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Organisation : National Board For Higher Mathematics
Document Type : PhD Scholarship Test Sample Question Paper
Website : nbhm.dae.gov.in

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NBHM Research Scholarships Screening Test Question Paper

National Board For Higher Mathematics
Saturday, January 23, 2010
Time Allowed: 150 Minutes
Maximum Marks: 40

Related : National Board for Higher Mathematics NBHM Research Scholarships Screening Test Sample Question Paper 2016 : www.pdfquestion.in/8516.html

Please read, carefully, the instructions on the following page.

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, Applied Mathematics 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 in the above questions only if all the correct choices are made. There will be no partial credit.

** N denotes the set of natural numbers, Z – the integers, Q – the rationals, R – the reals and C – the field of complex numbers. Rn denotes the ndimensional Euclidean space, which is assumed to be endowed with its ‘usual’ topology. The symbol Zn will denote the ring of integers modulo n. The symbol ]a, b[ will stand for the open interval {x 2 R | a < x < b} while [a, b] will stand for the corresponding closed interval; [a, b[ and ]a, b] will stand for the corresponding left-closed-right-open and leftopen- right-closed intervals respectively. The symbol I will denote the identity matrix of appropriate order. The space of continuous real valued functions on an interval [a, b] is denoted by C[a, b] and is endowed with its usual ‘sup’ norm.

Calculators are not allowed.

Section 1 Algebra

1.2 Let G be a group. A subgroup H of G is called characteristic if ‘(H) H for all automorphisms ‘ of G. Pick out the true statement(s):
(a) Every characteristic subgroup is normal.
(b) Every normal subgroup is characteristic.
(c) If N is a normal subgroup of a group G, and M is a characteristic subgroup of N, then M is a normal subgroup of G.

1.2 Let S7 denote the symmetric group of all permutations of the symbols f1; 2; 3; 4; 5; 6; 7g. Pick out the true statements:
a. S7 has an element of order 10;
b. S7 has an element of order 15;
c. the order of any element of S7 is at most 12.

1.4 Write the following permutation as a product of disjoint cycles:
{1 2 3 4 5 6}
{6 5 4 3 1 2}

Section 3 Topology

3.3 Which of the following statements are true?
a. If A is a dense subset of a topological space X, then XnA is nowhere dense in X.
b. If A is a nowhere dense subset of a topological space X, then XnA is dense in X.
c. The set R, identied with the x-axis in R2, is nowhere dense in R2.

3.6 Consider the set of all n×n matrices with real entries as the space Rn2 . Which of the following sets are compact?
(a) The set of all orthogonal matrices.
(b) The set of all matrices with determinant equal to unity.
(c) The set of all invertible matrices.

3.7 In the set of all n×n matrices with real entries, considered as the space Rn2 , which of the following sets are connected?
(a) The set of all orthogonal matrices.
(b) The set of all matrices with trace equal to unity.
(c) The set of all symmetric and positive definite matrices.

3.8 Let X be an arbitrary topological space. Pick out the true statement(s):
(a) If X is compact, then every sequence in X has a convergent subsequence.
(b) If every sequence in X has a convergent subsequence, then X is compact.
(c) X is compact if, and only if, every sequence in X has a convergent subsequence.

3.10 Classify the following alphabets into homeomorphism classes:
N, B, H, M

Section 4 Applied Mathematics

4.1 A body, falling under gravity, experiences a resisting force of air proportional to the square of the velocity of the body. Write down the differential equation governing the motion satisfied by the distance y(t) travelled by thebody in time t.

Section 5  Miscellaneous

5.1 Which of the following sets are countable?
a. The set of all sequences of non-negative integers.
b. The set of all sequences of non-negative integers with only a nite number of non-zero terms.
c. The set of all roots of all monic polynomials in one variable with rational coecients.

5.2 A magic square of order N is an N N matrix with positive integral entries such that the elements of every row, every column and the two diago- nals all add up to the same number. If a magic square is lled with numbers in arithmetic progression starting with a 2 N and common dierence d 2 N,what is the value of this common sum?

5.3 A committee consists of n members and a group photograph is to be taken by seating them in a row. If two particular members do not get along with each other, in how many ways can the committee members be seated so that these two are never adjacent to each other?

5.5 Five letters are addressed to ve dierent persons and the corresponding envelopes are prepared. The letters are put into the envelopes at random. What is the probability that no letter is in its proper envelope?

5.2 Find the number of ways 2n persons can be seated at 2 round tables, with n persons at each table.

5.5 Consider a circle of unit radius centered at O in the plane. Let AB be a chord which makes an angle with the tangent to the circle at A. Find the area of the triangle OAB.

5.10 Assume that the line segment [0, 2] in the x-axis of the plane acts as a mirror. A light ray from the point (0, 1) gets reflected off this mirror and reaches the point (2, 2). Find the point of incidence on the mirror.