Unit Test — Unit test Answers

Review the four points on the complex plane.

Review the graph of complex number z.

In the complex plane, the rectangular coordinates (x, y) represent a complex number. Which statement explains why polar coordinates (r, θ) represent the same complex number?

What is the modulus of 6 + 7i?

Let z = and .Which statement describes the geometric construction of the product zw on the complex plane?

Review the graph.

Consider w1 = 4 + 2i and w2 = –1 – 3i. Which graph represents the sum w1 + w2?

Which values of p and q would make the value of the following expression equal to 58i? (3 – 7i)(p + qi)i

What is the additive identity of the complex number 14 + 5i?

What is the midpoint of a segment in the complex plane with endpoints at 6 – 2i and –4 + 6i?

Review the graph.

Which graph represents points on the polar curve r = 2 + 5sin(θ)?

Review the proof of de Moivre’s theorem (not in order). 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 + θ) + i sin(kθ + θ)C= cos(kθ)cos(θ) − sin(kθ)sin(θ) + i [sin(kθ)cos(θ) + cos(kθ)sin(θ)]D= [cos(kθ) + i sin(kθ)] ∙ [cos(θ) + i sin(θ)]E= cos[(k + 1)θ] + i sin[(k + 1)θ]

Which equation represents the rectangular form of r = 4cos(θ)?

Let z = 9 – 2i and zw = 25 + 70i.What is w?

A circle has a diameter with endpoints at –2 + i and –6 –11i. What is the center of the circle?

Review the four points shown on the polar coordinate system.

Consider the limaçon with equation r = 3 + 4cos(θ). How does the quotient of a and b relate to the existence of an inner loop?

When asked to find the distance between the complex points 3 + 8i and 7 + 6i, Zach showed the work below. He asked his friend Zöe to find his mistake, and she correctly told him that he should have

Which expression represents a 5th root of –i?