Code: AE01/AC01/AT01 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: (2 x 10)
a.
The value
of 
is
(A) 3 (B) –3
(C) limit does not exist (D) –1
b.
If 
then
(A)
(B) 
(C)
(D) 
c.
If 
, then the value of 
is
(A) 1 (B) r
(C) 1/r (D) 0
d.
The value
of integral 
is equal to
(A) – 4 (B) 3
(C) 4 (D) –3
e.
The
solution of the differential equation 
under the condition y(1)=1 is given by
(A)
(B) 
(C)
(D) 
f.
The particular integral of the differential equation 
is
(A)
(B) 
(C)
(D) 
g.
The sum
of the eigen values of
is equal to
(A) 6 (B) – 8
(C) 7 (D) – 6
h. If 
. Then, the matrix A is equal
to
(A) 
(B)
(C) 
(D)
i.
The value of 
(m being an integer < n) is
equal to
(A) 1 (B) –1
(C) 2 (D) 0
j.
The value of the 
is
(A)
(B) 
(C)
(D) 
Answer any FIVE Questions out of EIGHT Questions.
Each Question carries 16 marks.
Q.2
a.
Compute 
for the function
Also discuss the
continuity of 
at (0,0).
(8)
b.
Find the minimum values
of 
subject to the condition 
.
(8)
Q.3
a.
The function 
is
approximated by a first degree
with centre at (2,3) such
that the error of approximation is less than or equal to 0.1 in magnitude for
all points within the
square.
(8)
b. Find the Volume of the ellipsoid
(8)
Q.4 a. Solve the differential equation
(8)
b.
Using the
method of variation of parameters, solve the differential equation 
.
(8)
Q.5
a.
Find the general solution of the equation 
.
(8)
b.
The eigenvectors of a 3 x 3 matrix A corresponding to the eigen
values1, 1, 3 are 
respectively. Find the matrix
A.
(8)
Q.6 a. Test for consistency and solve the system of equations
(8)
b.
Given
that 
show that 
is a unitary
matrix.
(8)
Q.7 a. Show that the transformation
is non-singular. Find
the inverse
transformation.
(8)
b.
If 
then show that 
.
(8)
Q.8 a. Find the power series solution about the origin of the equation
.
(11)
b.
Find the value of 
.
(5)
Q.9 a. Prove the orthogonal property of Legendre Polynomials. (8)
b.
Show that 
(8)