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复数练习题加答案
例
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
例
如图,平行四边形
OABC
,顶点
O
、
A
、
C
分别
表示
0
p>
,
3?2i
,
?2
?4i
,试求:
AO
p>
所表示的复数,
BC
所表示的复数
.
对角线
CA
所表示的复数.
p>
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
.
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
例
复数
z
1?1?2i
,
z2??2?i
,
p>
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
:设复数
z1
,
z2
p>
,
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
p>
的复数为
x?yi
则
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
p>
关于原点对称,所以原点
O
为正方形
的中心.
∴
点
O
p>
也是
B
与
D
点的中点,于是由
??0
∴ x?2,y??1.
故
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
例
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
的
值.
分析:
Z2Z1
对应
的复数为纯虚数,利用复数减法先求
出
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
例
z?r
,求
2z?3?4i
对应的点的轨迹方程.
解:
??2z?3?4i<
/p>
,则
2z???3?4i.
又
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.
p>
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.
当
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.
当
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
例
< 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
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
p>
即
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
时
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.
2
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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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/p>
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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
p>
-
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
则
a
+
2
=
0
,
∴a=-
2.
有
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
限
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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