解题方法
1 . 下列选项正确的有( )
A.若![]() ![]() ![]() |
B.复数![]() ![]() ![]() ![]() ![]() ![]() |
C.若复数![]() ![]() ![]() ![]() |
D.若复数![]() ![]() ![]() |
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2 . 复数p,q,r在复平面内对应的点分别为P,Q,R,下列说法正确的有( )
A.若![]() ![]() | B.若![]() ![]() |
C.若![]() | D.若![]() ![]() ![]() ![]() |
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名校
3 . 已知复数
的实部分别为
,虚部分别为
,其中
.
(1)求
的取值范围;
(2)
能否为纯虚数,若能,求
;若不能,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90d860cb86e1467ac24010aecfc7a425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94794614e69f70a5a1daccbe0f25d09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92923558626742c01a5c00f5e78673b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88d1585d08f854e80e8172dd968cd21f.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d8b3f66119c2ce542984d12eb2b6b77.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6da4e6752d8c8a0705194f2b2f16ab5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec04f844e8fd9d9b1ef835e23eaa54e2.png)
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4 . 下列说法中正确的有( )
A.梯形可以确定一个平面 |
B.设![]() ![]() |
C.存在一个四面体,四个面均是直角三角形 |
D.在![]() ![]() ![]() ![]() ![]() |
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名校
5 . 已知实系数方程
的两个复根分别为
,
,且
,
.
(1)求a,b的值;
(2)记集合
,判断
,
与集合M的关系.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/800c481c8d3d8e9c3d494ed31bc5de24.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/d7cffbabb3a5cc3c165c0137aa5e082e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2244ce743c2fc657a474450a03856333.png)
(1)求a,b的值;
(2)记集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd6919e95bb11c6cdaa5785189afbdf0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af68f652b4c13657ffddf3c9e7eb262b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa224ed9be8766a4d0b5138bd57de0f0.png)
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6 . 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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解题方法
7 .
被称为“欧拉公式”,之后法国数学家棣莫弗发现了棣莫弗定理:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b58a6a143689e5ed2b3c688d45e251e.png)
,则我们可以简化复数乘法
.
(1)已知
,求
;
(2)已知O为坐标原点,
,且复数
在复平面上对应的点分别为
,点C在
上,且
,求
;
(3)利用欧拉公式可推出二倍角公式,过程如下:
,所以
.
类比上述过程,求出
.(将
表示成
的式子,将
表示成
的式子)(参考公式:
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fdc0ab4d45a4bef21ba8ae793f2e76f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b58a6a143689e5ed2b3c688d45e251e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e6a7030364178c2ef0f6ce638b3ebda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebd10f0306210459baee301dd367ff59.png)
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbe7c60d94b95c996840172915eb6069.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6da4e6752d8c8a0705194f2b2f16ab5d.png)
(2)已知O为坐标原点,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2b08933abf71f9fcb7b284d0bbb5438.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90d860cb86e1467ac24010aecfc7a425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd98a41e273bf640e0d567365fd20077.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d54eacd5cf71d799a3a9e73e929795b.png)
(3)利用欧拉公式可推出二倍角公式,过程如下:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3b47c4f23b5bb2ef3865facaf628223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e82564733fce91b617f1199dae622fbc.png)
类比上述过程,求出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4255fe1b4ac0018a1270e18a6ac9ab31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/864590e14d56eac2957323152c6b4b29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a48345d239aaf8e9ca1ff2846c08a99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2d6c547202109017a8fd210e12b32ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66db91bb3be9e2b6ad567774e3699758.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cee85d22b9fd3c1afea0688617132365.png)
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解题方法
8 . 记
为虚数单位,
为正整数,若
位于复平面的第四象限,则
的最小值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7035cd4adda5d72a9fc9f9fda75995.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/207cc58375e2a75b45760f7dc84043a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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9 . 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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10 . 如图,设复平面上的点
表示复数
,将点
绕原点
旋转90°得到的点
表示哪一个复数?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b40b6895776e0807c2baecbc8f33a8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cee6765a83140d745a6de4c85d9b6b50.png)
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