数).例如
.若数列
是首项为3,公比为2的等比数列,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80c47847486c103fabb5b4ba4220c6a8.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbc89a53c03cb86fb653bb82128f6cba.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/8ac535d98f300ff35496c66fe3c66a5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8be646cd52d7f2f1714e7542e75810f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adad9633b73dfbbb3d84b4f15979e99e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5b9bd3d8d836eb723be002c86a53740.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b9fa588c3a8ac4df1b963a1f2850163.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80c47847486c103fabb5b4ba4220c6a8.png)
您最近一年使用:0次
2024-06-03更新
|
591次组卷
|
3卷引用:浙江省绍兴市第一中学2024届高三下学期5月模拟数学试题
4 . 盒子中装有大小形状相同的4个小球,其中2个白色2个红色. 每次取一球,若取出的是白球,则不放回;若取出的是红球,则取完放回.
(1)取两次,求恰好一红一白的概率;
(2)取两次,记取到白球的个数为随机变量
,求随机变量
的分布列及均值;
(3)在第2次取出的球是红球的条件下,求第1次取出的球是白球的概率.
(1)取两次,求恰好一红一白的概率;
(2)取两次,记取到白球的个数为随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(3)在第2次取出的球是红球的条件下,求第1次取出的球是白球的概率.
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5 . 若一个
位数,各位从高到低分别为
,且满足
,我们便将其称之为“递减数”.那么正整数之中任取”递减数”,则在其中取到一个偶数概率是____ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b33774314fa55f0959f39973528d5533.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da55ad7ec1ecf2575406ab4d295c789f.png)
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解题方法
6 . 书接上回.麻将学习小组中的炎俊同学在探究完问题后返回家中观看了《天才麻将少女》,发现超能力麻将和现实麻将存在着诸多不同.为了研究超能力麻将,他使用了一些”雀力值”和”能力值”来确定每位角色的超能力麻将水平,发现每位角色在一局麻将中的得分与个人值和该桌平均值之差存在着较大的关系.(注:平均值指的是该桌内四个人各自的“雀力值”和“能力值”之和的平均值,个人值类似.)为深入研究这两者的关系,他列出了以下表格:
(1)①计算
的相关系数
,并判断
之间是否基本上满足线性关系,注意:保留至第一位非9的数.
②求出
与
的经验回归方程.
③以下为《天才麻将少女》中几位角色的”雀力值”和”能力值”:
试估计此四位角色坐在一桌打麻将每一位的得分(近似至百位)
(2)在分析了更多的数据后,炎俊发现麻将中存在着很多运气的成分.为衡量运气对于麻将对局的影响,炎俊建立了以下模型,其中他指出:实际上的得分并不是一个固定值,而是具有一定分布的,存在着一个标准差.运气实际上体现在这一分布当中取值的细微差别.接下去他便需要得出得分的标准差.他发现这一标准差来源自两个方面:一方面是在(1)②问当中方程斜率
存在的标准差
;另一方面则是在不影响平均值的情况下,实际表现“个人值”X符合正态分布规律
.(
为评估得出的个人值.)已知松实玄实际表现个人值满足
,求(1)③中其得分的标准差.(四舍五入到百位)
(3)现在新提出了一种赛制:参赛者从平均值为10开始进行第一轮挑战,之后每一轮对手的”雀力值”和”能力值”均会提升至原来的
.我们设进行了i轮之后,在前i轮内该参赛者的总得分为
;若园城寺怜参加了此比赛,求![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3b244e1fc240f8af39cc432c0bdc688.png)
参考数据和公式:①
;
.
②相关系数
;
经验回归方程
,
,
;
,其中
为回归数据组数.
③对于随机变量
,
,
,
.
④
时,
,
;
⑤对间接计算得出的值
有标准差
满足
.
⑥
;
;
个人值与平均值之差 | 0 | 3 | 6 | 9 | |||
得分 | 0 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
②求出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
③以下为《天才麻将少女》中几位角色的”雀力值”和”能力值”:
角色 | 宫永照 | 园城寺怜 | 花田煌 | 松实玄 |
雀力值 | 24 | 9 | 10 | 4 |
能力值 | 24 | 16 | 3 | 6 |
(2)在分析了更多的数据后,炎俊发现麻将中存在着很多运气的成分.为衡量运气对于麻将对局的影响,炎俊建立了以下模型,其中他指出:实际上的得分并不是一个固定值,而是具有一定分布的,存在着一个标准差.运气实际上体现在这一分布当中取值的细微差别.接下去他便需要得出得分的标准差.他发现这一标准差来源自两个方面:一方面是在(1)②问当中方程斜率
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec9a5a756248e63ccb381391a536c7b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1290917c2c835b61384480b335cc1d13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1100379a4385b9ce064847bc21760adc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20ad52b5d7044b8628cac082b7c12fe8.png)
(3)现在新提出了一种赛制:参赛者从平均值为10开始进行第一轮挑战,之后每一轮对手的”雀力值”和”能力值”均会提升至原来的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d599cb4a589f90b0205f24c2e1fa021e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e304a36d2cdfe1735fb6996bb115b07d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3b244e1fc240f8af39cc432c0bdc688.png)
参考数据和公式:①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da086bd372ecb12ca1f10aa90b3f8719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/974ea4eb8cea88db4ef02e90ec0bd2a0.png)
②相关系数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/368976f08508d324aa73ec6a9ceca54f.png)
经验回归方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/929ef3bed0a4bdd22f39e036506dc481.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9218b61bbc7b5304adf61be07f0a98ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35d35f886f6b590a2db330269ea9d939.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d67dd3a4864e96f1dda457d4ea0a6e24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
③对于随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1290917c2c835b61384480b335cc1d13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8f8641d4e8bbabc1e726417ac3c8cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee1c9871a68a9f90d1a27d3559aa974a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9546031173beb4c429883aae0e16e03b.png)
④
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/116a2ed855825981b8a1192011965989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1f5b5832b3a66d6527d09d2cd2a1d22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9883e09b1ac40ccaebcaec21e2871c56.png)
⑤对间接计算得出的值
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3c6eb9fb2fb74de6696c3c1b90d56e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/331ddc2602421701eef926d55293d9fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b511bbadf9b7a211430994992cde584.png)
⑥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bcd4238a75699911d8cc12b6feb0da8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e744614f8d01a805b4c71a8623740393.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03405dd7ea215f7dd9d4f60dc41e441b.png)
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7 . 定义2*2数表
,
,
,其中
称为
中的元素.
和
属于复数集合. 我们定义
,其中
,
(共
个),
,
,
:
(1)证明:
(i)
.
(ii)
.
(2)若存在A,B,C,使得
,且
,
,
:
(ⅰ)请列出所有满足条件的有序数对
.
(ⅱ)对
,
,
,试验证其是否满足上述条件.
(3)(ⅰ) 在上一题的(ⅱ)中,
以及
合称为Pauli数表. 任意
均可以表示为:
,
为复数,试用
表示
.
(ⅱ) 设
,请证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4b32cd779049dfa4b55b955e8885dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67236be30275de9ab515dc663ad9cc92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b75a92b76b4b5ae3c4571525b102491d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4ba1bbe411bc71bca016d3fd82352f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4ba1bbe411bc71bca016d3fd82352f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86cd9eab3d56d2fe1c6611ccf5be1695.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9fa3364d3a162682f50528680cf4403.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af76198c7bcd4ba4661c8ecb1fbf7930.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/855e9b47dd2df4781211d458bbb2dd40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b0d82117f3aeb73a48ad24e9b563e6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0f9b77ac8d2930cf5bed9fd31aa3791.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c04d42748ae6abf54a9ba61fb833a8fd.png)
(1)证明:
(i)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82643b4b9e28d0227c419de4e66df227.png)
(ii)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb3e9de425ba79559cd0a2feb7d2109a.png)
(2)若存在A,B,C,使得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/163d61d1df6137585249c17bdf1aff0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce53a57261f19484bf9cda89208d87b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc07d8e94e79abf1a6f717096df42c21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43c904bea2eaec0153be54e2ca1abb75.png)
(ⅰ)请列出所有满足条件的有序数对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52aaa08d0d339d57f466de67d31f54c7.png)
(ⅱ)对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/021cb7c01071fa8fb9ae6bc4312d2e4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/425989826f834d15fe877946eb5e674f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce3ad3326af484fb2493e45b1d8a98b7.png)
(3)(ⅰ) 在上一题的(ⅱ)中,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e5d6bebc33fde49c8b270b1887de4a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b05e6ffe5ec0915b52dcaabe337a3c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3356935218122dea72e8a911cc4dd203.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc11dc86b57dab7d0a07bace5ac7a13a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc80e3e48f5109325489205c77f742c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc11dc86b57dab7d0a07bace5ac7a13a.png)
(ⅱ) 设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/225b723bf52d579f519c13c41074cbbd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fe37acacda21f0768703d5217cf14de.png)
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8 . 如图所示,四棱台
,底面
为一个菱形,且
. 底面与顶面的对角线交点分别为
,
.
,
,
与底面夹角余弦值为
.![](https://staticzujuan.xkw.com/quesimg/Upload/formula/192f4f9446c954a291f779d963f90257.png)
平面
;
(2)现将顶面绕
旋转
角,旋转方向为自上而下看的逆时针方向. 此时使得底面与
的夹角正弦值为
,此时求
的值(
);
(3)求旋转后
与
的夹角余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d07c321ebb740613ff53c1d6e496ee85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2a61472439de1ba85cfe33840b775f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d76ce5fcc2291de26b44b4b082df5cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb34d99a48c5d1d8f21af99c1b70ea49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/192f4f9446c954a291f779d963f90257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1633988fd62a652de726ee92a917b52d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)现将顶面绕
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/192f4f9446c954a291f779d963f90257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2eb89294b31ffdd2680b4361e8994d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2236f820e9625bcc1e81f101e9ab7713.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4967af4c76d238fea5695537dfc9e91e.png)
(3)求旋转后
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
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9 . 对于任意的两点
,
,定义
间的折线距离
,反折线距离
,
表示坐标原点. 下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a3efb79f35db8448f3391252ab7d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df332f01628130c084fd46aaca0a4b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8531b72eeaab9286ca4131e1aac2565.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9153b4e560c31f5e9aaca7077dd0df7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
A.![]() |
B.若![]() ![]() |
C.若![]() ![]() ![]() |
D.若存在四个点![]() ![]() ![]() ![]() ![]() |
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10 . 下列条件能确定唯一一个三角形
的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
A.![]() ![]() ![]() ![]() |
B.![]() ![]() |
C.![]() ![]() ![]() |
D.![]() |
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