Code: A-01/C-01/T-01 Subject: MATHEMATICS-I
Time: 3 Hours Max. Marks: 100
NOTE: There are 11 Questions in all.
· Question 1 is compulsory and carries 16 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied.
· Answer any THREE Questions each from Part I and Part II. Each of these questions carries 14 marks.
· Any required data not explicitly given, may be suitably assumed and stated.
Q.1 Choose the correct or best alternative in the following: (2x8)
a. The value of limit 
(A)
equals 0. (B) equals 
.
(C)
equals 
. (D)
does not exist.
b. The total differential of the
function 
at
the point (1, 1) is
(A)
dx +
dy. (B) 
.
(C) 
. (D)
.
c. Let
then 
equals
(A) 0. (B) 2 u.
(C) 
. (D) 
.
d. The value of 
so that 
is an integrating
factor of differential equation 
is
(A) 
. (B)
- 2.
(C)
. (D)
.
e. If
method of undetermined coefficients is used for finding a particular integral
of differential equation 
then the solution to be tried is
(A)
. (B)
.
(C) 
. (D) 
.
f. Let A be a
non-singular matrix. Then the inverse of the matrix 
(A) is symmetric. (B) is skew – symmetric.
(C) does not exist. (D)
equals 
.
g. The linear
transformation 
represents
(A) reflection
about 
-axis.
(B)
reflection
about 
-axis.
(C)
clockwise
rotation through angle 
.
(D)
Orthogonal
projection on to -axis.
h. For
the
Legendre’s polynomial of order n, then 
equals
(A) 0. (B)
.
(C) 1. (D) 
.
PART I
Answer any THREE questions. Each question carries 14 marks.
Q.2 a. If
where c
is a constant, then find 
in terms of x, y, z. (6)
b. Expand
in
powers of x and 
as
far as 3rd degree terms using Taylor’s series expansion. (8)
Q.3 a. Let
u and v be two functions of x, y. Show that 
where 
denotes the Jacobian of u, v with
respect to x, y. (6)
b. Find points of local
minima and local maxima and saddle points for the function 
. (8)
Q.4 a. Solve the following system of equations by using the Cramer’s Rule
(7)
b. Find the value of so that the
vectors (1, 2, 9, 8), 
are linearly independent. (7)
Q.5 a. Prove that similar matrices have the same eigenvalues. Also give the relationship between the eigenvectors of two similar matrices. (6)
b. Find
the eigenvalues and the eigenvectors for the matrix 
(8)
Q.6 a. If a matrix 
find the matrix 
using Cayley
Hamilton theorem. (8)
b. Let
a 
matrix
A have eigenvalues 
and matrix 
Find
(i) determinant of matrix B.
(ii) trace of matrix B. (6)
PART II
Answer any THREE questions. Each question carries 14 marks.
Q.7 a. Change
the order of integration in integral 
and then evaluate the integral. (7)
b. Find the volume of the
solid bounded by the surfaces z = 0, 
and 
. (7)
Q.8 a. Solve
the differential equation 
by transforming the equation by
substitution 
. (7)
b.
Find the differential equation whose general solution is 
where a, b are arbitrary
constants. (7)
Q.9 a. Using method of
variation of parameters, show that A can always be determined so that 
is a solution of
the differential equation 
. (7)
b. Find
the general solution of the equation 
. (7)
Q.10 a. Solve
the differential equation 
. (6)
b. Using
Frobenius method, show that the differential equation 
has a solution 
near the
origin. Suggest the form of 2nd solution 
linearly independent of 
. (8)
Q.11 a. Show that under
change of dependent variable y defined by the substitution 
the Bessel’s equation of
order 
becomes
.
Hence show that for large values of t, the solutions of Bessel’s equation are
described approximately by the expression of the form 
. (7)
b.
Using Rodrigues formula for Legendre polynomials show that
where
f is any function integrable on interval 
. Hence show that 
(7)