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VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur – 603 203. DEPARTMENT OF MATHEMATICS QUESTION BANK SUBJECT
: MA6351- TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS
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SEM / YEAR : III Sem / II year (COMMON TO ALL BRANCHES)
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UNIT I - PARTIAL DIFFERENTIAL EQUATIONS Formation of partial differential equations – Singular integrals -- Solutions of standard types of first order partial differential equations - Lagrange’s linear equation -- Linear partial differential equations of second and higher order with constant coefficients of both homogeneous and non-homogeneous types. Bloom’s Q.No. Question Taxonomy Domain Level PART – A Form a partial differential equation by eliminating the BTL -6 Creating 1. arbitrary constants ‘a’ and ‘b’ from z ax 2 by 2. Eliminate the arbitrary function from = and form the partial differential equation Construct the partial differential equation of all spheres whose centers lie on the z-axis by the elimination of arbitrary constants Form the partial differential equation by eliminating the arbitrary constants a, b from the relation = + √ − + Form the partial differential equation by eliminating the arbitrary constants a, b from the relation log( az 1) x ay b.
Creating
BTL -3
Applying
BTL- 6
Creating
BTL -6
Creating
6.
Form the partial differential equation by eliminating the arbitrary constants a, b from the relation 4(1 a 2 ) z ( x ay b) 2 .
BTL -6
Creating
7.
Form the partial differential equation from ( x a) 2 ( y b) 2 z 2 cot 2
BTL -6
Creating
8.
Form the partial differential equation by eliminating the arbitrary function from ( x 2 y 2 , z) 0
BTL -6
Creating
Form the partial differential equation by eliminating the x arbitrary function from z 2 xy f ( ) z
BTL -6
Creating
10. Find the complete solution of p q 1
BTL -3
Applying
11. Find the complete solution of q 2 px
BTL -3
Applying
4.
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5.
ata
3.
s.b
BTL -6
2.
9.
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BTL -3
Applying
2 2 2 13. Solve px qy z
BTL -3
Applying
2 2 14. Solve ( D 7 DD'6D' ) z 0
BTL -3
Applying
3 2 2 15. Solve ( D 4D D'4DD' ) z 0
BTL -3
Applying
BTL -3
Applying
16. Solve
2 z 2 z z 0 x 2 xy x
4 4 17. Solve ( D D' ) z 0
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12. Find the complete integral of p q pq
BTL -3
Applying
BTL -3
Applying
BTL -3
Applying
BTL -3
Applying
BTL -6
Creating
2 2 1. (b) Find the singular integral of z px qy 1 p q
BTL -2
Understanding
Form the partial differential equation by eliminating 2. (a) arbitrary function from ( x 2 y 2 z 2 , ax by cz ) 0
BTL -6
Creating
2 2 2.(b) Find the singular integral of z px qy p pq q
BTL -2
Understanding
Form the partial differential equation by eliminating 3. (a) arbitrary functions f and from z x 2 f ( y) y 2 g ( x) g
BTL -6
Creating
p q x) 2 ( y ) 2 1 2 2 2 2 4. (a) Solve x p y q z ( x y)
BTL -3
Applying
BTL -3
Applying
18. Solve ( D D'1)( D 2D'3) z 0 3 19. Solve ( D D' ) z 0
20. Solve ( D 1)( D D'1) z 0
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3.(b) Solve (
s.b
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PART – B Form the partial differential equation by eliminating the 1.(a) arbitrary functions f , f from z xf ( x t ) f ( x t ) 1 2 1 2
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Form the partial differential equation by eliminating arbitrary function f from the relation 4.(b) 1 z y 2 2 f ( log y) x
Creating BTL -6 BTL -3
Applying
2 2 5.(b) Solve 9( p z q ) 4
BTL -3
Applying
BTL -2
Understanding
2 2 6.(b) Solve ( y z ) p xyq xz 0
BTL -3
Applying
2 2 2 2 2 7. (a) Find the complete solution of z ( p q ) x y
BTL -4
Analyzing
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2 2 2 5. (a) Solve ( x yz ) p ( y xz )q ( z xy )
6. (a) Find the singular solution of z px qy
p p q
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BTL -2
Understanding
2 2 2 2 8. (a) Find the general solution of ( D D' ) z x y
BTL -2
Understanding
2 2 2 2 2 2 8.(b) Find the complete solution of p x y q x z
BTL -2
Understanding
2 2 x2 y 9. (a) Solve ( D 3DD'2D' ) z (2 4 x)e
BTL -3
Applying
Find the general solution of 9.(b) ( z 2 y 2 2 yz ) p ( xy zx)q ( xy zx)
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Find the general solution of 7. (b) ( D 2 2DD' D'2 ) z 2 cos y x sin y
BTL -2
Understanding
2 2 2 2 2 2 10.(a) Solve x( y z ) p y( z x )q z( x y )
BTL -3
Applying
2 2 10.(b) Solve ( D 3DD'2D' ) z sin( x 5 y)
BTL -3
Applying
BTL -3
Applying
BTL -3
Applying
BTL -3
Applying
Solve the partial differential equation 12(b) ( x 2 z) p (2 z y) q y x
BTL -3
Applying
2 2 x y 13(a) Solve ( D D' ) z e sin(2 x 3 y)
BTL -3
Applying
2 2 y 13(b) Solve (2D DD' D' 6D 3D' ) z xe
BTL -3
Applying
2 3 2 x y 14(a) Solve ( D 2DD' ) z x y e
BTL -3
Applying
3 2 3 14(b) Solve ( D 7 DD' 6D' ) z sin( x 2 y)
BTL -3
Applying
Solve the Lagrange’s equation 11(a) ( x 2 z) p (2 xz y)q x 2 y 2 2 2 x4 y 11(b) Solve ( D DD'2D' ) z 2 x 3 y e
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s.b
log
2 2 12(a) Solve ( D DD'6D' ) z y cos x
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UNIT II - FOURIER SERIES Dirichlet’s conditions – General Fourier series – Odd and even functions – Half range sine series – Half range cosine series – Complex form of Fourier series – Parseval’s identity – Harmonic analysis. PART-A
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Bloom’s Q.No. Question Domain Taxonomy Level State the sufficient condition for a function to be expanded BTL-1 Remembering 1. as a Fourier series Remembering BTL-1 2 State the Dirichlet’s condition for Fourier series. 3 4
If � −
=
�
that value of ∑∞=
+ ∑∞= .
�
�
<
< � then deduce
Does f(x) = tanx posses a Fourier expansion?
BTL-3
Applying
BTL-2
Understanding
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6 7 8
Determine the value of in the Fourier series expansion of = � −�, � . Find the constant term in the expansion of as a Fourier series in the interval (-�, � If f(x) is an odd function defined in (- , ) what are the value of ? If the function f(x) = x in the interval 0
BTL-5
Evaluating
BTL-2
Understanding
BTL-2
Understanding
BTL-2
Understanding
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5
BTL-2
Understanding
BTL-2
Understanding
BTL-2
Understanding
BTL-2
Understanding
BTL-2
Understanding
BTL-2
Understanding
15 Write down Parseval’s formula on Fourier coefficients
BTL-1
Remembering
16 Define RMS value of a function f(x) over the interval (a,b).
BTL-1
Remembering
BTL-2
Understanding
BTL-2
Understanding
BTL-1
Remembering
BTL-3
Applying
BTL-2
Understanding
BTL-2
Understanding
BTL-3
Applying
BTL-2
Understanding
9
�
+ ⋯ ]then find the sum of the series 1− + − +…
10 Find the sine series of function f(x) = 1. 0 11 Find the RMS value of f(x) = x in (0, ). 12 Find the root mean square value of f(x) =
in (0, )
s.b
14 Find the RMS value of f(x) =
in (0,�).
log
13 Find the RMS value of f(x) = x( -x) in 0
�.
Without finding the value of and the Fourier series, in the Interval (0, � find the 17 for the function f(x) = �0 ∞ + } value of { + ∑ = 19
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18 Find the R.M.S value of f(x) = 1-x in 0<
<
State Parseval’s identity for the half-range cosine expansion of f(x) in (0,1).
20 What is meant by Harmonic Analysis?
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PART-B Find the Fourier series expansion of � 1.(a) ={ �− � �. Find the Fourier series of = � −� < 1. (b) ∞ deduce the value of ∑ =
< �.Hence
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Obtain the Fourier series to represent the function
2. (a)
| |−� <
< � and Deduce ∑∞=
−
=
�
.
Find the Fourier series of the function −� and Hence Evaluate ={ 2.(b) � � + . + . +…… .
=
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3. (a)
={
+
−
�
�
, −� < ,
Fourier series in the interval
4. (a)
4.(b)
5. (a)
<
<�
as a full range BTL-3
Applying
BTL-2
Understanding
BTL-5
Evaluating
BTL-3
Applying
BTL-5
Evaluating
BTL-3
Applying
BTL-2
Understanding
=
BTL-2
Understanding
BTL-2
Understanding
BTL-2
Understanding
−�, � .Hence deduce that
+ + +⋯ ∞= Find a Fourier series with period 3 to represent = − � , Determine the Fourier series for the function = � � < < � Obtain the Fourier series for the function f(x) given by − , −� < < ={ Hence deduce that + , < <� � + + +⋯= Determine the Fourier series for the function | � −� < <� =| ={
5.(b) Obtain the sine series for
−
�
�
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3.(b)
�
<
log
Expand
�
�
s.b
Find the Fourier expansion the following periodic function of period 4. 6. (a) + , − ={ Hence deduce that − , � + + + ⋯∞ = Find the Fourier series of
6.(b) period 2 �. Hence deduce ∑∞=
+
�
=
in (-�, � with
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Find the half range sine series of = − in the 7. (a) interval (0,4).Hence deduce the value of the series − + − + ⋯∞ Find the Fourier series of = | � | � − � < < � of 7. (b) periodicity 2�
Compute the first two harmonics of the Fourier series of f(x) from the table given BTL-3 8. (a) �/ �/ � �/ �/ � x 0 f(x) 1 1.4 1.9 1.7 1.5 1.2 1
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Find the complex form of the Fourier series of � 8.(b) −�, � where “a” is not an integer Find the Fourier series of
9. (a)
hence deduce that +
+
=
+
�
−�, � and
+⋯∞ =
�
in
Applying
BTL-2
Understanding
BTL-2
Understanding
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The following table gives the variation of a periodic current over a period by harmonic analysis, Show that there is a direct current part of 0.75amps in the variable current. Also 9.(b) obtain the amplitude of the first harmonic.
11.(b)
12.(a)
�
By using Cosine series show that for
=
�
<
<�
=
+
+
Find the Fourier cosine series up to first harmonic to represent the function given by the following data:
x 0 1 2 3 4 5 y 4 8 15 7 6 2 Show that the complex form of Fourier series for the function = ℎ −� < <� =
+ � �
=
� ℎ� �
∑∞=−∞
−
Find the complex form of the Fourier series of =
�
� −� <
+�
+
BTL-2
Understanding
BTL-3
Applying
BTL-3
Applying
BTL-2
Understanding
BTL-3
Applying
BTL-2
Understanding
BTL-3
Applying
BTL-2
Understanding
BTL-2
Understanding
�
<�
ata
12.(b)
+⋯
log
11.(a)
Obtain the Fourier cosine series expansion of = � < < .Hence deduce the value of + + +⋯ ∞
s.b
10.(b)
Applying
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t secs 0 T/6 T/3 T/2 2T/3 5T/6 T A amps 1.98 1.30 1.05 1.30 -0.88 -0.25 1.98 Find the half range Fourier cosine series of = �− 10.(a) in the interval , � . Hence Find the sum of the series + +
BTL-3
Calculate the first 3 harmonics of the Fourier of f(x) from the following data 13.
60 90 120 150 180 210 240 270 300 330 0.3 0.16 0.5 1.3 2.16 1.25 1.3 1.52 1.76 2
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x 0 30 f(x) 1.8 1.1
Find the complex form of the Fourier series of
14.(a)
=
−
� −
<
<
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Find the Fourier series as far as the second harmonic to represent the function f(x) With period 6
14.(b)
x y
0 9
1 18
2 24
3 28
4 26
5 20
UNIT III -APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS Classification of PDE – Method of separation of variables - Solutions of one dimensional wave equation - One dimensional equation of heat conduction – Steady state solution of two MA6351_TPDE_QBank_ACY 2016-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH. PREPARED BY DEPARTMENT OF MATHEMATICS Visit : Civildatas.blogspot.in
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dimensional equation of heat conduction (excluding insulated edges). Bloom’s Taxonomy Q.No. Question Level PART - A Classify the PDE − − + − + BTL-4 1. + − = =0 by method of separation of variables
BTL-3
Applying
What are the various solutions of one dimensional wave equation 2 y 2 y In the wave equation 2 c 2 2 what does C2 stand t x for? What is the basic difference between the solutions of one dimensional wave equation and one dimensional heat equation with respect to the time? Write down the initial conditions when a taut string of length 2 is fastened on both ends. The midpoint of the string is taken to a height b and released from the rest in that position A slightly stretched string of length l has its ends fastened at x = 0 and x = l is initially in a position given by � , = � .If it is released from rest from this � position, write the boundary conditions A tightly stretched string with end points = & = is initially at rest in equilibrium position. If it is set vibrating giving each point velocity � − . Write the initial and boundary conditions If the ends of a string of length are fixed at both sides. The midpoint of the string is drawn aside through a height h and the string is released from rest, state the initial and boundary conditions State the assumptions in deriving the one dimensional equation Write down the various possible solutions of one dimensional heat flow equation? 2u u In the one dimensional heat equation C2 t x 2 2 what is C ? The ends A and B of a rod of length 20 cm long have their temperature kept 300 C and 800 C until steady state prevails. Find the steady state temperature on the rod An insulated rod of length 60 cm has its ends at A and B maintained at 200C and 800 C respectively. Find the steady
BTL-1
Remembering
BTL-2
Understanding
BTL-3
Applying
BTL-1
Remembering
BTL-2
Understanding
BTL-2
Understanding
BTL-2
Understanding
BTL-1
Remembering
BTL-1
Remembering
BTL-2
Understanding
BTL-2
Understanding
BTL-2
Understanding
3.
Solve
9.
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10.
�
log
8.
�
11. 12.
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13.
14. 15.
�
s.b
7.
�
−
=
ata
6.
�
�
sp ot. in
Analyzing
Classify the PDE
5.
�
Analyzing
BTL-4
2.
4.
�
Domain
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17. 18.
19.
s.b
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3.
PART-B A string is stretched and fastened to two points that are distinct string l apart. Motion is started by displacing the string into the form = − from which it is released at time t=0. Find the displacement of any point on the string at a distance of from one end at time t. A tightly stretched string of length 2 l is fastened at both ends. The Midpoint of the string is displaced by a distance transversely and the string is released from rest in this position. Find an expression for the transverse displacement of the string at any time during the subsequent motion. A slightly stretched string of length l has its ends fastened at = and = l is initially in a position given by � , = � . If it is released from rest from this � position, find the displacement at any distance from one end and at any time. A tightly stretched string with fixed end points = and = l is initially at rest in its equilibrium position. If it is set vibrating string giving each point a velocity � − . Find the displacement of the string at any distance from one end at any time . A tightly stretched string with fixed end points x=0 and x=l is initially at rest in its equilibrium position. If it is vibrating string by giving to each of its points a velocity
ata
2.
log
A rectangular plate with insulated surface is 10cm wide. The two long edges and one short edge are kept at00 C, while the temperature at short edge x =0 is given by 20. , ={ Find the steady state − , temperature at any point in the plate
1.
Ci
4.
5.
BTL-2
Understanding
BTL-1
Remembering
BTL-1
Remembering
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16.
state solution of the rod. An insulated rod of length cm has its ends at A and B maintained at 00C and 800 C respectively. Find the steady state solution of the rod. Write down the governing equation of two dimensional steady state heat equation. Write down the three possible solutions of Laplace equation in two dimensions A plate is bounded by the lines x=0, y=0, x= and y= . Its faces are insulated. The edge coinciding with x-axis is kept at 1000 C. The edge coinciding with y-axis at 500 C. The other 2 edges are kept at 00 C. write the boundary conditions that are needed for solving two dimensional heat flow equation
BTL-2
Understanding
BTL-2
Understanding
BTL-2
Understanding
BTL-2
Understanding
BTL-2
Understanding
BTL-3
Applying
BTL-2
Understanding
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l 2kx if 0 x l 2 Find the displacement of the string . v 2k (l x) if l x l 2 l
9.
A bar 10 cm long with insulated sides has its ends A and B maintained at temperature 500C and 1000C respectively. Until steady state conditions prevails. The temperature at A is suddenly raised to 900C and at the same time lowered to 600C at B. Find the temperature distributed in the bar at time t.
A square plate is bounded by the lines
= ,
= ,
=
= . Its faces are insulated. The temperature along the upper horizontal edge is given by , = − when < < while the other three edges are kept at 00 C. Find the steady state temperature in the plate.
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10.
BTL-2
Understanding
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log
8.
s.b
7.
ata
6.
at any distance x from one end at any time t. A tightly stretched string of length is initially at rest in this equilibrium position and each of its points is given the 3 x . Find the displacement y(x,t). velocity v 0 sin l u 2u Solve subject to the conditions (i) u(0,t)=0 C2 t x 2 for all t≥0 (ii) , = l if 0 x x 2 (iii) u ( x,0) . l x if l x l 2 A rod 30 cm long has its ends A and B kept at 200 and 800 respectively until steady state conditions prevail. The temperature at each end is then suddenly reduced to 00C and kept so. Find the resulting temperature function u(x, t) taking x = 0 at A.
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A square metal plate is bounded by the lines x=0, x=a, y=0, y=a .The edges x=a, y=0, y=a are kept at zero degree 11. temperature while the temperature at the edge x=0 is . Find the steady state temperature distribution at in the plate. A rectangular plate with insulated surface is 10 cm wide and so long compared to its width that it may be considered infinite in length without introducing appreciable error. The 12. temperature at short edge y=0 is given by , and all the other three ={ − , edges are kept at 00C. Find the steady state temperature at
BTL-3
Applying
BTL-2
Understanding
BTL-2
Understanding
BTL-2
Understanding
BTL-2
Understanding
BTL-2
Understanding
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any point in the plate. An infinitely long rectangular plate with insulated surface is 10cm wide. The two long edges and one short edge are kept at 00C, while the other short edge x=0 is kept at temperature 13. , ={ . Find the steady state − , temperature distribution in the plate.
Understanding
sp ot. in
A long rectangular plate with insulated surface is cm . If the temperature along one short edge y=0 is u(x,0)=K(lx-x2) degrees, for 0
BTL-2
BTL-2
Understanding
s.b
log
UNIT -IV FOURIER TRANSFORM Statement of Fourier integral theorem – Fourier transform pair – Fourier sine and cosine transforms – Properties – Transforms of simple functions – Convolution theorem – Parseval’s identity. Bloom’s Domain Q. No. Question Taxonomy Level PART-A BTL -1 Remembering 1. State Fourier integral Theorem Write Fourier Transform in Pairs
BTL -1
Remembering
3.
If F s denote the Fourier Transform of f x , Prove that 1 s F f ax F , a 0. a a
BTL -3
Applying
F s F f x , then showthat F f x a e ias F (s) .
BTL -3
Applying
BTL -2
Understanding
4.
ata
2.
If the Fourier Transform of f x is
a x
Find the Fourier Transform of e
6.
Find the Fourier Transform of e i k x , if a x b f x . 0 , if x a & x b
BTL -2
Understanding
7.
State and Prove Modulation theorem on Fourier Transforms
BTL -2
Understanding
8.
If F s F f x , then find F e iax f x .
BTL -2
Understanding
9.
Define self-reciprocal with respect to Fourier Transform
BTL -1
Remembering
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5.
.
10. Find the Fourier sine Transform of
1 . x
BTL -2
Understanding
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11. Find the Fourier sine Transform of f ( x) e 2 . Give an example of a function which is self- reciprocal 12. under Fourier Sine & Cosine Transform 13. Write down the Fourier cosine Transform pair of formulae If F(s) is the Fourier Transform of f x . Show that the 14. Fourier Transform of e ia x f ( x) is F (s a) . Show that the Fourier Transform of the derivatives of a dn n 15. function F n f ( x) is F ( s). dx
16. If F s F f x , then find F x n f x . 17. Find the Fourier cosine Transform of e 2 x .
BTL -2
Understanding
BTL -3
Applying
BTL -1
Remembering
BTL -3
Applying
sp ot. in
x
BTL -3
Applying
BTL -2
Understanding
BTL -2
Understanding
BTL -3
Applying
BTL -1 BTL -1
Remembering Remembering
BTL -2
Understanding
Let Fc s be the Fourier cosine Transform of f x . Prove 18.
1 Fc s a Fc s a . 2 19. State Convolution theorem in Fourier Transform 20. State Parseval’s Identity on Fourier Transform
log
that Fc f ( x) cos ax
PART-B
s.b
Find the Fourier Transform of 1 , x a and f x 0 , x a 0 1.(a) sin t hence evaluate dt. Also using Parseval’s t 0
ata
2 Identity Prove that sin2 t dt 0
t
2
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Find the Fourier Cosine Transform of the function e a x e b x 1. (b) f x ,x 0 x Find the Fourier Transform of the function 1 x , if x 1 f(x) = Hence deduce that 2. (a) 0 , if x 1 2
BTL -2
Understanding
BTL -2
Understanding
BTL -3
Understanding
4
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sin t sin t (i) dt (ii ) dt . t 2 t 3 0 0 x2 2
Show that the function e is self-reciprocal under the 2.(b) Fourier Transform by finding the Fourier Transform of
e a
2
x2
,a 0
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Show that the Fourier Transform of 2 1 cos as a x , if x a is . Hence f x 2 s 0 , if x a 0 3.(a)
Applying
BTL -3
2
Applying
BTL -3
sp ot. in
sin t deduce that dt . 2 t 0 Prove that d d 3.(b) Fc x f ( x) Fs f x and Fs x f ( x) Fc f x ds ds 1 x 2 , if x 1
Find the Fourier Transform of f ( x)
0 , if x 1 s 3 sin s s cos s , (ii) 4. (a) Hence Show that (i) cos ds 3 2 16 s 0 2
sin s s cos s ds . 3 0 15 s
x
Remembering
, 0 x 1
log
4.(b) Find the Fourier Sine Transform of f x 2 x , 1 x 2 0
BTL -1
BTL -2
Understanding
BTL -3
Applying
BTL -3
Applying
BTL -2
Understanding
BTL -5
Evaluating
, x2
Show that the Fourier transform of
s.b
2 2 cos as a x , x a is 2 2 sin as as . Hence 3 5. (a) f ( x) 0 s , 0 x a deduce that sin t 3t cos t dt . 4 t 0
sin sx dx s 0
vil d
ata
Find the Fourier sine transform of e a x (a>0). Hence find e a x Fs xe a x and Fs hence deduce the value of 5.(b) x
6. (a) Find the Fourier Transform of f x
1
.
x
Using Parseval’s Identity evaluate the following integrals.
6.(b) i
Ci
0
dx 2 a x2
2
, ii
a 0
x 2 dx 2
x
2 2
where a 0.
Find the Fourier Cosine and Sine Transforms of x n 1 . and 7. (a) 1 is self reciprocal with respect to both. hence show that x dx 7. (b) Evaluate 0 x 2 a 2 x 2 b 2 , using Fourier Transform
BTL -2
Understanding
BTL -5
Evaluating
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8. (a) Find the function whose Fourier Sine Transform is e a s ,a 0 s 8.(b) State and Prove Convolution Theorem on Fourier Transform 9. (a)
Find the Fourier Transform of e
x
and hence find the
cos 2 x. Fourier Transform of f x e Find the Fourier Cosine transform of e– 4x. Deduce cos 2 x x sin 2 x 9.(b) that 2 dx e 8 and 2 dx e 8 .
BTL -3
Applying
BTL -2 Understanding
8
0
x 16
log
Using Fourier Sine Transform prove that x 2 dx 10.( b) 0 ( x 2 a 2 )( x 2 b 2 ) 2(a b)
11.(b) Find Fs xe
s.b
Find the Fourier cosine & sine Transform of e x . Hence evaluate 11.(a) 2 i 2 1 2 dx and ii 2 x 2 dx . 0 x 1 0 x 1
ax
and F xe . c
ata
Find the Fourier Sine Transform of the function 12.(a) sin x , 0 x a f x 0 , x a
vil d
Find the Fourier Cosine Transform of f x e a
12.(b) hence find the Fourier Cosine Transform of e 2 x
2
Find the Fourier sine transform of
Fourier cosine transform of
+
.
2 x
2
2
x2
Ci
BTL -3
Applying
BTL -2
Understanding
BTL -2
Understanding
BTL -2
Understanding
BTL -2
Understanding
BTL -2
Understanding
BTL -5
Evaluating
BTL -5
Evaluating
and
and the
. +
and
dx . 25 x 2 9 0 Verify the Convolution Theorem for Fourier Transform if
13.(b) Using Parseval’s Identity evaluate
f x g ( x) e x
Analyzing
BTL -4
ax
Fourier Sine Transform of x e
BTL -2
2
Find the Fourier Transform of e and hence deduce that cos x t a x a x 2as (i) 2 2 dt e =i 2 , (ii ) F xe 2 a 10.(a) 2 2 2 a t a s 0 here F stands for Fourier Transform.
Understanding
sp ot. in
x 16
a x
14.(a)
Remembering
x
0
13.(a)
BTL -1
x
2
2
14.(b) Derive the Parseval’s Identity for Fourier Transform
BTL -6
Creating
UNIT -V Z - TRANSFORMS AND DIFFERENCE EQUATIONS MA6351_TPDE_QBank_ACY 2016-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH. PREPARED BY DEPARTMENT OF MATHEMATICS Visit : Civildatas.blogspot.in
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2.
Find the Z – Transform of
3.
an Find Z n!
4.
1 Find Z n !
5.
1 Find Z n 1
6.
sp ot. in
Z- Transforms - Elementary properties – Inverse Z - transform (using partial fraction and residues) – Convolution theorem - Formation of difference equations – Solution of difference equations using Z - transform. Bloom’s Domain Q.No. Question Taxonomy Level PART-A BTL -1 Remembering 1. Define Z – Transform of the sequence f n . BTL -2
Understanding
BTL -2
Understanding
BTL -2
Understanding
Remembering
Define the unit step sequence .write its z-transform
BTL -1
Remembering
7.
State initial value theorem and final value theorem
BTL -1
Remembering
8.
Find Z n 2 .
BTL -2
Understanding
9.
Find inverse Z transform of
BTL -2
Understanding
−
−
ata
1 10. Find Z n 1!
s.b
log
BTL -2
t 11. Find Z e sin 2t .
vil d
z n 12. Prove that Z a f (n) f ( ) a z n 13. Prove that Z a za
14. Find
−
[
−
]
[
+
]
Ci
15. Find Z transform of 16. Find
−
BTL -2
Understanding
BTL -2
Understanding
BTL -5
Evaluating
BTL -5
Evaluating Analyzing
BTL -2 ,
17. Solve yn+1 + 2yn =0 given that y(0)=2 18. State Convolution theorem in Z – Transforms Form the difference equation by eliminating arbitrary 19. constants from =� +
BTL -2
Understanding
BTL -2
Understanding
BTL -3 BTL -1
Applying Remembering
BTL -6
Creating
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20.
Form the difference equation by eliminating arbitrary constants from =� PART-B f n
1.(a) Find the z transform of 1 1. (b) Find the Z
1
n 1n 2
BTL -2 Understanding
10 z z 3 z 2
BTL -2
2. (a) Find the z-transform of (n+1)2 and sin(3n+5)
BTL -2
z z 2 z 2 . 2.(b) Find the inverse Z – Transform of 2
BTL -2
Understanding Understanding
sp ot. in
3. (a)
2
Creating
BTL -6
z 1z 1
Find i Z r n cos n , ii Z r n sin n iii )Z (e at cos bt )
BTL- 2
Understanding Understanding
Using convolution theorem find inverse Z transform of 3.(b)
z 4. (a) Find the Z 1 z 1z 2
log
z2 z a z b
z2 z 1z 3
4.(b) Using convolution theorem find Z 1
z2 by the method z 2 z 2 4
s.b
5. (a) Find inverse Z -Transform of
BTL -3
Applying
BTL -2
Understanding
BTL -3
Analyzing
BTL -2
Understanding
BTL -3
Applying
BTL -3
Applying
BTL -3
Applying
BTL -3
Analyzing
of Partial fraction Using convolution theorem find inverse Z transform of 5.(b) −
+
−
[
ata
6. (a) Using residue find
−
−
]
z 6.(b) Using convolution theorem find Z 1 z 4z 5 2
vil d
7.(a) Using Z transform Solve y n 2 3 y n 1 2 y n 0 given that y(0)=0,y(1)=1 z 2 z . 7.(b) Find inverse Z transform of z 13 Using Z transform solve + + = + + given that y(0)=0,y(1)=1 Using the inversion method (Residue theorem )find the
Ci
8.(a)
z2 inverse Z transform of U(z)= z 2z 4 z 1 9.(a) Using Residue method find Z 2 z 2z 2
8.(b)
BTL -2
Understanding
BTL -3
Applying
BTL -3
Applying
BTL -3
Applying
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z2
9.(b) Using convolution theorem evaluate Z 1 z 1z 3 Using convolution theorem evaluate 10.(a) − [ ] − −
BTL -3
Applying
BTL -3
Applying
BTL -3
Applying
BTL -3
Applying
BTL -3
Applying
BTL -6
Creating
BTL -3
Applying
BTL -2
Understanding
BTL -3
Applying
Solve the equation using Z – Transform 14.(a) y 5 y 6 y 36 given that y(0) = y(1) = 0 n2 n 1 n
BTL -3
Applying
z 1 14.(b) Find Z 2 by convolution theorem z 7 z 10
BTL -2
Understanding
Using Z transform solve y(0)=0,y(1)=1
11.(a) Find the Z transform {
11.(b)
+
Using Z transform solve
−
+
} and {
given that y(0)=0,y(1)=0
}
+
+
−
+
+
Form the difference equation
12.(b)
+
= ,
−
=
+
+
=
�ℎ
=
State and prove final value theorem and their inverse transformation �
}
s.b
13.(a) Find the Z transform of { } and{ 13.(b)
=
with
log
12.(a)
=
sp ot. in
10.(b)
Solve the difference equation y n 3 3 y n 1 2 y n 0
****ALL THE BEST****
Ci
vil d
ata
given that y(0)=4 ,y(1)=0,y(2)=8
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