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Mu Alpha Theta National Convention: Hawaii, 2005
Complex Numbers Topic Test – Alpha Division
1.
A root of a polynomial function is also the x-coordinate of an x-intercept of the function’s
graph (on the Cartesian plane) if and only if the root is ______.
A) complex
2.
E) NOTA
D) 5  3i
E) NOTA
C) 5  9i
B) II
C) III
D) IV
E) NOTA
B) 1
C)  i
D) i
E) NOTA
D) 12
E) NOTA
D) e
E) NOTA
B) 3
C) 4
Which of the following is a value of i i ?

2

B) ie 
C) ie 2
Which of the following expresses the number 2i  2 in polar form?
A) 2cis
9.
D) i
Let x12  i 3 . How many distinct values of x exist?
A) e
8.
C)  i
Evaluate: i 47118
A) 1
7.
B) 1
In which quadrant of the real-imaginary plane does (2i ) lie?
A) –1
6.
E) NOTA
B) 1  3i
A) I
5.
D) real
Let x  3  6i and y  2  3i . What is ( x  y ) ?
A) 1  9i
4.
C) rational
Evaluate: (i 3 )
A) –1
3.
B) integral

4
B) 2 2 cis

4
C) 2cis
3
4
D) 2 2 cis
3
4
E) NOTA
A ray with an endpoint at the origin of the real-imaginary plane passes through the point
(2  2i ) . Another ray with the same endpoint passes through the point (2  2i 3) . Which
of the following is the measure of the angle formed by the two rays?
A) 45
B) 75
C) 90
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D) 105
E) NOTA
Mu Alpha Theta National Convention: Hawaii, 2005
Complex Numbers Topic Test – Alpha Division
10. The polynomial h( x)  8x3  4 x 2  18x  9 has three distinct roots. What is the sum of the
squares of the real components of each of these roots?
A) 0
B)
1
4
C)
9
2
D)
19
4
E) NOTA
11. Let z  cis(3 ) , where 0    2 . Which of the following is equivalent to z 3 for all
values of  ?
A)  cis(3 )
B) cis(6 )
  i 
4
12. Evaluate:
n 0
A) –1
n
C) cis(9 )
D) cis(27 )
E) NOTA
C) 1
D) 1  i
E) NOTA
D) 2i
E) NOTA

B) 0
13. If f ( x, y )  y  xi , evaluate f (1, i) .
C) 1  i
B) 2
A) 0
14. The number
3  2i
can be expressed as a fraction with a real denominator in the form
1  4i
a  bi
, where c is positive; and a, b, and c are integers that are groupwise relatively prime.
c
What is (a  b  c) ?
A) –2
B) 14
C) 18
D) 34
E) NOTA
C)  i
D) i
E) NOTA
997
15. Evaluate:
A) –1
 i 
4n
n 1
B) 1
16. Let z  (2  i ) . Express
3
z
1
2
in the form a  bi , where a and b are rational, and find
2a  4b .
A) 9
B) 12
C) 18
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D) 24
E) NOTA
Mu Alpha Theta National Convention: Hawaii, 2005
Complex Numbers Topic Test – Alpha Division
17. In the quadratic equation y  kx 2  4kx  4 , k is an integer. For what value of k does this
equation have exactly one distinct real root?
A) –2
B) –1
C) 1
D) 2
E) NOTA
18. The leg adjacent to angle  in a right triangle measures half the hypotenuse. What is the

value of (i csc ) 2 , if   ?
2
A) 4
B) 
4
3
C)
4
3
D) 4
E) NOTA
19. Let  be half the sum of the interior angles of a convex heptagon. Evaluate:
(cot   i sin  )
A)  i
B) i
C)
2i 2
2
D)
2  i  1
2
E) NOTA
20. For all real values of  , which of the following is the equivalent of cis ?


A)  cis     B)  cis
2



C)  cis     D)  cis    
2

E) NOTA
21. Let f ( z)  z z 2i . What is f (3  4i) ?
A) 100  75i
B) 120  35i
C) 120 125i
D) 125 120i
E) NOTA
D) 2 10i
E) NOTA
 2i 1 i 
22. Evaluate the determinant of the matrix.  2
i 2i 

 2 1
i 
A) 2  6i
B) 2  2i
C) 2  6i
23. The expansion of (1  2i)5 can be expressed simply as a  bi , where a and b are real. What
is the value of b ?
A) 12
B) 21
C) 36
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D) 38
E) NOTA
Mu Alpha Theta National Convention: Hawaii, 2005
Complex Numbers Topic Test – Alpha Division
24. If f ( x)  1  x and g ( x)  x 2  1 , how many of the following four expressions have real
values?
  1 
f  g  
  2 
A) 1
  3 
f  g  
  2 
B) 2

g

 1 
f  
 2 
C) 3

g

 3 
f  
 2 
D) 4
E) NOTA
25. Let a and b be real numbers such that 0  a  b . Which is the conjugate of
A) a  b i
B) a  b i
C) a  b i
D) a  b i
 a  bi  ?
E) NOTA
26. Let f ( x)  ax5  bx 4  cx 2  2 , where a, b, and c are positive integers. If f ( x ) is never
tangent to the x-axis, how many real roots does f ( x ) have?
A) 0
27. Let m 
A) m
B) 1
C) 2
D) 5
E) NOTA
 1 i 3 
. Which of the following expressions is a value of ln 
 ?
6
 2 
i
B) 2m
C) 4m
D) 5m
E) NOTA
28. What is the probability that h(k ) is real, if h( x) is a function selected at random from the


 x
set  e x , cis   , ( xi)3  and k is a value selected at random from the set
2


  , 2 ,  i , 2 i  ?
A)
1
2
B)
7
12
C)
2
3
D)
3
4
E) NOTA


cos  3   sin(6 )cos  3  
2

29. If each solution to the equation
 0 is substituted into the
3
4cos ( )  3cos( )
expression  5cis   , all resultant numbers are N th roots of the complex number Z, where N
is a whole number. What is the units digit of the least possible positive value of N?
A) 2
B) 4
C) 6
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D) 8
E) NOTA
Mu Alpha Theta National Convention: Hawaii, 2005
Complex Numbers Topic Test – Alpha Division
30. The complex number cis( M ) , where sin  2M   0 , is a root of the cubic polynomial
function g ( x)  ax3  bx 2  cx  d , where {a, b, c, d} 
c
of the following is equal to ?
a
A) 5  2cos  M 
B) 1  10cos( M )
C) cos(2M )  10cos(M )
D) 12
Page 5 of 5
. If 5 is also a root of g ( x ) , which
E) NOTA