AnswersMO-Algebra II A-CRPerforming Operations with Complex Numbers

Powers and Roots of Complex Numbers Answers

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Two cell phone towers were used to determine the location of an individual calling for assistance. The signal from one tower is represented by r = + sin(θ), and the signal from the other tower by r = 2sin(θ). The points of intersection of the curves are the possible locations of the individual. Which coordinate is one of the possible locations?

Question illustration
A
(Negative StartRoot 3 EndRoot, StartFraction 7 pi Over 6 EndFraction)
Option A
B
(Negative StartRoot 3 EndRoot, StartFraction 5 pi Over 6 EndFraction)
Option B
C
(StartRoot 3 EndRoot, StartFraction 7 pi Over 6 EndFraction)
Option C
D
(StartRoot 3 EndRoot, StartFraction 5 pi Over 6 EndFraction)
Option D
2
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A choir is performing in front of an audience. A microphone is placed in front of the choir to capture sound. Which polar equation represents the microphone the choir is using?

A
r = 4cos(2θ)
B
r = 4cos(4θ)
C
r = 4 + cos(θ)
D
r = 4 + 4cos(θ)
3

Which equation represents the polar form of x – y = 0?

A
θ = 0
B
Theta = StartFraction pi Over 4 EndFraction
Option B
C
Theta = StartFraction 3 pi Over 4 EndFraction
Option C
D
θ = π
4

Which graph represents the polar curve r = 3 – 2cos(θ)?

A
On a polar coordinate plane, a circle crosses the x-axis at (0, 0) and (5, pi). It has an inner loop at (0, 0).
Option A
B
On a polar coordinate plane, a circle crosses the x-axis at (0, 0) and (5, StartFraction 3 pi Over 2 EndFraction). It has an inner loop at (0, 0).
Option B
C
On a polar coordinate plane, a circle crosses the x-axis at (0, 0) and (5, pi). It has a convex at (0, 0).
Option C
D
On a polar coordinate plane, a circle crosses the x-axis at (0, 0) and (6, StartFraction 3 pi Over 2 EndFraction). It has an inner loop at (0, 0).
Option D
5

Which graph shape represents the polar curve r = 1 + 4sin(θ)?

A
a cardioid
B
a rose curve
C
a limaçon with an inner loop
D
a limaçon without an inner loop
6

Which graph represents points on the polar curve r = 6cos(θ)?

A
On a polar coordinate plane, points are at (0, 0), (6, 0), (4, StartFraction pi Over 4 EndFraction), and (4, StartFraction 7 pi Over 4 EndFraction).
Option A
B
On a polar coordinate plane, points are at (0, 0), (6, StartFraction pi Over 2 EndFraction), (4.5, StartFraction pi Over 4 EndFraction), and (4.5, StartFraction 3 pi Over 4 EndFraction).
Option B
C
On a polar coordinate plane, points are at (0, 0), (6, pi), (4.5, StartFraction 5 pi Over 4 EndFraction), and (4.5, StartFraction 3 pi Over 4 EndFraction).
Option C
D
On a polar coordinate plane, points are at (0, 0), (6, StartFraction 3 pi Over 2 EndFraction), (4.5, StartFraction 5 pi Over 4 EndFraction), and (4.5, StartFraction 7 pi Over 4 EndFraction).
Option D
7

The equation r = 3cos(6θ) represents a rose curve. How many petals does the graph contain?

A
3
B
6
C
12
D
18
8

The equation r = 4sin(θ) represents a circle. What is the radius of the circle, and what is the symmetry of the graph?

A
The radius is 2 units, and the graph is symmetric about the line .
Option A
B
The radius is 4 units, and the graph is symmetric about the line .
Option B
C
The radius is 2 units, and the graph is symmetric about the polar axis.
D
The radius is 4 units, and the graph is symmetric about the polar axis.
9

Which equation represents the rectangular form of ?

Question illustration
A
StartRoot 3 EndRoot x minus y = 0
Option A
B
StartRoot 3 EndRoot x + y = 0
Option B
C
StartRoot 3 EndRoot x minus 3 y = 0
Option C
D
StartRoot 3 EndRoot x + 3 y = 0
Option D
10

An experiment determines that the polar equation r = 5 + 6cos(θ), where r is in meters, represents the greatest distance between two individuals that still allows them to hear each other speak. Which curve represents this scenario?

A
On a polar coordinate plane, a circle goes through (3.75, StartFraction pi Over 2 EndFraction) and (3.75, StartFraction 3 pi Over 2 EndFraction).
Option A
B
On a polar coordinate plane, a circle goes through (4, StartFraction pi Over 2 EndFraction) and (4, StartFraction 3 pi Over 2 EndFraction).
Option B
C
On a polar coordinate plane, a circle goes through (0, 0) and (11, 0). It has an inner loop at (0, 0).
Option C
D
On a polar coordinate plane, a circle goes through (0, 0) and (12, 0). It has a convex at (0, 0).
Option D
11

Which graph represents the polar curve r = 4sin(θ)?

A
On a polar coordinate plane, a rose curve has 8 petals. The petals are between the x- and y-axes.
Option A
B
On a polar coordinate plane, a circle goes through (0, 0) and (4, StartFraction pi Over 2 EndFraction).
Option B
C
On a polar coordinate plane, a rose curve has 8 petals. Four petals are on the x and y-axes.
Option C
D
On a polar coordinate plane, a circle goes through (0, 0) and (4, 0).
Option D
12

Which graph represents the polar curve r = 5 – cos(θ)?

A
On a polar coordinate plane, a circle goes through (4, StartFraction pi Over 2 EndFraction), (5, pi), and (6, StartFraction 3 pi Over 2 EndFraction).
Option A
B
On a polar coordinate plane, a circle goes through (6, StartFraction pi Over 2 EndFraction), (5, pi), and (4, StartFraction 3 pi Over 2 EndFraction).
Option B
C
On a polar coordinate plane, a circle goes through (4, 0), (5, StartFraction 3 pi Over 2 EndFraction), and (6, pi).
Option C
D
On a polar coordinate plane, a circle goes through (5, StartFraction pi Over 2 EndFraction), (6, 0), and (4, pi).
Option D
13

Which equation represents the polar form of x2 + (y + 4)2 = 16?

A
r = 8sin(θ)
B
r = 8cos(θ)
C
r = –8sin(θ)
D
r = –8cos(θ)
14

Which equation represents the polar form of x2 + (y – 6)2 = 36?

A
r = 6cos(θ)
B
r = 12cos(θ)
C
r = 6sin(θ)
D
r = 12sin(θ)
15

Which graph represents the polar curve r = 4cos(3θ)?

A
On a polar coordinate plane, a rose curve has 8 petals. Four petals are on the x and y-axes.
Option A
B
On a polar coordinate plane, a rose curve has 3 petals. One petal goes to (4, 0), another to (4, StartFraction 2 pi Over 3), and another goes to (4, StartFraction 4 pi Over 3 EndFraction).
Option B
C
On a polar coordinate plane, a rose curve has 8 petals. The petals are not on the x- or y-axes.
Option C
D
On a polar coordinate plane, a rose curve has 3 petals. One petal goes to (4, StartFraction pi Over 6 EndFraction), another to (4, StartFraction 5 pi Over 6), and another goes to (4, StartFraction 3 pi Over 2 EndFraction).
Option D
16

A company designs microphones for use in conference rooms. A new microphone needs to focus on the person speaking and not the voices of other people in the room. Which polar equation would best match the desired audible range of the microphone?

A
r = 3
B
r = 3cos(2θ)
C
r = 3cos(3θ)
D
r = 3 + 3cos(θ)

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