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University of Hyderabad, Entrance Examination, 2008 Ph.D. (Mathematics/Applied Mathematics)
Hall Ticket No.
Time: 2 hours Max. Marks: 75 Part A: 25 Marks Part B: 50 Marks
Instructions
- Calculators are not allowed.
- Part A carries 25 marks. Each cor- rect answer carries 1 mark and each wrong answer carries minus one third mark. So do not gamble. If you want to change any answer, cross out the old one and circle the new one. Over written answers will be ignored.
- Part B carries 50 marks. Instructions for answering Part B are given at the beginning of Part B.
- Do not detach any pages from this an- swer book. It contains 15 pages in addition to this top page. Pages 14 and 15 are for rough work.
Answer Part A by circling the correct letter in the array below: 1 a b c d 2 a b c d 3 a b c d 4 a b c d 5 a b c d 6 a b c d 7 a b c d 8 a b c d 9 a b c d 10 a b c d 11 a b c d 12 a b c d 13 a b c d 14 a b c d 15 a b c d 16 a b c d 17 a b c d 18 a b c d 19 a b c d 20 a b c d 21 a b c d 22 a b c d 23 a b c d 24 a b c d 25 a b c d
Part A
- Let f : R → R be a function given by f (x) = min(1, x, x^3 ). Then
(a) f is continuous but not differentiable on R. (b) f is continuous and differentiable on R. (c) f is not continuous but differentiable on R. (d) f is neither continuous nor differentiable on R.
- Let G be an infinite cyclic group. If f is an automorphism of G, then
(a) f n^6 = IdG for any n ∈ N. (b) f 2 = IdG. (c) f = IdG. (d) there exists an n ∈ N such that f (x) = xn, for all x ∈ G.
- 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.
- For each ∏ α ∈ I, let Xα be a non-empty topological space such that the product space
α∈I
Xα is locally compact. Then
(a) Xα must be compact except for finitely many α. (b) Xα must be a singleton except for finitely many α. (c) each Xα must be compact. (d) the indexing set I must be countable.
- Let f : R → R be a function. Then the set {x ∈ R : f is continuous at x} is always
(a) a Gδ set. (b) an Fσ set. (c) an open set. (d) a closed set.
- 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.
- The number of degrees of freedom of a rigid cube moving in space is (a) 1. (b) 3. (c) 5. (d) 6.
- Let A ⊂ 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.
- The equation uxx + x^2 uyy = 0 is
(a) elliptic. (b) elliptic everywhere except on x = 0 axis. (c) hyperbolic. (d) hyperbolic everywhere except on x = 0 axis.
- The solution of the Laplace equation in spherical polar co-ordinates (r, θ, φ) is (a) log(r). (b) r. (c) 1/r. (d) r and 1/r.
- A particle moves in a circular orbit in a force field F (r) = −K/r^2 , (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.
- 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 finite dimensional. (b)X and Y are finite dimensional. (c) Y = C. (d) Y is finite dimensional.
- 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 finite dimensional.
- For each n ∈ N, let fn : [0, 1] → [0, 1] be a continuous function and let f : [0, 1] → [0, 1] be defined as f (x) = lim sup n→∞
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.
- Let f, g : C → C be holomorphic and let A = {x ∈ R : f (x) = g(x)}. 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.
- The critical point of the system x′(t) = −y + x^2 , y′(t) = x is (a) a stable center. (b) unstable. (c) an asymptotically stable node. (d) an asymptotically stable spiral.
- An example of a subset of N which intersects every set of form {a + nd : n ∈ N}, a, d ∈ N, is (a) { 2 k : k ∈ N}. (b) {k^2 : k ∈ N}. (c) {k + k! : k ∈ N}. (d) {k + k^2 : k ∈ N}.
- The characteristic number of the integral equation φ(x)−λ
∫ (^2) π
0
sin(x) sin(t)φ(t) dt = 0 is (a) π. (b)^1 π
. (c) 2π. 1 2 π
- Let f (z) = z^6 − 5 z^5 +2z^4 +1 and K = {z ∈ C : |z − 2 i| ≤ 1 }. Show that min {|f (z)| : z ∈ K} is attained at some point on the boundary of K.
- Let f : W → R^3 be a linear transformation given by f (λ 1 v 1 + λ 2 v 2 ) = (λ 1 , λ 2 , 0) where W is the space generated by the vectors v 1 = (1, 1 , −1) and v 2 = (1, − 1 , 1). Describe how you would extend f to R^3 so that the determinant of f is 1. Define such an extended f.
- Consider the Banach space ` 1 of all complex sequences {αn} such that
∑^ ∞
n=
|αn| < ∞
with the norm || {αn} || 1 =
∑^ ∞
n=
|αn|. Let {λn} be a sequence of complex numbers such that {λnαn} ∈ 1 for all {αn} ∈ 1. Define T : 1 → 1 by T ({αn}) = {λnαn}. If T is a bounded linear operator on ` 1 then show that {λn} is bounded. In this case what will be the value of ||T ||?
- Determine the smallest m such that the field with 5m^ elements has a primitive 12th root of 1.
- Define a topology T on R by declaring a subset U ⊂ R to be open if U = φ or 0 ∈ U. Describe all finite subsets of R which are dense in (R, T ). Give a basis of (R, T ) each of whose element is a finite set.
- Let f : R → R be a differentiable function with a bounded derivative. Define
fn(x) = f
x + n^1
. Show that fn converges uniformly on R to f.
- Let fn(x) = xn^ for 0 ≤ x ≤ 1. Find the pointwise limit f of the sequence {fn}. Prove
that lim n→∞
0
fn(x) dx =
0
f (x) dx. Is the convergence uniform?
- Find the extremal of the functional J[y] =
0
x + 2y + y
′ 2 2
dx, y(0) = 0, y(1) = 0. Also test for extrema.
- Solve the integral equation φ(x) − λ
∫ (^2) π
0
|x − t| sin(x)φ(t) dt = x.
Rough Work
