名校
解题方法
1 . 在复数域中,对于正整数
,满足
的所有复数
称为
次单位根,若一个
次单位根满足对任意小于
的正整数
,都有
,则称该
次单位根为
次本原单位根,规定1次本原单位根为1,例如当
时存在四个
次单位根
,因为
,
,因此只有两个
次本原单位根
,对于正整数
,设
次本原单位根为
,则称多项式
为
次本原多项式,记为
,规定
,例如
,请回答以下问题.
(1)直接写出
次单位根,并指出哪些是
次本原单位根(无需证明);
(2)求出
,并计算
,由此猜想
(无需证明);
(3)设所有
次本原单位根在复平面内对应的点为
,复平面内一点
所对应的复数
满足
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65dc6548571fb407b11bd8e20fc9a860.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c6e88d54d09eb7a4c8e934e296f8357.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/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/874631e1de2f86a9c0c8465db03fc7e2.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/fac3649308b528fd56545ba102dc42d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5948aa4e0018b7e8e2d57f350ca5c718.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4291b447692fcd6becaeda53b10095c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6f79fedb9f7313e14fe9b7823011e5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bd52d1543e19aea6fd5742a4328ddf5.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/adc1b027c5aac5d97ee4eb33005fd9dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09a213315196fb915fe48505cc9f65a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e89220eb96a4757f2988362bc04e80c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ba63d9bf401b254e5857cab89cf27e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/721b4bc405a8fe427893f4656e5918dd.png)
(1)直接写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3304e23f3b0f9569c4140ca89b6498.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3304e23f3b0f9569c4140ca89b6498.png)
(2)求出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ac0b017e80bfa576ff04b9a3a934927.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b962b1bcf29fcfc66941ca4fc14c5ea1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/719446337e4e8f52cf56bba204db24ed.png)
(3)设所有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c748e40ba21ac5063d3bccaa57ef278.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/588283c9af6716f9f56adec76399863a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08b31f74f1bf8831816cede046b1bf50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee56eb9a6c76435dfec59163c289c9fe.png)
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名校
2 . 已知复数z满足
,则
中不同的数有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f494b71d4d63070f5bbe8c6a8f83a6b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da4d65b4267ee08da7d362039abbd4ce.png)
A.4个 | B.6个 | C.2019个 | D.以上答案都不正确 |
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3 . 现定义
,其中
为虚数单位,
为自然对数的底数,
,且实数指数幂的运算性质对
都适用,若
,
,那么复数
等于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7d70c067339e0c34782459c774c50a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e36c9c91220b0f2cbd4a48e8fa90e3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/321c3ad8fd5cd18229e7d996de448120.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c21e257572d59ae5626178d02f3adeb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6147ad0eebe0ef517453a1816700ae60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc228995efbba2e64d7f1de747024cc4.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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