JRF Mathematics ISI Admission Test Sample Question Paper : isical.ac.in
Name of the University : Indian Statistical Institute
Exam : ISI Admission Test
Document Type : Sample/Previous Year Question Paper
Name of the Subject : JRF Mathematics
Year : 2016
Website : http://www.isical.ac.in/~admission/IsiAdmission2017/PreviousQuestion/Questions-Jrf-Math.html
Download Sample/Previous Years’ Questions :
MTA 2016 : https://www.pdfquestion.in/uploads/11356-JRFMTA-2016.pdf
MTB 2016 : https://www.pdfquestion.in/uploads/11356-JRFMTB-2016.pdf
MTA 2015 : https://www.pdfquestion.in/uploads/11356-JRFMTA-2015.pdf
MTB 2015 : https://www.pdfquestion.in/uploads/11356-JRFMTB-2015.pdf
MTA 2014 : https://www.pdfquestion.in/uploads/11356-JRFMTA-2014.pdf
MTB 2014 : https://www.pdfquestion.in/uploads/11356-JRFMTB-2014.pdf
JRF Mathematics Sample Question :
Solve any six questions. :
1. Suppose that X is a Hausdor space and [0; 1] ! X is a continuous function. If is one-one, then prove that the image of is homeomorphic to [0; 1]. Give an example where is not one-one but the image of is homeomorphic to [0; 1].
Related : Indian Statistical Institute JRF Statistics ISI Admission Test Sample Question Paper : www.pdfquestion.in/11352.html
2. Prove that the following collection of subsets denes a topology on the set of natural numbers N ;;N; Un = f1; : : : ; ng; n 2 N: Is N compact in this topology? What are the continuous functions from this space to the space R of real numbers with standard topology?
3. Let fang be a sequence of non-zero real numbers. Show that it has a subsequence fankg such that lim ank+1 ank exists and belongs to f0; 1;1g.
4. Let D denote the open ball of unit radius about origin in the complex plane C. Let f be a continuous complex-valued function on its closure D which is analytic on D. If f(eit) = 0 for 0 < t < 2, show that f(z) = 0 for all z.
5. Let U be an open connected set in C and f : U ! C be a continuous map such that z 7! f(z)n is analytic for some positive integer n. Prove that f is analytic.
6. Let C be a closed convex set in R2. For any x 2 C, dene Cx = fy 2 R2 j x + ty 2 C; 8 t 0g: Prove that for any two points x; x0 2 C, we have Cx = Cx0 .
Test Code : MTA
Session : Forenoon
Time : 2 hours
Instructions :
** Answer as many questions as you can
** Answering +ve questions correctly would be considered adequate
** Write your Registration number, Test Centre, Test Code and the Number of this booklet in the appropriate places on the answer-booklet.
** All Rough Work Is To Be Done On This Booklet And/Or The Answer-Booklet.
** Calculators Are Not Allowed.
** R denotes the set of real numbers.
** C denotes the set of complex numbers.
** N denotes the set of positive integers.
Q 1. Find the sum of the series 1X n=0 n2 2n
Q 2. Let A be an m n real matrix.
(a) Show that N(A) \ Im(AT ) = f0g, where AT is the transpose of A, Im(A) is the image of A and N(A) = fv 2 Rn : Av = 0g.
(b) If for two suitable matrices B and C, AATB = AATC then show that ATB = ATC.
Q 3. Let V be a nite dimensional vector space over R. Suppose that a subset A V has the following property: For any nite set of scalars a1; a2; : : : ; an 2 R which satises Pn i=1 ai = 1 and any vectors v1; v2; : : : ; vn 2 A, a1v1 + a2v2 + + anvn 2
A.Show that A = x0 +W for some x0 2 V and some subspace W of V , where x0 +W = fx0 + v : v 2 Wg.
Q 4. Let Xn = number of heads obtained from n independent coin tosses with probability of head p. Let pn be the probability that Xn is an even number.
(a) Show that pn+1 = (1 -2p)pn + p.
(b) Show that lim n!1 pn exists and nd the limit.
MTB 2016 :
Solve any six questions :
1. Let n 3 be a natural number. Prove that the three cycle (1; 2; 3) is not a cube of any element in the symmetric group Sn.
2. Prove that any group of order 35 is cyclic.
3. Give an example of a degree 2 extension K of a eld F of characteristic two which is not obtained by attaching a square root of an element of F.
4. Prove that for any natural number n, there exist n consecutive integers each of which is divisible by a perfect square greater than one.
5. Let Xn = number of heads obtained from n independent coin tosses with probability of head p. Let pn be the probability that Xn is an even number.