Code: A-01/C-01/T-01 JUNE 2006 Subject: MATHEMATICS-I
Time: 3 Hours Max. Marks: 100
NOTE: There are 9 Questions in all.
· Question 1 is compulsory and carries 20 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.
· Out of the remaining EIGHT Questions answer any FIVE Questions. Each question carries 16 marks.
· Any required data not explicitly given, may be suitably assumed and stated.
Q.1 Choose the correct or best alternative in the following: (2x10)
a. The value of limit 
is
(A) 0 (B) 1
(C) 2 (D) does not exist
b. If 
,
then 
equals
(A) 0 (B) u
(C) 2u (D) 3u
c. Let 
. Then the value
of 
is
(A)
(B)
0
(C) 
(D)
d. The value of 
is
(A)
1 (B)
(C) 
(D)
3
e. The solution of 
is
(A)
(B)
(C) 
(D)
f. The solution of 
is
(A) sin x (B) cos x
(C) x sin x (D) x cos x
g. Let 
and 
be elements of 
. The set of
vectors 
is
(A) linearly independent (B) linearly dependent
(C) null (D) none of these
h. The eigenvalues
of the matrix 
are
(A)
and 1 (B)
0, 1 and 2
(C) –1, –2 and 4 (D) 1, 1 and –1
i. Let 
, 
, 
be the Legendre
polynomials of order 0, 1, and 2, respectively. Which of the following
statement is correct?
(A)
(B) 
(C) 
(D) 
j. Let 
be the Bessel function of order n.
Then 
is
equal to
(A) 
(B)
(C) 
(D) 
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. Consider
the function f (x, y) defined by 
Find 
and 
.
Is 
differentiable at
(0, 0)? Justify your answer. (8)
b. Find
the extreme values of 
subject to the constraints 
and 
. (8)
Q.3 a. Find
all critical points of 
and determine relative extrema at
these critical points. (8)
b. Find the second order Taylor expansion of 
about the point 
. (4)
c. Change
the order of integration in the following double integral and evaluate it : 
. (4)
Q.4 a. Solve
the differential equation 
. (4)
b. Solve the differential
equation 
. (6)
c. Find the
general solution of the differential equation 
. (6)
Q.5 a. Find the general solution of the differential equation
. (8)
b. Find the linear Taylor series polynomial approximation to the
function 
about
the point (1, 2). Obtain the maximum absolute error for the polynomial
approximation in the region 
, 
. (8)
Q.6 a. Find the general solution of the differential equation 
. (9)
b. Show that the eigenvalues of a Hermitian matrix are real. (7)
Q.7 a. Using
Frobenius method, find two linearly independent solutions of the differential
equation 
. (10)
b. Solve the following system of equations by matrix method:
(6)
Q.8 a. Express
the polynomial 
in
terms of Legendre polynomials. (8)
b. Let
be the
Bessel function of order 
. Show 
. (8)
Q.9 a. If A is a diagonalizable matrix and f (x) is a polynomial, then show that f (A) is also diagonalizable. (7)
b. Let
. Find the matrix P so that
is a
diagonal matrix. (9)