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To get more 'n' more just vist : www.examsadda.comCode No: R10202/R10 MATHEMATICS - II(Common to All Branches) Time: 3 hours Answer any FIVE Questions All Questions carry equal marks Max. Marks 75
Set No - 1
I B.Tech II Semester Regular Examinations, June/July - 2011
*****1 (a) (b) 2 (a)(b)
3 (a)
(b) 4 (a) (b)
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5 (a)
(b)
6
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find the value of
[8 M+7 M] Form the partial differential equation by eliminating arbitrary function from ( x + y + z, x 2 + y 2 z 2 ) = 0 Find the complete integral of z 2 ( p 2 + q 2 ) = x 2 + y 2 .
[8 M+7 M] Derive the complete solution for the one dimensional heat equation with zero boundary conditions problem with initial temperature u ( x, 0) = x ( L x ) in the interval (0, L ) . [15 M]
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Verify whether the function f (t ) = t 3 is exponential order and find its transform. Find the Laplace transform of t e3t sin2t . [8 M+7 M] 2 d x dx Solve the initial valueproblem 2 4 + 8 x = e 2t . Given that x (0) = 2 and dt dt x (0) = 2 . State convolution theorem and using it find the inverse Laplace transform of s2 . ( s 2 + 4)( s 2 + 9) [8 M+7 M] 0 < x < 0 Find the Fourier series of f ( x) = x 0 x < 1 1 1 and hence find the value of 2 + 2 + 2 + 1 3 5 Find the half range cosine series of f ( x ) = x (2 x ) in 0 x 2. [8 M+7 M] State Fourier Integral theorem and deduce Fourier sine and cosine integrals. 1 | x |< a Find the Fourier transform of f ( x) = and hence 0 | x |> a
sin ax dx . x 0
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To get more 'n' more just vist : www.examsadda.comCode No: R10202/R10Set No - 1
7 (a)
8 (a) (b)
0
Show that
1
0
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Show that x m (log x )n dx =
1
(1) n ! , where n is a positive integer and m > 1 . (m + 1) n +1 2.4.6. ( n 1) xn , where n is an odd integer. dx = 1.3.5. n 1 x2 [8 M+7 M]
n
m[8 M+7 M]
Define one sided Z-transform and show that Z {n k un } = z ( z ( z ( Z {un }) )) z z z z . (b) Find the inverse Z-transform of 2 z + 8 z + 15
To get more 'n' more just vist : www.examsadda.comCode No: R10202/R10 MATHEMATICS - II(Common to All Branches) Time: 3 hours Answer any FIVE Questions All Questions carry equal marks Max. Marks 75
Set No - 2
I B.Tech II Semester Regular Examinations, June/July - 2011
*****1 (a) (b) Using Laplace transforms prove that
2 (a) (b)
3 (a) (b) 4 (a) (b)
5 (a) (b) 6
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[8 M+7 M] Form the partial differential equation of family of coneshaving vertex at origin. Find the general solution of x( y 2 + z ) p y ( x 2 + z ) q = z ( x 2 y 2 ) . [8 M+7 M] 2 2 u y Derive the solution of the Laplace equation 2 + 2 = 0 . Given that x y u (0, y ) = u ( a, y ) = u ( x, b) = 0 and u ( x, 0) = u x 0 . [15 M]
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1 e t sin 2 t 0 t dt = 4 log5 . State the conditions under which a function is Laplace transformable. Give some examples. [8 M+7 M] t dy Solve + 3 y + 2 y (t ) dt = t by Laplace transform method. 0 dt ( s + 3) . Find the inverse Laplace transform of 2 ( s + 6s + 13) 2 [8 M+7 M] 1 Find the Fourier series of f ( x) = ( x) 2 inthe interval 0 < x < 2 . 4 Find the half range cosine series of f ( x) = x ( x) in 0 x . [8 M+7 M] 1 is self-reciprocal under Fourier sine transform. Prove that x If f ( w) is the Fourier transform of f (t ) , then show that
| f ( x) |2 dx = | f ( w) |2 dw.
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To get more 'n' more just vist : www.examsadda.comCode No: R10202/R10Set No - 2
7 (a)
State and prove the left and right shifting theorems and using them find the Z transform of (n + 1) 2 .
(b) State convolution theorem and using it find the inverse Z transform of( m) ( n ) . ( m + n)
(b)
Evaluate e x x11/3 dx .3
1
0
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8 (a)
Show that (m, n) =
m[8 M+7 M]
z2 . ( z 3)( z 4) [8 M+7 M]
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Code No: R10202/R10 MATHEMATICS - II(Common to All Branches) Time: 3 hours Answer any FIVE Questions All Questions carry equal marks
Set No - 3
I B.Tech II Semester Regular Examinations, June/July - 2011
Max. Marks 75
1 (a)
If L { f (t )} is the Laplace transform, then prove that L f (t ) = s L { f (t )}
{
}
a. f (0).
and hence show that L { f ( k ) (t )} = s k L { f (t )} s k 1 f ( k 1) (0) s k 2 f ( k 2) (0) (b) 2 (a) (b)
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State and Prove second shifting theorem of Laplace transforms.
Using Laplace transforms,solve ( D 2 + 1) x = t cos2t , given that x =
sa
4s + 5 Find L1 2 ( s 1) ( s + 2)
(b) 4 (a) (b)
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3 (a)
k < x < 0 Find the Fourier series for the function f ( x) = 0 x 0 and hence deduce the inversion
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5 (a)
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(b)
6
y Form the partial differential equation by eliminating from ( , x 2 + y 2 + z 2 ) = 0 . x x2 y2 + =z. Find the complete integral of p q [8 M+7 M] A homogeneous rod of length L with insulated sides has its ends A and B are maintained at temperatures 500 C and 1000 C until steady state conditions prevail. The temperature at end A is suddenly raised to 900 C and at the same time the temperature at B is lowered to 600 C . Show that the temperature at the midpoint of the rod remains unaltered for all times regardless of the material of the rod. [15 M] Page 1 of 2
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co[8 M+7 M] dx = 0 at t = 0 . dt
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m[8 M+7 M] [8 M+7 M]
To get more 'n' more just vist : www.examsadda.comCode No: R10202/R10Set No - 3
7 (a)
Define convolution and using it find the Z transform of
1 1 where n! n!
is the
8 (a) (b)
Evaluate
2 0
sin 9/ 2 cos5 d . /2
Show that
0
tan d =
a.
(1 / 4) (3 / 4) . 2
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convolution operator. (b) By using Z transforms solve the difference equation yn + 2 3 yn +1 + 2 yn = 0 subject to the conditions y0 = 1 and y1 = 2 . [8 M+7 M]
To get more 'n' more just vist : www.examsadda.comCode No: R10202/R10 MATHEMATICS - II(Common to All Branches) Time: 3 hours Answer any FIVE Questions All Questions carry equal marks Max. Marks 75Set No - 4
I B.Tech II Semester Regular Examinations, June/July - 2011
*****1 (a) (b) 2 (a)
(b) 3 (a) (b)
Find the inverse Laplace transform of
4 (a) (b) 5 (a)
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[8 M+7 M] State Dirchlet conditions and find the Fourier series for the function f ( x ) = x sin x defined in the interval 0 < x < 2 Find the half range sine series of f ( x ) = x defined in the interval (0, 2) . [8 M+7 M] cos x d . Prove that e x = 2 2 +1 0
m
Find the Fourier cosine transform of f ( x) =
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(b)
6
[8 M+7 M] Form the partial differential equation by eliminating constants a,b and c from x2 y2 z 2 + + = 1. a2 b2 c2 Find the general solution of ( y zx) p + ( x + yz ) q = x 2 + y 2 . [8 M+7 M] The points of trisection of a tightly stretched string of length L with fixed ends are pulled aside through a distance d on opposite sides of the position of equilibrium and the string isreleased from rest. Obtain an expression for the displacement of the string at a later time and show that the midpoint of the string is always at rest. [15 M]
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e ( s + 2) and sketch it. s+2
dd1 . 1 + x2
[8 M+7 M] Using Laplace transforms solve ( D + 5 D + 6) X = 5e . Given that X (0) = 2 and X (0) = 1 .2t
a.
1 e t Find L . t Define Dirac Delta function and find its Laplace transform.
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To get more 'n' more just vist : www.examsadda.comCode No: R10202/R10Set No - 4
7 (a)
If
3z 2 4 z + 7 is the Z-transform of f (n) , then find f (0) , f (1) and f (2) . ( z 1)3
(b) Find the inverse Z transform of
(b)
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8 (a)
1 Define Gamma function and show that ( ) = 2 y n 1 dy Show that (m, n) = 0 (1 + y ) m + n
m[8 M+7 M] [8 M+7 M]
z2 + z . ( z 1)2