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1 . 阅读材料一:“装错信封问题”是由数学家约翰·伯努利(Johann Bernoulli,1667~1748)的儿子丹尼尔·伯努利提出来的,大意如下:一个人写了
封不同的信及相应的
个不同的信封,他把这
封信都装错了信封,问都装错信封的这一情况有多少种?后来瑞士数学家欧拉(Leonhard Euler,1707~1783)给出了解答:记都装错
封信的情况为
种,可以用全排列
减去有装正确的情况种数,结合容斥原理可得公式:
,其中
.
阅读材料二:英国数学家泰勒发现的泰勒公式有如下特殊形式:当
在
处
阶可导,则有:
,注
表示
的
阶导数,该公式也称麦克劳林公式.阅读以上材料后请完成以下问题:
(1)求出
的值;
(2)估算
的大小(保留小数点后2位),并给出用
和
表示
的估计公式;
(3)求证:
,其中
.
![](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/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66d4e8502106802f1485c3b0f28f2664.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a8412f5256b2b370e421c07f18cc732.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4403d632f9a81e52c6cd135c6834bc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
阅读材料二:英国数学家泰勒发现的泰勒公式有如下特殊形式:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ce152ca98ac7e21237e00667f005b62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35993bd1db970330494665d925c0be7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)求出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/395c6efaa63dcd4ee513323d51c6a7eb.png)
(2)估算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2598975ac1edb754817eada15b9a473e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041a7c8fc017f596542c5e6ec7d1c40b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66d4e8502106802f1485c3b0f28f2664.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca08ded0d1136421f0a81517f5c2fc9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
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2 . 已知复数
,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcb5c7b22f5eaf647d28a5cc7bdaf1fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2d67321ace1e6b3be0fc0e5e8130022.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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3 . 若复数
,则
的共轭复数
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea8f500173e71572e707aeff49064bb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/545141bbe7cc66d18946a129179d8df1.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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4 .
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2cdefcca6c6eb7c9208b3de1c008aa9.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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5 . 已知复数
,
(
为虚数单位)在复平面上对应的点分别为
,则
的面积为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b95423c449f5d44dd949bf3b0130946f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a6cef94c1bce90f6573fd0315db6b8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7035cd4adda5d72a9fc9f9fda75995.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38033198bf936b904a8c74db67e4cdcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5927f1967a8f72e8fb887edb5023a921.png)
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6 . 向量
对应的复数为
,把
绕点
按逆时针方向旋转
,得到
,则
对应的复数为______ (用代数形式表示).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eeb0e25bbccbee4a1b9db38b49e87978.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48cb91d448faf9ad0e0c835b91329b61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eeb0e25bbccbee4a1b9db38b49e87978.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76cb1df84bc3fff3d3c59ad5eb761aec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a30deb1f343048675b9b231620369668.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a30deb1f343048675b9b231620369668.png)
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7 . 设
,则
=______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b1ddfd074012b1736ae721bd63de102.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce48af55c99256efdc68fac0767d944c.png)
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解题方法
8 . 已知复数
,z在复平面内对应的点记为M,则下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/134bc2317c4fc44892042a552464e514.png)
A.若![]() ![]() | B.若z为纯虚数,则![]() |
C.若点M在第一象限,则![]() | D.若![]() ![]() ![]() |
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9 .
,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66d09b4afbd9d413f53dd8cf668b4eb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfd4a842fda2524039430cac053ee979.png)
A.![]() | B.![]() | C.3 | D.![]() |
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10 . 若复数
为纯虚数,则复数
在复平面上的对应点的位置在( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/447f93d57ba462e92481fa2377a20b85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29b3a7faf7b3130c57bd932496bcd8d2.png)
A.第一象限内 | B.第二象限内 |
C.第三象限内 | D.第四象限内 |
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