PRE CALCULUS Pre-Calculus Complex Numbers 5 — Questions and Answers
Question 1: What is the distance between the complex numbers 2 + 3i and -1 + 7i in the complex plane?
- 3
- 5 (Correct answer)
- 7
- √125
Correct answer: 5
Distance = |(-1+7i) - (2+3i)| = |-3 + 4i| = √(9 + 16) = 5.
Question 2: Which form is most efficient for computing large powers of complex numbers?
- Standard form (a + bi)
- Conjugate form
- Polar form with De Moivre's Theorem (Correct answer)
- Rectangular matrix form
Correct answer: Polar form with De Moivre's Theorem
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°)?
- 4i
- -4i (Correct answer)
- 4
- -4
Correct answer: -4i
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?
- x = 2 ± 3i (Correct answer)
- x = -2 ± 3i
- x = 4 ± 3i
- x = 2 ± 9i
Correct answer: x = 2 ± 3i
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?
- A line
- A parabola
- The unit circle (Correct answer)
- An ellipse
Correct answer: The unit circle
All complex numbers at distance 1 from the origin form the unit circle centered at the origin.
Question 6: What is the principal argument of the complex number -4i?
- π/2
- -π/2 (Correct answer)
- π
- 3π/2
Correct answer: -π/2
The number -4i lies on the negative imaginary axis, so its principal argument is -π/2 (in the range (-π, π]).
Question 7: The product of two complex conjugates a + bi and a - bi always results in:
- A purely imaginary number
- A negative real number
- A non-negative real number (Correct answer)
- Zero
Correct answer: A non-negative real number
(a + bi)(a - bi) = a² + b², which is always a non-negative real number.
PRE CALCULUS Pre-Calculus Complex Numbers 5
What is the distance between the complex numbers 2 + 3i and -1 + 7i in the complex plane?