math.iisc.ac.in NBHM Research Scholarships Screening Test Question Paper : National Board For Higher Mathematics

Name of the University : National Board For Higher Mathematics
Name of the Exam : Research Scholarships Screening Test
Department : Mathematics
Document Type : Sample Question Paper
Year : 2013
Website : http://www.math.iisc.ac.in/degprog-phd.html
Download Model Question Paper :
January 2011 : https://www.pdfquestion.in/uploads/23743-nbhmra11.pdf
January 2012 : https://www.pdfquestion.in/uploads/23743-nbhmra12.pdf
January 2013 : https://www.pdfquestion.in/uploads/23743-nbhmra13.pdf

NBHM Research Scholarships Screening Test Question Paper

The Department of Mathematics offers excellent opportunities for research in both pure and applied mathematics. Visit the Research Areas page to get a sense of the research interests of the faculty in the department.

Related : Institute of Science Education and Research National Entrance Screening Test Question Paper 2017 : www.pdfquestion.in/13863.html

Time Allowed: 150 Minutes
Maximum Marks: 40

Instructions To Candidates

Please read, carefully, the instructions that follow.
** 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 ve 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 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.

Sample Question Paper

** Download NBHM Ph.D. Screening Test: past question papers & answer keys

Section 1 – Algebra

1.1 Find the number of elements of order two in the symmetric group S4 of all permutations of the four symbols f1; 2; 3; 4g.
1.2 Let G be the group of all invertible 2 2 upper triangular matrices (under matrix multiplication). Pick out the normal subgroups of G from the following
a. H = fA 2 G : a12 = 0g;
b. H = fA 2 G : a11 = 1g;
c. H = fA 2 G : a11 = a22g

1.3 Let G = GLn(R) and let H be the (normal) subgroup of all matrices with positive determinant. Identify the quotient group G=H.
1.4 Which of the following rings are integral domains?
a. R[x], the ring of all polynomials in one variable with real coecients.
b. Mn(R).
c. The ring of complex analytic functions dened on the unit disc of the complex plane (with pointwise addition and multiplication as the ring oper- ations).

1.5 Find the condition on the real numbers a; b and c such that the following system of equations has a solution:
2x + y + 3z = a
x + z = b
y + z = c:

1.6 Let Pn denote the the vector space of all polynomials in one variable with real coecients and of degree less than, or equal to, n, equipped with the standard basis f1; x; x2; ; xng. Dene T : P2 ! P3 by
T(p)(x) = Z x 0 p(t) dt + p0(x) + p(2):
Write down the matrix of this transformation with respect to the standard bases of P2 and P3.

1.7 Determine the dimension of the kernel of the linear transformation T defined in Question 1.6 above.

1.8 A symmetric matrix in Mn(R) is said to be non-negative denite if xTAx 0 for all (column) vectors x 2 Rn. Which of the following statements are true?
a. If a real symmetric n n matrix is non-negative definite, then all of its eigenvalues are non-negative.
b. If a real symmetric nn matrix has all its eigenvalues non-negative, then it is non-negative denite.
c. If A 2 Mn(R), then AAT is non-negative definite.

1.9 Only one of the following matrices is non-negative denite. Find it.
1.10 Let B be the real symmetric non-negative denite 2 2 matrix such that B2 = A where A is the non-negative denite matrix in Question 1.9 above. Write down the characteristic polynomial of B.

Section 2 – Analysis

2.1 Evaluate: lim n!1 sin 2n + 1 2n sin 2n +1 2n
2.2 Evaluate: lim n!1 1 n [(n + 1)(n + 2) (n + n)] 1n :
2.3 Which of the following series are convergent?
2.4 Which of the following functions are uniformly continuous?
a. f(x) = x sin 1
x on ]0; 1[.
b. f(x) = sin2 x on ]0;1[.
c. f(x) = sin(x sin x) on ]0;1[.

2.5 Find the points where the following function is dierentiable:
f(x) = ( tan??1 x; if jxj 1; x 4jxj + jxj??1 2 ; if jxj > 1:
2.6 Which of the following sequences/series of functions are uniformly convergent on [0; 1]?
a. fn(x) = (cos(n!x))2n.
2.7 Let f 2 C1[0; 1]. For a partition (P) : 0 = x0 < x1 < x2 < < xn = 1; define S(P) = Xn i=1 jf(xi) ?? f(xi??1)j: Compute the supremum of S(P) taken over all possible partitions P.

2.8 Write down the Taylor series expansion about the origin in the region fjxj < 1g for the function f(x) = x tan??1(x) ?? 1 2 log(1 + x2):
2.9 Write down all possible values of i??2i.
2.10 What is the image of the set fz 2 C : z = x + iy; x 0; y 0g under the mapping z 7! z2.

Section 3 – Topology

3.1 Let (X; d) be a metric space. For subsets A and B of X, dene d(A;B) = inffd(a; b) : a 2 A; b 2 Bg: Which of the following statements are true?
a. If A \ B = ;, then d(A;B) > 0.
b. If d(A;B) > 0, then there exist open sets U and V such that A U;B V; U \ V = ;.
c. d(A;B) = 0 if, and only if, there exists a sequence of points fxng in A converging to a point in B.

3.2 Let X be a set and let (Y; ) be a topological space. Let g : X ! Y be a given map. Define 0 = fU X : U = g??1(V ) for some V 2 g:
Which of the following statements are true?
a. 0 denes a topology on X.
b. 0 denes a topology on X only if g is onto.
c. Let g be onto. Dene the equivalence relation x y if, and only if, g(x) = g(y). Then the quotient space of X with respect to this relation, with the topology inherited from 0, is homeomorphic to (Y; ).

3.3 Find pairs of homeomorphic sets from the following:
A = f(x; y) 2 R2 : xy = 0g;
B = f(x; y) 2 R2 : x + y 0; xy = 0g;
C = f(x; y) 2 R2 : xy = 1g;
D = f(x; y) 2 R2 : x + y 0; xy = 1g.

3.4 Let (X; ) be a topological space. A map f : X ! R is said to be lower semi-continuous if for every 2 R, the set f??1(] ??1; ]) is closed in X. It is said to be upper semi-continuous if, for every 2 R, the set f??1([;1[) is closed in X. Which of the following statements are true?
a. If ffng is a sequence of lower semi-continuous real valued functions on X, then f = supn fn is also lower semi-continuous.
b. Every continuous real valued function on X is lower semi-continuous.
c. If a real valued function is both upper and lower semi-continuous, then it is continuous.

3.5 Let
S = fA 2 Mn(R) : tr(A) = 0g:
Which of the following statements are true?
a. S is nowhere dense in Mn(R).
b. S is connected in Mn(R).
c. S is compact in Mn(R).

Section 4 – Applied Mathematics

4.1 Find all the solutions (; u); u 6 0, of the problem: u00 + u = 0; in ]0; 1[; u(0) = 0 = u0(1):
4.2 Find the constant c such that the following problem has a solution: ??u00 = c in ]a; b[; u0(a) = ??1 ; u0(b) = 1:
4.5 Let L(y) denote the Laplace transform of a function y = y(x). If y and y0 are bounded, express L(y00) in terms of L(y); y and y0.
4.10 What is the highest value of n such that Simpson’s rule (see Question
4.9 above) gives the exact value of the integral of f on [a; b] when f is a polynomial of degree less than, or equal to, n?

Section 5 – Miscellaneous

5.1 Let m > n. In how many ways can we seat m men and n women in a row for a photograph if no two women are to be seated adjacent to each other?
5.2 Let n 2 N be xed. For r n, let Cr denote the usual binomial coecient (n r ) which gives the number of ways of choosing r objects from a given set of n objects.
Evaluate: C0 + 4C1 + 7C2 + + (3n + 1)Cn:

5.3 Let
A = the set of all sequences of real numbers,
B = the set of all sequences of positive real numbers,
C = C[0; 1] and D = R.
Which of the following statements are true?
a. All the four sets have the same cardinality.
b. A and B have the same cardinality.
c. A;B and D have the same cardinality, which is dierent from that of C.

5.6 Write down the equation (with leading coecient equal to unity) whose roots are the squares of the roots of the equation x3 ?? 6×2 + 10x ?? 3 = 0:
5.7 Let A = (0; 1) and B = (1; 1) in the plane R2. Determine the length of the shortest path from A to B consisting of the line segments AP; PQ and QB, where P varies on the x-axis between the points (0; 0) and (1; 0) and Q varies on the line fy = 3g between the points (0; 3) and (1; 3).