What is the modulus and principal argument of the complex number z = -1 + i*sqrt(3)?
Answer and explanation
Correct answer: C. |z| = 2, arg(z) = 2*pi/3
- Step 1: Modulus: |z| = sqrt((-1)^2 + (sqrt(3))^2) = sqrt(1 + 3) = 2.
- Step 2: The point (-1, sqrt(3)) lies in quadrant II.
- Step 3: Reference angle: arctan(sqrt(3)/1) = pi/3.
- Step 4: In quadrant II, arg(z) = pi - pi/3 = 2*pi/3.
Answer
|z| = 2, arg(z) = 2*pi/3.
Why not the others: arg(z) = pi/3 places z in quadrant I; |z| = 4 forgets the square root; |z| = sqrt(2) miscalculates the sum of squares.
Practise more Vectors, Analytic Geometry and Probability.
Free questions with an explanation for every one, and full timed Bagrut (Secondary School Matriculation) mock tests.
More Vectors, Analytic Geometry and Probability questions
- Compute the value of (1 + i)^10 using De Moivre's theorem.
- What is the sum of all complex solutions to the equation z^5 - 1 = 0?
- Find all four roots of the equation z^4 + 4 = 0 in the complex plane.
- Describe the geometric locus of points in the complex plane satisfying |z - (3 - 2*i)| = 5.
- Solve the complex equation z^2 + |z|^2 = 2 - 4i, where z = x + yi.
- Express the complex quotient (3 + 4i) / (1 - 2i) in standard form a + b*i.