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) limit does not exist (B) 0
(C) 1 (D) –1
b.
If
then
the value of 
is equal to
(A) 0 (B) 2
(C) 
(D) 
c.
If
, then the value of 
is
(A) 2u (B) u
(C) 0 (D) 1
d.
The
value of integral 
is
(A) 
(B) 2.
(C) –2. (D) 0.
e.
The
solution of the differential equation 
is given by
(A) 
(B) 
(C) 
(D) 
f.
is the solution
of
(A) 
(B) 
(C) 
(D) 
g.
If
, then
the eigen values of 
are
(A) 1,2,3 (B) –1,2,3
(C) 1,4,9 (D) –1,4,9
h.
Let
then
(A) A is row equivalent to B only if 
(B) A is row equivalent to B only if 
(C) A is not row equivalent to B
(D) A is row equivalent to B for all values of 
i.
The
value of 
is
(A) 0 (B) 
(C) 
(D) –1
j.
The
value of the integral 
if
(A) 
(B) 
(C) 
(D) none of the above
Answer any FIVE Questions out of EIGHT Questions.
Each Question carries 16 marks.
Q.2 a. If 
then evaluate
(8)
b. Find
the absolute maximum and minimum values of the function 
over the region 
(8)
Q.3 a. Evaluate the integral 
where T is region
bounded by the
surfaces
(8)
b. The diameter and altitude of a can in the shape of a right circular cylinder are measured as 4cm, 6cm respectively. The possible error in each measurement is 0.1cm. Find approximately the maximum possible error in the values computed for the volume and the lateral surface. (8)
Q.4 a. The initial value problem governing the current i flowing in a series RL
circuit
when a voltage v(t)=t is applied is given by 
where R and L are constants. Find the
current i(t) at time t.
(8)
b. Using
the method of undetermined coefficients, find the general solution of the
differential equation 
(8)
Q.5 a. Find the general
solution of the equation 
.
(8)
b.
Show that the matrix A is diagonalizable, where
. Obtain the matrix P such that 
is a diagonal
matrix. (8)
Q.6 a. Solve the equations 4x+2y+z+3w=0, 6x+3y+4z+7w=0, 2x+y+w=0. (8)
b. Using Gauss Jordan Method, find the inverse of the matrix
(8)
Q.7 a. Show that the transformation
is non-singular.
Find the inverse transformation. (8)
b. Prove that the following matrix is orthogonal: (8)
(8)
Q.8 a. Find the first five non-vanishing terms in the power series solution of the
initial
value problem 
(11)
b. Show that 
(5)
Q.9 a. Show that 
(8)
b. Show
that 
(8)