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

a. I and II are actual but III is false.
b. I and III are actual but II is false.
c. Only II and III are actual.
d. Only III is actual.
e. None of the above.
17. 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.
18. For the function f(z) = sin z
z2 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.
19. The number of roots of f(z) = z5 +5z3 +z ¡2 which lie inside the circle
of radius 5/2 centred at the origin is
a. 0 b. three c. five d. seven e. none of these.

20. The image of the unit circle under the map f(z) = one + z2 is
a. again the identical unit circle.
b. a different circle with a various centre but the identical radius.
c. a different circle with the identical centre but a various radius.
d. not a a circle.
e. none of the above.
21. If A and B are subsets of IR, describe the distance ranging from them by
d(A;B) = supn2IRjÂA(n) ¡ ÂB(n)j. (ÂA is the indicator function of A)
Then the metric space (P(IR); d) is
a. compact and connected.
b. compact and normal.
c. connected and normal.
d. 2nd countable.
e. discrete.
22. In the complex Hilbert space L2([0; 2¼])
a. the functions feinx j n two ZZg form an orthonormal basis.
b. the functions f one p2¼ einx j n two ZZg form an orthonormal basis.
c. the functions fenx j n two ZZg form an orthonormal basis.
d. the functions f one p2¼ enx j n two ZZg form an orthonormal basis.
e. none of the above.
23. The number of Sylow 2-subgroups in D7, the dihedral group of order 14,
is
a. one b. two c. three d. five e. 7.
24. If y1 and y2 are 2 solutions of y00 + x2y0 + (1 ¡ x)y = 0 on [-1, 1] such
that y1(0) = 0, y01 (0) = ¡1, y2(0) = ¡1 and y02 (0) = one then
a. y1, y2 are linearly independent on [-1,1].
b. y1, y2 are linearly dependent on [-1,1].
c. y1, y2 are linearly dependent on [0,1].
d. y1, y2 are linearly dependent on [-1,0].
e. none of the above.

25. The ODE x2(1 ¡ x)2y00 + (1 ¡ x)y0 + x2y = 0 has
a. both x = 0 and x = one as regular singular points.
b. both x = 0 and x = one as irregular singular points.
c. x = 0 as a regular singular point and x = one as an irregular singular
point.
d. x = 0 as an irregular singular point and x = one as a regular singular
point.
e. none of the above.
Part - B
There are 15 ques. in this part. every ques. carries five marks. ans
as many as you can. The maximum you can score is 50 marks. Justify your
answers. This part must be answered in a separate ans book given.
1. Let p : P(IN) ! IN be the function described 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= Á.
2. Show that the function f(x) = x + sin x describes a homeomorphism from
IR to IR.
3. What is the characteristic polynomial and minimal polynomial over QI of
the matrix A =26664
0 0 0 ¡1
1 0 0 ¡1
0 one 0 ¡1
0 0 one ¡1
37775
? obtain a vector v such thatfv; Av;A2v;A3vg
is a basis of IR4.
4. Let n ¸ three be an odd integer and ®1; ®2; :::; ®n¡1 the non-real nth roots
of 1. Show that (1 + ®2
1)(1 + ®2
2):::(1 + ®2
n¡1) = 1.
5. In a commutative ring R with 1, for any subset I of R describe V (I) =
fP j P is a prime ideal containing Ig. Show that if I1 and I2 are two
ideals of R then V (I1)SV (I2) = V (I1 TI2).
6. Show from 1st principles that a group of order 65 must be cyclic.

7. describe absolute continuity. provide an example of a continuous function
that is not absolutely continuous. Show why your example works.
8. For a real number p > one describe the space lp. Show that the dual space
(lp)¤ is isomorphic to lq where q = p
p¡1 .
9. Determine the Galois group of QI (e
2¼i
7 ) over QI .
10. Let V be a finite dimensional vector space, V = V1+V2, where V1 and V2
are 2 subspaces of V . Let T be a linear transformation on V such that
T(V1) µ V2 and T(V2) µ V1. Suppose that TjV1
and TjV2
are injective.
Show that T is invertible. (Hint: consider T2).
11. Investigate for solvability the integral formula
Á(x) ¡ ¸ Z 1
0
(2xt ¡ 4x2)Á(t) dt = one ¡ 2x
for various values of the parameter ¸.
12. obtain the extremals with corner point for the functional
J[y] = Z 2
0
(y0)2(y0 ¡ 1)2 dx; y(0) = 0; y(2) = 1:
13. Construct the Green’s function for the B.V.P. y00 = ¡f(x), y(0) = 0,
y(1) + y0(1) = two and hence write its solution in terms of the Green’s
function.
14. Consider the non-linear p.d.e. pq = 1. Show that 2 initial strips are
possible for the initial curve x = 2t, y = 2t, z = 5t. obtain a solution of
the formula containing the initial curve.
15. Show that the transformation Q = p + iaq, P = p¡iaq
2ia is canonical and
obtain a generating function.

Earning:   Approval pending.