Lisa's birthday is on October 18th. It's on Wednesday. She is going to have a birthday party. She wants to invite Peter, Gao Wei and Li Yan to her birthday party. They can celebrate after school. Today Lisa's parents go shopping with Lisa. They want to buy a My dream is to work in a hospital.I thin that a(n) __1__ is one of the most important jobs in the world.__2__ people are busy working every day 课时跟踪检测（十） 复数代数形式的乘除运算
层级一 学业水平达标
1．复数(1＋i)(2＋3i)的值为( )
D．－6＋4i
2
2
A．6－4i B．－6－4i
C．6＋4i
解析：选D (1＋i)(2＋3i)＝2i(2＋3i)＝－6＋4i.
2．(全国卷Ⅰ)已知复数z满足(z－1)i＝1＋i，则z＝( )
A．－2－i B．－2＋i
C．2－i
D．2＋i
1＋i
解析：选C z－1＝＝1－i，所以z＝2－i，故选C.
i
3．(广东高考)若复数z＝i(3－2i)(i是虚数单位)，则z＝( )
D．3－2i
2
A．2－3i B．2＋3i
C．3＋2i
解析：选A ∵z＝i(3－2i)＝3i－2i＝2＋3i，∴z＝2－3i.
4．(1＋i)－(1－i)的值是( )
A．－1 024 B．1 024
C．0
210
10
10
10
20
20
20
20
D．512
210
解析：选C (1＋i)－(1－i)＝[(1＋i)]－[(1－i)]＝(2i)－(－2i)＝(2i)－
(2i)＝0.
10
2＋ai
5．(全国卷Ⅱ)若a为实数，且＝3＋i，则a＝( )
1＋i
A．－4 B．－3
C．3
解析：选D
D．4
2＋ai(2＋ai)(1－i)a＋2a－2
＝＝＋i＝3＋i，1＋i(1＋i)(1－i)22
a＋2
??2＝3，所以?a－2
??2＝1，
解得a＝4，故选D.
a
6．(天津高考)已知a，b∈R，i是虚数单位，若(1＋i)(1－bi)＝a，则的值为________．
b
解析：因为(1＋i)(1－bi)＝1＋b＋(1－b)i＝a，
又a，b∈R，所以1＋b＝a且1－b＝0，得a＝2，b＝1，
Lisa's birthday is on October 18th. It's on Wednesday. She is going to have a birthday party. She wants to invite Peter, Gao Wei and Li Yan to her birthday party. They can celebrate after school. Today Lisa's parents go shopping with Lisa. They want to buy a My dream is to work in a hospital.I thin that a(n) __1__ is one of the most important jobs in the world.__2__ people are busy working every day
2
a所以＝2.
b答案：2
7．设复数z＝1＋2i，则z－2z＝________.
解析：∵z＝1＋2i，
∴z－2z＝z(z－2)＝(1＋2i)(1＋2i－2)＝(1＋2i)(－1＋2i)＝－3.
答案：－3
2
a
8．若＝1－bi，其中a，b都是实数，i是虚数单位，则|a＋bi|＝________.
1－i
a
解析：∵a，b∈R，且＝1－bi，
1－i
则a＝(1－bi)(1－i)＝(1－b)－(1＋b)i，
?a＝1－b，?∴???0＝1＋b.
??a＝2，∴???b＝－1.
∴|a＋bi|＝|2－i|＝22＋(－1)2＝5.
答案：5
(i－2)(i－1)－3－2i
9．计算：＋.(1＋i)(i－1)＋i2－3i
解：因为
(i－2)(i－1)(i－2)(i－1)(i－2)(i－1)－3－2i
＝＝＝i－1，＝
(1＋i)(i－1)＋ii2－1＋i－2＋i2－3i
(－3－2i)(2＋3i)－13i
＝＝－i，
(2－3i)(2＋3i)13
(i－2)(i－1)－3－2i
所以＋＝i－1＋(－i)＝－1.
(1＋i)(i－1)＋i2－3i
10．已知z为z的共轭复数，若z·z－3iz＝1＋3i，求z.
解：设z＝a＋bi(a，b∈R)，
则z＝a－bi(a，b∈R)，
由题意得(a＋bi)(a－bi)－3i(a－bi)＝1＋3i，
即a＋b－3b－3ai＝1＋3i，
??a2＋b2－3b＝1，则有?
?－3a＝3，?
2
2
Lisa's birthday is on October 18th. It's on Wednesday. She is going to have a birthday party. She wants to invite Peter, Gao Wei and Li Yan to her birthday party. They can celebrate after school. Today Lisa's parents go shopping with Lisa. They want to buy a My dream is to work in a hospital.I thin that a(n) __1__ is one of the most important jobs in the world.__2__ people are busy working every day
??a＝－1，解得?
?b＝0，?
??a＝－1，
或?
?b＝3.?
所以z＝－1或z＝－1＋3i.
层级二 应试能力达标
1．如图，在复平面内，点A表示复数z，则图中表示z的共轭复数
D．D的点是( )A．AB．BC．C
解析：选B 设z＝a＋bi(a，b∈R)，且a＜0，b＞0，则z的共轭复数为a－bi，其中
2．设a是实数，且
a＜0，－b＜0，故应为B点．
1＋ai
∈R，则实数a＝( )1＋i
A．－1 B．1
C．2
D．－2
1＋ai1＋ai
解析：选B 因为∈R，所以不妨设＝x，x∈R，则1＋ai＝(1＋i)x＝x＋xi，
1＋i1＋i
??x＝1，
所以有?
?a＝x，?
所以a＝1.
3．若a为正实数，i为虚数单位，?
?a＋i?＝2，则a＝( )??i?
A．2 B.3
C.2
D．1
a＋i?a＋i?＝|1－ai|＝1＋a2＝2，解得a解析：选B ∵＝(a＋i)(－i)＝1－ai，∴??i?i?
4．计算
＝3或a＝－3(舍)．
(－1＋3i)3－2＋i
＋的值是()
(1＋i)61＋2i
A．0 B．1
C．i
D．2i
(－1＋3i)3(－2＋i)(1－2i)(－1＋3i)3－2＋4i＋i＋2
解析：选D 原式＝＋＝＋[(1＋i)2]3(1＋2i)(1－2i)(2i)35
13
－＋i3221i＝＋i＝＋i＝＋i＝2i.
－i－i(－i)i
z1
5．若z1＝a＋2i，z2＝3－4i，且为纯虚数，则实数a的值为________．
z2
高中数学课时跟踪检测十复数代数形式的乘除运算新人教A版选修1_2
