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A hall ticket for the ph.d. Entrance examination in mathematics and applied mathematics held at the university of hyderabad in 2008. Instructions for the examination, including the duration, maximum marks, and marking scheme. It also includes a series of multiple-choice questions covering various topics in mathematics.
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University of Hyderabad, Entrance Examination, 2008 Ph.D. (Mathematics/Applied Mathematics)
Hall Ticket No.
Time: 2 hours Max. Marks: 75 Part A: 25 Marks Part B: 50 Marks
Instructions
Answer Part A by circling the correct letter in the array below: 1 a b c d 2 a b c d 3 a b c d 4 a b c d 5 a b c d 6 a b c d 7 a b c d 8 a b c d 9 a b c d 10 a b c d 11 a b c d 12 a b c d 13 a b c d 14 a b c d 15 a b c d 16 a b c d 17 a b c d 18 a b c d 19 a b c d 20 a b c d 21 a b c d 22 a b c d 23 a b c d 24 a b c d 25 a b c d
Part A
(a) f is continuous but not differentiable on R. (b) f is continuous and differentiable on R. (c) f is not continuous but differentiable on R. (d) f is neither continuous nor differentiable on R.
(a) f n^6 = IdG for any n ∈ N. (b) f 2 = IdG. (c) f = IdG. (d) there exists an n ∈ N such that f (x) = xn, for all x ∈ G.
(a) G is an abelian group. (b) G is a cyclic group. (c) there is a normal proper subgroup. (d) there is a subgroup of order 5 which is not normal.
α∈I
Xα is locally compact. Then
(a) Xα must be compact except for finitely many α. (b) Xα must be a singleton except for finitely many α. (c) each Xα must be compact. (d) the indexing set I must be countable.
(a) a Gδ set. (b) an Fσ set. (c) an open set. (d) a closed set.
(a) If A is dense then the Lebesgue measure of A is positive. (b) If the Lebesgue measure of A is zero then A is nowhere dense. (c) If the Lebesgue measure of A is positive then A contains a nontrivial interval. (d) All of (a), (b), (c) are false.
(a) elliptic. (b) elliptic everywhere except on x = 0 axis. (c) hyperbolic. (d) hyperbolic everywhere except on x = 0 axis.
(a) H is always separable. (b) If H has an orthogonal Schauder basis, then H is separable. (c) If H is separable, then H is locally compact. (d) If H has a countable Hamel basis, then H is finite dimensional.
fn(x). Then
(a) f is continuous and measurable. (b) f is continuous but need not be measurable. (c) f is measurable but need not be continuous. (d) f need not be either continuous or measurable.
∫ (^2) π
0
sin(x) sin(t)φ(t) dt = 0 is (a) π. (b)^1 π
. (c) 2π. 1 2 π
n=
|αn| < ∞
with the norm || {αn} || 1 =
n=
|αn|. Let {λn} be a sequence of complex numbers such that {λnαn} ∈ 1 for all {αn} ∈
1. Define T : 1 →
1 by T ({αn}) = {λnαn}. If T is a bounded linear operator on ` 1 then show that {λn} is bounded. In this case what will be the value of ||T ||?
fn(x) = f
x + n^1
. Show that fn converges uniformly on R to f.
that lim n→∞
0
fn(x) dx =
0
f (x) dx. Is the convergence uniform?
0
x + 2y + y
′ 2 2
dx, y(0) = 0, y(1) = 0. Also test for extrema.
∫ (^2) π
0
|x − t| sin(x)φ(t) dt = x.
Rough Work