PRE CALCULUS Pre-Calculus Complex Numbers 2 — Questions and Answers
Question 1: What is the modulus of the complex number 5 - 12i?
- 7
- 13 (Correct answer)
- 17
- 60
Correct answer: 13
The modulus is √(5² + (-12)²) = √(25 + 144) = √169 = 13.
Question 2: Which of the following is equal to i⁵⁰?
- i
- -i
- 1
- -1 (Correct answer)
Correct answer: -1
Since i repeats with period 4, i⁵⁰ = i^(4·12+2) = (i⁴)¹²·i² = 1·(-1) = -1.
Question 3: What is the argument (angle) of the complex number -1 + i in radians?
- π/4
- 3π/4 (Correct answer)
- -π/4
- 5π/4
Correct answer: 3π/4
The point (-1, 1) is in the second quadrant, so the argument is π - π/4 = 3π/4.
Question 4: If z = 3 + 4i, what is z · z̄ (z times its conjugate)?
- 7
- 25 (Correct answer)
- 14 + 24i
- 0
Correct answer: 25
z · z̄ = (3 + 4i)(3 - 4i) = 9 + 16 = 25 = |z|².
Question 5: Simplify: (2 + 3i)(2 - 3i)
- 4 - 9i²
- 13 (Correct answer)
- 4 + 9
- -5
Correct answer: 13
(2 + 3i)(2 - 3i) = 4 - 9i² = 4 + 9 = 13.
Question 6: What is the polar form of the complex number 1 + i√3?
- 2(cos 30° + i sin 30°)
- 2(cos 60° + i sin 60°) (Correct answer)
- √2(cos 45° + i sin 45°)
- 2(cos 90° + i sin 90°)
Correct answer: 2(cos 60° + i sin 60°)
The modulus is √(1+3) = 2 and the argument is arctan(√3/1) = 60°, giving 2(cos 60° + i sin 60°).
Question 7: Which expression equals (1 + i)² ?
- 2
- 2i (Correct answer)
- 1 + 2i
- 2 + 2i
Correct answer: 2i
(1+i)² = 1 + 2i + i² = 1 + 2i - 1 = 2i.
PRE CALCULUS Pre-Calculus Complex Numbers 2
What is the modulus of the complex number 5 - 12i?