Mathematics 4 Semester/Term 2010

Sem IV / COMP / 19.05.10 / APPEND MARKS IV Oct 2009-10 REVISED COURSE (3 Hours) (Total Marks : 100 NB: (1) Question No 1 is compulsory. (2) Attempt any four questions out of the remaining six questions. (3) Figures in right indicate full marks. (4) Assume any suitable data wherever required and justify the same. 1. a) Find c such that 1/2 log(x^2 + y^2) + i tan^-1 (ky/x) is analytic. (7) b) If u = [x/y, y/z, z/x] find curl u. (5) c) Find all the basic feasible, initial bfs, degenerate, non-degenerate solutions of x1 + 2x2 + 3x3 + 4x4 = 1; 2x1 + x2 + 3x3 + 2x4 = 4. (8) d) Evaluate the line integral ∫ (x^2 + y^2) dt along the straight line from (1,2) to (1,2) and then from (2,2) to (5,2). (8) II. a) Find the eigenvalues and eigenvectors of A = [8 -6 2; -6 7 -4; 2 -4 3]. Are the eigenvectors linearly independent? (8) b) If f(z) = u+iv is analytic and u - v = (cos x + sin x) / (e^y + e^-y - 2cos x) find f(z) in terms of z. (7) c) Solve by simplex method: Max z = 3x1 + 5x2 Subject to the constraints: x1 + 4x2 <= 24 3x1 + 2x2 <= 21 2x1 + x2 <= 9 x1, x2 >= 0 (7) III. a) Prove that the matrix A = [1 -1 1; 1 -1 1; -1 1 -1] is diagonalizable. Hence find the diagonal and the transforming matrix. (8) b) Use Big M method to solve: Max Z = 2x1 + 4x2 Subject to the constraints: 3x1 + x2 >= 3 4x1 + 3x2 >= 6 x1 + 2x2 <= 3; x1, x2 >= 0 (7) c) State Cauchy's residue theorem and hence evaluate: ∫ (z-1) / (z^2 + 2z + 5) dz ; |z| = 3 (7) d) ∫ (dθ) / (5 - 3cosθ)