Jawaharlal Nehru Technological University Hyderabad 2005-1st Sem B.Tech Aeronautical Engineering II Supplementary s, MATHEMATICS-II - Question Paper

Code No: NR210101 NR
II B.Tech I Semester Supplementary Examinations, May 2005
MATHEMATICS-II
( Common to Civil Engineering, Electrical & Electronic Engineering,
Mechanical Engineering, Electronics & Communication Engineering,
Computer Science & Engineering, Chemical Engineering, Electronics &
Instrumentation Engineering, Bio-Medical Engineering, info
Technology, Electronics & Control Engineering, Mechatronics, Computer
Science & Systems Engineering, Electronics & Telematics, Metallurgy &
Material Technology, Electronics & Computer Engineering, Production
Engineering and Aeronautical Engineering)
Time: three hours Max Marks: 80
ans any 5 ques.
All ques. carry equal marks
? ? ? ? ?
1. (a) describe the rank of the matrix and obtain the rank of the subsequent matrix.
2664
2 one three 5
4 two one 3
8 four seven 13
8 four -3 -1
3775
(b) obtain whether the subsequent equations are consistent, if so solve them.
x+y+2z = four ; 2x-y+3z= nine ; 3x-y-z=2
2. (a) obtain the characteristic roots of the matrix and the corresponding eigen values
24
6 -2 2
-2 three -1
2 -1 3
35
(b) If 1, 2,........, n are the eigen values of A, then prove that the eigen values
of (A -kI) are 1 - k, 2 - k, 3 - k, ..........., n - k.
3. (a) describe the following:
i. Hermitian matrix
ii. Skew-Hermitain matrix
iii. Unitary matrix
iv. Orthogonal matrix.
(b) Show that the eigen values of an unitary matrix is of unit modulus.
4. (a) describe a periodic function. obtain the Fourier expansion for the function
f (x) = x - x2, - one < x < 1.
(b) Prove that the function f(x) = x, 0x can be expanded in a series of sines
as x = two sin x
1 - sin 2x
2 + sin 3x
3 - ....... .
5. (a) Form the partial differential formula by eliminating the arbitrary constants
a, b from z = ax + by + (a / b) - b.
(b) Solve the partial differential formula x2(y2-z2)p+y2(z2-x2)q = z2(x2-y2).
(c) Solve the partial differential formula (2z - y ) p + (x + z ) q + (2 x + y ) =
0.
6. An elastic string is stretched ranging from 2 points at a distance ‘l’ apart. 1 end
is taken as origin and at a distance ( 2L
3 ) from this end, the string is displaced a
distance‘d’ transversely and is released from rest, when it is in this position. obtain
the formula of the following motion.
7. (a) Evaluate the subsequent using Parseval’s identity
1
R0
dx
(x2+a2)2
(b) obtain the Fourier transform of f(x) =
1 - |x| if |x| < 1
0 if |x| > 1
and hence obtain the value of
1
R0
sin4 t
t4 dt
8. (a) If Z(un)=[ 2z2+3z+4
(z-3)3 ]. obtain u1 and u2.
(b) obtain Z-1[ 8z2
(2z-1)(4z-1) ]

Earning:   Approval pending.