Code:
AE-01/AC-01/AT-01 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: (2x10)
a. The value of limit 
is
(A) 0 (B) 1
(C) limit does not exist (D) -1
b. If 
then
the value of 
is
equal to
(A)
0 (B)
(C) 
(D)
c. If 
, then the value
of 
is
(A) z (B) 2z
(C) tan(z) (D) sin(z)
d. The value of
integral 
is
equal to
(A)
(B)
(C) 
(D)
e. The differential equation of a family of circles having the radius r and the centre on the x-axis is given by
(A)
(B) 
(C) 
(D) 
f. The solution of
the differential equation 
satisfying the initial conditions
y(0) = 1, y(π/2) = 2 is
(A) y = 2 cos(x) + sin(x) (B) y = cos(x) + 2 sin(x)
(C) y = cos(x) + sin(x) (D) y = 2 cos(x) + 2 sin(x)
g. If the matrix 
then
(A) C=Acos(θ) – Bsin(θ) (B) C=Asin(θ) + Bcos(θ)
(C) C=Asin(θ) – Bcos(θ) (D) C=Acos(θ) + Bsin(θ)
h. The three vectors (1,1,-1,1), (1,-1,2,-1) and (3,1,0,1) are
(A) linearly independent (B) linearly dependent
(C) null vectors (D) none of these.
i. The value of 
is equal to
(A) 1 (B) 0
(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. For
the function
show that 
. (8)
b. Find
the absolute maximum and minimum values of the function 
over the rectangle in the
first quadrant bounded by the lines x = 2, y = 3 and the coordinate axes. (8)
Q.3 a. If
, find
an approximate value of f(1.1,0.8) using the Taylor’s series quadratic
approximation. (8)
b. Evaluate the integral 
by changing to polar coordinates,
where R is the region in the x-y plane bounded by the circles 
and 
=9. (8)
Q.4 a. Find the solution of the differential equation
(y-x+1)dy – (y+x+2) dx = 0. (6)
b. Solve the differential equation
(6)
c. Show that the functions 1, sin x, cos x are linearly independent. (4)
Q.5 a. Using method of
undetermined coefficients, find the general solution of the equation 
. (8)
b. Solve 
. (8)
Q.6 a. In an L-C-R circuit, the charge q on a plate of
a condenser is given by 
. The circuit is tuned to resonance
so that 
.
If initially the current I and the charge q be zero, show that, for small
values of R/L, the current in the circuit at time t is given by (Et/2L)sinpt. (8)
b. Find a linear transformation T from 
into such that
(8)
Q.7 a. Examine,
whether the matrix A is diagonalizable. 
. If, so, obtain the matrix P such
that 
is
a diagonal matrix. (8)
b. Investigate the values of
µ and λ so that the equations 
, has (i) no solutions (ii) a unique
solution and (iii) an infinite number of solutions. (8)
Q.8 a. Find the power
series solution about the point 
of the equation 
. (11)
b. Express
f(x)= 
in
terms of Legendre Polynomial. (5)
Q.9 a. Express
in
terms of 
and
. (8)
b. If
show that 
. (8)