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Posted By: Ravindranath Member Level: Gold Posted Date: 23 May 2008
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2005 University of Hyderabad ENTRANCE EXAMINATION,2005 (Ph.D. Mathematics) University Question paper
ENTRANCE EXAMINATION,2005
Ph.D. Mathematics/ Applied Mathematics
TIME: 2 hours MAX. MARKS: 75
Part A: 25 Part B: 50
HALL TICKET No.
INSTRUCTIONS
1. Calculators are not allowed.
2. 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.
3. Instructions for answering Part B are given at the beginning of Part B.
4. Do not detach any pages from this answer book. It contains 8 pages. A
separate answer book will be provided 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.
Part-A
Answer Part A by circling the correct answer. A correct answer
gets 1 mark and a wrong answer gets ¡(1=4) mark.
1. If A =26664
1 1 0 0
0 1 1 0
0 0 1 0
0 0 0 2
37775
then the rank of (A ¡ I) (I is the 4 £ 4 identity
matrix) is
a. 4 b. 3 c. 2 d. 1 e. 0
2. The minimal polynomial of
26666664
2 1 0 0 0
0 2 1 0 0
0 0 2 0 0
0 0 0 2 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 9 vectors in IR6. Then
a. S contains a basis of IR6.
b. there exist 6 linearly independent vectors in S.
c. S must span IR6.
d. there exist 3 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.
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 three statements are true.
b. None of these statements is true.
c. I and II are true but III is false.
d. Only I is true.
e. none of the above.
6. Let f(x) = max(sin x; cos x) for all x 2 IR. Then
a. f is differentiable on IR.
b. f is nowhere differentiable.
c. f is differentiable except at 0.
d. f is differentiable except at a countable set of points.
e. none of the above.
7. Let, for each x 2 [0; 1), x = 0:x1x2x3:::: be the decimal expansion of x
not eventually all 9’s. Define f : [0; 1) ! IR by f(x) = x1, the first digit
in the expansion. Then R1
0 f(x) dx =
a. 41
2 b. 10 c. 0 d. 1 e. does not exist.
8. The function f(x; y) = ( xy
x2+y2 if(x; y) 6= (0; 0)
0 at (0; 0)
is
a. continuous at (0,0) but partial derivatives do not exist at (0,0).
b. continuous at (0,0) and partial derivatives exist at (0,0).
c. discontinuous at (0,0) and partial derivatives do not exist at (0,0).
d. discontinuous at (0,0) but partial derivatives exist at (0,0).
e. none of the above.
9. 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.
10. Let T = fÁ; IRgSf(x;1) j x 2 IRg. 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.
11. The number of homomorphisms from C2 £ C2 ! C2 is (Cn is the cyclic
group of order n)
a. 5 b. 4 c. 3 d. 2 e. 1
12. 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.
13. 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.
14. The number of proper subfields of F32 is
a. 16 b. 8 c. 4 d. 2 e. 1.
15. An example of a function on IR whose graph does not intersect the x-axis
is
a. f(x) = x3 ¡ 3x + 2
b. f(x) = x4 + x2 + 1
c. f(x) = x7 ¡ 2
d. f(x) = x11 ¡ x
2 + 1
e. none of the above.
16. 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 L3([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.
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. 3 c. 5 d. 7 e. none of these.
20. The image of the unit circle under the map f(z) = 1 + z2 is
a. again the same unit circle.
b. another circle with a different centre but the same radius.
c. another circle with the same centre but a different radius.
d. not a a circle.
e. none of the above.
21. If A and B are subsets of IR, define the distance between 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. second countable.
e. discrete.
22. In the complex Hilbert space L2([0; 2¼])
a. the functions feinx j n 2 ZZg form an orthonormal basis.
b. the functions f 1 p2¼ einx j n 2 ZZg form an orthonormal basis.
c. the functions fenx j n 2 ZZg form an orthonormal basis.
d. the functions f 1 p2¼ enx j n 2 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. 1 b. 2 c. 3 d. 5 e. 7.
24. If y1 and y2 are two 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) = 1 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 = 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.
1. 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= Á.
2. Show that the function f(x) = x + sin x defines 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 1 0 ¡1
0 0 1 ¡1
37775
? Find a vector v such thatfv; Av;A2v;A3vg
is a basis of IR4.
4. Let n ¸ 3 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 define 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 first principles that a group of order 65 must be cyclic.
7. Define absolute continuity. Give an example of a continuous function
that is not absolutely continuous. Show why your example works.
8. 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 .
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 two 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 equation
Á(x) ¡ ¸ Z 1
0
(2xt ¡ 4x2)Á(t) dt = 1 ¡ 2x
for different values of the parameter ¸.
12. Find 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) = 2 and hence write its solution in terms of the Green’s
function.
14. 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.
15. Show that the transformation Q = p + iaq, P = p¡iaq
2ia is canonical and
find a generating function.
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