AMIETE – ET/CS/IT (OLD SCHEME)
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 the best alternative
in the following: (2 
10)
a.
The value of limit 
is
(A) limit does not
exist (B) 0
(C) 1 (D) -1
b. If 
,
then the value of 
is
(A)
u (B) 2u
(C)
3u (D) 0
c.
The solution of
the differential equation 
is given by
(A)
(B)
(C)
(D)
d. The solution of
the differential equation 
is
(A) 
(B) 
(C) 
(D) 
e. If 3x+2y+z= 0, x+4y+z=0, 2x+y+4z=0, be a
system of equations then
(A) System is inconsistent.
(B) It has only trivial
solution.
(C) It can be reduced to a single equation thus
solution does not exist.
(D)
Determinant
of the coefficient matrix is zero.
f.
If λ is an eigenvalue of a non-singular matrix A then
the eigenvalue of A-1 is
(A) 1/ λ (B) λ
(C) –λ (D) –1/ λ
g.
The value of 
is
(A) –1
(B) 1
(C) 
(D) 0
h.
The value of
integral 
is equal to
(A) 
(B) 
(C) 
(D) 
i.
If 
then
(A) 
(B) 
(C) 
(D) 
j.
The value of the
integral 
is
(A) 
(B) 
(C) 
(D) 
Answer
any FIVE Questions out of EIGHT Questions.
Each
Question carries 16 marks.
Q.2 a. Find the extreme value of the function
f(x,y,z) = 2x + 3y + z such
that x2+y2 = 5
and x + z =1. (8)
b. Show that the function 
is continuous at (0,0) but its
partial derivatives of first order does not exist at (0,0). (8)
Q.3 a. Evaluate
by changing to polar
coordinates. Hence show that 
. (8)
b. Show that the approximate change in the angle
A of a triangle ABC due to small changes
in the sides a, b, c respectively, is given by 
where ∆ is the
area of the triangle. Verify that 
(8)
Q.4 a. If 
Show
that
(8)
b. Using the method of variation of parameter
method, find the general solution of the differential equation
. (8)
Q.5 a. Find the general solution of the equation
. (8)
b. Find the general solution of the equation 
. (8)
Q.6 a. Solve 
(8)
b.
If 
, show that AA* is a Hermitian matrix, where
A* is the conjugate transpose of A.
(8)
Q.7 a. Show that the matrix A is diagonalizable. 
. If so, obtain
the matrix P such that 
is a diagonal matrix. (8)
b. Investigate the values of λ for which
the equations
are consistent, and hence
find the ratios of x:y:z when λ has the smallest of these values. (8)
Q.8 a. Use elementary
row operations to find inverse of 
(5)
b. Find the first five non-vanishing terms in the
power series solution of the initial value problem
(11)
Q.9 a. Show that 
(8)
b. Show that 
(8)