University of Hyderabad (UoH) 2009 Ph.D Maths Entrance - Question Paper
PART A
Each question carries 1 mark. | mark will be deducted for each wrong answer. There will be no penalty if the question is left unanswered.
The set of real numbers is denoted by R, the set of complex numbers by C, the set of rational numbers by Q and the set of integers by Z.
1. Let X be an inner-product space over the complex field C. Which of the following statements are not equivalent to any other three statements? For x, y X
(a) xy, i.e., x is orthogonal to y.
(b) \\x + y\\2 = \ \x\\2 + ||y||2, where || || is the norm on X induced by the inner product of X.
(c) \\x + ay\\ = \ \x ay\\ for all a C.
(d) \\x + ay\\ > \\x\\ for all a C.
2. Let f : R R. Then
(I) if f is a differentiable function, then f is absolutely continuous.
(II) if f is an absolutely continous function then f is uniformly continuous.
(III) if f is a differentiable function then f is uniformly continuous.
(a) Only (I) is true. (b) (II) and (III) are true.
(c) (I) and (III) are true. (d) Only (II) is true.
3. The number of roots of z1 + z2 + 8z3 + 2z + 1 = 0 between the circles \z\ = 1 and \z\ = 2 are
(a) 3. (b) 4. (c) 5. (d) 6.
if x < 0 if x > 0.
(b) is \x\.
2
1+x2
2
(a) does not exist.
(c) is x. (d) is \x\.
6. How many automorphisms are there on Zg x Z16?
(b) 72.
(c) 48.
(d) 16.
(a) 144.
7. For the function f (z) = sin - J ,z = 0 is
(a) an essential singularity.
(b) a branch point.
(c) a removable singularity.
(d) a simple pole.
8. Let V be the set of all homogeneous polynomials of degree d in n-variables over a field F. Then dim Vf is
n\ , \ f d\ f \ ( n + d 1 \ , . f n + d d - (bM n) (cH 1 . (d) d
(a)
9. The space lp is a Hilbert space if and only if
(a) p < 1. (b) p is even. (c) p = to. (d) p = 2.
dx 2 dy 3
= x + y + x y, = 3x y + 2xy is
dt dt y y
(a) a saddle point. (b) a stable node.
(c) asymptotically stable. (d) not a simple critical point.
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13. aijxj = bi, i = 1,2,..., 6 is a set of 6 real equations in seven j=i
variables. The condition that the system have infinitely many solutions is
(a) rank of [aij] = rank of [aij ,bi] where [aij, bi] is the augmented matrix.
(b) a 6 x 6 minor of [aij] must have a non-zero determinant.
(c) a 6 x 6 minor of [aij] must have a zero determinant.
(d) the matrix [aij] must have rank 6.
14. Let E be a Lebesgue measurable subset of R and a be a non-zero real number. Suppose aE = {ax : x E}. Then the Lebesgue measure m(aE) is equal to
(a) m(E). (b) a m(E). (c) |a| m(E). (d) none of these.
15. The locus represented by the equation in complex variables z = x + iy, lz 2| + |z + 4| = 10 is
(a) a circle. (b) a parabola. (c) a straight line. (d) an ellipse.
16. Suppose G is a finite group and N is a normal subgroup and H is a subgroup. Let n = /HI If gcd(n,O(N)) = 1, then
(a) H C N. (b) N C H. (c) H = N. (d) O(H) divides O(N).
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(a) elliptic, x > 0. (c) hyperbolic, x > 0. |
(b) hyperbolic, x < 0. (d) none of these. |
19. Let X be a compact Hausdorff space. Then
(a) X is metrizable and separable.
(b) X is metrizable but need not be separable.
(c) X is normal but need not be metrizable.
(d) X is completely regular but need not be normal.
20. Let X be a connected metric space and let Y C X.
(a) If Y is open, then Y is connected.
(b) If Y is closed, then Y connected.
(c) If Y is compact, then Y is connected.
(d) All of (a), (b), (c) are false.
21. Let f : [0, 2] [0,1] be any continuously differentiable function. Then
(a) \f '(x)\ < 1 V x e [0, 2].
(b) \f '(x)\ < 2 V x e [0, 2].
(c) \f '\ is bounded but need not be bounded by 2.
(d) \f '\ need not be bounded.
22. Let d be the Euclidian metric and d be another metric on R. Let A C R be closed and bounded with respect to d. Then
(a) A is compact.
(b) A is compact only if d = d.
(c) A is compact if the topologies induced by d and d are the same.
(d) A is compact if there exists a > 0, /3 > 0 such that a d(x, y) < d(x, y) < /3 d(x, y) V x,y e R.
23. Let X = R and T = {U \ X U is countable} U {$}. Then (X, T) is
(a) not connected but compact.
(b) connected and compact.
(c) connected but not compact.
(d) neither connected nor compact.
24. Let f(x) = x2, g(x) = x\x\ be defined in [1,1], and let W be the Wronskian of f and g on [-1,1]. Then on [-1,1]
(a) f and g are linearly independent and W = 0.
(b) f and g are linearly dependent and W = 0.
(c) f and g are linearly independent and W = 0.
(d) f and g are linearly dependent and W = 0.
25. F(z xy, x2+y2) = 0 is the solution of the partial differential equation
(a) yzx xzy = y2 x2. (b) yzx + xzy = y2 x2.
(c) yzx + xzy = y2 + x2. (d) yzx xzy = y2 + x2
1 2 2
1. Evaluate the integral r.-0. , 0dx, a > 0,b > 0 using
Laplace's inversion formula.
3. Does there exist a function f : R R (need not be continuous) whose graph {(x, f (x)) : x R} is dense in R2? Justify your answer.
4. Show that the sequence of functions fn(x) = n2xn(1 x) defined on
I = [0,1] converges point wise to zero on I but not uniformly.
5. Let G be a group with a topology on it such that for every x E G the map from G G, y xy is a homeomorphism. Let H be a subgroup. Suppose H is open. Show that H is also closed.
6. Consider the Banach space C[1,1] of all real valued continuous functions defined on [1,1] with the norm \\f|| = sup \f(x)| where
-1<x<1
f E C[1,1]. For f E C[1,1], denote by f(x) = f(\x\), x E [1,1]. Define T : C[1,1] C[1,1] by T(f) = f. Show that T is a bounded linear operator on [1,1]. Determine the value of ||T||.
7. Determine all real numbers L > 1 so that the boundary value problem x2y''(x) + y(x) =0, 1 < x < L; y(1) = y(L) = 0 has a non-trivial solution.
8. Using Greens function, find the solution of the ordinary differential
equation y" + y = 1, y(0) = 0, y (2) = 0- m x2dx 7n .
9. Show that / r,-0-- = , by contour integration
J-oo (x2 + 1)2(x2 + 2x + 2) 50 J 6
along a closed contour consisting of the semi-circular arc of radius R passing through (R, 0), (R, 0) and the real axis from (R, 0) to (R, 0) including the real axis.
10. Show that a finite field cannot be algebraically closed.
aq\ aq2 ( p2
11. Discuss whether the transformation Q = arctan ( , p = 1 +
is canonical, where a is an arbitrary constant.
12. Let A be a 5 x 5 matrix with a minimal polynomial (x 1)(x + 2)2. What are the possible Jordan canonical forms for A?
13. Determine the inverse of | 2 3 1 | by computing its echelon form.
1 1 2
14. Let p : R R be a non-constant real polynomial. If an = p(n) for n = 0,1,2,..., then determine the radius of convergence of the
complex power series anzn.
n=0
15. Find the general integral of the first order partial differential equation xux + (1 + y)uy = x(1 + y) + xu.
16. Solve the following Cauchy problem z = xp + yq p2q with the initial data x0(s) = s, yo(s) = 2, z0(s) = s + 1.
r 2n
0(x) \ \x n\ 0(t)dt = x.
0
18. Consider the following Linear Programming problem
P : max 2xi + x2 x3 such that x1 + 2x2 + x3 < 8 x1 + x2 2x3 < 4 x1, x2, x3 > 0
The optimal solution to this problem has x1 =8 and x5 = 12 in the basis. Now if you have to choose between increasing the right hand sides of the first constraint and second constraint, which one would you choose? Explain your choice. What is the effect of this increase on the optimal value of the objective function?
P : min -x2 such that xi + x2 > 2 -x2 > -4 x1,x2 > 0.
Solve P graphically to obtain the primal optimal solution. State the dual D and solve it graphically. Write down the complementary slackness condition for the the problem and illustrate the same for the primal and dual variables here.
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20. Consider the following transportation problem with 3 origins and 4 destinations and the cij cost matrix given by | ||||||||||||||||||||||||||||||
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(a) Is the problem balanced? If not, reformulate it as a balanced transportation problem.
(b) Find an initial basic feasible solution using any method.
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(d) such a linear transformation is not possible.
(X2 + 1)2. Then
The cardinality of G, |G| = pq, where p, q are prime p < q. Show that if q 1mod(p) then G has at least two non-trivial normal abelian subgroups.
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Attachment: |
| Earning: Approval pending. |
