1 . 对于函数
,分别在
处作函数
的切线,记切线与
轴的交点分别为
,记
为数列
的第n项,则称数列
为函数
的“切线-
轴数列”,同理记切线与
轴的交点分别为
,记
为数列
的第n项,则称数列
为函数
的“切线-
轴数列”
(1)设函数
,记
“切线-
轴数列”为
,记
为
的前n项和,求
.
(2)设函数
,记
“切线-
轴数列”为
,猜想
的通项公式并证明你的结论.
(3)设复数
均为不为0的实数,记
为
的共轭复数,设
,记
“切线-
轴数列”为
,求证:对于任意的不为0的实数
,总有
成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0ed953d6e0bd80a5da66552c7bfbcbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd3d48fa9493a86f262569df235a82ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/086e9b14c35ef3c57b20f5e952ebf9c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/086e9b14c35ef3c57b20f5e952ebf9c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8292395dc894796602a60486e575a808.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a79b9eaa5e7ab7a1e5c512b571914dc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36460040ddea4761eee10d537b14a1f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36460040ddea4761eee10d537b14a1f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
(1)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8ef5756fec38d1b4dc62358b45c3352.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(2)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10e845964df4b271bd7b4cf99ede79be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.png)
(3)设复数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80bcf306a6aa8554d1d7fc8317f4e946.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fcfebd9f5a57036e6df6b6e14865da3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec5f303f666e164582da05968d9d8cfd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18c49ca8562b98657ca9c499093f7233.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38ef4c4439b36c2847b0056a116d56d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe6f044a9b06a7a769e613c977cbfb87.png)
您最近一年使用:0次
2024-01-01更新
|
443次组卷
|
7卷引用:上海市普陀区桃浦中学2022-2023学年高二下学期期中数学试题
上海市普陀区桃浦中学2022-2023学年高二下学期期中数学试题(已下线)模块一专题1【练】《导数的概念、运算及其几何意义》单元检测篇B提升卷(人教A2019版)(已下线)模块二 专题1 与曲线的切线相关问题(已下线)模块二 专题3 与曲线的切线相关问题(人教B版)(已下线)模块一 专题1 《导数的概念、运算及其几何意义》B提升卷(苏教版)(已下线)模块二 专题1 与曲线的切线相关问题(苏教版高二)(已下线)模块二 专题4 与曲线的切线相关问题(高二北师大版)
2 . 已知复数
满足
.
(1)求证:
;
(2)若
的虚部为正数,求
,
,
,
,根据
的规律,求出
的值(不需要证明).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8646eaa05bfde39d27813c301a076420.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5c6b89b69721e00cdfb30bef61a4916.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0a89e3c30f6e4d4c5db4378b05d987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d800f03de80068a1172beac3a2c75587.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3336c8ed5361c10c37300e41e03f9f2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ccdb989a252ffac6e46c5a80979043c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ace3db19b2dd8d613cfbbb1abdfddc3.png)
您最近一年使用:0次
3 . 现新定义两个复数
(
、
)和
(
、
)之间的一个新运算
,其运算法则为:
.
(1)请证明新运算
对于复数的加法满足分配律,即求证:
;
(2)设运算
为运算
的逆运算,请推导运算
的运算法则.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4734ba5631121fac6a8488d56ab05d58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebfabd3af39419289bfeb15e2ffc672c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d14d00cc1a2b2d219e9140c9c61ae01b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a5103387b9d41429d9882ea1f2e9e3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e16415b61722f9961e412386e6819f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ab1f1bc414abe3f0c2ccbc537af2a5e.png)
(1)请证明新运算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e16415b61722f9961e412386e6819f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c214bf6ceecb02886ca24322fe2ddb15.png)
(2)设运算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3334218505dc667679059f10d5c5b93a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e16415b61722f9961e412386e6819f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3334218505dc667679059f10d5c5b93a.png)
您最近一年使用:0次
2020-07-16更新
|
326次组卷
|
6卷引用:上海市静安区2019-2020学年高二下学期期末数学试题
上海市静安区2019-2020学年高二下学期期末数学试题(已下线)第三章 数系的扩充与复数的引入【专项训练】-2020-2021学年高二数学(理)下学期期末专项复习(人教A版选修2-2)沪教版(2020) 必修第二册 同步跟踪练习 第9章 复数 9.1~9.2 阶段综合训练苏教版(2019) 必修第二册 必杀技 专练1 新定义、新情境专练沪教版(2020) 必修第二册 同步跟踪练习 第9章 9.1~9.2阶段综合训练(已下线)7.2.2 复数的乘、除运算 (精讲)(2)-【精讲精练】2022-2023学年高一数学下学期同步精讲精练(人教A版2019必修第二册)
名校
4 . 在复平面内复数
,
所对应的点为
,
,
为坐标原点,
是虚数单位.
(1)
,
,计算
与
;
(2)设
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/368f9e12546277731776041c73dbe58c.png)
,求证:
,并指出向量
,
满足什么条件时该不等式取等号.
![](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/4bb8477bcc87b1401970171bf57b9ecb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2c1fd680a5d355178273c6d6025eb80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7035cd4adda5d72a9fc9f9fda75995.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c552c2a57bd5d204c2d700ef1112554.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1435039f4745c7e6fbc266636e414705.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6da4e6752d8c8a0705194f2b2f16ab5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f75805768bce2c1699aa5f9e33adbf4.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0bf9b2a7378e73e9fd06c693bfda07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/368f9e12546277731776041c73dbe58c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32d3995815fb78d6abdacc08145a89b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b49f7ebf36aba9ca166881222ca6aa71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a30deb1f343048675b9b231620369668.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1640d3fff861f45c5eb4019943b000f.png)
您最近一年使用:0次
名校
解题方法
5 . 已知复数
,
(1)求证:
;
(2)化简:
;
(3)若
是方程
的一个根,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed3a7e1ea06b2eb0061ad24605eb7fdd.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0377e5c91846b3a0e71f4cc03ca1c9c4.png)
(2)化简:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdf5260cb65d60c90a2b833bee113589.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/074c228ffc7b1e306f8410afe7bc4b5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0aa02320487aa599298f13c2cab97879.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaacfaef44a654c0a1c283ef03fc0550.png)
您最近一年使用:0次
解题方法
6 . 已知复数z满足
.
(1)求z;
(2)若
,
.证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e850a7e02ff18fb7273584abf91cce6f.png)
(1)求z;
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40e72f6b2ef3329828cb8fc873eeba7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3b0d64a53b18a14b5afc092d058f45e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aac8b24eef48a18cb3a01ec5c315bb3c.png)
您最近一年使用:0次
7 . 我们把
(其中
)称为一元
次多项式方程.代数基本定理:任何一元
次复系数多项式方程(即
为实数)在复数集内至少有一个复数根;由此推得,任何一元
次复系数多项式方程在复数集内有且仅有
个复数根(重根按重数计算).那么我们由代数基本定理可知:任何一元
次复系数多项式在复数集内一定可以分解因式,转化为
个一元一次多项式的积.即
,其中
,
为方程
的根.进一步可以推出:在实系数范围内(即
为实数),方程
有实数根,则多项式
必可分解因式.例如:观察可知,
是方程
的一个根,则
一定是多项式
的一个因式,即
,由待定系数法可知,
.
(1)在复数集内解方程:
;
(2)设
,其中
,且
.
(i)分解因式:
;
(ii)记点
是
的图象与直线
在第一象限内离原点最近的交点.求证:当
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e84c81e3949025da4732289b47bc967e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05df956e0764b674a2bbe08189665e5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6368fec0c2c25db7c29b014d60270e97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/212c82587ab19801f2646fd69abc79e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6368fec0c2c25db7c29b014d60270e97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6368fec0c2c25db7c29b014d60270e97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09cc8bcc4cc8654f658e72201394f749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9183f361e3939156795c74554ff0c00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef89860e2407a46b9ffcee5bd92de776.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e84c81e3949025da4732289b47bc967e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/212c82587ab19801f2646fd69abc79e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e84c81e3949025da4732289b47bc967e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86e94c9d2a456af86591f79e6f76563f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2527822fd5ee6ded770ffc9857c41bff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b924d856924e8cf2b172d5cacffe0610.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f2c82aa40a712f2ef6fda7eaeb88a48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7344f58d5f08fab08d4e99baa13fa652.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd7126d6d76248996a222631cc9ea93c.png)
(1)在复数集内解方程:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0604e6606c4d5869aca3ca1e0295ffef.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/536bbd87dd4193314aec2e214e5f05b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83416409fe7c91e72b59afcea75d65ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/653588ca473b428b4a437d6a8ed7a76c.png)
(i)分解因式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e42787c800e5f9c7ac483bea80d8440.png)
(ii)记点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7775aa57ca0e62216f3039ed88dceed0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c520c63104bb6669c3591bd100b10e1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51969fc1a8030cef11cab59267689e89.png)
您最近一年使用:0次
名校
解题方法
8 . 现定义“
维形态复数
”:
,其中
为虚数单位,
,
.
(1)当
时,证明:“2维形态复数”与“1维形态复数”之间存在平方关系;
(2)若“2维形态复数”与“3维形态复数”相等,求
的值;
(3)若正整数
,
,满足
,
,证明:存在有理数
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9dc4e868a310c371ff88075d8a966a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef9d830212489b316bb052455098108e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7035cd4adda5d72a9fc9f9fda75995.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edc8299790d98621b87e73212a2ebb91.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/905dd10639c9fef5ef8d66a124756140.png)
(2)若“2维形态复数”与“3维形态复数”相等,求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c136aaf9b5dedec254a92ce302f4a70c.png)
(3)若正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94742ebbb028c50d7a58e3e8f4ab329c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35490c12e57ecd91af9934cb17b5c927.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ed110fbfeb14003270a1039ba174d0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f02f2606180ffeda602ff9ae747af6f.png)
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2024-05-11更新
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733次组卷
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3卷引用:安徽省合肥市第一中学2023-2024学年高一下学期5月期中联考数学试题
9 . 复平面是人类漫漫数学历史中的一副佳作,他以虚无缥缈的数字展示了人类数学最纯粹的浪漫.欧拉公式可以说是这座数学王座上最璀璨的明珠,相关的内容是,欧拉公式:
,其中
表示虚数单位,
是自然对数的底数.数学家泰勒对此也提出了相关公式:
其中的感叹号!表示阶乘
,试回答下列问题:
(1)试证明欧拉公式.
(2)利用欧拉公式,求出以下方程的所有复数解.
①
;②
;
(3)求出角度
的
倍角公式(用
表示,
).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5aa584db159b0f9bfae801d0134393b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7035cd4adda5d72a9fc9f9fda75995.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041a7c8fc017f596542c5e6ec7d1c40b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/574f94ac7dfd3477b58799e0251bb6a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a260aee25664815506d2720174b03829.png)
(1)试证明欧拉公式.
(2)利用欧拉公式,求出以下方程的所有复数解.
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bde2a8df1f0418c41a6e077c7f5de21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1150e58bbcb15a349fb5b9b5ef708d41.png)
(3)求出角度
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f9d7bbcbeb05fbbb06463120f9a6811.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aefd06c239145a2b6ae87a955aa51414.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8cd112c1cb203187e3c9554617c45b8.png)
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