AMIETE
– ET/CS/IT (NEW SCHEME) – Code: AE51/AC51/AT51
Subject: ENGINEERING 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
period of a simple pendulum is 
maximum error in T due
to the possible error up to 1% in 
and 2.5% in g, equals
(A) 1.63% (B) 1.59%
(C) 1.75% (D) None of these
b. The value of integral 
is
equal to
(A) 
(B) 
(C) 
(D) 
c. The solution of
the differential equation 
satisfying the initial conditions y(0) = 1, y(π/2) = 2
is
(A) y = 2cos(x) + sin(x) (B) y = cos(x) + 2 sin(x)
(C) y = cos(x) + sin(x) (D) y = 2cos(x) + 2 sin(x)
d. 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)
e. The general
solution of 
is
(A) 
(B) 
(C) 
(D) 
f. The function
whose first difference 
, is
(A) 
(B) 
(C) 
(D) 
g.
The Bessel’s equation of order 0 is given as
(A) 
(B) 
(C) 
(D) 
h. In terms of
Beta function 
is
(A) 
(B) 
(C) 
(D) None of these
i. 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
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. If 
, show that 
. (8)
b. If 
where x-1+y-1+z-1=1,
show that the stationary value of u is given by 
. (8)
Q.3 a. Evaluate
the integral 
by changing to polar coordinates, R is the
region in the x-y plane bounded by the circles 
and 
=9.
(8)
b. Evaluate the integral 
where
T is region bounded by the cone 
and the planes
z=0 to z=h in the first octant. (8)
Q.4 a. Investigate
the values of λ for which the equations are consistent,
Hence find the ratios of x:y:z when λ has the
smallest of these values. (8)
b. Find the eigen
value and eigen vector of the matrix 
. (8)
Q.5 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, sinx, cosx are linearly independent. (4)
Q.6 a. Prove
that 
. (8)
b. Solve 
. (8)
Q.7 a. Find the power series solution about the
point 
of the equation 
. (11)
b. Express f(x)= 
in terms of Legendre Polynomial. (5)
Q.8 a. Express
in terms of 
and 
. (8)
b. Solve 
. (8)
Q.9 a. Solve by Gauss-Seidel method, the following
system of equations: (8)
b. Using
Runge-Kutta method of fourth order,
solve for y(0.1), y(0.2) given that 
. (8)