Download Alpha- Complex Numbers

Alpha: Complex Numbers
FAMAT State Convention 2016
Answer choice E. NOTA means None of the Above. Otherwise, let 𝑖 = √−1, and let
𝑐𝑖𝑠(𝑥) = cos(𝑥) + 𝑖𝑠𝑖𝑛(𝑥) = 𝑒 𝑖𝑥 . The notation 𝑧̅ means the complex conjugate of z. Good luck!
1. Let 𝑧 = 2 − 3𝑖. Find (1 − 𝑧)(1 + 𝑧).
A. 13 − 12𝑖
B.
6 + 12𝑖
C.
14 + 12𝑖
D.
−4 − 12𝑖
E. NOTA
5
1
2. Which of the following is equivalent to the complex number [2 (1 + √3𝑖)] ?
A.
1 𝜋𝑖
𝑒3
32
B.
5𝜋
32𝑒 3 𝑖
C.
𝜋
D.
𝑒 3𝑖
𝜋
32𝑒 3 𝑖
E. NOTA
3. Let 𝑧1 , 𝑧2 , 𝑧3 , 𝑧4 belong to the set of complex numbers. What of the following is/are true?
i. |𝑧1 𝑧2 𝑧3 𝑧4 | = |𝑧1 ||𝑧2 ||𝑧3 ||𝑧4 |
ii. |𝑧1 + 𝑧2 | = |𝑧1 | + |𝑧2 |
iii. ||𝑧1 | + |𝑧2 || = |𝑧1 | + |𝑧2 |
A. i only
B.
iii only
C.
i and ii only
D.
i and iii only
E. NOTA
−4 + 8𝑖
C.
8
D.
8 + 8𝑖
E. NOTA
4. Let 𝑧 = 1 + 𝑖. Find (𝑧̅)4 .
A. -4
B.
5. 𝑧1 = 3𝑥 + 𝑦 + (𝑥 + 2𝑦)𝑖, 𝑧2 = 𝑥 + 3 − (3 − 𝑦)𝑖. If 𝑧1 = 𝑧2 , what is 𝑥 + 𝑦, if both x and y belong to
the set of real numbers?
A. -1
B.
0
C.
-3
D.
6
E. NOTA
6. A cubic polynomial 𝑝(𝑥) with integer coefficients has a root at 1 and 1 − 2𝑖. Find 𝑝(−1) if 𝑝(2) = 25.
A. 12
B.
5
C.
-16
7. Which of the following are solutions to the quadratic equation
A.
C.
E.
1
3
𝜋
−22 ± 24 𝑐𝑖𝑠 (− )
8
1
3
𝜋
−22 ± 24 𝑐𝑖𝑠 (− )
6
NOTA
D.
1 2
𝑥
2
-80
E. NOTA
+ √2 𝑥 + 𝑖 = 0?
1
3
𝜋
22 ± 24 𝑐𝑖𝑠 (− )
8
1
3
𝜋
D.
22 ± 24 𝑐𝑖𝑠 (− )
6
B.
For the next two questions, it will be useful to know the following tidbit. In complex analysis, it is
often helpful to rewrite functions of the form 𝒛 = 𝒙 + 𝒚𝒊 in terms of their real and imaginary
components. If you rewrite z as suggested:
8. For the function 𝑓(𝑧) = 𝑧 2 + 2𝑖𝑧, what is the real part of 𝑓(𝑧), in terms of 𝑥 and 𝑦?
A.
𝑥 2 + 𝑦 2 − 2𝑦
B.
𝑥 2 + 𝑦 2 + 2𝑦
C.
𝑥 2 − 𝑦 2 − 2𝑦
D.
𝑥 2 − 𝑦 2 + 2𝑦
E. NOTA
Alpha: Complex Numbers
9. For the function 𝑓(𝑧) =
A.
𝑥2
𝑥
− 𝑦2
10. Let 𝑓(𝑧)
B.
=
𝑧̅
𝑧̅ −2𝑖
1
,
𝑧
FAMAT State Convention 2016
what is the imaginary part (not including the i) of 𝑓(𝑧) in terms of x and y?
𝑥2
𝑦
+ 𝑦2
C.
𝑥2
𝑥
+ 𝑦2
D.
𝑥2
𝑦
− 𝑦2
E. NOTA
. Find 𝑓 −1 (𝑧).
A. 𝑧̅ − 2𝑖
𝑧̅
B.
𝑧̅ + 2𝑖
𝑧̅
C.
2𝑖𝑧̅
𝑧̅ − 1
D.
2𝑖𝑧̅
1 − 𝑧̅
E. NOTA
11. Keeping in mind that the period of the function 𝑔(𝑥) = 𝑐𝑖𝑠(𝑥) is 2𝜋, what is the period of the
function 𝑓(𝑧) = 𝑒 𝑧 ?
A. 2𝜋
2𝜋𝑖
C.
𝜋
D.
𝜋𝑖
E. NOTA
B.
24𝑖
C.
-24
D.
−24𝑖
E. NOTA
B.
0
C.
2
D.
0.5
E. NOTA
B.
12. (√−12)(√−48) =?
A. 24
13. 𝑖 2014 + 𝑖 −2014 =?
A. -2
14. A Gaussian integer is one such that for a complex number of the form 𝑎 + 𝑏𝑖, both a and b belong to
the set of integers. If k is an integer, how many different values of k would make the complex number
10
3−𝑘𝑖
a Gaussian integer?
A. 0
B.
1
C.
2
D.
3
E. NOTA
15. Which of the following is equivalent to 𝑖𝑥 2 + 𝑥 + 12𝑖?
A.
B.
C.
D.
E.
(𝑥 − 3𝑖)(𝑥 + 4𝑖)
(𝑥 + 3𝑖)(𝑥 − 4𝑖)
𝑖(𝑥 − 3𝑖)(𝑥 + 4𝑖)
𝑖(𝑥 + 3𝑖)(𝑥 − 4𝑖)
NOTA
16. Which of the following creates an equivalent expression to (𝑎 + 𝑏𝑖)(𝑐 + 𝑑𝑖)?
A.
C.
E.
𝑐
𝑏𝑖 ] [𝑑𝑖 ]
D. 𝑑𝑒𝑡 ([ 𝑎 −𝑎]) + 𝑑𝑒𝑡 ([𝑏𝑖
𝑑𝑖 𝑐
𝑑𝑖
𝑎 𝑏𝑖
𝑑𝑒𝑡 ([
])
𝑐 𝑑𝑖
𝑎 𝑎 𝑐 𝑐
[
][
]
𝑏𝑖 𝑏𝑖 𝑑𝑖 𝑑𝑖
NOTA
B.
[𝑎
−𝑏𝑖
])
𝑐
17. What complex number has the same angle as the complex number 6 + 8𝑖 but with a magnitude of
one instead?
A. 4 3
+ 𝑖
5 5
B.
2 √5
+
𝑖
3
3
C.
6 √13
+
𝑖
7
7
D.
1 3
+ 𝑖
2 4
E. NOTA
Alpha: Complex Numbers
FAMAT State Convention 2016
(2𝑛+1)
18. ∑2014
𝑛=1 𝑖
A. 𝑖
B.
1
C.
0
D.
−𝑖
E. NOTA
B.
0
C.
2015𝜋
𝑐𝑖𝑠 (
)
1007
D.
1337𝜋
𝑐𝑖𝑠 (
)
1007
E. NOTA
D.
-1
E. NOTA
2𝑛𝜋
19. ∏2014
𝑛=1 𝑐𝑖𝑠 (1007) =?
A. 1
20. If 𝑧 5 − 1 = 0, which of the following is a solution of z?
A.
2𝜋
𝑐𝑖𝑠 ( )
5
B.
3𝜋
𝑐𝑖𝑠 ( )
5
C.
2𝑐𝑖𝑠 (
8𝜋
)
5
21. If the set described in the Argand plane by the equality |𝑧 − 𝑖| = 2 were to be instead sketched on
the Cartesian plane, what would be the formula to model it?
A.
B.
C.
D.
E.
(𝑥 − 1)2 + 𝑦 2
𝑥 2 + (𝑦 − 1)2
(𝑥 + 1)2 + 𝑦 2
𝑥 2 + (𝑦 + 1)2
NOTA
=4
=4
=2
=2
𝜋
22. What is 4𝑐𝑖𝑠 3 represented in rectangular form?
A.
2√3 + 2𝑖
B.
√3 + 2𝑖
C.
2 + 2𝑖√3
D.
√3 + 𝑖
E. NOTA
23. The Julia dragon curve can be created by the formula 𝑧𝑛 = (𝑧𝑛−1 )2 − 2𝑖. If the first iteration is 𝑧1 =
√2
2
+
√2
𝑖
2
, what would be the fourth iteration, 𝑧4 ?
A. −1
B.
−3 + 2𝑖
24. Consider the rational function f ( z ) 
C.
12 + 5𝑖
D.
−5 + 𝑖
2i
. In the partial fraction decomposition
z3  z
E. NOTA
𝐴
𝐵
+𝑧
𝑧−1
𝐶
+ 𝑧+1,
what is 𝐵?
A.
−2 + 3𝑖
B.
−2 − 𝑖
C.
1+𝑖
D.
1−𝑖
E. NOTA
Alpha: Complex Numbers
25. For the rational function f ( z ) 
FAMAT State Convention 2016
z  iz  2
, state all of the discontinuities and whether they
( z  1)( z  1)4
2
2
are removable or non-removable.
A.
B.
C.
D.
E.
Removable: 𝑧 = −𝑖, Non-removable: 𝑧 = 𝑖, 1
Non-removable: 𝑧 = −1
Removable: 𝑧 = 𝑖, Non-removable: 𝑧 = −𝑖, −1
Non-removable: 𝑧 = 1
NOTA
26. Find the principal argument of (1 + √3 𝑖)6 .
𝜋
3
7𝜋
D. 11𝜋
E. NOTA
6
6
𝑖𝜋
27. A polar bear and a dragon are on the complex plane. The polar bear is located at 𝑒 3+ 4 . The dragon is
A. 0
B.
C.
𝑖𝜋
located at 𝑒 4− 3 . Assuming the beasts have no width or length, who is further from the origin, and what
is the ratio of the distance of the further creature to the distance of the nearer creature?
A. Dragon, 4 times
3
B.
Dragon, 𝑒 times
C.
4
Bear, 3 times
D.
Bear, 𝑒 times
E. NOTA
28. In the Argand plane, what figure will the equation |𝑧 + 𝑖| = |𝑧 + 3 − 𝑖| create?
A. A circle.
B.
An ellipse.
C.
A parabola.
D.
A line.
E. NOTA
212 + 572𝑖
D.
4825
E. NOTA
29. If 𝑧 = 3 + 4𝑖 and 𝑤 = 7 + 12𝑖, what is 𝑧𝑤𝑧̅𝑤
̅?
A. 106 + 286𝑖
B.
4225
C.
30. Describe the geometric transformation that would always take a complex 𝑧 to a corresponding
complex number i𝑧.
A.
B.
C.
D.
E.
Reflection about the x axis.
Rotation of 90 degrees clockwise.
Vertical translation of 𝑖 units
Horizontal translation of 𝑖 units.
NOTA