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Let z = (pi/4) * (1 + i)⁴ * [(1 - sqrt(pi)*i) / (sqrt(pi) + i) + (sqrt(pi) - i) / (1 + sqrt(pi)*i)]. Compute the ratio |z| / arg(z).

1. 1
2. pi
3. 3*pi
4. 4

Correct answer: 4

Solution

(1+i)⁴ = -4. Each fraction equals -i, so their sum = -2i. Then z = (pi/4)*(-4)*(-2i) = 2*pi*i, giving |z| = 2*pi and arg(z) = pi/2, so |z|/arg(z) = 2*pi/(pi/2) = 4.

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