MI 4 Complex Practice Exam
Name __________________________
You may use a TI-30 calculator on this exam.
True and False
_____ 1. 18cis 180 has six 6th roots in the set of complex numbers, none of which are real.
_____ 2. The complex conjugate of 2 cis(40) is 2 cis( 40) .
  
 3 
_____ 3. The complex number 4  4i can be written 4 2 cis 
.
 or 4 2 cis 
 4 
 4 
_____ 4.  2cis 2   8cis8.
3
1 
 1
5. Calculate 

i
2 
 2
2012
. Put your answer in rectangular form.
6. Let z1  12cis40 and z2  3cis10 . Find the following. State your answers in polar form.
a) z12
b) z1 ·z2
c) z1 / z22
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MI 4 Complex Practice Exam
7. One of the solutions to the equation z 4  7  24i is 1  2i . With this information, graph all
four roots on the grid below.
8.
Convert each rectangular representation into its corresponding cis representation:
 3  3i
a.
b.
2i  3
9.
Convert each cis representation into its corresponding rectangular representation:
a.
4cis(45 )
b.
6 2 cis(150o)
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MI 4 Complex Practice Exam
10. Find a complex number that you can multiply z with so that z gets rotated 120 degrees
clockwise, if z  2 cis(50 ) . Give your answer in rectangular form.

3 1 
 i  . Leave your answers in cis form.
11. Solve the equation z 6  64  
 2 2 
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MI 4 Complex Practice Exam
12.
Write a polar equation for each graph below.
r  5sin(7 )
a.
b.
2 
4 


r  3sin   
 or r  3cos   

3 
3 


2 

or r  3cos   
 or
3 

13. Change each polar function to rectangular form.
2
a. r  3cos   r 2  32 r cos 
 x  y  3x
3
r

 2r cos   3r sin   3
b.
2cos   3sin 
 2 x  3 y  3
 
c.

 
 tan   tan   
3
 3
y
   3 or y   3 x
x
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MI 4 Complex Practice Exam
14.
Sketch the graph of r  2  4sin
Label all points (using polar coordinates) where the graph intercepts the horizontal and
vertical axes.
Find all values of  with 0    2 for which the graph passes through the pole (origin).
 
 6, 
 2
3 

 2, 
2 

 2,  
 7 
 0,

 6 
 2, 0 
If the graph passes through the pole, r  0  2  4sin   sin( )  
Then  
1
.
2
7
11
or
.
6
6
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