IHall Ticket No.
Time: 2 hours
Max. Marks: 75 Part 25 Part 50
Instructions
1.
Calculators are not allowed.

2.
Part A carries 25 marks. Each cor­rect answer carries 1 mark and each wrong answer carries 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.

3.
Part B carries 50 marks. Instructions for answering Part B are given at the beginning of Part B.

4.
Use a separate booklet for Part B.


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
Each question carries 1 mark., 0.33 mark will be deducted for each wrong answer.
There will be no penalty if the question is left:Uilanswered.-·
The set of real numbers is denoted by JR, the set of complex numbers by
the set of rational numbers by Q and the set of integers by Z.

1. Let V be a real vector space and S V2, . .. Vk} be a linearly independent subset of V. Then

dim V k.


dim V k.


dim V k.


nothing can be said about dim V.


2. The value of limx-+o is
x eX
O. 1.
3. Consider the function on JR defined by
3 if x2 1
if x2 1.
Then

f is continuous at each point of JR.


f is continuous at each point of except at x ±1.


f is differentiable at each point of JR.


f is not continuous at any point of JR.


4. Let be defined on JR2 by Ixl Iyl. Then

the partial derivatives of f at exist.


f is differentiable at


f is continuous at


none of the above hold.


U-S4

5. Let f lR lR be continuous taking values in the set of rational numbers. Then

f is strictly monotone.


f is unbounded.


f is differentiable.


the image of f is infinite


6. For the set n the element i is

both an element in the set and a limit point of the set.


neither an element in the set nor a limit point.


an element in the set, but not a limit point.


a limit point of the set, but not an element in the set.


7. If Itan zl then
7r n7r
Re z
42

Re z ="47r +n7r.


Re z 2" n7r.


1r
7r n1r

8. The number of zeroes of z9 Z5 -8z3 2z 1 in the annular region
1 Izi 2 are
3. 6. 9. 14.
9. The residue of cot z at any of its poles is

O. 1. .;2. 2y'3.


10. Let be a metric space and A c X. Then A is totally bounded if and only if

every sequence in A has a Cauchy subsequence.


every sequence in A has a convergent subsequence.


every sequence in A has a bounded subsequence.


every bounded sequence in A has a convergent subsequence.


11. Suppose N is a normal subgroup of G. For an element x in a group, let denote the order of x. Then

divides O(aN).


divides






O(aN).


12. If G is a group such that it has a unique element a of order n. Then

n 2.


n is a prime.


n is an odd prime.


n.
13. Consider the ring Z. Then

all its ideals are prime.


all its non-zero ideals are maximal.


Z/I is an integral domain for any ideal I of Z.


any generator of a maximal ideal in Z is prime.


14. Fn denotes the finite field with n elements. Then
F4 C Fg•
F4 c F12 •
F4 C FIB·
F4 c F32 .

V-S

15. Let A be an n x n matrix which is both Hermitian and unitary. Then

A2 I.


A is real.


The eigenvalues of A are -1.


The minimal and characteristic polynomials are same.


16. For 0 the matrix
sin cos

has real eigenvalues.


is symmetric.


is skew-symmetric.


is orthogonal.


17. Let C R3 be a linearly independent set and let A E R3 IIwll 1 and is linearly independent for some n EN}. Then

A is a singleton.


A is finite but not a singleton.


A is countably finite.


A is uncountable.


18. A: A is a 5 x 5 complex matrix of finite order, that is, Ak I for some kEN, must be diagonalizable.
B A diagonalizable 5 x 5 matrix must be of finite order.
Then


A and B are both true.


A is true but B is false.


B is true but A is false.


Both A and B are false.



19. Let V be the vector space of continuous functions on Let 11/111 Jo 1 and 11/1100 sup{lf(t)1 0 I}. Then

II lid and 111100) are Banach spaces.


111100) is complete but 11111) is not.


II lit) is complete but 111100) is not.


Neither of the spaces 11111), 111100) are complete.


20. The space lp is a Hilbert space if and only if
p 1. p is even. p 00. p 2.
21. Which of the following statements is correct?

On every vector space V over IR or there is a norm with respect to which V is a Banach space.


If 11.11) is a normed space and Y is a subspace of then every bounded linear functional 10 on Y has a unique bounded linear extension I to X such that I11I1 11/011.


Dual of a separable Banach space is separable.


A finite dimensional vector space is a Banach space with respect to any norm on it.


dx dy.
22. The critical point for the system dt 2x, dt 3y IS

a stable node


an unstable node.


a stable spiral.


an lillstable spiral.


d4y d2y
23. The set of linearly independent solutions of dx-dx= 0 is
42
eX,




eX, xeX}.


eX, xe-X}.


24. The set of all eigenvalues of the Sturm-Liouville problem
11"
y" Ay 0
is given by
A A 2n, n .
A A 4n2, n ..
A 2n, n ..
A 4n2, n ..
25. A complete integral of zpq p2q(X +pq2(y is

z ax by -2ab.


xz ax -by 2ab.


xz by -ax 2ab.


xz ax by 2ab.



PARTB Answer any ten questions. Each question carries 5 marks.
1.
List all possible Jordan Canonical forms of a 7 x 7 real matrix whose minimal polynomial is and characteristic poly­nomial is 3)3.

2.
A isanm x n matrix and B is an n x m matrix, n then prove that AB is never invertible.

3.
Let I C C be a non-constant entire function with the point at infinity as a pole. Show that I is a polynomial.

4.
Show that L2 and the inclusion map I I from L2 to is a bounded linear operator.

5.
Let I C be a complex analytic function such that for all z E C with Izl 1. Show that I is either constant or identity.


n
6.
Let anx+ an_lXn-1 ... alX ao be a polynomial whose

co-efficients satisfy 0. Then has a real root between and 1.

7.
Give an example of a decreasing sequence of measurable functions defined on a measurable set E of IR such that In I pointwise a.e. on E but IE limn .......co IE In­

8.
Show that all 3-Sylow subgroups in 84 are conjugate.

9.
Let a be algebraic over a field K such that the degree is odd. Show that K(a2


22
10.
Let E be the ellipse H be the hyperbola xy 1 and P be the parabola y x2• Show that no two of these are homeomorphic.

11.
Verify whether


Ql qlq2, Q2 ql q2, PI -P2 P. q2P2 -QlPl
Ql, ql -Q2 Q2 -Ql
is a canonical transformation for a system having two degrees of free­dom.

12. Determine the Green's function for the boundary value problem
xy" x 00

lim ly(x)1 00


13. Find the critical point of the system
dx dy 2
dt x y dt y 3y
and discuss its nature and stability.

14.
Reduce the following partial differential equation to a canonical form and solve, if possible.

15.
Determine the two solutions of the equation pq 1 passing through the straight line C Xo 2s, Yo 2s, Zo 5s.

16.
Let x denote the optimal solution to the following linear programming problem PI



min cTx
s.t. Ax b
x

lRmxn
where A b E lRm and x E lRn . Now, a new constraint aTx where a E lRn and E lR, is added to the feasible region and we get the following linear programming problem P2
min cTx
s.t. Ax b
aTx x 0.
Discuss about the optimality of x for the two cases x is feasible to the new LP and x is not feasible to the new LP
8

17. Consider the following linear programming problem
min Xl X2
S.t. SXl tX2 1 Xl
X2 unrestricted.
Find conditions on S and t to make the linear programming problem have multiple optimal solutions and an unbounded solution.
18. Five employees are available for four jobs in a firm. The time (in minutes) taken by each employee to complete each job is given in the table below.
Person1 1 2 3 4
1 2 3 4 5 24 20 28 18 23 18 -22 24 32 29 30 -27 18 24 30 16 29

The objective of the firm is to assign employees to jobs so as to minimize the total time taken to perform the four jobs. Dashes indicate a person cannot do a particular job.

Formulate the above problem as a linear programming problem and state the Dual of the problem.


What is the optimal assignment to the problem?


9