imsc.res.in NBHM PhD Scholarship Test Model Question Paper 2017 :Institute of Mathematical Sciences
Organisation : The Institute of Mathematical Sciences
Exam : NBHM PhD Scholarship Test
Document Type : Model Question Paper
Category or Subject : Research Awards Screening Test
Year : 2005 – 2017
Website : http://www.imsc.res.in/nbhm_model_questions
Download Model Question Paper :
1. June 2005 : https://www.pdfquestion.in/uploads/10007-nbhmra05.pdf
2. February 2006 : https://www.pdfquestion.in/uploads/10007-nbhmra06.pdf
3. January 2007 : https://www.pdfquestion.in/uploads/10007-nbhmra07.pdf
4. February 2008 : https://www.pdfquestion.in/uploads/10007-nbhmra08.pdf
5. January 2009 : https://www.pdfquestion.in/uploads/10007-nbhmra09.pdf
6. January 2010 : https://www.pdfquestion.in/uploads/10007-nbhmra10.pdf
7. January 2011 : https://www.pdfquestion.in/uploads/10007-nbhmra11.pdf
8. January 2012 : https://www.pdfquestion.in/uploads/10007-nbhmra12.pdf
9. January 2013 : https://www.pdfquestion.in/uploads/10007-nbhmra13.pdf
10. January 2014 : https://www.pdfquestion.in/uploads/10007-nbhmra14.pdf
11. January 2015 : https://www.pdfquestion.in/uploads/10007-nbhmra15.pdf
12. January 2016 : https://www.pdfquestion.in/uploads/10007-nbhmra16.pdf
13. January 2017 : https://www.pdfquestion.in/uploads/10007-nbhmra17.pdf
NBHM PhD Scholarship Test :
Mathematics :
Time Allowed : 90 Minutes
Maximum Marks : 40
Related : Institute of Mathematical Sciences NBHM MA/MSc Scholarship Test Old Question Papers : www.pdfquestion.in/7251.html
Instructions To Candidates :
** Please ensure that this booklet contains 11 numbered (and printed) pages. The back of each printed page is blank and can be used for rough work.
** There are five sections, containing ten questions each, entitled Algebra, Analysis, Topology, Calculus & Didifferential Equations, and Miscellaneous.
** Answer as many questions as possible.
** The assessment of the paper will be based on the best four sections. Each question carries one point and the maximum possible score is forty.
** Answer each question, as directed, in the space provided in the answer booklet, which is being supplied separately.
** This question paper is meant to be retained by you and so do not answer questions on it.
** In certain questions you are required to pick out the qualifying statement(s) from multiple choices. None of the statements, or one or more than one statement may qualify.
** Write none if none of the statements qualify, or list the labels of all the qualifying statements (amongst (a),(b), and (c)).
** Points will be awarded for questions involving more than one answer only if all the correct answers are given. There will be no partial credit.
** Calculators are not allowed.
Algebra :
1.1 Find the value of a G such that 2 + /3 is a root of the polynomial x3 – 5×2 + ax – 1
1.2 £ et10 be a 3%2 3 matrix whose eigenvalues are – 1 , 1 , 2. Find$3 ,54 and76 such that
1.3 What is the number of groups of orderRQ (upto isomorphism)
1.4 Pick out the integral domains from the following list of rings ¢ (a) {a + b/5 ? a, b G ?}
(b) The ring of continuous functions from [0, 1] into ?.
(c) The ring of complex analytic functions on the disc {yx GC ? ?x ? < 1}.
(d) The polynomial ring ¯[x].
1.5 Pick out the abelian groups from the following list ¢
(a) Any group of order1w .
(b) Any group of order 3 Q .
(c) Any group of order?w 7.
(d) Any group of order1w??
2.1 What is the radius of convergence of the following series
2.2 What is the least value ofrqs V 0 such that ? sin2 x – sin2ut ??v q ? x – t ?
2.3 Pick out the functions from the following list which are analytic in C ¢
January 2017 :
Section 1: Algebra
1.1 Let G be a group. Which of the following statements are true?
a. Let H and K be subgroups of G of orders 3 and 5 respectively. Then H \ K = feg, where e is the identity element of G.
b. If G is an abelian group of odd order, then ‘(x) = x2 is an automorphism of G.
c. If G has exactly one element of order 2, then this element belongs to the centre of G.
1.2 Let n 2 N; n > 2. Which of the following statements are true?
a. Any nite group G of order n is isomorphic to a subgroup of GLn(R).
b. The group Zn is isomorphic to a subgroup of GL2(R).
c. The group Z12 is isomorphic to a subgroup of S7.
1.4 Let p be an odd prime. Find the number of non-zero squares in Fp.
1.5 Find a generator of F7 7 , the multiplicative group of non-zero elements of F7.
1.7 Let A 2 M3(R) be a symmetric matrix whose eigenvalues are 1; 1 and 3. Express A-11 in the form I + A, where ; 2 R.
1.8 Let A 2 Mn(R); n > 2. Which of the following statements are true?
a. If A2n = 0, then An = 0.
b. If A2 = I, then A = I.
c. If A2n = I, then An = I.
1.9 Which of the following statements are true?
a. There does not exist a non-diagonal matrix A 2 M2(R) such that A3 = I.
b. There exists a non-diagonal matrix A 2 M2(R) which is diagonalizable over R and which is such that A3 = I.
c. There exists a non-diagonal matrix A 2 M2(R) such that A3 = I and such that tr(A) = -1.