MATHEMATICAL SCIENCES
PAPER-II
1. Let {xn} and {yn} be two sequences of real numbers. Prove or disprove each of the
statements :
1. If {xnyn} converges, and if {yn} is convergent, then {xn} is convergent.
2. {xn + yn} converges to x+y if {xn} converges to x and {yn} converges to
y.
3. If{xn/yn} is convergent, then both {xn} and {yn} are convergent.
4. If {xn} is convergent and {yn} is divergent, then {xnyn} is divergent.
2. Let f : [a,b] ? ¡ be differentiable.
(a) Prove that
2( ) 0 iff 0 on [ , ].
b
a
? f t dt = f = a b .
(b) If 3( ) 0,
b
a
? f t dt = then for some t0e[a,b], either '
0 f (t )=0 or 0 f (t )=0 .
3. Let {fn} be a sequence of real-valued Lebesgue integrable functions on ¡ such that
1
. n
n
f
8
=
S? ¡ <8 Prove that for any
1
, in ( )
n
n
a e a f x
8
=
?¡ S converges a.e. to a function
ga(x) and that ga is Lebesgue integrable.
4. Let a = (a1, a2, L , an) ? ¡ n and let f(x) = ex.a where x = (x1, x2 L , xn) ? ¡ n , and
1
. ..
n
i i
i
xa xa
=
=S Compute the directional derivative of f at a point p ? ¡ n in the
direction of h ? ¡ n .
5. (a) Let A = {(x,y) ? ¡ 2 ? x2 + 2y2 < 37} n {(x,y) ? ¡ 2 ? ex > y}
Prove that A is compact.
(b) Show that any convex subset C of ¡ n is connected.
6. Show that the set {log p? p prime number} is linearly independent over ¤ .
7. Let V be the vector space of all polynomial functions of degree < n, n > 2, and let
D : V ? V denote the derivative map P a P' on V. Show that D is nilpotent and
that D is not diagonalizable.
8. Let A and B be two 3 × 3 complex matrices. Show that A and B are similar if and
only if ?A = ?B and µA = µB, where ?A, ?B are characteristic polynomials of A, B,
respectively and µA, µB are minimal polynomials of A, B, respectively.
9. Determine whether the quadratic form q(X1,X2) = 7X1 + 88X1X2 + 88p 2
2 X is
degenerate or not.
10. Let ? : O ? £ be a continuous function and suppose {z : ?z? < 1} ? O . Prove that
the function defined as
1
( ) ( )
w
f z w dw
w z
?
=
=
? - is an analytic function on the open unit
disc.
11. Let f : £ ? £ be an entire function and that f ' (z) < f (z) for all z?£ . Identify
all such functions.
12. (a) Let f :£ ?£ be an analytic function and g:£ ?¡ a harmonic function.
Prove that gof is a harmonic function.
(b) Prove that the function
u(x, y)=e2 xy .cos (x2 - y2 ), (x, y)?¡ 2 is harmonic, and find the harmonic
conjugate.
13. Compute
1
2
e z
z
e dz
=
? , where the circle is parametrized by t ? 2eit, 0 < t < 2p.
14. Let ? and µ denote the Euler totient function and Möbius function respectively.
Show that
/ /
( ). Henceshow that ( ) ( )
d n k n
n d n k
n k
? µ
= S? =S .
15. Define conjugacy class in a finite group G and show that the cardinality of any
conjugacy class divides the order of G. Use this to show that if p is a prime and G is
a group of order pn, then the centre of G contains elements other than the identity.
16. Let I, I', J be ideals in a commutative ring A. If I, J are comaximal, i.e., I + J = A and
I’, J are comaximal, i.e., I' + J = A, then show that I I' and J are also comaximal.
17. Let L K be a finite field extension of prime degree p. Show that L = K[a] for any a
? L\K.
18. Solve the BVP by determining the appropriate Green’s function, expressing the
solution as a definite integral.
-y? = f(x), y(0) + y'(0) = 0, y(1) + y'(1) = 0.
19. Consider the initial value problem :
y' = f(x,y), y(0) = 1
where f(x,y) : = ?xy? + y2, (x,y) ? D with D : = [-2,2] X [-1,3].
Show that f (x,y) is bounded and statisfies a Lipschitz condition with respect to y on
D and determine a bound and Lipschitz constant on D. Further, determine h, as
required in the Picard’s Theorem, for a unique solution of the initial value problem to
exist on ?x? < h.
20. Outline briefly the three classes of integrals of the non-linear first order partial
differential equation f(x,y,z,p,q) = 0, where p z , q z .
x y
? ?
= =
? ?
For the partial differential equation
pqz = p2(3p2 +qx) + q2(py + 4q2), obtain one of the integrals and indicate the
procedure for determining the remaining two integrals.
21. Classify and reduce the second order partial differential equation
uxx – 4x2uyy = 1
x
ux into canonical form and hence, find the general solution.
22. Derive Simpson’s 1
3
rd rule to evaluate the integral
2
( )
a h
a
f x dx
+
? . Estimate the error.
23. Find the eigenvalues and the eigenfunctions of the functional
1 1
2 '2 2
0 0
J ( y)= ?( y + y )dx subject to the conditions y(0)= y(1)= 0, ? y dx =1.
24. Find the resolvent kernel for the integral equation
1
2 2
1
? (s) f (s) ? (st s t )? (t)dt
-
= + ? +
25. Show that the transformation
1 2 2 tan 2 , 1
4
Q q P q p
p
- ? ?
= ? ? = +
? ?
is canonical. Find a generating function.
26. Let X and Y be two independent random variables such that X is uniformly
distributed on [0, 1] and Y has a discrete uniform distribution on
{0, 1, 2,L , n–1}, that is,
1, if 0,1, , 1,
( )
0, otherwise.
k n
P Y k n
? = - ? = =???
L
Define Z = X + Y. Show that Z is uniformly distributed on [0, n].
27. Let M(g) denote the moment generating function of the standard normal
distribution. Let I(a) = sup{ta – log M(t): t? ¡ }.
(i) Find I( g)
(ii) Express log M( g) in terms of I( g)
28. Using the central limit theorem for appropriate Poisson random variables show that
0
lim 1 1
! 2
n
n j
n j
e n
j
-
?8
=
S = .
29. Let {Xn} be a Markov chain with transition probability matrix P given by
n n
ij n 0 ij
1/ 2 1/ 4 1/ 4 0
2 /3 1/ 3 0 0
.
0 0 1/5 4/5
0 0 1/2 1/2
Let p = P(X = j X =i). Find limp for all i,j
n
P
?8
? ?
? ?
=? ?
? ?
? ?
? ?
? .
30. A coin with probability p for head is tossed. If a tail turns up, a random number of
balls are added to an urn. (Assume that the urn is initially empty). This procedure
is repeated till a head appears at which stage it is stopped. Let N denote the number
of stages when balls are added, and Xi = number of balls added at ith stage. Assume
that {Xi} are i.i.d. Poisson (?) random variables, and that N and {Xi} are
independent. Find the expected number of balls in the urn when the procedure
terminates.
31. Let x1, x2 L , xn be the values of a variable x. Define xmax = max{x1, L , xn},
xmin = min {x1, L , xn}, max min R = x - x and 2 2
1
( )/ .
n
i
i
s x x n
=
=S -
2 2
Showthat 2
2 4
R s R
n
= = .
32. Let T be the minimum variance unbiased estimator (MVUE) of ?. Then prove that
TK ( K a +ve integer) is the MVUE for E(TK) provided E(T2K) < 8.
33. Suppose (x1, y1),L ,(xn, yn) represent a random sample from 2 2
2 1 2 N (0,0,s ,s ,? ).
Suppose 0 ? =? (known), then find a confidence interval of 1 2 s /s with confidence
coefficient (1 – a) that incorporates the information that 0 ? =? .
34. Let X1, X2, L , Xn be i.i.d. with density
2 f (x, ) , x , 0
x
?
? = >? ? > .
(a) Find MLE of ?
(b) Derive the likelihood ratio test for H0: ?=1 vs H1 : ? ? 1.
(c) If n=4 and the observations are X1 = 3.2, X2 = 4.0, X3 = 2.0, X4 = 5.6, find the
P-value of the test derived in (b).
35. Let X1,L , Xn be independent random variables with common probability distribution
function
0 0
[ ; , ] ( ) 0
1
i
if x
P X x x if x
if x
a ß a ß
ß
ß
? <
??
= = = = ???
? >
where a, ß > 0.
(a) Find a two dimensional sufficient statistic for (a, ß)
(b) Find an unbiased estimator of 1
a +1
when ß=1.
36. Consider a regression model Yi = ?0 + ?1 xi + ei, i=1, …n, where
1
1
1 1, , ,
i 0, 1, ,
if i n
x
if i n n
? =
= ? = + ?
L
L
and ei are uncorrelated random errors with mean 0 and common variance s2. Let T1
and T2 be the two estimators of ?1 given by T1 = Y1 - Yn and
T2 =
1
1 1
1 1
where 1
n
n i
i
Y Y Y Y
n =
- = S .
(a) Verify whether T1 and T2 are unbiased and find their
variances.
(b) If possible, propose an unbiased estimator of ?1, which has variance smaller
than that of T1 and T2, with justification.
37. Consider a linear model Y = X? +e where Y is a 4 ×1 vector of observations,
1 2 3 ? =(? ,? ,? )T is a vector of unknown parameters,
4 3
1 1 0
1 0 0
1 0 1
1 0 0
X ×
? - ?
? ?
= ? ?
? ?
? ?
? ?
and e is a 4 ×1 vector of uncorrelated random errors with mean 0 and variance s2.
(a) Verify whether the following parametric functions are estimable
(i) ?1+ ?2, (ii) ?1+ ?2 + ?3
(b) Find the best linear unbiased estimator(s) of the estimable parametric
function(s) in (a) above and obtain the variance of the estimator(s).
38. Suppose 1
2
1 ( 1) 1
( 1)
2 ( 1) 2
~ , p
p p
p
x
x N
x
µ
µ
×
×
×
? ? ? ? ? ?
=?? ?? ? ?? ?? S? ? ? ? ? ? ? ? ?
% %
%
% %
with 11 12
21 22
0.
?S S ?
S=? ? > ?S S ?
Prove that the necessary and sufficient condition for 1 x
%
and 2 x
%
to be
independent is 12 S =0 . You may assume ~ ( , ), 1, 2. i pi i ii x N µ S i =
%
39. Suppose the problem is to classify an observation x
%
into one of the populations
, 1,2. i P i= Suppose fi( x
%
) denotes the density of x
%
corresponding to population Pi.
Also we attach the prior probability pi (i = 1, 2) for an observation x
%
to belong to
population Pi. Find the total probability of misclassification (TPM) and prove that
the classification rule minimizing TPM is given by:
for an x
%
, if 1 2
2 1
( )
( )
f x p
f x p
%=
%
classify it as an observation belonging to population P1 and
otherwise belonging to population P2.
40. An unknown number N of taxis plying in a town are supposed to be serially
numbered from 1 to N. If the n different taxis you have come across in the town can
be assumed to form a simple random sample with replacement, find an unbiased
estimator of the total number of taxis in the town. Also find the variance of your
estimator.
41. Show that in a randomized block design the estimates of the elementary block and
treatment contrasts are orthogonal observational contrasts.
42. Suppose in a 25–factorial experiment with factors A, B, C, D and E, a replicate is
divided into four blocks of size eight each. How many effects will be confounded?
Is it possible to confound the effects AB, BC and ABC? Justify your answer.
43. The daily demand for a commodity is approximately 100 units. Every time an order
is placed, a fixed cost of Rs.10,000/- is incurred. The daily holding cost per unit
inventory is Rs.2/-. If the lead time is 15 days, determine the economic lot size and
the reorder point. Further suppose that the demand is actually an approximation of a
probabilistic distribution in which the daily demand is normal with mean µ = 100 and
s.d. s = 10. How would you determine the size of the buffer stock such that the
probability of running out of stock during lead time is at most 0.01?
44. Consider the following linear program (LP)
max z = 4x1 + 14x2
Subject to
2x1 + 7x2 + x3 = 21
7x1 + 2x2+ x4 = 21
x1, x2, x3, x4 =0.
Each of the following cases provides an inverse matrix and its corresponding basic
variables for the LP above. Determine whether or not each basic solution is feasible.
Interpret these basic feasible solutions and hence find an optimal solution. Is the
optimal solution unique?
(a) (x2, x4);
1 0
7
2 1
7
? ?
? ?
? ?
?? - ??
? ?
(b) (x1, x4);
1 0
2
7 1
2
? ?
? ?
? ?
?? - ??
? ?
(c) (x2, x1);
7 2
45 45
2 7
45 45
? - ? ? ?
? ?
?? - ??
? ?
45. Consider an M/M/c queuing system with parameters ? and µ. Draw its statetransition
rate diagram and find the steady-state probability distribution for number
of customers in the system.
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