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In the Argand plane, find the area of the region satisfying all three conditions: (i) |z| >= 4, (ii) (1-i)*z + (1+i)*z_bar <= 32*sqrt(2), (iii) z lies in the first quadrant (Re(z) >= 0, Im(z) >= 0). The area equals:

1. 256 - pi/4
2. 256 - 16*pi
3. 256
4. 256 - 4*pi

Correct answer: 256 - 4*pi

Solution

Condition (ii) reduces to x + y <= 16*sqrt(2). The first-quadrant region below this line is a right isoceles triangle of area 256, and removing the quarter-circle of radius 4 (area 4*pi) gives 256 - 4*pi.

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