phd_06 - Instructions (1) This question paper consists of...

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Unformatted text preview: Instructions (1) This question paper consists of two parts: Part A and Part B and carries a total of 100 Marks. (2) There is no negative marking. (3) Part A carries 20 multiple choice questions of 2 marks each. Answer all questions in Part A. (4) Answers to Part A are to be marked in the OMR sheet provided. (5) For each question, darken the appropriate bubble to indicate your answer. (6) Use only HB pencils for bubbling answers. (7) Mark only one bubble per question. If you mark more than one bubble, the question will be evaluated as incorrect. (8) If you wish to change your answer, please erase the existing mark completely before marking the other bubble. (9) Part B has 24 questions. Answer any 12 in this part. Each question carries 5 marks. (10) Answers to Part B are to be written in the separate answer book provided. (11) Candidates are asked to fill in the required fields on the sheet attached to the answer book. (12) Let Z , R , Q and C denote the set of integers, real numbers, rational numbers and complex numbers respectively. (13) If G is a group, then O ( G ) denotes the order of G . 2 MATHEMATICS PART A (1) The ordinary differential equation g = 2 g with g (0) = a has (A) the solution g ( x ) = 2 exp ( ax ), (B) the solution g ( x ) = (exp ( ax )- exp (- ax )) / 2, (C) the solution g ( x ) = a exp (2 x ), (D) no solution. (2) Let x ( t ) and y ( t ) be C functions on R and let z ( t ) = x ( t ) y ( t ) ! . Let A be a 2 2 real constant matrix such that z ( t ) = Az ( t ) for all t R . Let be an eigenvalue of A with corresponding eigenvector v . Then a solution for z ( t ) is (A) exp ( t ) v , (B) exp ( t ) v , (C) exp (- t ) v , (D) exp ( it ) v . (3) Let f be a non-constant entire function such that | f ( z ) | = 1 for every z with | z | = 1. Then (A) f has a zero in the open unit disc. (B) f always has a zero outside the closed unit disc. (C) f need not have any zero. (D) any such f has exactly one zero in the open unit disc. (4) Let f have a pole of order 2 at 0 and let g be an analytic function in a neigh- bourhood of 0 having a zero of order 3 at 0. Then the function f ( z ) g ( z ) has (A) a pole of order 2 at 0, (B) a zero of order 2 at 0, (C) a pole of order 1 at 0 (D) a zero of order 1 at 0....
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phd_06 - Instructions (1) This question paper consists of...

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