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(a) Let z = x + iy with x, y real. Define A = { z: |z| <= 2 } and B = { z: (1 - i)z + (1 + i)conj(z) >= 4 }. Find the area of A intersect B. (b) For a non-real complex number z, find the minimum value of Im(z⁵)/Im(z).

1. (a) 2pi - 2; (b) -1
2. (a) pi - 2; (b) -4
3. (a) 2pi - 4; (b) -5
4. (a) 4pi - 2; (b) -2

Correct answer: (a) pi - 2; (b) -4

Solution

Region B is the half-plane x + y >= 2; intersected with the disk of radius 2 it is a circular segment of area pi - 2. For part (b) the minimum of Im(z⁵)/Im(z) is -4.

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