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 the limit 
is
(A) limit does not exist (B) 0
(C) 1 (D) -1
b.
If
then
the value of 
is equal to
(A)
0 (B) 
(C)
(D) 
c.
If
then the value of 
is
(A) u (B) 2u
(C) 3u (D) 0
d.
The
value of integral 
is equal to
(A) 22 (B) 26
(C) 5 (D) 25
e.
The
solution of the differential equation 
is given by
(A)
(B) 
(C)
(D) 
f. The
solution of the differential equation 
is
(A)
(B) 
(C)
(D) 
g. 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.
h. If λ is an eigen value of a non-singular matrix A then the eigen value of A-1 is
(A) 1/ λ (B) λ
(C) –λ (D) –1/ λ
i. The product of the eigen values of the matrix
is
(A) 3 (B) 8
(C) 1 (D) –1
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 do not exist at (0,0).
(8)
Q.3 a. Evaluate the integral 
where T is region bounded
by the cone
and the planes z=0 to z=h
in the first octant. (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. The set of vectors {x1, x2}, where x1 = (1,3)T, x2 = (4,6)T is a basis in R2. Find a linear transformations T such that Tx1 = (-2,2,-7)T and Tx2 = (-2,-4,-10)T (8)
Q.7 a. Show that the matrix A is
diagonalizable. 
. Hence,
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. 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)