1 . 判断题
(1)判断:实数集在复数集中的补集是虚数集.( )
(2)判断:满足
的数x只有i.( )
(3)判断:形如
的数不一定是纯虚数.( )
(4)判断:两个复数不相等的一个充分条件是它们的虚部不相等.( )
(5)判断:复数由实数、虚数、纯虚数构成.( )
(1)判断:实数集在复数集中的补集是虚数集.
(2)判断:满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51a11c538a2fdc64c1f118d3d0d7261b.png)
(3)判断:形如
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63abe8503c60f45db13b3fa2e28c73bc.png)
(4)判断:两个复数不相等的一个充分条件是它们的虚部不相等.
(5)判断:复数由实数、虚数、纯虚数构成.
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2024高三·全国·专题练习
2 . 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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3 . 阅读以下材料,判断下列命题的真假
在复数域内,大小成为了没有意义的量,那么我们能否赋予它一个定义呢.在实数域内,我们通常用绝对值来描述大小,而复数域中也相应的有复数的模长来代替绝对值,于是,我们只需定义复数的正负即可.我们规定复数的“长度”即为模长,规定在复平面x轴上方的复数为正,在x轴下方的复数为负,在x轴上的复数即为实数大小.“大小”用符号+“长度”表示,我们用[z]来表示这个复数的“大小”
例如
,
,
,
.
①在复平面上的复数的大小一定大于在它正下方的复数大小;
②在复平面内做一条直线
,
对应的点在该直线上,则
的最小值为
;
③复数
;
④
在复平面上表现为一个半圆;
⑤无法在复平面上找到满足方程
的点.
其中,正确的序号为__________
在复数域内,大小成为了没有意义的量,那么我们能否赋予它一个定义呢.在实数域内,我们通常用绝对值来描述大小,而复数域中也相应的有复数的模长来代替绝对值,于是,我们只需定义复数的正负即可.我们规定复数的“长度”即为模长,规定在复平面x轴上方的复数为正,在x轴下方的复数为负,在x轴上的复数即为实数大小.“大小”用符号+“长度”表示,我们用[z]来表示这个复数的“大小”
例如
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e1a4212ab8791873d6d0ddbfc88265c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14eb682e1e712218094425e49f4eab6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/baf2cb29f6f6700c6dc1683de1b2cbed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8ab778d8ce4074ed517a7b0df099283.png)
①在复平面上的复数的大小一定大于在它正下方的复数大小;
②在复平面内做一条直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c31c4f39399ec245a67db2933ed639f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20b8e7c0a6f877ef89d6e9a78f1dcbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
③复数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ff25025839c012b7136df2f3e8254ca.png)
④
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0da924b9e57c11a99e550a9d9b05cd0.png)
⑤无法在复平面上找到满足方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c36a4e9b7a22c1bc159f2ed1b53b1e2.png)
其中,正确的序号为
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4 . 在复平面内,菱形
对角线交点为原点
,且两条对角线长度之比为2:1,顶点
对应的复数是
,设
,
,
三点对应的复数分别为
,
,
,求
,
,
,并计算出
,
,
三点所对应的复数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.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/246d747e26000397caa6a2d0b88c80d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/795493a60182973eaa42b4a65b29bb19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/111739c84217132b98585838c6bf0c67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80e97bce30ba5879aa86338724584fe4.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/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
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5 . 化简:
,
,
,
,
,
,
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af95b61a7e348dac2ba0e9663288da6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d74cd0d356bab546ffd038cabd3ebc34.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f27404cf78fcd9089ed4aa2a4e8dc436.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61edd463546c93e4a58db673fda6095c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95d53f57e7721519692bd7dffeb372b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8af8342682e2fd493673352623e1a3de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5bf573e5f74d884d6ef940b87df0897.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/addee0167a29fb8b1e0fbee889f87981.png)
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6 . 图中四边形ABCD,DCEF,FEGH都是正方形,用复数方法证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd6e7b446e3329dfe2894751f93cf0c1.png)
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解题方法
7 . (1)在复平面上画出与以下复数
,
,
,
分别对应的点
,
,
,
.
,
,
,
.
(2)求向量
,
,
,
的模.
(3)点
,
,
,
中是否存在两个点关于实轴对称?若存在,则它们所对应的复数有什么关系?
![](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/3b29a77cfdb8d2a0b684389921e1496c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2708fa6298e52f617383efc175b71ddc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b9cb8e6ff801523b0304576cd69fd2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/797e67927616b141ed7c6b83f8b6f4fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fee50575e3ebd56c4f46dd0bbf8e55d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a976c188b072c749662aedf482193dce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f11988aa7569790689bea75a91925eeb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca0c37c074c1b18db19e9a315abe7815.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64a1508dfc7acb431d0ffc31e549e353.png)
(2)求向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96cb7ee166966ed0b1605c263e433cf8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcc77d6da6e8f3ad5c239e7cc6c9930c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7aed22ce98b22693a90269d933bb6c02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3a7cb1e5ee3fe9de342bb85371db7e5.png)
(3)点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2708fa6298e52f617383efc175b71ddc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b9cb8e6ff801523b0304576cd69fd2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/797e67927616b141ed7c6b83f8b6f4fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fee50575e3ebd56c4f46dd0bbf8e55d3.png)
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8 . 下列几个命题,其中正确的命题的个数有( )
(1)实数的共轭复数是它本身;
(2)复数的实部是实数,虚部是虚数
(3)复数与复平面内的点一一对应;
(4)一个复数的共轭复数的共轭复数是它本身.
A.1 | B.2 | C.3 | D.4 |
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解题方法
9 . (1)求复数
的模的最小值;
(2)复数
,若
,
,
,求复数
对应的点的集合形成的图形的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f9770d52839a3d7c4034badba0a2c2a.png)
(2)复数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa701dc94f56378824f7b5ec2afa491c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61a0bcbd3c304589176a837ba79fb7ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/606ef9cb8c9c4f61ab2acc4c11fec693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af1d7f79603414b0bea06846041631af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
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10 . 下列四种说法正确的是( )
A.如果实数![]() ![]() |
B.实数是复数. |
C.如果![]() ![]() |
D.任何数的偶数次幂都不小于零. |
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2023-06-05更新
|
397次组卷
|
9卷引用:人教B版(2019) 必修第四册 北京名校同步练习册 第十章 复数 10.1 复数及其几何意义 10.1.1 复数的概念
人教B版(2019) 必修第四册 北京名校同步练习册 第十章 复数 10.1 复数及其几何意义 10.1.1 复数的概念(已下线)模块四 专题1 小题入门夯实练2(北师大版)(已下线)第05讲 复数的概念-【寒假预科讲义】(人教A版2019必修第一册)(已下线)专题7.1 复数的概念-举一反三系列-(已下线)专题7.5 复数全章九大基础题型归纳(基础篇)-举一反三系列-(已下线)第01讲 7.1.1 数系的扩充和复数的概念-【帮课堂】(人教A版2019必修第二册)(已下线)第7.1.1讲 数系的扩充和复数的概念-同步精讲精练宝典(已下线)12.1 复数的概念-【帮课堂】(苏教版2019必修第二册)(已下线)专题07 复数综合题归类(1) -期末考点大串讲(苏教版(2019))