University of Hyderabad (UoH) 2007 Ph.D (Maths/Applied Maths) 2008-Entrance - Question Paper

University of Hyderabad,
Entrance Examination, 2008
Ph.D. (Mathematics/Applied Mathematics)



Part A
1. Let f : R ! R be a function provided by f(x) = min(1; x; x3). Then
(a) f is continuous but not di®erentiable on R.
(b) f is continuous and di®erentiable on R.
(c) f is not continuous but di®erentiable on R.
(d) f is neither continuous nor di®erentiable on R.

2. Let G be an in¯nite cyclic group. If f is an automorphism of G, then
(a) fn 6= IdG for any n two N.
(b) f2 = IdG.
(c) f = IdG.
(d) there exists an n two N such that f(x) = xn, for all x two G.

3. Let G be a group of order 10 . Then
(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 five which is not normal.
4. For every ® two I, let X® be a non-empty topological space such that the product space Y®2I
X® is locally compact. Then
(a) X® must be compact other than for ¯nitely many ®.
(b) X® must be a singleton other than for ¯nitely many ®.
(c) every X® must be compact.
(d) the indexing set I must be countable.
5. Let f : R ! R be a function. Then the set fx two R : f is continuous at xg is always
(a) a G± set. (b) an F¾ set.
(c) an open set. (d) a closed set.


6. Let A = R £ R and B = Q £ Q. 2 distinct points in A n B can be joined together
within A n B
(a) always by a line segment.
(b) always by a smooth path.
(c) not always by a smooth path but always by a continuous path.
(d) cannot be joined together always by a continuous path.

7. Let G be a group of order 255. Then
(a) the number of Sylow - three subgroups cannot be more than 1.
(b) the number of Sylow - 11 subgroups is at lowest 1.
(c) the number of Sylow - three subgroups is one or 85.
(d) the number of Sylow - five subgroups is 51.

8. The number of ideals in the ring
R[x]
(x2 ¡ 1)
is
(a) 1. (b) 2. (c) 3. (d) 4.

9. All the eigenvalues of the matrix264
1 two 0
2 one 0
0 0 ¡1
375
lie in the disc
(a) j¸ + 1j · 1. (b) j¸ ¡ 1j · 1.
(c) j¸ + 1j · 2. (d) j¸ ¡ 1j · 2.

10. For the ordinary di®erential formula sin(x)y00(x) + y0(x) + y(x) = 0,

Earning:   Approval pending.