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Posted Date: 28 Mar 2009 Posted By: msureshmca06 Member Level: Gold
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2007 University of Hyderabad Ph.d(Maths/Applied Maths) 2008-Entrance Exam University Question paper
University of Hyderabad,
Entrance Examination, 2008
Ph.D. (Mathematics/Applied Mathematics)
Part A
1. Let f : R ! R be a function given 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 2 N.
(b) f2 = IdG.
(c) f = IdG.
(d) there exists an n 2 N such that f(x) = xn, for all x 2 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 5 which is not normal.
4. For each ® 2 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 except for ¯nitely many ®.
(b) X® must be a singleton except for ¯nitely many ®.
(c) each X® must be compact.
(d) the indexing set I must be countable.
5. Let f : R ! R be a function. Then the set fx 2 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. Two 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 - 3 subgroups cannot be more than 1.
(b) the number of Sylow - 11 subgroups is at least 1.
(c) the number of Sylow - 3 subgroups is 1 or 85.
(d) the number of Sylow - 5 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 2 0
2 1 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 equation sin(x)y00(x) + y0(x) + y(x) = 0,
(a) every point is an ordinary point.
(b) every point is a singular point.
(c) x = n¼ is a regular singular point.
(d) x = n¼ is an irregular singular point.
11. If in a group, an element a has order 65, then the order of a25 is
(a) 5. (b) 12. (c) 13. (d) 65.
12. The number of sub¯elds of F227 (distinct from F227 itself) is
(a) 1. (b) 2. (c) 3. (d) 4.
13. The number of Jordan canonical forms for a 5 £ 5 matrix with minimal polynomial
(x ¡ 2)2(x ¡ 3) is
(a) 1. (b) 2. (c) 3. (d) 4.
14. The number of degrees of freedom of a rigid cube moving in space is
(a) 1. (b) 3. (c) 5. (d) 6.
15. Let A ½ R be a measurable set. Then
(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.
16. The equation uxx + x2uyy = 0 is
(a) elliptic.
(b) elliptic everywhere except on x = 0 axis.
(c) hyperbolic.
(d) hyperbolic everywhere except on x = 0 axis.
17. The solution of the Laplace equation in spherical polar co-ordinates (r; µ; Á) is
(a) log(r). (b) r. (c) 1=r. (d) r and 1=r.
18. A particle moves in a circular orbit in a force ¯eld F(r) = ¡K=r2, (K > 0). If K
decreases to half its original value then the particle's orbit
(a) is unchanged. (b) becomes parabolic.
(c) becomes elliptic. (d) becomes hyperbolic.
19. Let T : X ! Y be a linear map between normed spaces over C. Then the minimum
requirement ensuring the continuity of T is
(a)X is ¯nite dimensional. (b)X and Y are ¯nite dimensional.
(c) Y = C. (d) Y is ¯nite dimensional.
20. Let H be a Hilbert space. Which of the following is true?
(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 ¯nite dimensional.
21. For each n 2 N, let fn : [0; 1] ! [0; 1] be a continuous function and let f : [0; 1] ! [0; 1]
be de¯ned as f(x) = lim sup
n!1
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.
22. Let f; g : C ! C be holomorphic and let A = fx 2 R : f(x) = g(x)g : The minimum
requirement for the equality f = g is
(a) A is uncountable. (b) A has a positive Lebesgue measure.
(c) A contains a nontrivial interval. (d) A = R.
23. The critical point of the system x0(t) = ¡y + x2, y0(t) = x is
(a) a stable center. (b) unstable.
(c) an asymptotically stable node. (d) an asymptotically stable spiral.
24. An example of a subset of N which intersects every set of form fa + nd : n 2 Ng,
a; d 2 N, is
(a) f2k : k 2 Ng. (b) fk2 : k 2 Ng.
(c) fk + k! : k 2 Ng. (d) fk + k2 : k 2 Ng.
25. The characteristic number of the integral equation Á(x)¡¸Z 2¼
0
sin(x) sin(t)Á(t) dt = 0
is
(a) ¼. (b)1¼
(c) 2¼.12¼
Part B
Answer any Ten questions
1. Let f be a map from R to R such that f(a+b) = f(a)f(b). If f 6= 0 and it is continuous
at 0 then show that there exists a nonzero c 2 R such that f(x) = cx for all x 2 R.
2. Give an entire function whose image omits only the value 2¼. Also ¯nd a MÄobius map
whose only ¯xed point is 2¼.
3. Let f(z) = z6¡5z5+2z4+1 and K = fz 2 C : jz ¡ 2ij · 1g : Show that min fjf(z)j : z 2 Kg
is attained at some point on the boundary of K.
4. Let f : W ! R3 be a linear transformation given by f (¸1v1 + ¸2v2) = (¸1; ¸2; 0) where
W is the space generated by the vectors v1 = (1; 1;¡1) and v2 = (1;¡1; 1). Describe
how you would extend f to R3 so that the determinant of f is 1. De¯ne such an
extended f.
5. Consider the Banach space `1 of all complex sequences f®ng such that
1 Xn=1
j®nj < 1
with the norm jj f®ng jj1 =
1 Xn=1
j®nj. Let f¸ng be a sequence of complex numbers such
that f¸n®ng 2 `1 for all f®ng 2 `1. De¯ne T : `1 ! `1 by T (f®ng) = f¸n®ng. If T
is a bounded linear operator on `1 then show that f¸ng is bounded. In this case what
will be the value of jjTjj?
6. Determine the smallest m such that the ¯eld with 5m elements has a primitive 12th
root of 1.
7. Let A = f® 2 R j a®2 + b® + c = 0 for some integers a; b; cg. Then prove that A is a
countably in¯nite set.
8. Let RN be the set of all sequences of real numbers. Two members (an) and (bn)
are said to be asymptotic if lim sup
n!1
(jan ¡ bnj) = 0; they are said to be proximal if
lim inf
n!1
(jan ¡ bnj) = 0. Prove that asymptoticity is an equivalence relation on RN where
as proximality is not. Give an example of a proximal pair that is not asymptotic.
9. De¯ne a topology T on R by declaring a subset U ½ R to be open if U = Á or 0 2 U.
Describe all ¯nite subsets of R which are dense in (R; T ). Give a basis of (R; T ) each
of whose element is a ¯nite set.
10. Let f : R ! R be a di®erentiable function with a bounded derivative. De¯ne
fn(x) = f µx +
1
n¶. Show that fn converges uniformly on R to f.
11. Let fn(x) = xn for 0 · x · 1. Find the pointwise limit f of the sequence ffng. Prove
that lim
n!1Z 1
0
fn(x) dx = Z 1
0
f(x) dx. Is the convergence uniform?
12. Find the extremal of the functional J[y] = Z 1
0 µx + 2y +
y02
2 ¶ dx, y(0) = 0, y(1) = 0.
Also test for extrema.
13. Construct the Green's function for the boundary value problem y00 + y = 0 subject to
the boundary conditions y(0) + y0(¼) = 0, y0(0) ¡ y(¼) = 0.
14. Find the complete integral of p2q2 + x2y2 = x2q2(x2 + y2).
13
15. Solve the integral equation Á(x) ¡ ¸Z 2¼
0 jx ¡ tj sin(x)Á(t) dt = x:
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