PRE CALCULUS Pre-Calculus Complex Numbers 4 — Questions and Answers
Question 1: Solve for x: x² + 25 = 0
- x = ±5
- x = ±5i (Correct answer)
- x = 5i only
- No solution
Correct answer: x = ±5i
x² = -25, so x = ±√(-25) = ±5i.
Question 2: What is the sum of a complex number and its conjugate?
- 2bi
- 2a (Correct answer)
- 0
- 2|z|
Correct answer: 2a
(a + bi) + (a - bi) = 2a, which is always a real number.
Question 3: In the complex plane, which direction does multiplying by i rotate a vector?
- 90° clockwise
- 90° counterclockwise (Correct answer)
- 180°
- 45° counterclockwise
Correct answer: 90° counterclockwise
Multiplying by i = cos90° + i sin90° rotates a vector 90° counterclockwise.
Question 4: What is i²⁰²⁶?
- 1
- -1 (Correct answer)
- i
- -i
Correct answer: -1
2026 mod 4 = 2 (since 4 × 506 = 2024), so i²⁰²⁶ = i² = -1.
Question 5: The cube roots of unity are the solutions to which equation?
- z³ = -1
- z³ = 1 (Correct answer)
- z² = 1
- z³ = i
Correct answer: z³ = 1
The cube roots of unity satisfy z³ = 1 and are equally spaced at 120° apart on the unit circle.
Question 6: Which of the following is NOT a field property preserved by complex numbers?
- Commutativity of addition
- Ordering (like real numbers) (Correct answer)
- Distributive law
- Multiplicative inverse
Correct answer: Ordering (like real numbers)
Complex numbers cannot be ordered (no consistent total ordering exists), unlike the real numbers.
Question 7: If z₁ = 1 + i and z₂ = 1 - i, what is z₁/z₂?
- 1
- i (Correct answer)
- 0 + i
- -i
Correct answer: i
(1+i)/(1-i) · (1+i)/(1+i) = (1+i)²/2 = 2i/2 = i.
