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Consider the following two statements : Statement I : For any two non-zero complex numbers z1, z2, (|z1| + |z2|) |z1/|z2| + z2/|z2|| ≤ 2(|z1| + |z2|) and Statement II : If x, y, z are three distinct complex numbers and a, b, c are three positive real numbers such that a/|y - z| = b/|z - x| = c/|x - y|, then a^2/(y - z) + b^2/(z - x) + c^2/(x - y) = 1. Between the above two statements,

1. both Statement I and Statement II are incorrect.
2. Statement I is incorrect but Statement II is correct.
3. Statement I is correct but Statement II is incorrect.
4. both Statement I and Statement II are correct.

Correct answer: Statement I is correct but Statement II is incorrect.

Solution

Statement I is correct as it adheres to the triangle inequality in complex numbers, demonstrating that the sum of magnitudes is bounded by a linear combination of the magnitudes. Statement II, however, is incorrect because the equality does not hold in general for arbitrary distinct complex numbers and positive real coefficients.

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