University of Hyderabad (UoH) 2005 Ph.D ENTRANCE , , Mathematics/ Applied Mathematics - Question Paper

ENTRANCE EXAMINATION,2005
Ph.D. Mathematics/ Applied Mathematics
TIME: two hours MAX. MARKS: 75
Part A: 25 Part B: 50
HALL TICKET No.
INSTRUCTIONS
1. Calculators are not allowed.
2. ans all the 25 ques. in Part A. every accurate ans carries one mark
and every wrong ans carries minus quarter mark. Note that this means
that wrong answers are penalised by negative marks. So do not gamble.
3. Instructions for answering Part B are provided at the beginning of Part B.
4. Do not detach any pages from this ans book. It contains eight pages. A
separate ans book will be given for Part B.
5. 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.
1
Part-A
ans Part A by circling the accurate ans. A accurate ans
gets one mark and a wrong ans gets ¡(1=4) mark.
1. If A =26664
1 one 0 0
0 one 1 0
0 0 one 0
0 0 0 2
37775
then the rank of (A ¡ I) (I is the four £ four identity
matrix) is
a. four b. three c. two d. one e. 0
2. The minimal polynomial of
26666664
2 one 0 0 0
0 two one 0 0
0 0 two 0 0
0 0 0 two 1
0 0 0 0 2
37777775
is
a. (X ¡ 2) b. (X ¡ 2)2 c. (X ¡ 2)3 d. (X ¡ 2)4 e. (X ¡ 2)5
3. Let S = fv1; v2; :::; v9g be nine vectors in IR6. Then
a. S contains a basis of IR6.
b. there exist six linearly independent vectors in S.
c. S must span IR6.
d. there exist three linearly independent vectors in S.
e. none of the above.
4. Let Pan be a convergent series of complex numbers but let Pjanj be
divergent. Then it follows that
a. an ! 0 but janj does not converge to 0.
b. the sequence fang 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.
2
5. I. A bounded sequence in IR need not be convergent.
II. A bounded sequence in IR need not have a convergent subsequence.
III. A bounded sequence in IR need not have a constant subsequence.
a. All 3 statements are actual.
b. None of these statements is actual.
c. I and II are actual but III is false.
d. Only I is actual.
e. none of the above.
6. Let f(x) = max(sin x; cos x) for all x two IR. Then
a. f is differentiable on IR.
b. f is nowhere differentiable.
c. f is differentiable other than at 0.
d. f is differentiable other than at a countable set of points.

Earning:   Approval pending.