2024高三·全国·专题练习
1 . 下面是应用公式
,求最值的三种解法,答案却各不同,哪个解答错?错在哪里?已知复数
为纯虚数,求
的最大值.
解法一:∵
,
又∵
是纯虚数,令
(
且
),
∴
.
故当
时,即当
时,所求式有最大值为
.
解法二:∵
,∴
.
故所求式有最大值为
.
解法三:∵
,
又∵
为纯虚数,∴
,
∴
.
故所求式有最大值为
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b24b1f5fe3cf65914e79532f4d2b23d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e3958fbc45ee3e72d9a6dc37a8f9474.png)
解法一:∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8201b70b4e9a66d8843dff2e728199c.png)
又∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cdecf72a044cbeb148db4e743c52514.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dd0914dc4d4c7f75710ff460a286fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b03b011f69dfc5262a3d82f64676739b.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7794d59445d4545e6fd58d484fef86d3.png)
故当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a3c442579603164f3fc19458677d307.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/128730d6a25a11ed9b6b0f0e7f4f0433.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b7e9bd225e22d3c95a681720114056f.png)
解法二:∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a61f82d3db0076d8d07b901691021f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c186e2d43b0f52ff872a3613d56f8b1.png)
故所求式有最大值为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/934c6d8e32b31bdcfa263c705b95182b.png)
解法三:∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62981b55a133db7d326bef9d3e73b4c2.png)
又∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed98359cf005d2b49ec68f55d1f87c6e.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c56d0e6036bae5d5a30c2a1f9fff19a0.png)
故所求式有最大值为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c296c04fc52c07364a234c0ac6233022.png)
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名校
解题方法
2 . 在复平面内有一个平行四边形
,点
为坐标原点,点
对应的复数为
,点
对应的复数为
,点
对应的复数为
,则下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3241d7fedd89d85711acd7a2635298af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/038c8559f47ce952773250e879b2bb75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/463df83d50695b3171ad9cda102bb171.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a67a742d2a43e907fb1c3a1bdf1d6a9.png)
A.点![]() | B.![]() | C.![]() | D.![]() |
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2021-07-10更新
|
578次组卷
|
4卷引用:专题7.4 复数的四则运算(重难点题型检测)-2022-2023学年高一数学举一反三系列(人教A版2019必修第二册)
(已下线)专题7.4 复数的四则运算(重难点题型检测)-2022-2023学年高一数学举一反三系列(人教A版2019必修第二册)浙江省浙东北联盟(ZDB)2020-2021学年高一下学期期中数学试题(已下线)第七章 复数(选拔卷)-【单元测试】2021-2022学年高一数学尖子生选拔卷(人教A版2019必修第二册)安徽省合肥市六校联盟2021-2022学年高一下学期期中数学试题
解题方法
3 . 下面命题中错误的是( )
A.![]() ![]() |
B.若两个复数的差是纯虚数,则它们一定互为共轭复数 |
C.若![]() ![]() ![]() ![]() |
D.若两个虚数的和与积都为实数,则它们互为共轭复数 |
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4 . “
”是“复数
”的______ 条件.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3612d008f91a5f75b61ae3c73c76ac81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d32d37e72181718279600fc43d5aa532.png)
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5 . 如图所示,已知复数
,
所对应的向量
,
,它们的和为向量
.请根据两个向量相加的运算写出对应的复数运算过程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7769a9ed79bc3d423437e1ed98b033af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc358e9ddfe2e61a2f3ebd4273de848c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/612af62da1c0563e32852f2e8122ee1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43c6f2f26d1cd1232a8902de0177e395.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4407c0b2e5febf70f610bd00067f105.png)
![](https://img.xkw.com/dksih/QBM/2021/12/1/2863252955602944/2863403750727680/STEM/59180b8be78041e4af71565ea581733d.png?resizew=261)
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6 . 已知复数
,
,
,分别记作
,
,
,即
,
,
,求证:
(1)
;
(2)
;
(3)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e840ea0100bed41c779ba4d30fee7670.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4160c2acea59936bd06a7d7a7aa4514.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/491ff1d34a2e0b5c35ec5b482f9e6dc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af68f652b4c13657ffddf3c9e7eb262b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa224ed9be8766a4d0b5138bd57de0f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a67a742d2a43e907fb1c3a1bdf1d6a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7769a9ed79bc3d423437e1ed98b033af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/935a9ada9a7372c5d296243c435a3c73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf111e7aa8784c1c67b30e1a2ad3671c.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f06ca9d2cc05ecc6fbe8022f12002632.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e866b8803b38e56c63e811cde5885cf.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dc05d7f017b7a96b162888d4162254c.png)
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2023-01-06更新
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151次组卷
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4卷引用:沪教版(2020) 必修第二册 新课改一课一练 第9章 9.1 复数及其四则运算
沪教版(2020) 必修第二册 新课改一课一练 第9章 9.1 复数及其四则运算(已下线)专题7.4 复数的四则运算(重难点题型检测)-2022-2023学年高一数学举一反三系列(人教A版2019必修第二册)(已下线)12.2 复数的四则运算(2)-2022-2023学年高一数学《考点·题型·技巧》精讲与精练高分突破系列(苏教版2019必修第二册)(已下线)专题7.4 复数运算的综合应用大题专项训练-举一反三系列-
21-22高一·全国·课后作业
7 . 复数的加、减法运算法则
设
,
则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffe82a3f7f76832113ecf0e00803a589.png)
____________ ,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35aa5256411e4bfe86bab7783a9022a6.png)
_____________ .
复数加法的运算律
(1)交换律:____________ .
(2)结合律:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12e3811125e6000e04efaaa746377905.png)
___________ .
复数加、减法的几何意义
如图,设在复平面内复数
对应的向量分别为
,以
为邻边作平行四边形,则与
对应的向量是_________ ,与
对应的向量是_____________ .
设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f0223af3b0dacb162f5a1fb7a01ee89.png)
则
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffe82a3f7f76832113ecf0e00803a589.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35aa5256411e4bfe86bab7783a9022a6.png)
复数加法的运算律
(1)交换律:
(2)结合律:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12e3811125e6000e04efaaa746377905.png)
复数加、减法的几何意义
如图,设在复平面内复数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90d860cb86e1467ac24010aecfc7a425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a48ccc06cf3f8dc889e2cfc6b39977b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95aa323fbc002e36c732051858419868.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e34b88f343ca5a4c29057465541b9cf4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e4e2b866b0043a32fc78326553841d3.png)
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8 . 判断正误,正确的填“正确”,错误的填“错误”.
①复数加减乘除的混合运算法则是先乘除,再加减,有括号的先算括号里面的.( )
②复数与复数相加(或相减)后的结果只能是实数.( )
②若
,
,且
,则
.( )
④实数a的共轭复数仍是a本身.( )
⑤
.( )
⑥若
(
为虚数单位),则
.( )
①复数加减乘除的混合运算法则是先乘除,再加减,有括号的先算括号里面的.
②复数与复数相加(或相减)后的结果只能是实数.
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af68f652b4c13657ffddf3c9e7eb262b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d726936fc92297ac4056b3491ce5a26a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1cb45531be54c6d7f02de0c19e6f775.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db6a996e043f3d89b8f90f3e55145872.png)
④实数a的共轭复数仍是a本身.
⑤
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b633664df86c2d29567e79c1473162fd.png)
⑥若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/908638c6598314c0a28faf38df931987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7035cd4adda5d72a9fc9f9fda75995.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba1a29a88a4ac9e429cac9600adaede1.png)
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9 . 类比复数加法的几何意义,请写出复数减法的几何意义.
您最近一年使用:0次
2023-10-09更新
|
30次组卷
|
3卷引用:2.1 复数的加法与减法