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Let z1 and z2 be complex numbers such that arg(z1 - (3*z1 - 2*z2)/4) = pi/12. Find a possible value of arg((z1² + 4*z1*z2 + 4*z2²)¹⁰).

1. 5*pi/3
2. -pi/3
3. 11*pi/3
4. 2*pi/3

Correct answer: -pi/3

Solution

Simplifying: z1 - (3z1-2z2)/4 = (z1+2z2)/4, so arg(z1+2z2) = pi/12. The expression (z1²+4z1z2+4z2²)¹⁰ = (z1+2z2)²⁰, so its argument is 20*(pi/12) = 5*pi/3. The principal value (in (-pi, pi]) is 5*pi/3 - 2*pi = -pi/3.

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