CS 2203 – OBJECT ORIENTED PROGRAMMING QUESTION BANK

CS 2203 – OBJECT ORIENTED PROGRAMMING

QUESTION BANK

Part- A: 2marks each: write with illustrations and syntax:

01.List out the features of a POP.

02.Give the differences between POP and OOP.

03.What are the basic concepts of OOP. Explain a class.

04.Define the term Data abstraction and data encapsulation.

05.Explain the term polymorphism. What are its types?

06.Give the benefits of OOP over POP.

07.What do you know about “Name space”.

08.Write a c++ program to get any two numbers and output the greater one.

09.What are manipulators? What function do they serve?

10.Explain with use, the term In-line functions.

11.Give the situations where in-line functions does not work.

12.What is meant by “function overloading”? What is its syntax?

13.Define a friend function. When is a class called a friend class?

14.Define the term “abstract class”.

15.Give the ways in which the member functions can be declared.

16.What are static member functions? Give the syntax.

17.Explain nested classes.

18.Write a note on the local classes.

19.Define a constructor. Give its syntax.

20.What is a default constructor? What purpose does it serve?

21.Explain-Dynamic constructors.

22.Define operator overloading. Give the syntax.

23.State the operator that cannot be overloaded in a normal function inside class.

24.When we use a friend function for overloading operators, state the restrictions.

25.Give the source and the destination class when the following conversions are

made:

(a)class type to basic type

(b)class A type to class B type

(c)Basic type to class type.

26.Define the term Inheritance. Give its types.

27.Give the derivations (or) visibility for the derived class based on that of the base

class. (private, public and protected derivations).

28.What are Virtual functions? Give the typical prototype.

29.For the following inheritance mention the order of execution of constructors:

(a)class A : public B, virtual public C

(b)class A: public B, public C.

Is there any change in the order of execution in these cases?

30.What is “this” pointer? What purpose does it serve?

31.Give the syntax for overloading << and >> operators?

32.List the ios format functions and their functions.

33.What is fill() used for?

34.List the manipulators, their functions and respective ios equivalents.

35.Give the syntax for designing your own manipulators.

36.What are the file stream classes used for creating input and output files?

37.What are the ways of opening a file? Give the syntax.

38.How will you detect the end-of-file?

39.Give the functions of seekg(), seekp(), tellg() and tellp().

40.What are template? Give its uses and syntax.

41.Explain the differences between function and class templates with syntaxes.

42.What is exception handling? Give a schematic sketch of its mechanism.

43.How will you specify exceptions?

44.When is the handler catch(. . .) used? Give an example.

45.Give the components of STL. Give a schematic sketch of it.

46.What are containers? What are its types?

47.What are iterators? Mention them with the access method, directions of

movement and I/O capability.

48.Distinguish between the various types of containers in pairs.

49.How are STL algorithms implemented?

50.How will you manipulate string objects?

Part- B: 4/8 marks each: write with programs and syntax:

01.Explain in detail the various types involved in Type conversions.

02.In detail discuss the blocks in handling exceptions. Also describe the following:

(a)Re throwing (b)Catch all (c)Restrictions in throwing.

03.(a).Explain the constructors. Give the rules.

(b).What is a copy constructor? How is it implemented?

04.Elucidate the various forms of Inheritance with examples.

05.What are static members and member functions? Give the rules. Illustrate them.

06.Elucidate the types of Containers supported by STL.

07.Explain with examples the basic concepts of OOP.

08.What are friend functions and friend classes? How will use them in overloading

the operators? Show as how will you overload New and Delete operators.

09.Briefly discuss the various types of Iterators.

10.In brief explain the following concepts (a) RTTI (b)Templates (c)Virtual functions.

Concepts based on which the programs are asked:

Friend functions, operator overloading, files opening and closing, templates, inheritance, exception handling, virtual functions & RTTI, type conversion.

Analysis of Questions:

Part – A: All are simple definitions, uses, rules and properties of the various

concepts as given here.

Part – B: Each of (a) and (b) carries two questions. One is a theory worth 4 marks

(or) maximum 8 marks and other one a program for 8-12 marks.

1.Go through all the notes you have for the programs.

2.The programs are familiar. Mainly know the syntax of all concepts.

3.Give a brief introduction of the concepts used and explain your

program with comments and a paragraph about its flow.

4.Give diagrammatic representations wherever possible.

ALL THE BEST AND BEST OF EFFORTS!!!

by,

RANGARAJAN T.R.

MAXIMUM POSSIBLE QUESTIONS--CS2201- DATA STRUCTURES IN “C”

                                      MAXIMUM POSSIBLE QUESTIONS
                             CS2201- DATA STRUCTURES IN “C”

PART—B cum PART—A QUESTIONS:

01.Explain the prim's algorithm and kruskal's algorithm to find the minimum
spanning tree. Elucidate with routines and examples.

02.What is topological sorting? Give a diagram, sort the elements in topological
order. State the routines also.

03.Given a graph, apply the two algorithms to find the minimum cost. Write
their routines.

04.Write a note on
(a) Euler's paths and circuits.
(b)Graph traversals.

05.Explain Articulation points and routines to find them with examples.

06.(a).Write a note on weighted and non-weighted graphs. How will find the shortest path of them?
           Write the routines and explain with examples.
     (b).Given a weighted graph, find the shortest path of it and write the routines.

07.Elucidate with routines and diagrams the operations performed on a doubly
linked list.PART—B cum PART—A QUESTIONS:

08.What are heaps? Explain the properties with routines. How will you perform Heap sort?

09.Elucidate with routines and diagrams the operations performed on a singly linked list.

10.(a).Explain the mechanism of Radix sort and TOH.
     (b).How will you add, subtract and multiply two polynomials? Explain .

11.Elucidate the cursor implementation of linked lists in detail.

12.Describe the applications and operations performed on a stack.

13.With routines, explain the operations performed on a DEQUEUE and a
circular Queue.

14.Explain the (a)dynamic equivalence problem
                        (b)smart union algorithms.
                        (c)path compression.

15.Explain the analysis of insertion and deletion operations on a BST.

16.Elucidate the Tree traversals. Given an expression, form the tree for that
and perform these three traversals.

17.Elucidate the operations performed on a BST and a B-tree.

18.Describe the operations performed on an AVL tree.

19.What is Hashing? Explain open addressing and separate chaining with
the collision resolving techniques.

20.Explain the following:
(a)Extendible hashing (b)Double hashing (c)Rehashing.

ANALYSIS from previous years' question paper...so far the questions are asked
like the following order:
Unit-1:
11.(a) lists,stacks.
11.(b)lists,Queues.

Unit-2:
12.(a)Binary tree and binary search trees.
     (b)B- trees and binary search trees.

Unit-3:
13.(a)AVL trees.
     (b)Heaps.

Unit-4:
14.(a)Hashing.
     (b)Sets.

Unit-5:
15.(a)Shortest path algorithms, topological sort, graph traversals.
(b)Minimum spanning tree, euler's concepts, articulation points.







Prepared by,
RANGARAJAN T.R.

ENVIRONMENTAL SCIENCE-MAXIMUM POSSIBLE QUESTIONS

                                         ENVIRONMENTAL SCIENCE
                            Question bank in full (max ques):


**please note the following**
1.Some questions may be repeated. So go through all the
questions.
2.Hope you use it effectively like a checklist.


                                                         Part—A :
01.What are the characteristics of the Desert ecosystem?
02.''India is a mega--diversity nation''--Account.
03. What is meant by keystone species?
04.What do you understand by the terms flora and fauna?
05.How does Biome differ from an ecosystem?


06. Define genetic diversity and species diversity.
07.What are the classifications of biotic components of ecosystem?
08. What is food chain and food web?
09.Explain endangered species.
10.What do you mean by environmental impact.


11. What is meant by biosphere?
12. Distinguish between primary and secondary consumer.
13.What is ecology?
14. Why do we need environmentalist?
15.Write short note on biodiversity.


16. Name various ecosystems
17. Suggest three important features of forest ecosystem.
18. Define environmental impact statement.
19.What is meant by food webs?
20.What are the main kinds of organism in aquatic life zones.


21. Define primary and secondary production.
22.What is ecological pyramid? Why is pyramid of energy always take upright?
23.Define 'hot spots of Biodiversity'
24. What are the roles of women in environmental protection.
25.What is called thermal pollution?


26.State the role and responsibility of an individual in the prevention of pollution.
27.Mention the causes and effects of acid rain
28.What is meant by non point pollution?
29.What is meant by CDM?
30.What are the various sources of radioactive pollution?


31.Differentiate between recycling and reuse.
32.Give examples for primary and secondary air pollutants.
33.What is Dobson unit?
34.Write any four major water pollutants.
35.Write briefly how human activities can introduce thermal pollution in to streams


36.How CFC''s are accumulated in atmosphere?
37.Define hazardous waste.
38.Define photochemical smog.
39.What are the types of solid wastes?
40.What is acid precipitation?


41.State the significance and scope of environmental education.
42.Mention the various causes of desertification.
43.What are the roles of women in environmental protection?
44.What is meant by ISO 14000?
45.Differentiate between deforestation and forest degradation


46.Enumerate the desired qualities of an ideal pesticide.
47.Write any two functions of forests.
48.What are the advantages in conjuctive use of water?
49.what are the causes of deforestation?
50.Explain overgrazing.




51.What is desertification?
52.What is water logging?
53.What is meant by eutrophication?
54.Why world environment day is observed 5th June every year?
55.State the significance and scope of environmental education.


56.State few drawbacks of pollution related Acts.
57.Define Doubling Time with reference to population growth.
58.What is the effect of HIV//AIDS on the environment.
59.What are the roles of women in environmental protection?
60.What is meant by NIMBY syndrome?


61.What are the reasons behind the increased population growth in the less developed nations compared with developed nations?
62.Differentiate between HIV and AIDS..
63.List the problems of population growth..
64.What do you mean by environmental impact?


65.Define population eqm.
66.Define population explosion.
67.What is meant by environmental audit?
68.Define the term sustainable development.


                                          Part-- B: maximum possible 16/8 marks:
01.Write note on ecological pyramids and their types
02.Substantiate the value of biodiversity with suitable illustrations and
statistical evidences


03. What is an ecosystem? Explain the structure, characteristics and types of the Aquatic Ecosystem.
04.(a) How is biodiversity conserved? Explain
     (b)Explain characteristic features and function of forest ecosystem


05.(a).What are various hot spots of Biodiversity in India?
     (b).Discuss endangered and endemic species of India.


06.With a neat sketch explain the flow of energy through the various components of the ecosystem(producers, consumers and decomposers).


07.(a).Explain: The grass land ecosystem
     (b).Write notes on the following:
           ((i)) Energy flow in ecosystem
          ((ii)) Pond ecosystem
          ((iii)) Threats to biodiversity.
          ((iv))State four components of ecosystem
          ((v))how fat-soluble pollutants like DDT get bio magnified?
         ((vi))Four kinds of biodiversity
         ((vii))measuring biodiversity
        ((viii)Why biodiversity is rich in the tropics?
         ((ix)causes for loss of biodiversity?
  (c)Explain the food chain and food web.


08.(a)Define and explain the term ecosystem.
     (b)Decline its types,characteristics,structure and function.
     (c)Describe the biotic component of an ecosystem


09.(a).Discuss the importance, hierarchical concept and values of biodiversity.
     (b).Write informative notes on 'In-situ conservation.


10.Explain causes,, effects and control measures of air pollution.


11.Give a comparative account of urban and industrial wastes in terms of their sources,characteristics and management and disposal methods.


12. Explain the causes and effects and control measures of marine pollution. Also Write a short note on disaster management.


13.Explain the various sources,, effects and methods of control of water pollution. Also Discuss about the significance of hazardous waste management.


14.Explain the sources,, effects and control of industrial noise pollution. Discuss about ozone hole and ozone depleting substances.


15.(a).What are the effects of oil pollution on the ocean?
     (b).With the flow diagram explain the Activated sludge process for waste water treatment.


16.What is the significance of Dissolved Oxygen in rivers? Explain.


17.Explain the concept of source, path receiver in the control of noise pollution.
18.Explain the causes and effects of
     (a)nuclear and radiation pollution.
     (b) Global warming and Acid Rain
     (c)Discuss the major air pollutants and their impact.


19.Discuss briefly the disposal of municipal solid waste.


20.(a).What are the sources of Radioactive pollution?
     (b).What is thermal pollution and explain its effects.
     (c)What are the sources of greenhouse gases in atmosphere?
     (d)Write briefly on Bhopal disaster and Chernobyl disaster.


21. critically review these statements in the present day context.
       (a).''Almost all the rivers in India have become sewage carriers''.
       (b).'The invention of plastic a boon or a bane--
       (c)Environmental damages caused by mining last long after mine has closed.
       (d).Interlinking of rivers is the solution for drought as well as flood situation in India.


22.With neat block diagrams explain the various unit operations and processes in a municipal sewage treatment plant.


23.How is solid waste it classified?Explain its disposal by incineration.


24.(a).Discuss the significance of any five parameters of drinking water quality standards.
     (b).Explain any two conflicts over water,confining to our nation.


25.Discuss the Indian Ambient air quality and noise standards.


26.Discuss the impacts of handling the mineral resources for extraction and subsequent utilization on the environment.


27.Discuss in detail the causes and consequences of overexploitation of forest resources.


28.Give a brief account of renewable energy resources and their significance.


29.Discuss the various factors influencing soil degradation.


30.Discuss the various sources of energy and the linkage between energy and environment.


31.Explain the role of an individual on environmental protection.


32.Discuss the environmental effects due to agriculture.


33.Discuss the possible solutions to improve the acceptability of dam projects in Indian conditions.


34.(a).Explain over utilization of surface and ground water.
     (b).Write briefly on the Hydrological cycle.


35.Explain the adverse effects of agricultural practices away from the farm. Also explain What is deforestation and explain its impacts on the environment.


36. (a).Write a note on solar production of electricity.
      (b).Explain the methods of harnessing tidal power.
      (c)Discuss the impacts of ''Green Revolution'' on ecosystem.
      (d).What is meant by soil erosion? What are its causes? Suggest in—situ remedies.


37.How is environment protected through Legislation? Explain


38.Explain the terms with statistical support. Exponential Growth and population Explosion. Also explain the theories behind them.


39.Explain the role of information Technology in Environment and Human Health.


40.(a).Explain the role of NGO s in environmental protection and health.
     (b).Outline various family welfare plans in the post independent India.


41.Discuss about the environmental and social problems faced by large Indian city like Chennai..


42.(a)What are the salient features of following Acts?
        i))The Air((Prevention and control of pollution)) Act,, 1981
       ii))The Environment((Protection)) Act 1986
     (b) Discuss briefly on the Indian Environmental Acts.


43.Draw a typical population pyramid of developing country and discuss how it is likely differ from that of a developed country wide.


44.Define Human Rights and discuss the salient features of Universal Declaration of Human Rights by UN


45.What are the modes of transmission of HIV and how it can be prevented? What is AIDS? How to prevent it? Write in relation to the developing countries also.


46.Write notes on the following in relation to human population and environment
     ((i))Women and Child Welfare
    ((ii)) Human Rights
   ((iii))Value Education
   ((iv))population dispersion.
   ((v))Reason for the population explosion
   ((vi)) impacts of growing population.
   ((vii))factors influencing family size..
  ((viii))methods and implementation of family planning.


47.Write note on Pollution Control Boards and the effect of increasing affluence on environment.


48.Bring out the various details of Wasteland Reclamation Practices.


49.(a).Discuss about Environmental ethics..
     (b).Write a note on integrated wasteland development program.


50.(a).What is sustainable development and explain is concepts?
     (b).Write a note on watershed management.


ANALYSIS FOR PART-B:
QUES.       (A)                                      (B)                    
11          Resources                       Renewable sources
12          Ecosystems                     Biodiversity
13          Pollutions                       Management
14          Environmental issues     Acts and incidents
15          Population                     Welfare schemes, diseases 
                                                      and issues


                               ALL THE BEST AND BEST OF EFFORTS!!!
by,
Rangarajan T.R.

ENGG. MATHS-III-TRANSFORMS AND PDE-PRACTICE QUESTIONS FOR Z-TRANSFORMS

                                          ”Z”-TRANSFORMS

Marks:100                                                                                        Time:3hours

                                            PART—A (16marks)                                          

                   ATTEMPT ANY 8 OF THE FOLLOWING:(8*2=16):                      

01.Define a two sided Z-transform. State the applications of it.

02.Find the Z-transform of aⁿ

                                              n!

03.Evaluate the Z-transforms of rⁿcosnф and rⁿsinnф.

04.Define the following functions and derive their Z-transforms:

(a)Unit impulse function

(b)Unit function

05.state and prove the scaling in Z-domain property.

06.Find the Z-transforms of the following functions:

(a)cos³t

(b)sin³nП/6

07.Find the inverse Z-transform of F(z) given by

               F(z) = log(1 divided by (1 – az^-1))

08.State and prove the time shifting property of Z-transforms.

09.Derive an expression to find the Z-transform of the derivative of a function

from the Z-transform of the actual function.

10.Find the Z-transform of (n+1)(n+2).

11.Form the difference from the following by eliminating the arbitary constants:

(a) U_n=a2ⁿ⁺¹

(b) y_n=a+b3ⁿ

12.Evaluate the Z-transform of { t ^k}.

                                            PART—B:(84 marks):                                           

    ANSWER ANY 12 FROM THE FOLLOWING QUESTIONS:(12*7=84):   

13.Find the inverse Z-transform of the following using the residue theorem:

                                                            z

                                      (z – 1)(z² + 1) (7marks)

14.Use the power series method to evaluate inverse Z-transform of the

following:

(a) 10z               (b)         z                           (4+3)marks

(z - 1)(z - 2)           z² + 7z + 10

15.Solve the following equation using Z-transforms only:

y(n+2) + 6y(n+1) + 9y(n) = 2ⁿ provided y₀=y₁=0. (7marks)

16.By the method of partial fractions find the inverse Z-transform of the

following:

         z(z2- z + 2)                                                               (7marks)

       (z + 1)( z - 1)²

17.State and prove the following theorems: ((2+3+3)marks)

(a)Initial value theorem

(b)Final value theorem

(c)Frequency shifting theorem

18.(a)State and prove the convolution theorem for Z-transforms.

     (b)Using the above concept alone, find the Z^-1 of the following:

                             8Z²                (3+4marks)

                      (2z – 1)(4z + 1)

19.Evaluate the Z-transforms of the following: (2+2+3marks)

(a) 3ⁿ cosh 2n

(b)(k-1)a^(k-1)

(c)    2n+3

    (n+1)(n+2)

20.Find the inverse Z-transform of the following using the residue theorem:

                             z(3z² - 6z + 4)

                              (z + 2)(z – 1)²             (7marks)

21.Solve the following difference equation using Z transformation method:

x(n+2) - 4x(n+1)+ 3x(n) = 2ⁿ.n² ,x(0)=0 and x(1)=0. (7marks)

22.Find the inverse Z-transform of the following. Ensure that the final answer

is a pre - defined function:

                           4z² – 2z

                      z³ – 5z² + 8z – 4             (7marks)

23.Find the Z-transform of f(n)*g(n) if (4+3marks)

(a) f(n)=U(n) and g(n)= δ(n) + (1/2)ⁿ U(n).

(b)f(n)=2ⁿ u(n) and g(n)=3ⁿ u(n).

24.(a)Find the inverse Z-transform of

                      z – 4

               (z – 1)(z - 2)²                   (4+1+2marks)

(b)Find the initial and final value of { f(n) } if

F(z)  =                   0.4 z²

                  (z-1)(z² – 0.736z +0.136)

25.Find the Z-transform of the following: (2+2+1+2marks)

(a)aⁿ and naⁿ

(b)cosnθ and ncosnθ

(c)e^-at sinbt

(d)(t+T) e^-(t+T)

26.Solve the following difference equation by (4+3marks)

(a)Residue

(b)partial fractions:

y(n) – y(n-1) = u(n) + u(n-1)

27.(a)Discuss the differences between L and Z-transforms. (2+3+2marks)

     (b)Find the inverse Z-transform of

                                       z2 – 3z

                                    (z – 5)(z + 2)

(c)Find the initial value of the following function:

         F(z)   =       ze^aT (ze^aT – cosbT)

                         z²e2aT – 2ze^aTcosbT + 1

NOTE:

1.Apply the conditions in the correct order.

2.Write down all the corresponding formulas.

ALL THE BEST AND BEST OF EFFORTS!!!

set by,

RANGARAJAN T.R.

ENGG. MATHS-III-TRANSFORMS AND PDE-PRACTICE QUESTIONS FOR PDE

                                      PARTIAL DIFFERENTIAL EQUATIONS
                                       MODEL QUESTION PAPER
Marks:100                                                                                                                   Time:3hours
                                                    PART—A (16marks)
                               ATTEMPT ANY 10 OF THE FOLLOWING:(10*2=20):


01.Find the PDE of all planes having the equal intercepts on the x and y axis.


02.Find the complete solution of x2p2 + y2q2 = z2.


03.Find the general solution of
                                 4 ∂2z/∂x2 – 12∂2z/∂x∂y + 9∂2z/∂y2 = 0.


04.Eliminate the arbitrary constants and write down the PDE for
                                     (x –  a)2 + (y – b)2 = z2 cot2β


05.Solve the following PDE::Use direct integration method.
                                             ∂2z/∂x2 = a2z
    given that:   ∂z/∂x = asiny
                        ∂z/∂y = 0; when x=0;
06.Obtain the general, complete, singular solutions for the following PDE:
                        pq + p + q = 0.


07.Solve the following PDE by the method of grouping:
                               px + qy = z.


08.Solve the following higher order Partial differential equation:
∂2z/∂x2 – 6 ∂2z/∂y2 + ∂2z/∂x∂y = cos(3x + 4y).


09.Solve the following higher order non-homogenous PDE:
[(D – D' - 1)(D – D' – 2)]z = e2x-y.


10.Find the solution of the following partial differential equation:
y2p – yxq = x( z – 2y).


11.Form the PDE from the following function:
f ( y/x , x2+y2+z2 ) =0.


12.Find the complete integral of the following equation:
z/pq = x/q + y/p + √pq.


                                                   PART—B:(80 marks):
              ANSWER ANY 10 FROM THE FOLLOWING QUESTIONS:(10*8=80):


13.Solve the following equations:
(a) z = px + qy + √ p2 + q2 +1 (3+3+2 marks)
(b)p2 + q2 = x2 + y2
(c)z = p2 + q2.


14.Obtain the solutions of the following equations: (4+4 marks)
(a)x4p2 + y2zq = 2z2.
(b)x2p2 + xpq = z2.


15.Find the solutions of : (4+4marks)
(a)x4p2 – yzq = z2.
(b)z(p2 – q2) = x2 – y2.


16.Use the
(a) Langrange's multipliers
(b) method of grouping (5+3 marks)
to find the solution of :
(x2 + y2 +yz)p + (x2 +y2 – xz)q = z(x + y).


17.Solve the following:
(a) (D3 – 7DD'2 - 6D'3)z = sin(x+2y) + e2x+y . (5+3 marks)
(b) ∂2z/∂x2 + ∂2z/∂y2 = cos 2x cos 2y.


18.Obtain the solution of the following PDE: (5+3marks)
(a) (D2 – 6DD' + 5D'2)z = ex sinhy + xy.
(b)(D3 + D2D' – DD'2 – D'3)z = 3sin(x + y)


19.Find the solution of the following differential equations: (8 marks)
(2D4 – 3D2D' + D'2)z = e2x+y.


20.Solve each of the following: (5+3marks)
(a)D(D + D' – 1)(D +3D' – 2)z = e2x+3y .
(b)(D + 3D')(4D' + 3)z = cos(x + 4y).


21.Form the partial differential equations by eliminating the arbitrary functions
“ f “ and “ g “ and “Ψ” from the following: (2+2+2+2 marks)
(a)z = f( x3 + 2y) + g( x3 – 2y)
(b)z = x2 f(y) + y2 g(x)
(c)Ψ( x2 +y2 +z2 , xyz) = 0
(d)xy + z2 = Ψ(x + y +z).


22.(a)By changing the independent variables by the relations r = x+at , s = x-at,
show that the equation ∂2y/∂t2 = a2 ∂2y/∂x2 gets transformed to
∂2y/∂t ∂s = 0. hence find a general solution of the PDE.
(b) solve : z2 = p2 + q2 + 1. (5+3marks)




23.Solve the following PDE s : (4+2+2 marks)
(a)z2 (p2 + q2) = x2 + y2.
(b)z2 = xypq
(c)ap + bq + cz = 0.


24.Obtain the solutions of the following differential equations:
(a) (D2 – D'2 -3D')z = xy. (4+4marks)
(b) (D2 – DD' +D' -1)z = cos(x+2y) + ey-x.


25.Solve the following PDE completely using (3+5 marks)
(a)method of separation
(b)Multipliers
(x + 2z)p + (4xz – y)q = x2 +y .


ALL THE BEST AND BEST OF EFFORTS!!!


set by,
RANGARAJAN T.R.

ENGG. MATHS-III-TRANSFORMS AND PDE-PRACTICE QUESTIONS FOR FOURIER TRANSFORMS

                    MODEL PAPER FOR FOURIER TRANSFORMS
Marks:100                                                             Time:3hours
                                             PART—A (16marks)
                   ATTEMPT ANY 8 OF THE FOLLOWING:(8*2=16):
01.State the fourier integral theorem. Also state the sine and the cosine integral theorems of it with the necessary conditions and expressions.


02.Find the fourier cosine integral of e-ax. Hence deduce the value of the following integral
                                    ∞
                                    ∫ cosλx dλ.
                                    0 (1+λ2)
03.Derive the fourier cosine and sine transforms.


04.Find the fourier sine transforms of  f(x) = sinx for 0≤x<a
                                                                        =0 for x>a

05.Find the fourier cosine transforms of (e-ax – e-bx)
                                                                           x .
06.Deduce whether xe-x2/2 is self reciprocal with respect to FST . Given it is  self reciprocal with respect to FCT. Use the appropriate property.

07.State and prove the shifting property of the fourier transforms.

08.Show that the fourier transforms obey the modulation theorems.

09.Relate the fourier cosine and sine transforms when the function is first subjected to differentiation and them respective transformations.

10.Find the fourier transformation of f(x) if

                                              f(x) = 1, for |x|<a

                                                    = 0, for |x|>a>0

11.Evaluate the following using transforms:
                                            ∞
                                             ∫        dx    
                                             0 (x2+a2)(x2+b2)
12.State and prove the change of scale property for a function f(x) in a fourier transformation.

                                      PART—B:(84 marks):
      ANSWER ANY 12 FROM THE FOLLOWING:(12*7=84):
13.State and prove the following theorems for F-transforms
(a)Convolution theorem
(b)Rayleigh's theorem

14.Find the FCT of the following functions:
(a) f(x) = e-ax cosax
(b) h(x) = e-ax sinax

15.Find F_s[ x^n-1] and F_c[x^n-1] for 0<n<1. Hence show that 1/√x is self reciprocal under both. Hence deduce the value of F[ 1/√|x|].

16.Evaluate the Fourier transform of

                       f(x) = 1-x² in |x| ≤ 1

                               = 0 in |x| > 1

Hence deduce the value of

                                           ∞

                                          

                                            ∫ (sin s – s cos s)(cos(s/2)) ds

                                            0              s³

17.Find the Fourier Sine transformation of f(x) using the property if

             f(x) =  1      

                  x(a²+x²)

18.Find the fourier cosine transformation of f(x)=e ^-a2x2 and

hence find Fs[ x f(x) ].

19.Find the fourier transformation of

f(x) = a² – x² |x|<a

        = 0           |x|>a>0

Hence deduce the value of

                             ∞

                              ∫ (sin t – t cost )dt

                             0         t³

20.Find the F-transform of f(x) if

                          f(x) = 1-|x| for |x|<1

                                  = 0      for |x|>1

    Hence deduce the value of  

                                              ∞

                            

                                             ∫ sin ⁴t dt

                                            0    t ⁴ 

21.Find the F-transform of f(x) for

                          f(x) = 1, |x|<a

                                 = 0, |x|>a>0

Hence deduce the value of

    ∞

(i) ∫ sin t dt

    0   t

     ∞

(ii) ∫ sin ²t dt

      0   t²

22.Find the F-transform of f(x)= e^-a|x| and hence deduce the following:

    ∞

(i) ∫ cos xt dt

   0 (a² + t²)

(ii) F[x f(x)]

23.Using fourier integral show that

                  ∞

                 ∫ (1- cosΠλ )sin xλ dλ = П/2 in 0<x<П and 0 in x>П.

                0           λ

24.Using fourier integral find the integral value for e^-ax - e^-bx .

25.Use the Parseval's identity to calculate the following integrals:

                                   ∞

(i)                               ∫      dx            for a>0 for both the problems.

                                  0   (a²+x²)²

(ii)                              ∞
∫     x² dx
                                  0  (a²+x²)²

26.Find the F-transforms for f(x) provided:

(a) f(x) is “x” when |x|<a and “0” when |x|>a.

(b)f(x) is “x²” when |x|≤a and “0” when |x|>a.

27.Evaluate the following:

(a)The fourier Sine transform of f(x) = e^-ax

                                                              x

(b)The fourier Cosine transform of f(x) when

        f(x) is “cos x” when 0<x<a and “0” when x>a.

           ALL THE BEST AND BEST OF EFFORTS!!!

set by,

RANGARAJAN T.R.