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Let z = x - i y and z³ = p + i q. Then the value of (x/p + y/q)/(p² + q²) is

1. -2
2. -1
3. 2
4. 1

Correct answer: -2

Solution

Writing z=(p+iq)^3 gives x=p(p^2-3q^2) and -y=q(3p^2-q^2), so x/p=p^2-3q^2 and y/q=q^2-3p^2. Their sum is -2(p^2+q^2), hence (x/p + y/q)/(p^2+q^2) = -2.

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