Section A
Q1. Let R be the ring of n × n matrices over reals. Show that R has only two ideals namely 0 and R. Show that the series (1)/((1 + a)p) - (1)/((2 + a)p) + (1)/((3 + a)p) - \dots, a > 0 is If f'(x) = (x - a)2n (x - b)2m+1, where m, n are positive integers, show that f has neither a maximum nor a minimum at a and f has a local minimum at b. Let f(z) = u(r, θ) + iv(r, θ) be an analytic function. If u = -r3 sin 3θ, then construct the corresponding analytic function f(z) in terms of z. Find all optimal solutions of the following linear programming problems graphically:
(b) (i) absolutely convergent if p > 1.
(b) (ii) conditionally convergent if 0 < p ≤ 1.
(e) (i) Maximize z = 3x1 + 6x2 subject to x1 + x2 ≤ 8, x1 - x2 ≤ 4, 2x1 - x2 ≥ 4, x1, x2 ≥ 0
(e) (ii) The LPP in part (i) with the first constraint x1 + x2 ≤ 8 changed to x1 + 2x2 ≤ 12.
