Section A
Q1.
(a) Prove that a subgroup of a cyclic group is cyclic. Let G be a cyclic group with generator a. If the order of G is infinite, then prove that G is isomorphic to (\mathbbZ, + ).
(b) Find the relative extrema of the function f(x, y) = 4y3 + x2 - 12y2 - 36y + 2.
(c) Prove that in the interval 0 < x < 1, the function f(x) = x2 is uniformly continuous while f(x) = (1)/(x) is not uniformly continuous.
(d)
Prove that x1 = 2, x2 = 1, x3 = 0 is a feasible solution to the following set of equations :
2x1 - x2 + 3x3 = 3
-6x1 + 3x2 + 7x3 = -9
Is the solution basic? Justify your answer. If the solution is not basic, reduce it to a basic feasible one.
(e) Find a bilinear transformation which maps the points z = 0, -i, -1 into w = i, 1, 0 respectively.
