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1. Decide which of the following are true or false. 10 If true, furnish the proof, and if false, give a counter example. (a) A U (B fl C) = (A U B) n C for any subset A, B, C of a set U. (b) 1 .2 1I z2 , = Iz 1 Z21 for any two complex numbers z1 and z2, where Izi denotes the absolute value (or modulus) of z. (c) A linear equation with coefficients in C\ R in one unknown can have a root which is real. (d) The system of equations 4x + 6y = 9 6x + 9y = 1 3 is inconsistent. (e) For a 3 x 3 matrix A det (a A) = a (det A) holds for every scalar a where det A denotes the determinant of A. 2. (a) Prove that (ax + by) (ay + bx) 4abxy where 2 a, b, x, y are positive real numbers. (b) Find all the four complex fourth roots of 3 ( — 1). Show that these roots are in G.P.
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