The Institution of Engineers,India 2006 Diploma Electronics and Communication MATHEMATICS - II - - Question Paper
Code: D-23 / DC-23 Subject: MATHEMATICS - II
Time: three Hours June 2006 Max. Marks: 100
NOTE: There are nine ques. in all.
• ques. one is compulsory and carries 20 marks. ans to Q. 1. must be written in the space given for it in the ans book supplied and nowhere else.
• Out of the remaining 8 ques. ans any 5 ques.. every ques. carries 16 marks.
• Any needed data not explicitly given, may be suitably presumed and said.
Q.1 select the accurate or best option in the following: (2x10)
a. Let and . Express in the form a + bi, a , b R.
(A) (B)
(C) (D)
b. The complex numbers , and satisfying are vertices of the a triangle which is
(A) acute-angled and isosceles (B) right-angled and isosceles
(C) obtuse-angled and isosceles (D) equilateral
c. A unit vector parallel to 3i+4j-5k is
(A) (B)
(C) (D)
d. Let = (1, 2, 0), = (-3, 2, 0), = (2, 3, 4). Then equals
(A) 33 (B) 30
(C) 31 (D) 32
e. If is complex cube root of unity, and , then is equal to
(A) 0 (B) -A
(C) A (D) none of these
f. If A and B are symmetric matrices, then AB + BA is a
(A) diagonal matrix (B) null matrix
(C) symmetric matrix (D) Skew-symmetric matrix
g. The function is
(A) odd (B) even
(C) neither (D) none of these
h. The function cos x + sin x + tan x + cot x + sec x + cosecx is
(A) both periodic and odd (B) both periodic and even
(C) periodic but neither even nor (D) not periodic
odd
i. The Laplace Transform for sin at is
(A) (B)
(C) (D)
j. The Inverse Laplace Transform for is
(A) (B)
(C) (D)
ans any 5 ques. out of 8 ques..
every ques. carries 16 marks.
Q.2 a. If a, b, c are real numbers such that and b + ic = (1 + a)z, where z is a complex number, then show that . (8)
b. provided that and where is a cube root of unity. Express in terms of A, B, C and . (8)
Q.3 a. Show that for all real , . (8)
b. For any 4 vectors prove that . Hence prove that (8)
Q.4 a. In let , . Then obtain the vector representing AB and OM, where M is the midpoint of AB. (4)
b. Prove that the straight line joining the mid-points of 2 non-parallel sides of a trapezium is parallel to the parallel sides and is half their sum. (12)
Q.5 a. For reals A, B, C, P, Q, R obtain the value of determinant
(8)
b. Using matrix method obtain the values of and so that the system of equations:
has infinitely many solutions. (8)
Q.6 a. Solve the system of equations
by using inverse of a suitable matrix. (8)
b. Using Cayley-Hamilton theorem obtain for . (8)
Q.7 State whether the function f(x) having period two and described by
is even or odd. obtain its Fourier Series. (16)
Q.8 a. obtain the Laplace transform of . (8)
b. obtain the Inverse Laplace transform for . (8)
Q.9 a. Solve the differential formula
provided that y = -0.9 and , when x=0 (8)
b. Using the Laplace transform solve the differential formula
with initial conditions . (8)
Code: D-23 / DC-23 Subject: MATHEMATICS - II
Time: 3 Hours June 2006 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.
Let 
and 
. Express 
in the
form a + bi, a , b 
R.
(A) 
(B) 
(C) 
(D) 
b.
The complex numbers 
, 
and 
satisfying 
are
vertices of the a triangle which is
(A) acute-angled and isosceles (B) right-angled and isosceles
(C) obtuse-angled and isosceles (D) equilateral
c. A unit vector parallel to 3i+4j-5k is
(A)
(B) 
(C) 
(D) 
d. Let 
= (1, 2, 0), 
= (-3,
2, 0), 
=
(2, 3, 4). Then 
equals
(A) 33 (B) 30
(C) 31 (D) 32
e. If 
is complex cube root of
unity, and 
, then 
is equal to
(A) 0 (B) -A
(C) A (D) none of these
f. If A and B are symmetric matrices, then AB + BA is a
(A) diagonal matrix (B) null matrix
(C) symmetric matrix (D) Skew-symmetric matrix
g. The function 
is
(A) odd (B) even
(C) neither (D) none of these
h. The function cos x + sin x + tan x + cot x + sec x + cosecx is
(A) both periodic and odd (B) both periodic and even
(C) periodic but neither even nor (D) not periodic
odd
i. The Laplace Transform for sin at is
(A) 
(B) 
(C) 
(D) 
j. The Inverse Laplace Transform for 
is
(A) 
(B) 
(C) 
(D) 
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2
a. If a, b, c are real numbers such that 
and b + ic = (1
+ a)z, where z is a complex number, then show that 
.
(8)
b. Given that 
and 
where
is
a cube root of unity. Express 
in terms of A, B, C and .
(8)
Q.3
a. Show that for all real 
, 
. (8)
b. For any four vectors 
prove that 
. Hence
prove that
(8)
Q.4
a. In 
let 
, 
. Then find the
vector representing AB and OM, where M is the midpoint of
AB.
(4)
b. Prove that the straight line joining the mid-points of two non-parallel sides of a trapezium is parallel to the parallel sides and is half their sum. (12)
Q.5 a. For reals A, B, C, P, Q, R find the value of determinant
(8)
b. Using matrix method find the
values of 
and
so
that the system of equations:
has
infinitely many
solutions.
(8)
Q.6 a. Solve the system of equations
by using inverse of a suitable matrix. (8)
b. Using Cayley-Hamilton theorem find 
for 
.
(8)
Q.7 State whether the function f(x) having period 2 and defined by
is even or odd. Find its Fourier Series. (16)
Q.8
a. Find the Laplace transform of 
.
(8)
b. Find the Inverse Laplace transform for 
.
(8)
Q.9 a. Solve the differential equation
given that y = -0.9 and 
, when
x=0
(8)
b. Using the Laplace transform solve the differential equation
with initial conditions 
. (8)
| Earning: Approval pending. |
