Name of the Organisation : Chennai Mathematical Institute
Exam : Entrance Exam
Subject : MSc/PhD Mathematics
Year : 2016
Document Type : Previous Years’ Question Papers
Website : http://www.cmi.ac.in/admissions/syllabus.php
Download Model/Sample Question Paper :
MSc/PhD Mathematics 2010 : https://www.pdfquestion.in/uploads/13299-mscappmath2010.pdf
MSc/PhD Mathematics 2011 : https://www.pdfquestion.in/uploads/13299-mscappmath2011.pdf
MSc/PhD Mathematics 2012 : https://www.pdfquestion.in/uploads/13299-mscappmath2012.pdf
MSc/PhD Mathematics 2013 : https://www.pdfquestion.in/uploads/13299-mscappmath2013.pdf
MSc/PhD Mathematics 2014 : https://www.pdfquestion.in/uploads/13299-mscappmath2014.pdf
MSc/PhD Mathematics 2015 : https://www.pdfquestion.in/uploads/13299-mscappmath2015.pdf
MSc/PhD Mathematics 2016 : https://www.pdfquestion.in/uploads/13299-mscappmath2016.pdf
MSc/PhD Mathematics Question Papers :
Instructions :
** Enter your Registration Number here: CMI
** Enter the name of the city where you write this test
Related : Chennai Mathematical Institute MSc/PhD Computer Science Question Paper : www.pdfquestion.in/13295.html
** The allowed time is 3 hours.
** This examination has two parts. You may use the blank pages at the end for your rough-work.
** Part A is worth 40 marks and will be used for screening. There will be a cut-o for Part A, which will not be more than twenty (20) marks (out of 40).
** Your solutions to the questions in Part B will be marked only if your score in Part A places you over the cut-o.
** In particular, if your score in Part A is at least 20 then your solutions to the questions in Part B will be marked.
** However, note that the scores in both the sections will be taken into account while making the final decision.
** Record your answers to Part A in the attached bubble-sheet.
** Part B is worth 60 marks. You should answer six (6) questions in Part B.
** In order to qualify for the PhD Mathematics interview, you must obtain at least fteen (15) marks from among the starred questions (17){(20).
** Indicate the six questions to be marked in the boxes in the bubble-sheet. Write your solutions to Part B in the page assigned to each question.
** Please read the further instructions given before Part A and inside each part carefully.
Part A :
Instructions :
Each of the questions 1{8 has one or more correct answers. Record your answers on the attached bubble-sheet by filling in the appropriate circles. Every question is worth four (4) marks. A solution receives credit if and only if all the correct answers are chosen, and no incorrect answer is chosen.
(1) We say that two subsets X and Y of R are order-isomorphic if there is a objective map : X-> Y such that for every x1 x2 2 X, (x1) (x2), where `’ denotes the
usual order on R. Choose the correct statement(s) from below:
(A) N and Z are not order-isomorphic;
(B) N and Q are order-isomorphic;
(C) Z and Q are order-isomorphic;
(D) The sets N, Z and Q are pairwise non-order-isomorphic.
2. Let f : R -> R be defined as
Choose the correct statement(s) from below:
(A) f is continuous;
(B) f is discontinuous at 0;
(C) f is differential;
(D) f is continuously differential.
3. What is the cardinality of the centre of O2(R)? (Definition: The centre of a group G is fg 2 G j gh = hg for every h 2 Gg. Hint: Reaction matrices and permutation matrices are orthogonal.)
(A) 1;
(B) 2;
(C) The cardinality of N;
(D) The cardinality of R.
4. Let U R be a non-empty open subset. Choose the correct statement(s) from below
(A) U is uncountable;
(B) U contains a closed interval as a proper subset;
(C) U is a countable union of disjoint open intervals;
(D) U contains a convergent sequence of real numbers.
5. Let R be a commutative ring. The characteristic of R is the smallest positive integer n such that a+a+ +a (n times) is zero for every a 2 R, if such an integer exists, and zero, otherwise. Choose the correct statement(s) from below:
(A) For every n 2 N, there exists a commutative ring whose characteristic is n;
(B) There exists a integral domain with characteristic 57;
(C) The characteristic of a field is either 0 or a prime number;
(D) For every prime number p, every commutative ring of characteristic p contains Fp as a sub ring.
Instructions :
The answers to questions 9{10 are integers. You are required to write the answers in decimal form in the attached bubble-sheet. Every question is worth four (4) marks. (9) Consider the Q-vector-space ff : R -> R j f is continuous and Image(f) Qg: What is its dimension?
Part B :
Instructions :
Answer six (6) questions from below. Provide suffcient justification. Write your solutions on the page assigned to each question. Each of the questions is worth ten (10) marks. In order to qualify for the PhD Mathematics interview, you must obtain at least fifteen (15) marks from among the starred questions . Clearly indicate which six questions you would like us to mark in the six boxes in the bubble sheet. If the boxes are unfilled, we will mark the six solutions that appear in your answer-sheet. If you do not want a solution to be considered, clearly strike it out.
(11) Let U = f(x; y) 2 R2 j 1 < x2 + y2 < 4g. Let p; q 2 U. Show that there is a continuous map : [0; 1]-> U such that
(0) = p and
(1) = q and such that
is difierentiable
on (0; 1).
(12) If I; J are two maximal ideals in a PID that is not a field, then show that IJ is never a prime ideal.
(13) Let f : C ->C be an entire function. Suppose that f(z) 2 R if z is on the real axis or on the imaginary axis. Show that f0(z) = 0 at z = 0.