1 . 阅读以下材料并回答问题:
①单位根与本原单位根:在复数域,对于正整数
,满足
的所有复数
称为
次单位根,其中,满足对任意小于
的正整数
,都有
,则称这种复数为
次本原单位根.例如,
时,存在四个4次单位根
,
,因为
,
,因此只有两个4次本原单位根
;
②分圆多项式:对于正整数
,设
次本原单位根为
,则多项式
称为
次分圆多项式,记为
;例如
;
回答以下问题:
(1)直接写出6次单位根,并指出哪些为6次本原单位根(无需证明);
(2)求出
,并计算
,由此猜想
的结果,(将结果表示为
的形式)(猜想无需证明);
(3)设所有12次本原单位根在复平面上对应的点为
,两个4次本原单位根在复平面上对应的点为
,复平面上一点
所对应的复数
满足
,求
的取值范围.
①单位根与本原单位根:在复数域,对于正整数
![](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/f8823a4054e9766523a9882823753499.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/fac3649308b528fd56545ba102dc42d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e0262791cc14598c6fc5ed0937d0124.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bd52d1543e19aea6fd5742a4328ddf5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4291b447692fcd6becaeda53b10095c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47f0791e93b8f3616e06f9367820e249.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bd52d1543e19aea6fd5742a4328ddf5.png)
②分圆多项式:对于正整数
![](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/9390241d0bf0dc0efa86359e125bc107.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56c06caf7ed1d7cabf8fa1d956ef1dd5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9eea920288aa45211e17a9224327554f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3c4e28cfb407732561ecdfa9f91b228.png)
回答以下问题:
(1)直接写出6次单位根,并指出哪些为6次本原单位根(无需证明);
(2)求出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac784c7091192687523d2fdf403e4718.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/278d261a40d0f23185bedd3ec5476761.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a60cba22d39ec13d93d4b5f51c335fa6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c87605b42336721647fedb5174883a0b.png)
(3)设所有12次本原单位根在复平面上对应的点为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1e57fcd5fb8f222b56f449662144b6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04d468be20b4d43f5de75416de20e8ee.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/eaa72cef7840f8e14c5495a5fbcbcf49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd27b4c4080c7bde2009f7002f53d53b.png)
您最近一年使用:0次
解题方法
2 . 设
是虚数,
是实数,且
,
.
(1)求
;
(2)证明:
为纯虚数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af68f652b4c13657ffddf3c9e7eb262b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/174ce5fa8bd9b7c64f634c73f8e1c238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b815f20fe59376e85e812a5adbcffa0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cc67cd6651d99a82cb90c21b88294c7.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d8b3f66119c2ce542984d12eb2b6b77.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/074c228ffc7b1e306f8410afe7bc4b5c.png)
您最近一年使用:0次
解题方法
3 . 设
是虚数,
是实数,且
.
(1)求
的值以及
的实部的取值范围;
(2)若
,求证:
为纯虚数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af68f652b4c13657ffddf3c9e7eb262b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/589f07815f9268c302e826a4a99025fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/191f90b57850a507e63c707a056a5c83.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d8b3f66119c2ce542984d12eb2b6b77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af68f652b4c13657ffddf3c9e7eb262b.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23cb69ae45d2ffcdca812021b65f7533.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/074c228ffc7b1e306f8410afe7bc4b5c.png)
您最近一年使用:0次