Code: DE23/DC23 Subject: MATHEMATICS - II
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. If 
and
represent conjugate complex
numbers then the value of x and y is
(A) 
.
(B) 
.
(C) 
.
(D) 
.
b. Imaginary part of 
is
(A) – cos x cosh y (B) – cos x sinh y
(C) – sin x cosh y (D) – sin x sinh y
c. Three vectors 
are coplanar, the value of
their scalar triple product is
(A) 0 (B) 1
(C) –1 (D) i
d. If 
is the angle between the
vectors 
and 
such
that 
then is
(A) 
(B) 
(C)
(D) 
e. The value of the determinant 
is
(A) 1 (B) 2
(C) –1 (D) 0
f. If the product of two eigen values of the matrix 
is
16, then the third eigen value is
(A) 0 (B) 5
(C) 2 (D) –2
g. If f(x) is defined in (0, L), then the period of f(x) to expand it as a half range sine series is
(A) L. (B) 0.
(C)
2L.
(D) 
.
h. The inverse Laplace transform 
is possible
only when n is
(A) 0 (B) –ve integer
(C) –ve rational number (D) +ve integer
i. The differential equation of a family of circles having the radius r and centre on the x axis is
(A) 
(B) 
(C) 
(D) 
j. If y satisfies 
with 
then
Laplace transform 
is
(A) 
(B) 
(C) 
(D) 
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. Find the moment
of the force 
about a line
through the origin having direction of 
, due to a 30 Kg
force acting at a point (– 4, 2,
5) in the direction of 
.
(8)
b. Prove that the right bisectors of the sides of a triangle intersect at its circum centre. (8)
Q.3 a. Show that the components of a
vector 
along and perpendicular to 
in
the plane of and are 
and
.
(8)
b. If 
show that 
and
.
(8)
Q.4 a. If 
then
.
(8)
b. Show that the origin and the complex numbers represented
by the roots of the equation 
, where a, b are real, form
an equilateral triangle if 
.
(8)
Q.5 a. Prove that
.
(8)
b. Determine the values of 
when 
is
orthogonal.
(8)
Q.6 a. Find the values of k such that the system of
equations
, 
, 
has
non-trivial
solution.
(8)
b. Find the characteristic equation of the matrix 
.
Hence find 
.
(8)
Q.7
Find the Fourier series for 
.
(16)
Q.8 a. Find 
.
(8)
b. Find the inverse Laplace transform of 
.
(8)
Q.9 a. Using Laplace transformation, solve the following differential equation:
if x(0)
= 1, 
.
(8)
b. Solve 
(8)