DECEMBER 2006
Code: D-23 / DC-23 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. The smallest positive integer
n for which 
is
(A) 8 (B) 12
(C) 16 (D) None of these
b. A square root of 3 + 4i is
(A)
(B)
(C) 
(D)
None of these
c. Any vector a is equal to
(A)
(B)
(C) 
(D) 
d. If a and b are
two unit vectors inclined at an angle 
and are such that a + b is a unit
vector, then is
equal to
(A)
(B)
(C) 
(D)
e. The value of the
determinant 
,
where 
is
an imaginary cube root of unity is
(A)
(B)
3
(C) 
(D)
4
f. The value of the
determine 
is
equal to
(A) -4 (B) 0
(C) 1 (D) 4
g. The inverse of a diagonal matrix is
(A) not defined (B) a skew-symmetric matrix
(C) a diagonal matrix (D) a unit matrix
h. The period of function sin 2x + cot 3x + sec 5x is
(A)
(B)
(C) (D)
i. The
Laplace transform of 
is
(A)
(B)
(C) 
(D)
j. The solution of
the differential equation 
is
(A) 
(B) 
(C) 
(D) 
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. If
n is a positive integer, prove that 
. (8)
b. Find
all the values of 
and show that the product of all
these values is 1. (8)
Q.3 a. If the roots of 
represent
vertices of a triangle in the Argand plane, then find area of the triangle. (8)
b. Find the value of 
if
. (8)
Q.4 a. Prove that the sum of all the vectors drawn from the centre of a regular octagon to its vertices is the zero vector. (8)
b. Find the moment about the
point 
of
the force represented in magnitude and position by 
, where the point A and B
have the co-ordinates 
and 
respectively. (8)
Q.5 a. Show that 
. (8)
b. Write the following system of equations in the matrix form AX = B and solve this for X by finding A-1.
(8)
Q.6 a. Using matrix methods, find the values of 
and 
so that the
system of equations
.
has (i) unique solution and (ii) has no solution (8)
b. Verify Cayley Hamilton theorem for the matrix
.
Use Caley Hamilton theorem to evaluate A-1 and hence solve the equations
(8)
Q.7 Find the Fourier series for the functions
(16)
Q.8 a. Find the Laplace transform 
(8)
b. Find
the inverse Laplace transform 
(8)
Q.9 a. Solve the differential equation
(8)
b. By using Laplace transform, solve the differential equation
(8)