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复数练习题加答案

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2021-03-03 07:59
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2021年3月3日发(作者:lane)


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复数练习题加答案






计算



?



?




??



分析:根据复数加、减法运算法则进行运算。




解:


???i?6?i.


???[2?]i??7?7i.


????i??11i.



确定向量所表示的复数






如图,平行四边形


OABC


,顶点


O



A



C


分别

< p>



表示


0



3?2i



?2 ?4i


,试求:



AO


所表示的复数,


BC


所表示的复数


.



对角线


CA


所表示的复数.




对角线


OB


所表示的复数及


OB


的长度.



< /p>


分析:要求某个向量对应的复数,只要找出所求的向


量的始点和终 点。或者用向量的相等直接给出所求的结论.




解:


AO??OA


?AO


所表示的复数为


?3?2i




?BC?AO




?BC< /p>


所表示的复数为


?3?2i


< p>


CA?OA?OC




?CA


所表示的复数为


??5?2i


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对角线


OB?OA?AB?OA?OC


,它所对应的复 数为



??1?6i


|OB|??622?37



求正方形的第四个顶点对应的复数






复数


z 1?1?2i



z2??2?i



z3??1?2i


,它们在复平


面上的对应 点是一个正




方形的三个顶 点,求这个正方形的第四个顶点对应的


复数。




分析


1

:利用


AD?BC


或者


AB?DC


求点


D


对应的复数。

< br>



解法


1

< p>
:设复数


z1



z2



z3


所对应的点分别为

< br>A



B



C


,正方形的第四个顶点


D


对应




的复数为

x?yi




AD?OD?OA??


??i


BC?OC?OB???1?3i



∵ A


D?BC




∴?i?1?3i.



?x?1?1



?y?2??3?x?2?y??1∴ ? 解得


?



故点


D

对应的复数


2?i.



分析


2


:利用正方形的性质,对角钱相等且互相平分,


相对顶点连线段的




中点重合,即利用正方形的两条对角线交点是其对称


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中心求解.




解法


2


:设复数


z1



z2



z3


所对应的点分别为


A



B



C


,正方形的第四个顶点


D


对应




的复数为


x?yi



因为点


A


与点


C


关于原点对称,所以原点


O


为正方形


的中心.







O


也是


B



D


点的中点,于是由


??0



∴ x?2,y??1.





D


对应的复数为

2?i.



小结:解题


1


一定要善于发现问题中可能被利用的条


件,寻找最佳的解题方 法,解法


2


利用正方形是如


C


对称固


形,解题思路较巧.




根据条件求参数的值






已知


z1?a2?3?i



z2?a ?1?i


分别对应向量,



OZ1,OZ2


,若向量


Z2Z1


对应 的复数为纯虚数,求


a



值.



分析:


Z2Z1


对应 的复数为纯虚数,利用复数减法先求



Z2Z1


对应的复数,再利用复数为纯虚数的条件求解即得.




解:设向量


Z2Z1


对应复数


z ∵Z2Z1?OZ1?OZ2




∴z?z1?z2?a2?3?i?[a2?1?i]



?[?]?[?]i


??i



2???0?a?a?2?0z∵ 为纯虚数,



? 即


??0????a?a?6?0


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∴ a??1.




求复数的轨迹方程





z?r


,求


2z?3?4i


对应的点的轨迹方程.




解:


??2z?3?4i< /p>


,则


2z???3?4i.



z?r


,故有


2z?2r.



∴ ??2r




∴ ?对应点的轨迹是以


3 ?4i


为圆心,


2r


为半径的圆.




小结:


由减法的几何意义知


z?z1


表示复平面上两点


z



z1


间的距离.




z?z1?r


, 表示复数


z


对应的点的轨迹是以


z1< /p>


对应的点为圆心,半径为


r


的圆.




z?z1?z?z2

,表示


以复数


z1



z2


的对应点为端点的线段的垂直平分线.




求复数的最大值与最小值





< br>设复数满足


z?4?3i?2?2?z?4?3i


,求< /p>


z


的最大值


和最小值.

< br>



分析:仔细地观察、分析等式


z?4?3i?2?2?z?4?3i


,实


质是一实数 等式,由其特点,根据实数的性质知若


a??a


,则

< p>
a?0


,因此已知等式可化为


z?4?3i?2? 0



解:


由已知等式得


z??2?0



z??2?0



它表示的以点


P


为圆心,半径


R?2< /p>


的圆面.



如图可知

z?OQ


时,


z


有最大值


OP?R?5?2?7



z?OM



z


有最小值


OP?R?5 ?2?3



小结:求复数的模的最值常常根据其几何 意义,利用


图形直观来解.




名词的数



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一。单向选择题



1. Have you got __________? I want to write a


letter.


A. any paperB. papers C. any papers D. a paper


. Have you seen such big ________?


A. a tomato B. tomato C. tomatoes D. tomatos


. They got much __________ from those books.


A. ideas B. photos C. information D. stories


. There are a lot of _________ down there but


hardly any __________.


A. sheeps, people B. sheep, people C. sheeps,


peoples D. sheep, peoples


.



What can I do for you?


-- I



d like two ____________.


A.


bottle


of


orange


B.


bottle


of


orangesC.


bottles


of orangeD. bottles of oranges


.


Several


___________


are


talking


under


the


tree.


And their __________ are swimming in the lake.


A.


woman,


childrenB.


woman,


childC.


women,


children D. women, child


. Uncle Wang bought two ____________ yesterday.


A. watchs B. watches C. watch D. watched


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. Tom has made __________ with Jim.


A. a friend B. friendsC. friendD. some friends


.


What


can


I


do


for


you?


I



d


like


two


__________.


A. box of appleB. boxes of applesC. box of apples


D. boxes of apple


10. Let me give you ____________.


A. an adviceB. some advice C. advices D. the


advice


11. He has spent a great _________ of money on


his new house.


A. deal B. many C. number D. piece


12.


I


only


realized


age


when


I


found


I


had


__________.


A. much great hair B. a little great hair C. a


few grey hairs D. some grey hair


13. That farmer has many _________ on his farm.


A. geeesB. goosen C. gooses D. geese


14. When we saw his face, we knew ________ was


sad.


A. some news B. the news C. a newsD. news


15.


The


dentist


didn



t


tell


me


how


many


_________


I should pay for pulling out the tooth.


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A. money B. dollars C. funds D. moneies


16. Before we moved in the new house we bought


many ___________.


A.


furnituresB.


furniture C.


pieces


of


furniture


D. equipments


17. His friend lost _________ nylon stockings


yesterday morning.


A. two B. a C. a couple of D. a pair of


18.


My


family


raise


a


lot


of


_________,


including


two ___________.


A.


cattles,


cows


B.


cows,


cattle C.


cattle,


cows


D. cow, cattles


19.


All


the


________


are


made


of


___________,


not


plastics.


A.


glass,


glass



B.


glasses,


glass


C.


glass,


glasses D. glasses, glasses 0. A group of __________


are eating ________ and_________ at the foot of the


hill.


A.


sheep,


grass,


leaves



B.


sheeps,


grasses,


leaves C.


sheep,


grass,


leaf


D.


sheeps,


grass,


leaf1.


About0


________


of


the


population


in


our


country


________ on farms. < /p>


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A.


percent,


lives


B.


percents,


lives C.


percent,


liveD. percents, live


2. The number of people invited ________ fifty,


but a


number of


them ________


absent


for different


reasons.


A. were, was B. was, wasC. was, were D. were, was


3. My father



s ________ is grey, but my mother


has a few grey _________.


A. hair, hairB. hairs, hairs C. hairs, hair D.


hair, hairs


4. ----How many ____________ does a cow have?


----- Four.


A.


stomaches


B.


stomachC.


stomachs D.


stomachies


5.


Father


went


to


his


doctor


for


____________


about his heart trouble.


A. an advice B. advice C. advices D. the advice


6. The _________ of buildings are covered with


lots of _____________.


A.


roofs,


leaves



B.


rooves,


leafs



C.


roof,


leafD. roofs, leafs


7. As is known to us all, _________ travels much


faster than__________.


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A.


lights,


sounds



B.


light,


soundC.


sound,


lightD. sounds, lights


8. ---What would you like, sir?


---- _____________.


A. Two cups of coffees B. Two coffees C. Two cup


of coffeeD. Two cups coffee


9. These ___________ want to be __________ when


they grow up.


A. girl-students, woman doctorsB. girl-student,


woman doctors


C.


girls-students,


women


doctors



D.


girl-students, women doctors


0. What ________! Where did you get them?


A.


a


big


fish B.


big


fish C.


a


piece


of


fish D.


a piece of big fish


1. I saw _________ running about at the foot of


the mountain.


A.


a


cattle B.


two


cattlesC.


much


cattle


D.


a


head


of cattle


2. ______________! Let



s go climbing!


A. What a fine weatherB. How fine weatherC. What


a fine dayD. How fine day


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3.



Thanks for giving me _________ I wanted.


A.


the


information



B.


an


informationC.


the


informations D. information



二。填入所给单词的正确形式。



1. I have two .


2. There are many here.


3. I received a lot of Christmas. There are many.


Those are


enjoying


the


suise.


.


A


few are


drawing


on


the wall. . . The 10. Yangtze River in one of the


longest in China.



三。写出下列单词的复数形式。



match


_____________


____________foot____________half


fish


piano


___________wish


___________


_____________


____________fox__________


glass____________


cattle


story___________


radio____________


scarf_________


party_____________


life


____________photo______________ leaf ___________ boy


_____________


box


__________


goose_____________


mouse__________ sheep _____________ shop __________


city______________


hero


_________


German___________


zoo


____________


deer


______________man___________


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Negro_____________


baby_________


woman____________


thief____________ tooth____________ child___________


Chinese___________


policeman__________


wife


____________


Japanese___________


brush__________


tomato__________


knife____________


factory__________


monkey___________


wolf


bamboo


_____________


___________


map


_____________


stomach___________


ox__________


girl


student


________________


pencil-box


_______________


man


doctor


_______________


woman


teacher


______________



一、选择题



3



i


1


.复数等于



1



i


A



1


2iB



1


2iC



2


iD



2



i


答案:


C


3



i4

< br>+


2i


解析:=


2



i.


故选


C.


21



i


3



2i3



2i


2


.复数=



2



3i2



3i


A



0 B



C


.-


2iD



2i


答案:


D


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3



2i3



2i13i



13i


解析:==-

i



i



2i.


132



3i2< /p>



3i13


z



2


3


.已知


z


是纯虚数,


z


等于



1



i


A



2i B



i C


.-

iD


.-


2i


答案:


D



解析:由题意得


z



ai.< /p>



z



22< /p>



a


+i∴,



21



i




a



2



0


, ∴a=-


2.



z

=-


2i


,故选


D.


.若


f



x3



x2



x



1


,则


f

< br>=


A



2i B



0 C


.-

2iD


.-答案:


B



解析:依题意,


f


< br>i3



i2


< br>i



1


=-

i



1



i



1



0< /p>



选择


B.


2



i


5


.复数


z


=在复平面内对应的点位于

< br>


1



i


A


.第一象限


B


.第二象限


C


.第三象限


D


.第四象




答案:


D


2



i13



解析:


zi


,它对应的点在第四象限,故选


D.


1



i22


2



ib


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