Section A
Q1.
(a) Let G be a finite commutative group. Let n ∈ \mathbbZ be such that n and order of G are relatively prime. Show that the function \phi: G → G defined by \phi(a) = an, for all a ∈ G, is an isomorphism of G onto G.
(b) Apply Cauchy’s Principle of Convergence to prove that the sequence $ defined by fn = 1 + 1/4 + 1/7 + \dots + (1)/(3n-2)$ is not convergent.
(c) Find (dy)/(dx), when f(x, y) = log(x2 + y2) + tan-1((y)/(x)) = 0, on using derivatives of Implicit Functions.
(d)
An automobile dealer wishes to put four repairmen R1, R2, R3 and R4 to four different jobs J1, J2, J3 and J4. But R3 cannot do the job J2. The dealer has estimated the number of man-hours that would be required for each job-man on one-one basis as given in the following table :
R1 R2 R3 R4
J1 6 2 3 4
J2 9 7 - 5
J3 6 4 7 5
J4 6 8 8 9
Formulate the above as a Linear Programming Problem.
(e) If f(z) = u + iv is any analytic function of the complex variable z and u - v = ex (cos y - sin y), find f(z) in terms of z.
