Code: A-01/C-01/T-01 Subject: MATHEMATICS-I
Time: 3 Hours DECEMBER 2006 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 
(A)
0 (B) 
(C)
(D)
does not exist
b. Let a function f(x, y) be continuous and possess first and second
order partial derivatives at a point (a, b). If 
is a critical point and 
, 
, 
then the point P
is a point of relative maximum if
(A)
(B)
(C) 
(D) 
c. The triple
integral 
gives
(A) volume of region T (B) surface area of region T
(C) area of region T (D) density of region T
d. If 
then matrix A is
called
(A) Idempotent Matrix (B) Null Matrix
(C) Transpose Matrix (D) Identity Matrix
e. Let 
be an eigenvalue
of matrix A then 
,
the transpose of A, has an eigenvalue as
(A)
(B)
(C) (D)
f. The system of equations is said to be inconsistent, if it has
(A) unique solution (B) infinitely many solutions
(C) no solution (D) identity solution
g. The differential
equation 
is
an exact differential equation if
(A)
(B)
(C) 
(D)
h. The
integrating factor of the differential equation 
is
(A)
(B)
(C) xy (D)
i. The functions 
defined on an
interval I, are always
(A) linearly dependent (B) homogeneous
(C) identically zero or one (D) linearly independent
j. The value of 
, the second derivative of Bessel
function in terms of 
and 
is
(A) 
(B)
(C) 
(D) 
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. Show that the function
is
continuous at (0, 0) but its partial derivatives 
and 
do not exist at (0, 0). (8)
b. Find
the linear and the quadratic Taylor series polynomial approximation to the
function 
about
the point (1, 2). Obtain the maximum absolute error in the region 
and 
for the two
approximations. (8)
Q.3 a. Find
the shortest distance between the line 
and the ellipse 
. (8)
b. Evaluate the double integral 
, where R is the region bounded by the
x-axis, the line y = 2x and the parabola 
. (8)
Q.4 a. Evaluate
the integral 
where
R is the parallelogram with successive vertices at 
, 
, 
and 
. (8)
b. Show that 
where 
is the Bessel
function of 
order. (8)
Q.5 a. Show that 
. (6)
where 
are
the Legendre polynomials.
b. Find the power series solution about x =2, of the initial value problem
.
Express the solution in closed form. (10)
Q.6 a. Solve the initial value problem 
y(0) = 0, 
, 
. (8)
b. Solve 
.
(8)
Q.7 a. Show
that set of functions 
forms a basis of the differential
equation 
.
Obtain a particular solution when 
. (6)
b. Solve the following differential equations:
(i) 
(ii)
(2
5 = 10)
Q.8 a. Let
be a
linear transformation defined by 
. Taking 
as a basis in 
determine the
matrix of linear transformation. (8)
b. If
then
show that 
,
for 
.
Hence find 
. (8)
Q.9 a. Examine
whether matrix A is similar to matrix B, where 
, 
. (8)
b. Discuss the consistency
of the following system of equations for various values of 
and if consistent, solve it. (8)