AnswersMO-Algebra II APerforming Operations with Complex Numbers

Performing Operations with Complex Numbers — Unit test Answers

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The sum of two complex numbers, where the real numbers do not equal zero, results in a sum of 22i. Which statement must be true about the complex numbers?

A
The complex numbers have equal imaginary coefficients.
B
The complex numbers have equal real numbers.
C
The complex numbers have opposite imaginary coefficients.
D
The complex numbers have opposite real numbers.
2
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Review the four points on the complex plane.

Question illustration
A
point A
B
point B
C
point C
D
point D
3

Which region on the unit circle contains a cube root of – 8 – 8i?

A
in quadrant I
B
in quadrant II
C
on the imaginary axis
D
on the real axis
4

Which expression is equivalent to (3 + 5i)2?

A
–16 + 30i
B
6 + 34i
C
–16 – 34i
D
16 + 30i
5

Review the graph of complex number z.

Question illustration
A
(cos(135°) + isin(135°))
Option A
B
(cos(135°) + isin(135°))
Option B
C
(cos(225°) + isin(225°))
Option C
D
(cos(225°) + isin(225°))
Option D
6

Consider the polar curve r = 4sin(2θ). What is the length of each petal, and how many petals does the graph have?

A
The length of each petal is 2, and the number of petals is 2.
B
The length of each petal is 2, and the number of petals is 4.
C
The length of each petal is 4, and the number of petals is 2.
D
The length of each petal is 4, and the number of petals is 4.
7

The radar system in an airport determines the location of inbound and outbound planes once they are within range. The range, based on the signal from the tower, is represented by the polar equation r = 10cos(θ). Which graph represents the signal range of the airport’s radar system?

A
On a polar coordinate plane, a circle goes through (0, 0) and (10, 0).
Option A
B
On a polar coordinate plane, a circle goes through (0, 0) and (10, StartFraction 3 pi Over 2 EndFraction).
Option B
C
On a polar coordinate plane, a circle goes through (0, 0) and (10, pi).
Option C
D
On a polar coordinate plane, a circle goes through (0, 0) and (10, StartFraction pi Over 2 EndFraction).
Option D
8

Which property of multiplication is shown below? If x = 3 + 4i and y = 5 + 6i, x × y = y × x.

A
commutative property
B
identity property
C
distributive property
D
associative property
9

Which values of x and y would make the following expression represent a real number? (6 + 3i)(x + yi)

A
x = 6, y = 0
B
x = –3, y = 0
C
x = 6, y = –3
D
x = 0, y = –3
10

Review the proof of de Moivre’s theorem. Proof of de Moivre's Theorem [cos(θ) + i sin(θ)]k + 1A= [cos(θ) + i sin(θ)]k ∙ [cos(θ) + i sin(θ)]1B= [cos(kθ) + i sin(kθ)] ∙ [cos(θ) + i sin(θ)]C= cos(kθ)cos(θ) − sin(kθ)sin(θ) + i [sin(kθ)cos(θ) + cos(kθ) sin(θ)]D= cos(kθ + θ) + i sin(kθ + θ)E= cos[(k + 1)θ] + i sin[(k + 1)θ]

A
distributive property
B
factoring
C
multiplication rule
D
assumption (for n = k step)
11

What are the conjugate and modulus of –14 – 2i?

A
14 + 2i and
Option A
B
14 + 2i and
Option B
C
–14 + 2i and
Option C
D
–14 + 2i and
Option D
12

Which equation represents the rectangular form of r = 8?

A
x2 + y2 = 16
B
x2 + y2 = 64
C
StartRoot x squared + y squared EndRoot = 16
Option C
D
StartRoot x squared + y squared EndRoot = 64
Option D
13

Consider z = 3 + 3i. What happens to the modulus and argument when z is raised to the 4th power?

Question illustration
A
The modulus increases by a factor of 216, and the argument increases by .
Option A
B
The modulus increases by a factor of 216, and the argument increases by .
Option B
C
The modulus increases by a factor of 1,296, and the argument increases by .
Option C
D
The modulus increases by a factor of 1,296, and the argument increases by .
Option D
14

If z = 1 + i, what is z5?

Question illustration
A
16 + 16i
Option A
B
–16 + 16i
Option B
C
16 – 16i
Option C
D
–16 – 16i
Option D
15

Which graph represents points on the polar curve r = –4sin(3θ)?

A
On a polar coordinate plane, points are at (4, StartFraction 2 pi Over 4 EndFraction), (4, 0), and (4, StartFraction 4 pi Over 3 EndFraction).
Option A
B
On a polar coordinate plane, points are at (4, pi), (4, StartFraction pi Over 3 EndFraction), and (4, StartFraction 5 pi Over 3 EndFraction).
Option B
C
On a polar coordinate plane, points are at (4, StartFraction pi Over 6 EndFraction), (4, StartFraction 5 pi Over 6 EndFraction), and (4, StartFraction 3 pi Over 2 EndFraction).
Option C
D
On a polar coordinate plane, point are at (4, StartFraction pi Over 2 EndFraction), (4, StartFraction 11 pi Over 6 EndFraction), (4, StartFraction 7 pi Over 6 EndFraction).
Option D

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