AnswersPGHS Precalculus BUnit Test

Multiply and Divide Complex Numbers Answers

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1
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Let z = and .What is zw?

Question illustration
A
StartFraction 11 Over 7 EndFraction (cosine (StartFraction pi Over 6 EndFraction) + i sine (StartFraction pi Over 6 EndFraction))
Option A
B
77 (cosine (StartFraction pi Over 6 EndFraction) + I sine (StartFraction pi Over 6 EndFraction))
Option B
C
StartFraction 11 Over 7 EndFraction (cosine (StartFraction pi Over 12 EndFraction) + I sine (StartFraction pi Over 12 EndFraction))
Option C
D
77 (cosine (StartFraction pi Over 12 EndFraction) + I sine (StartFraction pi Over 12 EndFraction))
Option D
2
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Which statement describes how to geometrically determine the product of z = 25 – i and w = + 3i on the complex plane?

Question illustration
A
Stretch z by a factor of and rotate 30° counterclockwise.
Option A
B
Stretch z by a factor of and rotate 60° counterclockwise.
Option B
C
Stretch z by a factor of and rotate 30° counterclockwise.
Option C
D
Stretch z by a factor of and rotate 60° counterclockwise.
Option D
3

Review the graphs of complex numbers z and w.

Question illustration
A
–4 + 16i
B
6 + 10i
C
7 + 11i
D
8 + 2i
4

For the complex numbers z = and , which geometric transformation of z on the complex plane describes the quotient ?

Question illustration
A
Scale z by a factor of , then rotate clockwise by radians.
Option A
B
Scale z by a factor of 6, then rotate clockwise by radians.
Option B
C
Scale z by a factor of , then rotate counterclockwise by radians.
Option C
D
Scale z by a factor of 6, then rotate counterclockwise by radians.
Option D
5

Which statement describes how to geometrically divide a complex number, z, by a second complex number, w?

A
Scale z by the modulus of w, then rotate clockwise by the argument of w.
B
Scale z by the modulus of w, then rotate counterclockwise by the argument of w.
C
Scale z by the reciprocal of the modulus of w, then rotate clockwise by the argument of w.
D
Scale z by the reciprocal of the modulus of w, then rotate counterclockwise by the argument of w.
6

For z = 13 + 13i and w = + i , which geometric transformation describes the product of z and w on the coordinate plane?

Question illustration
A
Scale z by a factor of 13 and rotate 30° counterclockwise.
Option A
B
Scale z by a factor of 13 and rotate 45° counterclockwise.
Option B
C
Scale w by a factor of 13 and rotate 30° counterclockwise.
Option C
D
Scale w by a factor of 13 and rotate 45° counterclockwise.
Option D
7

Let z = 0.3(cos(31°) + i sin(31°)) and w = 20(cos(18°) + i sin(18°)).Which statement describes the geometric construction of the product zw on the complex plane?

A
Stretch z by a factor of 6, then rotate 18° counterclockwise.
B
Stretch z by a factor of 6, then rotate 49° counterclockwise.
C
Stretch z by a factor of 20, then rotate 18° counterclockwise.
D
Stretch z by a factor of 20, then rotate 49° counterclockwise.
8

For z = –3 – 5i, which graph shows z and the product of z · i?

A
On a coordinate plane, the y-axis is labeled imaginary and the x-axis is labeled real. Point z is (negative 3, negative 5) and point z times I is (5, negative 3).
Option A
B
On a coordinate plane, the y-axis is labeled imaginary and the x-axis is labeled real. Point z is (negative 3, negative 5) and point z times I is (3, negative 5).
Option B
C
On a coordinate plane, the y-axis is labeled imaginary and the x-axis is labeled real. Point z is (negative 3, negative 5) and point z times I is (negative 5, 3).
Option C
D
On a coordinate plane, the y-axis is labeled imaginary and the x-axis is labeled real. Point z is (negative 3, negative 5) and point z times I is (negative 3, 5).
Option D
9

Which phrase describes the geometric interpretation of dividing a complex number z by i?

A
Reflect z across the real axis.
B
Reflect z across the imaginary axis.
C
Rotate z by 90° clockwise about the origin.
D
Rotate z by 90° counterclockwise about the origin.
10

Let z = 5 – 9i and w = 6 + i. Which expression shows the first step in finding the quotient of ?

Question illustration
A
StartFraction 5 minus 9 I Over 6 + I EndFraction times StartFraction 6 minus I Over 6 minus I EndFraction
Option A
B
StartFraction 5 minus 9 I Over 6 + I EndFraction times StartFraction 6 + I Over 6 + I EndFraction
Option B
C
StartFraction 5 minus 9 I Over 6 + I EndFraction times StartFraction 5 minus I Over 5 minus I EndFraction
Option C
D
StartFraction 5 minus 9 I Over 6 + I EndFraction times StartFraction 5 + I Over 5 + I EndFraction
Option D

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