Complex Numbers Polar Form Test – Precalculus Math (Printable PDF)

Question 1 Reading the Plane (2 points)

The figure shows the complex number z.

1. Write z in the form a + bi and find |z|. (1 point)
2. Write the conjugate of z. (1 point)

Question 2 Rectangular to Polar (2 points)

Consider the number −2√2 + 2√2i.

1. Find its modulus. (1 point)
2. Write it in polar form with 0° ≤ θ < 360°. (1 point)

Question 3 Polar to Rectangular (2 points)

Write 10(cos 210° + i sin 210°) in the form a + bi. Give exact values.

Question 4 Product and Quotient (2 points)

Let z₁ = 6(cos 100° + i sin 100°) and z₂ = 3(cos 40° + i sin 40°). Write each result in polar form.

1. z₁z₂ (1 point)
2. z₁ ÷ z₂ (1 point)

Question 5 Distance and Midpoint (2 points)

Find the distance between −2 + 7i and 3 − i, and the midpoint of the segment joining them.

Question 6 Using De Moivre (3 points)

Use De Moivre’s theorem.

1. Evaluate [3(cos 12° + i sin 12°)]⁵ in polar form. (1 point)
2. Evaluate (√3 − i)⁷ in the form a + bi. Give exact values. (2 points)

Question 7 Cube Roots of −8 (3 points)

Find all three cube roots of −8. Write each in the form a + bi.

Question 8 Scaling and Turning (2 points)

The figure shows the point P = 5 + 2i. Multiplying P by 3i gives a new point Q.

1. Find Q in the form a + bi. (1 point)
2. Compare the distance of Q from the origin with the distance of P from the origin. (1 point)

Question 9 Modulus of a Product (2 points)

Let z = 3 + i and w = 1 − 4i. Compute zw, then verify that |zw| = |z| · |w|.