1 . 现有以下三个式子:①
;②
;③
(
为虚数单位),某同学在解题时发现以上三个式子的值都等于同一个常数.
(1)从三个式子中选择一个,求出这个常数;
(2)根据三个式子的结构特征及(1)的计算结果,将该同学的发现推广为一个复数恒等式,并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2636734f83d9695c884bc6425dfddefc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99a25dc4b651f21fb5300acfbd219fda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/941d28d27193013ce38648b53997d731.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/579743813856e2a9183f5ec6eaaefbb2.png)
(1)从三个式子中选择一个,求出这个常数;
(2)根据三个式子的结构特征及(1)的计算结果,将该同学的发现推广为一个复数恒等式,并证明你的结论.
您最近一年使用:0次
2022-08-19更新
|
185次组卷
|
3卷引用:高考新题型-复数
2 . 设复数数列
满足:
,且对任意正整数n,均有:
.若复数
对应复平面的点为
,O为坐标原点.
(1)求
的面积;
(2)求
;
(3)证明:对任意正整数m,均有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/078c417ea54a5065c1f72941b9e4b0be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b670aca396b96eaf2c553b1ca84486dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a383b03b4869ea984d58b8d87c35402.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/012d2d40a71783e79d67e7fbb01bc93a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8312ce4d9e9f0aff13e64d93fbea921e.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5927f1967a8f72e8fb887edb5023a921.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9fec8f8ce956e4621c34db6218ed072.png)
(3)证明:对任意正整数m,均有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea99696f6df9d98c2dcc87832002874.png)
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真题
3 . 设复数
和
满足关系式
,其中A为不等于0的复数.证明:
(1)
;
(2)
.
![](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/092fb0a02a678a65ce112d93bd3ba68d.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/987ff50ba187fc62a37198e3d1bf5631.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da4aea955a2a0d4a2c11d3acb5121efa.png)
您最近一年使用:0次
2022-11-09更新
|
189次组卷
|
2卷引用:1987年普通高等学校招生考试数学(理)试题(全国卷)
4 . 已知复数
的共轭复数为
,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/495e7188f9b646384d5edfce8165d84f.png)
(1)证明:
是一个定值,并求出这个定值;
(2)是否存在实数
,使得对于任意的复数
,
总是实数?若存在,求出
的值;若不存在,试说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fcfebd9f5a57036e6df6b6e14865da3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/495e7188f9b646384d5edfce8165d84f.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de971553ea8a66d7849b138a4a0625c5.png)
(2)是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28db77e4dd104b2533d50a571986de75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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5 . 若
为虚数,且
,求证
为纯虚数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6db0e748196e89b9d821e0289c751d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fd149f07fc77d1a32b9ed4d40737e71.png)
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