解题方法
1 . 1799年,哥廷根大学的高斯在其博士论文中证明了如下定理:任何复系数一元
次多项式方程在复数域上至少有一根(
).此定理被称为代数基本定理,在代数乃至整个数学中起着基础作用.由此定理还可以推出以下重要结论:
次复系数多项式方程在复数域内有且只有
个根(重根按重数计算).对于
次复系数多项式
,其中
,
,
,若方程
有
个复根
,则有如下的高阶韦达定理:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68be203b2490ecce4c0e2eadeb5d911b.png)
(1)在复数域内解方程
;
(2)若三次方程
的三个根分别是
,
,
(
为虚数单位),求
,
,
的值;
(3)在
的多项式
中,已知
,
,
,
为非零实数,且方程
的根恰好全是正实数,求出该方程的所有根(用含
的式子表示).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e167b43045b3297248e334c41c621b8f.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/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b024d78f428194127b5534f948810def.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7230de53663c75658c58bbf206a0085.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bed25da42194b5a81d123933d5704f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd3759b3561834cdc5b499b91f3850d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b92b70365c63607daecdc8deb73ecf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c83590c4a7ea5636843dd4b60c67cb40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68be203b2490ecce4c0e2eadeb5d911b.png)
(1)在复数域内解方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4800c5aa0e5b70b2141541cbd3853e34.png)
(2)若三次方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac603c0b3d1d7fd42bd50222b6ab94d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6755cd39b121a0dd2a14da8d43c1fff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ddb97874a62bb5530514a467d64af13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8079c5a2d8674d322f7abe6d4ef4a3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7035cd4adda5d72a9fc9f9fda75995.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
(3)在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5818ede14d21f6df9ef9c2bfe09286c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b024d78f428194127b5534f948810def.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cb3db0a99f86232e0cf3e55c789ea99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e2e2674707c28eddd3f3ab60f73f54f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c37d6353f394a5704a92113908a5c3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b92b70365c63607daecdc8deb73ecf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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2 . 定义域是复数集的子集的函数称为复变函数,
就是一个多项式复变函数.给定多项式复变函数
之后,对任意一个复数
,通过计算公式
,
可以得到一列值
.如果存在一个正数
,使得
对任意
都成立,则称
为
的收敛点;否则,称为
的发散点.则下列选项中是
的收敛点的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cd1b0f6b19f0e80f4157644e6cb8794.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5c01e03a93ade8659780af659f12e09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adcc78b50e4eb2ef6076b0ef4fab732d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ddd911777ff68a15267547fa66e1f91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd4613271f782a90ab580131d09d03d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/846098520aedb969a09cf9d7991fd9ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f61defc1a0455663c533df06d20ea40a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd4613271f782a90ab580131d09d03d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adcc78b50e4eb2ef6076b0ef4fab732d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5c01e03a93ade8659780af659f12e09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5c01e03a93ade8659780af659f12e09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cd1b0f6b19f0e80f4157644e6cb8794.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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3 . 对于非空集合
,定义其在某一运算(统称乘法)“×”下的代数结构称为“群”
,简记为
.而判断
是否为一个群,需验证以下三点:
1.(封闭性)对于规定的“×”运算,对任意
,都须满足
;
2.(结合律)对于规定的“×”运算,对任意
,都须满足
;
3.(恒等元)存在
,使得对任意
,
;
4.(逆的存在性)对任意
,都存在
,使得
.
记群
所含的元素个数为
,则群
也称作“
阶群”.若群
的“×”运算满足交换律,即对任意
,
,我们称
为一个阿贝尔群(或交换群).
(1)证明:所有实数在普通加法运算下构成群
;
(2)记
为所有模长为1的复数构成的集合,请找出一个合适的“×”运算使得
在该运算下构成一个群
,并说明理由;
(3)所有阶数小于等于四的群
是否都是阿贝尔群?请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38bdb8d4c486c37ac64517ed8d60888b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1240721fdf6d8e1ed9c1158ae723637.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c88c74a9c2efc329f92aa4203f0f780.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c88c74a9c2efc329f92aa4203f0f780.png)
1.(封闭性)对于规定的“×”运算,对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c9cbad1e8b405feac6e8fe403f024b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/830afd1befcf1a92874b5e0bc214578d.png)
2.(结合律)对于规定的“×”运算,对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d4ef2c168b3dba086f2485c3c9cc7c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44d2d88c195317bf5827a1304068f26a.png)
3.(恒等元)存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25e6faeeed98a19d7012c921ca71a046.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/572f76c63e3a74a90a1e6ca5ae401cf8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96e4c22a6a498e197149ce29d9e98fce.png)
4.(逆的存在性)对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/572f76c63e3a74a90a1e6ca5ae401cf8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5463eaf01a62bc6a772301d9e2ad19c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1487cef1d4227621d9311541dec87156.png)
记群
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c88c74a9c2efc329f92aa4203f0f780.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9f2eb65f2fe6546a5e318343d25fe66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c88c74a9c2efc329f92aa4203f0f780.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9f2eb65f2fe6546a5e318343d25fe66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c88c74a9c2efc329f92aa4203f0f780.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d22c891ccf3768b616c5ddaad575aca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c679fe86736064c65a292db59cb739c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c88c74a9c2efc329f92aa4203f0f780.png)
(1)证明:所有实数在普通加法运算下构成群
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74ba4c1c25324a8875b09d0c855a82ad.png)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d719c0c82f38a886db86c71bbaf8db32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d719c0c82f38a886db86c71bbaf8db32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de4ef5fbf807246591e03d07ba4e3a4e.png)
(3)所有阶数小于等于四的群
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c88c74a9c2efc329f92aa4203f0f780.png)
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4 . 在复平面内,复数
对应向量
(O为坐标原点),设
,以射线Ox为始边,OZ为终边逆时针旋转的角为
,则
,法国数学家棣莫弗发现棣莫弗定理:
,
,则
,由棣莫弗定理导出了复数乘方公式:
,则复数
所对应的点位于( ).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75079e5106e0ebd8fb942c59a67ccc03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eeb0e25bbccbee4a1b9db38b49e87978.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a168ae3862a080547bb71f47244eb8e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f56daf4df0f2bfb7e665bd623cd6f17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45eef4221f949bbea8498b39ac1c136a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c825b7acba8f9997d38806be7b3b87eb.png)
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A.第一象限 | B.第二象限 | C.第三象限 | D.第四象限 |
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2024-03-03更新
|
1012次组卷
|
7卷引用:宁夏银川一中、昆明一中2024届高三下学期3月联合考试(一模)理科数学试卷
宁夏银川一中、昆明一中2024届高三下学期3月联合考试(一模)理科数学试卷(已下线)第2套 重组模拟卷(模块二 2月开学)(已下线)考点7 复数的四则运算 --2024届高考数学考点总动员【练】(已下线)7.1.2复数的几何意义(第1课时)(已下线)第七章 本章综合--方法提升应用【第三练】“上好三节课,做好三套题“高中数学素养晋级之路四川外语学院重庆市第二外国语学校2023-2024学年高一下学期期中考试数学试题(已下线)核心考点4 复数及其运算 B提升卷 (高一期末考试必考的10大核心考点)
5 . 形如
我们称为“二阶行列式”,规定运算
,若在复平面上的一个点A对应复数为
,其中复数
满足
,则点A在复平面内对应坐标为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa5440a1b5d9338efd6976a56432e100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0c8894e0b37af5da23a1c1bffb32017.png)
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A.![]() | B.![]() | C.![]() | D.![]() |
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名校
6 . 随着我国国民教育水平的提高,越来越多的有志青年报考研究生.现阶段,我国研究生入学考试科目为思政、外语和专业课三门,录取工作将这样进行:在每门课均及格(
分)的考生中,按总分进行排序,择优录取.振华同学刚刚完成报考,尚有11周复习时间,下表是他每门课的复习时间和预计得分.设思政、外语和专业课分配到的周数分别为
,则自然数数组![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55b2ce97e1fd47e7a312391a6fd959ab.png)
________ 时,振华被录取的可能性最大.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b3779b4ea5477aebfe85113b0de1d60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3fb0c9c7e30bbc0ec8c3521577ee4fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55b2ce97e1fd47e7a312391a6fd959ab.png)
科目 | 周数 | ||||||||||
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
思政 | 20 | 40 | 55 | 65 | 72 | 78 | 80 | 82 | 83 | 84 | 85 |
外语 | 30 | 45 | 53 | 58 | 62 | 65 | 68 | 70 | 72 | 74 | 75 |
专业课 | 50 | 70 | 85 | 90 | 93 | 95 | 96 | 96 | 96 | 96 | 96 |
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2023-12-13更新
|
349次组卷
|
3卷引用:上海市宝山区2024届高三上学期期末教学质量监测(一模)数学试题
7 . 如图,在
的长方形棋盘的每个小方格中各放一个棋子.如果两个棋子所在的小方格共边或共顶点,则称这两个棋子相连.现从这56个棋子中取出一些,使得棋盘上剩下的棋子没有五个在一条直线(横、竖、斜方向)上依次相连.则最少取出______ 个棋子才可能满足要求.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7684debc1ba5eebfbcfaf790e6757140.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/2/3243bca9-270d-44fb-b055-0d841ffc2351.png?resizew=165)
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8 . “康威生命游戏(Game of Life)”是由剑桥大学约翰•何顿•康威教授设计的一款计算机程序,模拟生命之间既协同又竞争的生存定律.程序界面是一个无限大的网格,程序开始时,在每个方格放置一个生命细胞,用黑色方格表示该细胞为“存活”状态,白色方格(空格)表示该细胞为“死亡”状态,初始时每个细胞随机地设定为“存活”或“死亡”之一的某个状态,然后根据一定的规则计算出下一代每个细胞的状态,画出其细胞的生死分布图,再计算出下一代每个细胞的状态,画出其细胞的生死分布图,以此类推,每个细胞迭代后的状态由该细胞本身的状态及周围8个细胞的状态决定,规则如下表所示:
若某种初始状态在迭代过程中细胞的生死分布图发生改变,并在迭代了若干代之后能够回到初始状态,则称该初始状态对应的图形为“振荡器”.下列四种初始状态中(图中未画出的网格外侧均视为空格),对应的图形为“振荡器”的是______ (填序号).
当代细胞状态 | 存活 | 存活 | 存活 | 死亡 | 死亡 |
周围存活细胞数 | 0或1 | 2或3 | 3 | ||
迭代后细胞状态 | 死亡 | 存活 | 死亡 | 存活 | 死亡 |
模拟规律 | 个体由于得不到同伴的照应而走向死亡 | 既有充足的资源,又有同伴的扶持,保持存活 | 种群过度繁殖,争夺资源,导致个体数量下降 | 模拟繁殖 | ![]() |
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/12/c49088ac-f1a1-4646-9d14-d7562dd853ed.png?resizew=381)
若某种初始状态在迭代过程中细胞的生死分布图发生改变,并在迭代了若干代之后能够回到初始状态,则称该初始状态对应的图形为“振荡器”.下列四种初始状态中(图中未画出的网格外侧均视为空格),对应的图形为“振荡器”的是
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9 . 给定奇数
,设
是
的数阵.
表示数阵第
行第
列的数,
且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de075cbe45f637a11f53685a018e340a.png)
.定义变换
为“将数阵中第
行和第
列的数都乘以
”,其中
.设
.将
经过
变换得到
,
经过
变换得到
,
,
经过
变换得到
.记数阵
中
的个数为
.
(1)当
时,设
,
,写出
,并求
;
(2)当
时,对给定的数阵
,证明:
是
的倍数;
(3)证明:对给定的数阵
,总存在
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/767f5a4746f04db68386fac3970b1ed1.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7600d2cfbdc6146db96cc545706004f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/604a3f7f0c00236993c4659ff12fd63c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de075cbe45f637a11f53685a018e340a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40f4f7f26c112216d9a548b5ad082ea4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca81e328c01121c81869e7d304f01054.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0393f387f255f7ade0e5c7bf1a8a9a0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39e524eaa9507cc5c8c81a0831d853ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f8e7d8908eb361e60af9da39fc1b1ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10d2c0b8dcec87c9655bbb1a37d9884e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08c9bcf77059762fc46fc437ca8b060c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84221a34d3afe2b194c604aa642aa922.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b003c008551d85cda4fe287d0742216.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bd55ae46a41a37f90a3d745b9e8f879.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88d6bc8e1dc170db94a6caf502b9b187.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90c0d07535b2eca42f44b8109c6ccd11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbf9bc3b31e16bd645b864a346187f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00442d96d695db2c58bf1fb7165fca94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d13fffd82b0a6b66580a17a6e0d2802.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae537905eee4d73c55298fa4280794b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84406a04370c0c0834550b1f22b49d50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
(3)证明:对给定的数阵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/666a2f30e071ff37c5545baa18276fe7.png)
您最近一年使用:0次
10 . 某工艺品修复工作分为两道工序,第一道工序是复型,第二道工序是上漆.现甲,乙两位工匠要完成A,B,C三件工艺品的修复工作,每件工艺品先由甲复型,再由乙上漆.每道工序所需的时间(单位:h)如下:
则完成这三件工艺品的修复工作最少需要( )
原料 时间 工序 | A | B | C |
复型 | 9 | 16 | 10 |
上漆 | 15 | 8 | 14 |
A.43 h | B.46 h | C.47 h | D.49 h |
您最近一年使用:0次
2023-04-22更新
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90次组卷
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3卷引用:江西省名校协作体2023届高三二轮复习联考(二)(期中)数学(文)试题