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ENTRANCE EXAMINATION,
Ph.D. Mathematics/ Applied Mathematics
TIME: 2 hours MAX. MARKS: 75 Part A: 25 Part B: 50 HALL TICKET No.
INSTRUCTIONS
- Calculators are not allowed.
- Answer all the 25 questions in Part A. Each correct answer carries 1 mark and each wrong answer carries minus quarter mark. Note that this means that wrong answers are penalised by negative marks. So do not gamble.
- Instructions for answering Part B are given at the beginning of Part B.
- Do not detach any pages from this answer book. It contains 8 pages. A separate answer book will be provided for Part B.
- IR always denotes the set of real numbers, ZZ the set of integers, IN the set of natural numbers and QI the set of rational numbers. For any set X, P(X) is the power set of X.
Part-A Answer Part A by circling the correct answer. A correct answer gets 1 mark and a wrong answer gets −( 1 / 4 ) mark.
- If A =
then the rank of (A − I) (I is the 4 × 4 identity
matrix) is
a. 4 b. 3 c. 2 d. 1 e. 0
- The minimal polynomial of
is
a. (X − 2) b. (X − 2)^2 c. (X − 2)^3 d. (X − 2)^4 e. (X − 2)^5
- Let S = {v 1 , v 2 , ..., v 9 } be 9 vectors in IR^6. Then a. S contains a basis of IR^6. b. there exist 6 linearly independent vectors in S. c. S must span IR^6. d. there exist 3 linearly independent vectors in S. e. none of the above.
- Let
∑ an be a convergent series of complex numbers but let
∑ |an| be divergent. Then it follows that a. an → 0 but |an| does not converge to 0. b. the sequence {an} does not converge to 0. c. only finitely many an’s are 0. d. infinitely many an’s are positive and infinitely many are negative. e. none of the above.
- If IR is given the cofinite topology then a. IR is compact and connected. b. IR is connected but not compact. c. IR is compact but not connected. d. IR is neither compact nor connected. e. IR has a countable base.
- Let T = {φ, IR}
⋃ {(x, ∞) | x ∈ IR}. Then in the topological space (IR, T ) the set of integers ZZ is a. an open set. b. a closed set. c. a dense set. d. an uncountable set. e. none of the above.
- The number of homomorphisms from C 2 × C 2 → C 2 is (Cn is the cyclic group of order n)
a. 5 b. 4 c. 3 d. 2 e. 1
- The number of zero-divisors in the ring of integers modulo 24 is
a. 20 b. 15 c. 12 d. 8 e. none of the above.
- If R is a unique factorization domain then a. R is a Euclidean domain. b. R is a principal ideal domain. c. R[X] is a unique factorization domain. d. R[X] is a principal ideal domain. e. none of the above.
- The number of proper subfields of F 32 is
a. 16 b. 8 c. 4 d. 2 e. 1.
- An example of a function on IR whose graph does not intersect the x-axis is a. f (x) = x^3 − 3 x + 2 b. f (x) = x^4 + x^2 + 1 c. f (x) = x^7 − 2 d. f (x) = x^11 − x 2 + 1 e. none of the above.
- I. Every Lebesgue measurable function on IR is continuous. II. Every Lebesgue measurable subset of IR is Borel. III. The space of continuous functions on [a, b] is dense in L^3 ([a, b]). a. I and II are true but III is false. b. I and III are true but II is false. c. Only II and III are true. d. Only III is true. e. None of the above.
- The indicator function of the irrationals is a. differentiable everywhere. b. Riemann integrable. c. differentiable nowhere. d. differentiable only at 0. e. none of the above.
- For the function f (z) = sinz 2 z the point z = 0 is a. an essential singularity. b. a removable singularity. c. a pole of order 2. d. a pole of order 1. e. none of the above.
- The number of roots of f (z) = z^5 + 5z^3 + z − 2 which lie inside the circle of radius 5/2 centred at the origin is
a. 0 b. 3 c. 5 d. 7 e. none of these.
- The ODE x^2 (1 − x)^2 y′′^ + (1 − x)y′^ + x^2 y = 0 has a. both x = 0 and x = 1 as regular singular points. b. both x = 0 and x = 1 as irregular singular points. c. x = 0 as a regular singular point and x = 1 as an irregular singular point. d. x = 0 as an irregular singular point and x = 1 as a regular singular point. e. none of the above.
Part - B
There are 15 questions in this part. Each question carries 5 marks. Answer as many as you can. The maximum you can score is 50 marks. Justify your answers. This part must be answered in a separate answer book provided.
- Let p : P(IN) → IN be the function defined by p(A) = minimal element of A. Show that (a) p(A ∪ B) = min(p(A), p(B)) and (b) p(A ∩ B) ≥ min(p(A), p(B)) if A ∩ B 6 = φ.
- Show that the function f (x) = x + sin x defines a homeomorphism from IR to IR.
- What is the characteristic polynomial and minimal polynomial over QI of
the matrix A =
? Find a vector v such that{v, Av, A^2 v, A^3 v}
is a basis of IR^4.
- Let n ≥ 3 be an odd integer and α 1 , α 2 , ..., αn− 1 the non-real nth roots of 1. Show that (1 + α^21 )(1 + α^22 )...(1 + α^2 n− 1 ) = 1.
- In a commutative ring R with 1, for any subset I of R define V (I) = {P | P is a prime ideal containing I}. Show that if I 1 and I 2 are two ideals of R then V (I 1 )
⋃ V (I 2 ) = V (I 1
⋂ I 2 ).
- Show from first principles that a group of order 65 must be cyclic.
- Define absolute continuity. Give an example of a continuous function that is not absolutely continuous. Show why your example works.
- For a real number p > 1 define the space lp. Show that the dual space (lp)∗^ is isomorphic to lq^ where q = (^) p−p 1.
- Determine the Galois group of QI (e 2 πi 7 ) over QI.
- Let V be a finite dimensional vector space, V = V 1 + V 2 , where V 1 and V 2 are two subspaces of V. Let T be a linear transformation on V such that T (V 1 ) ⊆ V 2 and T (V 2 ) ⊆ V 1. Suppose that T |V 1 and T |V 2 are injective. Show that T is invertible. (Hint: consider T 2 ).
- Investigate for solvability the integral equation
φ(x) − λ
∫ (^1)
0
(2xt − 4 x^2 )φ(t) dt = 1 − 2 x
for different values of the parameter λ.
- Find the extremals with corner point for the functional
J[y] =
∫ (^2)
0
(y′)^2 (y′^ − 1)^2 dx, y(0) = 0, y(2) = 1.
- Construct the Green’s function for the B.V.P. y′′^ = −f (x), y(0) = 0, y(1) + y′(1) = 2 and hence write its solution in terms of the Green’s function.
- Consider the non-linear p.d.e. pq = 1. Show that two initial strips are possible for the initial curve x = 2t, y = 2t, z = 5t. Find a solution of the equation containing the initial curve.
- Show that the transformation Q = p + iaq, P = p− 2 iaiaq is canonical and find a generating function.
