Test:
Complex Numbers IV - Ambitious
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Question 1:
Find
3
cube roots of
z
=
8
(
cos
81
°
+
i
sin
81
°
)
A
{
z
1
=
8
27
°
z
2
=
8
54
°
z
3
=
8
81
°
B
{
z
1
=
2
27
°
z
2
=
2
147
°
z
3
=
2
267
°
C
{
z
1
=
2
27
°
z
2
=
2
54
°
z
3
=
2
81
°
D
{
z
1
=
8
27
°
z
2
=
8
147
°
z
3
=
8
267
°
Question 2:
Find
5
fifth roots of
z
=
14
305
°
A
{
z
=
1
2
61
°
z
=
2
2
111
°
z
=
3
2
161
°
z
=
4
2
201
°
z
=
5
2
251
°
B
{
z
=
1
2
61
°
z
=
2
2
133
°
z
=
3
2
203
°
z
=
4
2
274
°
z
=
5
2
345
°
C
{
z
=
1
1.695
60
°
z
=
2
1.695
120
°
z
=
3
1.695
180
°
z
=
4
1.695
240
°
z
=
5
1.695
300
°
D
{
z
=
1
1.695
61
°
z
=
2
1.695
133
°
z
=
3
1.695
205
°
z
=
4
1.695
277
°
z
=
5
1.695
349
°
Question 3:
Find the principal root of the
3
cube roots of the
z
=
5
(
cos
270
°
+
i
sin
270
°
)
A
1.71
330
°
B
5
3
90
°
C
1.71
210
°
D
1.71
90
°
Question 4:
Find the expression for
sin
4
θ
in terms of
sin
θ
(De Moivre’s theorem)
A
8
cos
4
θ
−
8
cos
3
θ
+
2
B
3
cos
4
θ
+
8
cos
3
θ
+
1
C
8
cos
4
θ
−
8
cos
2
θ
+
1
D
2
cos
θ
−
3
cos
θ
+
5
Question 5:
Find the expression for
sin
3
θ
A
sin
3
θ
=
1
7
i
sin
5
θ
+
1
6
i
sin
3
θ
B
sin
3
θ
=
−
2
5
i
sin
3
θ
−
1
4
i
sin
2
θ
C
sin
3
θ
=
1
4
i
sin
3
θ
−
3
4
i
sin
θ
D
sin
3
θ
=
−
2
i
sin
3
θ
−
1
2
i
sin
θ
Question 6:
Find the locus defined as
|
z
|
=
8
, where
z
=
x
+
i
y
A
2
x
2
+
3
y
2
−
2
x
+
y
=
0
B
x
2
+
y
2
−
64
=
0
C
x
2
+
y
2
+
8
=
0
D
x
2
+
y
2
−
8
=
0
Question 7:
Find the locus
arg
z
=
π
3
, where
arg
z
=
tan
−
1
{
y
x
}
A
y
=
6
x
2
+
7
B
y
=
1.5
x
2
−
2
x
C
y
=
3
x
D
y
=
−
1
2
x
Question 8:
Find the equation of the locus
|
z
|
=
x
2
+
y
2
, where
|
z
+
2
z
−
2
|
=
3
A
8
x
2
+
9
y
2
−
40
x
+
32
=
0
B
2
x
2
−
2
y
2
+
36
x
−
9
=
0
C
4
x
2
+
4
y
2
−
12
x
+
9
=
0
D
7
x
2
+
8
y
2
−
12
x
+
13
=
0
Question 9:
Determine the equation of the loci
|
z
−
3
z
|
=
4
A
−
3
x
2
−
y
2
+
8
x
+
1
=
0
B
5
x
2
+
5
y
2
+
2
x
−
3
=
0
C
−
7
x
2
+
3
y
2
+
5
x
−
3
=
0
D
2
x
2
−
3
y
2
+
5
x
+
2
=
0
Question 10:
Find the locus of the
z
=
x
+
i
y
, where
arg
(
z
+
2
)
=
−
π
4
A
y
=
−
x
+
1
B
y
=
−
x
−
2
C
y
=
x
+
1
D
y
=
x
+
2
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