PRE CALCULUS Pre-Calculus Complex Numbers 5

Question 1

What is the distance between the complex numbers 2 + 3i and -1 + 7i in the complex plane?

Accepted Answer

5

Answer

Distance = |(-1+7i) - (2+3i)| = |-3 + 4i| = √(9 + 16) = 5.

Question 2

Which form is most efficient for computing large powers of complex numbers?

Accepted Answer

Polar form with De Moivre's Theorem

Answer

Polar form combined with De Moivre's Theorem directly gives rⁿ(cos nθ + i sin nθ) without repeated multiplication.

Question 3

What is the rectangular form of 4(cos 270° + i sin 270°)?

Accepted Answer

-4i

Answer

cos 270° = 0 and sin 270° = -1, so 4(0 + i(-1)) = -4i.

Question 4

For the quadratic x² - 4x + 13 = 0, what are the solutions?

Accepted Answer

x = 2 ± 3i

Answer

Using the quadratic formula: x = (4 ± √(16 - 52))/2 = (4 ± √(-36))/2 = (4 ± 6i)/2 = 2 ± 3i.

Question 5

If |z| = 1, what geometric figure does z trace in the complex plane?

Accepted Answer

The unit circle

Answer

All complex numbers at distance 1 from the origin form the unit circle centered at the origin.

PRE CALCULUS Practice Test