GRE Exam Numerical Ability Geometry

In the complex z-plane, the set of points satisfying the equation z^2 = |z|^2 is a
(A) pair of points
(B) circle
(C) half-line
(D) line
(E) union of infinitely many different lines

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GRE Other Question

Let M be a 5*5 real matrix. Exactly four of the following five conditions on M are equivalent to each other. Which of the five conditions is equivalent to NONE of the other four?
(A) For any two distinct column vectors u and v of M, the set {u, v} is linearly independent.
(B) The homogeneous system Mx = 0 has only the trivial solution.
(C) The system of equations M x = b has a unique solution for each real 5*1 column vector b.
(D) The determinant of M is nonzero.
(E) There exists a 5*5 real matrix N such that NM is the 5*5 identity matrix. For which of the following rings is it possible for the product of two nonzero elements to be zero?
(A) The ring of complex numbers
(B) The ring of integers modulo 11
(C) The ring of continuous real-valued functions on [0, 1]
(D) The ring {a + b Sqrt 2 : a and b are rational numbers}
(E) The ring of polynomials in x with real coefficients