AnswersMO-Algebra II A-CRPerforming Operations with Complex Numbers

Performing Operations with Complex Numbers — Unit test Answers

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1
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Which expression represents a cube root of 1 + i?

A
3 power root 2 ( cosine ( StartFraction pi over 24 EndFraction) plus i sine (StartFraction pi over 24 EndFraction))
Option A
B
3 power root 2 ( cosine ( StartFraction 3pi over 4 EndFraction) plus i sine (StartFraction 3pi over 4 EndFraction))
Option B
C
6 power root 2 ( cosine ( StartFraction pi over 24 EndFraction) plus i sine (StartFraction pi over 24 EndFraction))
Option C
D
6 power root 2 ( cosine ( StartFraction 3pi over 4 EndFraction) plus i sine (StartFraction 3pi over 4 EndFraction))
Option D
2
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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= ?E= cos[(k + 1)θ] + i sin[(k + 1)θ]

A
cos(kθ – θ) + i sin(kθ – θ)
B
cos(kθ – θ) + i sin(kθ + θ)
C
cos(kθ + θ) + i sin(kθ – θ)
D
cos(kθ + θ) + i sin(kθ + θ)
3

Which expression represents a cube root of –8?

A
2 (cosine ( StartFraction pi over 6 EndFraction) plus i sine (StartFraction pi over 6 EndFraction))
Option A
B
2 (cosine ( StartFraction pi over 4 EndFraction) plus i sine (StartFraction pi over 4 EndFraction))
Option B
C
2 (cosine ( StartFraction pi over 3 EndFraction) plus i sine (StartFraction pi over 3 EndFraction))
Option C
D
2 (cosine ( StartFraction pi over 2 EndFraction) plus i sine (StartFraction pi over 2 EndFraction))
Option D
4

What is (1 +i)5?

A
–4 – 4i
B
–4 + 4i
C
4 – 4i
D
4 + 4i
5

If z = 2 + 2i, what is z4?

A
2 square root 2 (cosine ( StartFraction pi over 4 EndFraction) plus i sine (StartFraction pi over 4 EndFraction))
Option A
B
2 square root 2 (cosine (pi) plus i sine (pi))
Option B
C
64 (cosine ( StartFraction pi over 4 EndFraction) plus i sine (StartFraction pi over 4 EndFraction))
Option C
D
64 (cosine (pi) plus i sine (pi))
Option D
6

What is the modulus after (–1 + ) gets raised to the 6th power?

Question illustration
A
2
B
12
C
32
D
64
7

Given z = , what is z3?

Question illustration
A
12 (cosine ( StartFraction pi over 2 EndFraction) plus i sine (StartFraction pi over 2 EndFraction))
Option A
B
64 (cosine ( StartFraction pi over 2 EndFraction) plus i sine (StartFraction pi over 2 EndFraction))
Option B
C
12 (cosine ( StartFraction 3pi over 2 EndFraction) plus i sine (StartFraction 3pi over 2 EndFraction))
Option C
D
64 (cosine ( StartFraction 3pi over 2 EndFraction) plus i sine (StartFraction 3pi over 2 EndFraction))
Option D
8

Review the graph.

Question illustration
A
A
B
B
C
C
D
D
9

Review the graph of z3.

Question illustration
A
A
B
B
C
C
D
D
10

Which number represents a square root of ?

Question illustration
A
3 (cosine ( StartFraction pi over 8 EndFraction) plus i sine (StartFraction pi over 8 EndFraction))
Option A
B
3 (cosine ( StartFraction pi over 4 EndFraction) plus i sine (StartFraction pi over 4 EndFraction))
Option B
C
3 (cosine ( StartFraction 5pi over 6 EndFraction) plus i sine (StartFraction 5pi over 6 EndFraction))
Option C
D
3 (cosine ( StartFraction 3pi over 2 EndFraction) plus i sine (StartFraction 3pi over 2 EndFraction))
Option D
11

Which of the following is a square root of ?

Question illustration
A
square root 3 (cosine ( StartFraction 4pi over 45 EndFraction) plus i sine (StartFraction 4pi over 45 EndFraction))
Option A
B
square root 3 (cosine ( StartFraction 4pi over 9 EndFraction) plus i sine (StartFraction 4pi over 9 EndFraction))
Option B
C
square root 3 (cosine ( StartFraction 49pi over 60 EndFraction) plus i sine (StartFraction 49pi over 60 EndFraction))
Option C
D
square root 3 (cosine ( StartFraction 11pi over 9 EndFraction) plus i sine (StartFraction 11pi over 9 EndFraction))
Option D
12

Which expression is equivalent to ?

Question illustration
A
6 (cosine ( StartFraction pi over 2 EndFraction) plus i sine (StartFraction pi over 2 EndFraction))
Option A
B
6 (cosine ( StartFraction pi over 6 EndFraction) plus i sine (StartFraction pi over 6 EndFraction))
Option B
C
8 (cosine ( StartFraction pi over 2 EndFraction) plus i sine (StartFraction pi over 2 EndFraction))
Option C
D
8 (cosine ( StartFraction pi over 6 EndFraction) plus i sine (StartFraction pi over 6 EndFraction))
Option D
13

If z = 1 – i, what is z3?

Question illustration
A
–8
B
8
C
–8i
D
8i
14

What is the distance on the unit circle between successive fourth roots of ?

Question illustration
A
StartFraction pi Over 3 EndFraction
Option A
B
StartFraction pi Over 2 EndFraction
Option B
C
StartFraction 2 pi Over 3 EndFraction
Option C
D
Pi
Option D
15

Suppose z = . What is z4?

Question illustration
A
1cosine ( StartFraction 2pi over 12 EndFraction) plus i sine (StartFraction 2pi over 12 EndFraction)
Option A
B
1cosine ( StartFraction 8pi over 3 EndFraction) plus i sine (StartFraction 8pi over 3 EndFraction)
Option B
C
4 (cosine ( StartFraction 2pi over 12 EndFraction) plus i sine (StartFraction 2pi over 12 EndFraction))
Option C
D
4 (cosine ( StartFraction 8pi over 3 EndFraction) plus i sine (StartFraction 8pi over 3 EndFraction))
Option D
16

Where are the 4th roots of 16i located?

A
on a circle with radius 1.7
B
on a circle with radius 2
C
on a circle with radius 2.5
D
on a circle with radius 4

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