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A complex number z satisfies |z| = 1 and arg(z - 1) = 2 * pi / 3. Which of the following is/are correct?

1. arg(z² + z⁴) = pi
2. z = 1/2 + i * sqrt(3) / 2
3. z = -1/2 + i * sqrt(3) / 2
4. |z - 1| = 1

Correct answer: arg(z² + z⁴) = pi

Solution

Since |z|=1, write z = e^(i*theta). Then z-1 = (cos(theta)-1) + i*sin(theta). arg(z-1) = arctan(sin(theta)/(cos(theta)-1)). Using half-angle: sin(theta)/(cos(theta)-1) = -cot(theta/2) = -1/tan(theta/2). Setting this equal to tan(2*pi/3) = -sqrt(3): -1/tan(theta/2) = -sqrt(3) => tan(theta/2) = 1/sqrt(3) => theta/2 = pi/6 => theta = pi/3. So z = cos(pi/3) + i*sin(pi/3) = 1/2 + i*sqrt(3)/2 (option B). |z-1| = |(-1/2) + i*(sqrt(3)/2)| = sqrt(1/4 + 3/4) = 1 (option D also true). z² = e^(i*2*pi/3), z⁴ = e^(i*4*pi/3). z² + z⁴ = e^(i*2*pi/3) + e^(i*4*pi/3) = (-1/2 + i*sqrt(3)/2) + (-1/2 - i*sqrt(3)/2) = -1. arg(-1) = pi (option A true). Option C: z = -1/2 + i*sqrt(3)/2 is NOT the solution since that would have |z|=1 but theta=2*pi/3, giving arg(z-1) = arg(-3/2 + i*sqrt(3)/2) = arctan(-sqrt(3)/3 / (3/2))... not 2*pi/3. So A, B, D are correct.

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