GE2211 Environmental Science and Engineering question bank anna university

Subject Code : GE2211
Subject Name : Environmental Science and Engineering

UNIT – I

PART – A (2 MARKS)

1. Define Environmental Science.
2. What is meant by deforestation?
3. State the environmental effects of extracting & using mineral resources.
4. State the need for public awareness for solving environmental problems.
5. What is geothermal energy?
6. State any two problems caused by the construction of dams.
7. Write any two functions of forest.
8. What are the advantages of conjunctive uses of water?
9. Define environmental impact statement.
10. What is soil fertility?
11. Explain soil leaching.
12. What is soil erosion? How can it be controlled?
13. Define desertification.
14. Define Overgrazing. What are the effects of overgrazing?
15. Distinguish between water logging & Salinity.

PART – B (16 marks)

1. Explain Renewable & non-Renewable energy resources with examples. (16)

2. (a) Compare nuclear power with coal. (6) (b) Write a note on energy conservation. (4)
© What is Solar Space heating? Explain (6)

3. (a) What are the causes of soil erosion & deforestation? Explain in detail (8)
(b) Discuss the consequences of overdrawing surface and Ground water. (8)

4. a)Write briefly on any four alternate source of energy (8)
b) Explain effects of construction of dam on tribal people (8)

5. (a) Write the effects of extracting and using mineral resources. (4)
(b) Explain the scope & importance of Environment. (8)
© Enumerate the benefits & draw backs of building of dams. (4)

6. (a) What are the effects of modern Agriculture? (8)
(b) Discuss the causes of land degradation. (8)

UNIT – II

PART – A (2 MARKS)

1. Define Producers.
2. Name the four ecosystems.
3. What is Ecological succession?
4. What are food chain & food Webs?
5. What is the classification of biotic component of ecosystems?
6. Define Biodiversity.
7. What are the advantages& disadvantages of in-situ& ex-situ conservation of biodiversity?
8. Enumerate the human activities which destroy the biodiversity.
9. Write note on endemic species.
10. Define Biosphere.
11. Define consumers & Classify them.
12. What is Biomes? Give Examples.
13. Define Estuaries.
14. Define poaching.
15. Define Species & Genetic diversity.

PART – B (16 MARKS)

1. (a) What is an ecosystem? Describe the structure & function of various components of an ecosystem. (10)
(b) Explain the various threats to biodiversity. (6)

2. (a) Define Ecological pyramids and explain different types of ecological pyramids. (6)
(b) Describe the types, characteristics features, structure& function of
i). Forest ecosystem ii). Aquatic ecosystem (10)

3. (a) Explain In-Situ & Ex-Situ Conservation of Biodiversity (8)
(b) What is meant by value of biodiversity? Explain different values of biodiversity (8)

4. Write briefly on
1. Ecological succession
2. Energy flow through an ecosystem
3. Hot spots of biodiversity ( 4+4+8)

5. (a) What is the biodiversity what are the reasons for decline for biodiversity (10)
(b) Write short notes on: (1) Producers (2) Consumers (3) Decomposers (3 X 2 = 6)

UNIT – III

PART – A (2 MARKS)

1. Define pollution. 
2. Name any four air pollutants, and their sources and impacts.
3. What are point and non-point sources of water pollution?
4. Define hazardous wastes.
5. Write any four major water pollutants.
6. Define photochemical smog.
7. Define floods.
8. Define disasters
9. Write any two causes of soil pollution.
10. Define noise pollution.
11. When a sound causes noise pollution.
12. What are the types of solid waste?

PART – B (16 MARKS)

1 (a).Discuss the major air pollutants and their impacts. (8)
(b) Explain the various methods of controlling air pollution. (8)

2. (a) Discuss briefly the disposal of Municipal solid waste management. (8)
(b) Explain the causes, effects and control measures of water pollution. (8)

3. (a) Discuss major air pollutants and their impacts. (6)
(b) What is Thermal pollution and explain their impacts? (6).
©What are the sources of radioactive pollution? (4)

4. (a) Explain in detail the role of individual in conservation of natural resources. (8)
(b).Explain the effects nuclear and Radiation pollution. (8)

5. (a) Write a brief note on of thermal pollution. (6)
(b) Discuss the causes, effects & control measures of marine pollution. (10)

UNIT – IV

PART – A (2 MARKS)

1. What are the causes and effects of global warming?
2. What are causes and effects of ozone layer depletion?
3. What is acid rain?
4. Explain the term sustainable development.
5. Define resettlement and rehabilitation.
6. Write note on climate change.
7. Explain green house effect.
8. Write reason for wetland reclamation.
9. Write any four effects of climate change.
10. What is watershed management?

PART – B (16 MARKS)

1. (a) Explain forest conservation act.. (8)
(b) Write the factors influence the unsustainable to sustainable development. . (8)

2. (a) Explain nuclear accidents and holocaust. . (8)
(b) Write notes on global warming. . (8)

3. (a) Write a note on watershed management. (4)
(b) Write briefly Bhopal disaster and Chernobyl disaster. (8)
© Discuss briefly on environment protection act 1986. (4)

4. (a) Discuss the agenda for sustainable development (8)
(b) Write briefly on nuclear accidents (8)

5. Write briefly on
a. Green House Effect (4) b. Acid Rain (4) c. Rain water Harvesting (6) d. Climate Change (2)

UNIT – V

PART – A (2 MARKS)

1. Define Population explosion
2. Define ZPG (Zero Population Growth)
3. Briefly account on human rights.
4. Define Population equilibrium
5. Explain total fertility rate
6. Define population equation
7. State how environment and human health are related
8. State the role of information technology in environment.
9. List the problems of population growth.
10. What is AIDS? How to prevent it?

PART – B (16 MARKS)

1. (a) Explain the role of IT in environment and human health. (12)
(b) Write short note on Value Education (4)

2. (a) Write briefly on implementation of family planning programme. (6)
(b) Write a note on AIDS in developing country (4)
© Discuss the factors influencing the family size (6)

3. (a) Population explosion affects the environment seriously – discuss. (8) (b) Deterioration of environment leads to deterioration of human health – Justify(8)

4. (a) Write a note on the following in relation to human population & environment
i). Woman Child Welfare. (4)
ii). Human Rights. (4)

(b) Write briefly on:
i) Population momentum. (4)
ii) Population Profile. (4)

5. (a) Discuss the growth of population can be successfully implemented . (6)
(b) Discuss the factors influencing the family size. (4)
© Discuss on the steps taken in India to prevent AIDS. (6)

COMPUTER GRAPHICS CS2401 ANNA UNIVERSITY QUESTION PAPER |CS2401 CG NOVEMBER/DECEMBER 2011 PREVIOUS YEAR QUESTION PAPER

B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2011.

Seventh Semester

Computer Science and Engineering

CS 2401 — COMPUTER GRAPHICS

(Common to Information Technology)

(Regulation 2008)

Time : Three hours Maximum : 100 marks

Answer ALL questions.

PART A — (10 × 2 = 20 marks)

1. Write down any two line attributes.
2. Differentiate window and viewport.
3. What are spline curves?
4. Define quadric surfaces.
5. What is animation?
6. Define keyframes.
7. What do you mean by shading of objects?
8. What is texture?
9. Define fractals.
10. Differentiate Mandelbrot and Julia sets.

PART B — (5 × 16 = 80 marks)

11. (a) Write down and explain the midpoint circle drawing algorithm. Assume 10 cm as the radius and co-ordinate origin as the centre of the circle.

Or

(b) Explain in detail the Cohen-Sutherland line clipping algorithm with an example.
12. (a) Differentiate parallel and perspective projections and derive their projection matrices.

Or

(b) With suitable examples, explain all 3D transformations.
13. (a) Write notes on RGB and HSV color models.

Or

(b) Discuss the following:
(i) Methods to draw 3D objects. (8)
(ii) Basic OPENGL operations. (8)
14. (a) Explain the following:
(i) Adding texture to faces. (8)
(ii) Adding shadows of objects. (8)

Or

(b) Write down and explain the details to build a camera in a program.
15. (a) Write notes on Peano curves.

Or

(b) Write about random fractals in detail.
—————————

DATABASE MANAGEMENT SYSTEMS CS2255 MODEL QUESTION PAPER | CS 2255 DBMS QUESTION PAPER NOVEMBER/DECEMBER 2011

B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2011.

Fourth Semester

Computer Science and Engineering

CS 2255 — DATABASE MANAGEMENT SYSTEMS

(Common to Information Technology)

(Regulation 2008)

Time : Three hours                                                             Maximum : 100 marks

Answer ALL questions.

PART A — (10 × 2 = 20 marks)

1. What is a data model?
2. With an example explain what a derived attribute is?
3. Consider the following relation :
EMP (ENO, NAME, DATE_OF_BIRTH, SEX, DATE_OF_JOINING, BASIC_PAY, DEPT) Develop an SQL query that will find and display the average BASIC_PAY in each DEPT.
4. List the two types of embedded SQL SELECT statements.
5. Consider the following relation :
R (A, B, C, D, E)
The primary key of the relation is AB. The following functional
dependencies hold :
A →C
B →D
AB → E 
Is the above relation in second normal form?
6. Consider the following relation : R(A, B, C, D)
The primary key of the relation is A. The following functional dependencies
hold :
A →B,C 
B →D 
Is the above relation in third normal form?
7. List the two commonly used Concurrency Control techniques.
8. List the SQL statements used for transaction control.
9. What are ordered indices?
10. Distinguish between sparse index and dense index.

PART B — (5 × 16 = 80 marks)

11. (a) (i) Construct an E-R diagram for a car-insurance company whose customers own one or more cars each. Each car has associated with it zero to any number of recorded accidents. State any assumptions you make. (6)
(ii) A university registrar’s office maintains data about the following entities :
(1) Courses, including number, title, credits, syllabus, and prerequisites;
(2) Course offerings, including course number, year,
semester, section number, instructor, timings, and classroom;
(3) Students, including student-id, name, and program; and
(4) Instructors, including identification number, name,
department, and title. Further, the enrollment of students in courses and grades awarded to students in each course they are enrolled for must be appropriately modeled. Construct an E-R diagram for the registrar’s office. Document all assumptions that you make about the mapping constraints. (10)

Or

(b) (i) With a neat sketch discuss the three-schema architecture of a DBMS. (8)
(ii) What is aggregation in an ER model? Develop an ER diagram using aggregation that captures the following information : 
Employees work for projects. An employee working for a particular project uses various machinery. Assume necessary attributes. State any assumptions you make. Also discuss about the ER diagram you have designed. (2 + 6)
12. (a) (i) Explain the distinctions among the terms primary key, candidate key, and super key. Give relevant examples. (6)
(ii) What is referential integrity? Give relevant example. (4)
(iii) Consider the following six relations for an Order-processing Database Application in a Company :
CUSTOMER (CUSTNO, CNAME, CITY)
ORDER (ORDERNO, ODATE, CUSTNO, ORD_AMT)
ORDER_ITEM (ORDERNO, ITEMNO, QTY)
ITEM (ITEMNO, ITEM_NAME, UNIT_PRICE)
SHIPMENT (ORDERNO, ITEMNO, WAREHOUSENO,
SHIP_DATE)
WAREHOUSE (WAREHOUSENO, CITY)
Here, ORD_AMT refers to total amount of an order; ODATE is the date the order was placed; SHIP_DATE is the date an order is shipped from the warehouse. Assume that an order can be shipped from several warehouses. Specify the foreign keys for
this schema, stating any assumptions you make. (6)

Or

(b) With relevant examples discuss the various operations in Relational Algebra. (16)
13. (a) Define a functional dependency. List and discuss the six inference rules for functional dependencies. Give relevant examples. (16)
Or
(b) (i) Give a set of Functional dependencies for the relation schema R(A,B,C,D,E) with primary key AB under which R is in 2NF but not in 3NF. (5)
(ii) Prove that any relation schema with two attributes is in BCNF.(5)
(iii) Consider a relation R that has three attributes ABC. It is decomposed into relations R1 with attributes AB and R2 with attributes BC. State the definition of lossless-join decomposition with respect to this example. Answer this question concisely by
writing a relational algebra equation involving R, R1, and R2. (6)
14. (a) (i) Define a transaction. Then discuss the following with relevant examples : (8)
(1) A read only transaction
(2) A read write transaction
(3) An aborted transaction
(ii) With a neat sketch discuss the states a transaction can be in. (4)
(iii) Explain the distinction between the terms serial schedule and serializable schedule. Give relevant example. (4)

Or

(b) (i) Discuss the ACID properties of a transaction. Give relevant example. (8)
(ii) Discuss two phase locking protocol. Give relevant example. (8)
15. (a) (i) When is it preferable to use a dense index rather than a sparse index? Explain your answer. (4)
(ii) Since indices speed query processing, why might they not be kept on several search keys? List as many reasons as possible.(6)
(iii) Explain the distinction between closed and open hashing. Discuss the relative merits of each technique in database applications. (6)

Or

(b) Diagrammatically illustrate and discuss the steps involved in processing a query.(16)

SOFTWARE ENGINEERING AND QUALITY ASSURANCE IT2251 ANNA UNIVERSITY MODEL QUESTION PAPER | SEQA IT 2251 PREVIOUS YEAR QUESTION PAPER NOVEMBER/DECEMBER 2011

B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2011.

Fourth Semester

Information Technology

IT 2251 – SOFTWARE ENGINEERING AND QUALITY ASSURANCE

(Regulation 2008)

Time : Three hours Maximum : 100 marks

Answer ALL questions.

PART A — (10 × 2 = 20 marks)

1. What is meant by Independent Verification and Validation?
2. Differentiate Waterfall model with V-Model.
3. Name the components of CASE tools for structured methodology.
4. List the steps involved in requirements elicitation and analysis.
5. What is meant by heuristic evaluation?
6. List out the steps involved in real time software design.
7. What is meant by Boundary value analysis?
8. Define the term process maturity.
9. Name any four quality control tools.
10. State the benefits of QFD.

PART B — (5 × 16 = 80 marks)

11. (a) (i) Describe the process model which defines a network of activities. (8)
(ii) State as to why the first system is always a throw away system? Explain the concept with advantages and disadvantages. (8)

Or

(b) (i) Draw a system engineering hierarchy diagram and explain the concept. (8)
(ii) Explain the process model that combines the elements of waterfall and iterative fashion. (8)
12. (a) Give a brief description of software prototyping and briefly discuss the various prototyping techniques. (16)

Or

(b) With an example describe the three process steps for transforming a dataflow diagram to a structure chart. (16)
13. (a) (i) Draw a translating diagram for analysis model into a software design Specification. (8)
(ii) Give a complete template for documentation design specification. (8)

Or

(b) (i) Which is a measure of interconnection among modules in a program structure?Explain. (8)
(ii) What is the difference between Level-0 and Level-1 DFD? Draw a Level-0 and Level-1 DFD for safe Home Security System. (8)
14. (a) (i) What are all the formulas for cyclomatic complexity? Calculate cyclomatic Complexity for greatest of three numbers. (8)
(ii) How would you derive test cases for the given project? Explain in detail. (8)

Or

(b) (i) Narrate the path testing procedure in detail with a sample code.(8)
(ii) Explain the different integration testing approaches. (8)
15. (a) Explain how software process assessment helps software organizations to improve themselves. (16)
Or
(b) Write detailed notes on ISO9000 series of quality management standards. (16)

DESIGN AND ANALYSIS OF ALGORITHMS CS2251 ANNA UNIVERSITY PREVIOUS YEAR QUESTION PAPER | CS 2251 DAA NOVEMBER/DECEMBER 2011 QUESTION PAPER

B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2011.

Fourth Semester

Computer Science and Engineering

CS 2251 — DESIGN AND ANALYSIS OF ALGORITHMS

(Regulation 2008)

Time : Three hours Maximum : 100 marks

Answer ALL questions.

PART A — (10 × 2 = 20 marks)

1. What do you mean by linear search?
2. What is the properties of big-Oh notation.
3. What greedy algorithms?
4. What Knapsack problem?
5. What is traveling salesperson problem?
6. What do you mean by multistage graphs?
7. State the general backtracking method?
8. What is graph cloning?
9. What is spanning tree? Give an example.
10. What is NP Completeness?

PART B — (5 × 16 = 80 marks)

11. (a) (i) Define Asymptotic notations. Distinguish between Asymptotic notation and conditional asymptotic notation. (10)
(ii) Explain how the removing condition is done from the conditional Asymptotic notation with an example. (6)

Or

(b) (i) Explain how analysis of linear search is done with a suitable illustration. (10)
(ii) Define recurrence equation and explain how solving recurrence equations are done. (6)
12. (a) What is divide and conquer strategy and explain the binary search with suitable example problem.

Or

(b) Distinguish between Quick sort and Merge sort, and arrange the following numbers in increasing order using merge sort. (18, 29, 68, 32, 43,37, 87, 24, 47, 50)
13. (a) (i) Explain the multistage graph problem with an example. (8) 
          (ii) Find an optimal solution to the knapsack instance n = 7, m= 15 (p1, p2, p3, ….p7) = (10, 5, 15, 7, 6, 18, 3) and (w1, w2, w3, ... w7) (2, 3, 5, 7, 1, 4, 1) (8)

Or

(b) Describe binary search tree with three traversal patterns? Give suitable example with neat diagram for all three traversal of binary search tree.. (16)
14. (a) (i) How does backtracking work on the 8 Queens problem with suitable example? (8)
(ii) Explain elaborately recursive backtracking algorithm? (8)

Or

(b) What is Hamiltonian problem? Explain with an example using backtracking. (16)


15. (a) Write a complete LC branch-and-bound algorithm for the job sequencing with deadlines problem. Use the fixed tuple size formulation. (16)

Or

(b) Write a non-deterministic algorithm to find whether a given graph contains a Hamiltonian cycle. (16)

GE2211 ENVIRONMENTAL SCIENCE AND ENGINEERING EVS Anna University Question paper for Nov/Dec 2009

B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2009

Third Semester

Electrical and Electronics Engineering

GE 2211 — ENVIRONMENTAL SCIENCE AND ENGINEERING

(Regulation 2008)

(Common to Chemical Engineering / Textile Technology / Instrumentation and

Control Engineering / BioTechnology / Rubber and Plastics / Electronics and

Instrumentation Engineering / Polymer Technology / Textile Technology

(Fashion Technology))

Time : Three hours Maximum : 100 Marks

Answer ALL Questions

PART A — (10 × 2 = 20 Marks)

1. What are renewable and non-renewable energy sources? Give examples.
2. Mention the causes of desertification.
3. What are food chains and food webs?
4. Define primary production and secondary production.
5. What is meant by thermal pollution?
6. Mention the sources of solid wastes.
7. What is meant by sustainable development?
8. Mention the causes of ozone layer depletion.
9. What is meant by population explosion?
10. List the objectives of watershed management.

PART B — (5 × 16 = 80 Marks)

11. (a) (i) Explain the major causes and consequences of deforestation. (8)
(ii) Discuss the various techniques for harnessing solar energy. (8)
Or


(b) (i) Explain the environmental impacts of mineral extraction and use.(8)
(ii) Discuss the impact of modern agriculture on the environment. (8)
12. (a) (i) Discuss the major features of grass land ecosystem. (8)
(ii) Write a note on endangered and endemic species of India. (8)
Or
(b) (i) Describe the structural features of ecosystem. (8)
(ii) Write a note on conservation of biodiversity. (8)
13. (a) (i) Enumerate various methods for control of air pollution. (8)
(ii) Discuss the causes, effects and control of marine pollution. (8)
Or
(b) (i) Describe the sources, effects and control of noise pollution. (8)
(ii) Discuss the role of an individual in preventing pollution. (8)
14. (a) (i) Describe the measures to conserve water. (8)
(ii) Write a note on wasteland reclamation. (8)
Or
(b) (i) What is meant by acid rain? How does it form? What are the major
impacts? (8)
(ii) Discuss the causes and effects of global warming. (8)
15. (a) (i) Discuss the influence of environmental parameters on human
health. (8)
(ii) Narrate the role of information technology in environmental
management. (8)
Or
(b) (i) Discuss the various issues and measures for women and child
welfare. (8)
(ii) Explain the objectives and elements of value education. (8)

ENVIRONMENTAL SCIENCE AND ENGINEERING GE2211 Anna University Previous Year Question Paper Nov/Dec 2010 for CSE Students

B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2010

Third Semester

Electrical and Electronics Engineering

GE 2211 — ENVIRONMENTAL SCIENCE AND ENGINEERING

(Common to Chemical Engineering, Textile Technology, Instrumentation and

Control Engineering, Biotechnology, Plastic Technology, Electronics and

Instrumentation Engineering, Polymer Technology and

Textile Technology (Fashion Technology))

(Regulation 2008)

Time : Three hours Maximum : 100 Marks

Answer ALL questions

PART A — (10 × 2 = 20 Marks)

1. What are food chains? Mention their types.
2. What are the abiotic components of an ecosystem?
3. Give a comprehensive definition for air pollution.
4. Mention four causes of floods.
5. What are the present food problems of the world?
6. What are the effects of over utilization of ground water?
7. Define sustainable development.
8. What are the main causes of global climatic changes?
9. What is ‘Crude birth rate’ and ‘total fertility rate’?
10. Mention any four fundamental rights of the individuals.

PART B — (5 × 16 = 80 Marks)

11. (a) (i) Explain the features of ecological pyramids including inverted
pyramids. (8)
(ii) What are the threats to biodiversity? Discuss. (8)
Or
(b) (i) Explain the components, structure and functions of a desert
ecosystem. (8)
(ii) What are the values of biodiversities? Explain. (8)


12. (a) (i) Elaborate the various ways of controlling the noise pollution. (8)
(ii) Define and give two example each for gaseous air pollutant
suspended particulate matter, photochemical oxidant, and
hazardous pollutant. (8)
Or
(b) (i) Discuss the effects and control of thermal pollution. (8)
(ii) List the role of an individual in prevention of pollution. (8)
13. (a) (i) What are the ecological services rendered by forests? Discuss. (8)
(ii) Discuss the different ways of harnessing solar energy? (8)
Or
(b) (i) Discuss any four factors responsible for land degradation (8)
(ii) Discuss the problems of fertilizers and pesticides in modern
agriculture. (8)
14. (a) (i) How can we attain sustainable development? Explain. (8)
(ii) Mention the salient features of The Wildlife Protection Act, 1972
and Forest Conservation Act, 1986. (8)
Or
(b) (i) How can we make energy resources sustainable? Explain. (8)
(ii) What are the components of watershed management? Discuss. (8)
15. (a) (i) List the reasons for high population growth in India. (8)
(ii) Discuss the role of information technology in environment. (8)
Or
(b) (i) List various women welfare programmes of India. (8)
(ii) Discuss the variation age structure of population among nation
using Age - pyramids. (8)

MA2161 MATHEMATICS – II ANNA UNIVERSITY QUESTION PAPER QUESTION BANK IMPORTANT QUESTIONS 2 MARKS AND 16 MARKS

B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2009

Second Semester

Civil Engineering

MA 2161 — MATHEMATICS – II

(Regulation 2008)

(Common to all branches)

Time : Three hours Maximum : 100 Marks

Answer ALL Questions

PART A — (10 × 2 = 20 Marks)

1. Find the particular integral of ( ) x y D 2 sin 4 2 = + .
2. Convert ( )
x
y xD D x
2
7 2 2 = + + into an equation with constant coefficients.
3. Define solenoidal and irrotational vector point functions.
4. State the Gauss divergence theorem.
5. Show that the families of curves θ α n rn sec = and θ β n rn ec cos = intersect
orthogonally where α and β are arbitrary constants.
6. Find the critical points of the transformation z w sin = .
7. State Cauchy’s integral theorem.
8. Find the residue of ( )
z z z
z
z f
2 2
4
2 3
2
+ +
+
= at 0 = z .
9. Find the Laplace transform of ( )
t
t
t f
2 1 +
= .
10. Find the Laplace transform of the unit step function ( ) a t u − .


PART B — (5 × 16 = 80 Marks)
11. (a) (i) Solve x e x y
dx
dy
dx
y d x 2 sin 8 4 4 2 2
2
2
= + − . (8)
(ii) Solve t y x
dt
dx
5 3 2 = − + ; t e y x
dt
dy 2 2 2 3 = + − . (8)
Or
(b) (i) Solve ( ) x x y
dx
dy
x
dx
y d
x log sin 4 2
2
2
2 = + − . (8)
(ii) Solve by the method of variation of parameters
x e y
dx
dy
dx
y d − = − − 25 3 2
2
2
. (8)
12. (a) (i) Find the values of the constants c b a , , so that
( ) ( ) ( )k y xz j cz x i bz axy F − + − + + = 2 2 3 3 3 may be irrotational. For
these values of c b a , , find also the scalar potential of F . (8)
(ii) Verify Stoke’s theorem for k xz j yz i xy F − − = 2 , where S is the
open surface of the rectangular parallelepiped formed by the planes
0 = x , 1 = x , 0 = y , 2 = y and 3 = z above the XoY plane. (8)
Or
(b) (i) Find the angle between the surfaces 11 2 2 2 = − − z y x and
18 = − + xz yz xy at the point ( ) 3 , 4 , 6 . (8)
(ii) Verify Gauss Divergence theorem for k yz j z i x F + + = 2 , where S is
the surface of the cube formed by the planes 1 − = x , 1 = x , 1 − = y ,
1 = y , 1 − = z and 1 = z . (8)
13. (a) (i) Show that an analytic function with
(1) constant real part is a constant and
(2) constant modulus is a constant. (8)
(ii) Discuss the transformation
z
w
1 = . (8)
Or


(b) (i) Prove that ( ) ( ) [ ] 2 2 2 1 log − + − = y x v is harmonic in every region
which does not include that point ( ) 2 , 1 . Find the corresponding
analytic function and real part. (8)
(ii) Find the bilinear map which maps the points 1 , , 1 − = i z onto the
points i i w − = , 0 , . Also find the image of the interior region of the
unit circle of the z plane. (8)
14. (a) (i) Evaluate ( ) ∫ + C z
dz
2 2 4
, where C is the circle 2 = − i z , by Cauchy’s
integral formula. (8)
(ii) Evaluate θ
θ
θ π
d
a a ∫ + −
2
0
2 cos 2 1
2 cos
, using contour integration, where
1 2 < a . (8)
Or
(b) (i) Show that
2
5
9 10
2
2 4
2 π
=
+ +
+ − ∫
∞
∞ −
dx
x x
x x
. (8)
(ii) Find the residues of ( )
( ) ( )2
2
2 1 + −
=
z z
z
z f at its isolated
singularities using Laurent series expansion. (8)
15. (a) (i) Evaluate ( ) 


 

+ +
+ + −
13 4
26 16 3
2
2
1
s s s
s s
L . (8)
(ii) Using Laplace transforms solve t e y y y 2 8 4 = + ′ − ′ ′ , ( ) 2 0 = y and
( ) 2 0 − = ′ y . (8)
Or
(b) (i) Use convolution theorem to find ( ) 



 

+
−
2 2 2
1
a s
s
L . (4)
(ii) Find the Laplace transform of ( )
p
kt
t f = , for p t < < 0 and
( ) ( ) t f t p f = + . (4)
(iii) Using Laplace transforms solve t t y y 2 2 + = ′ + ′ ′ , ( ) 4 0 = y and
( ) 2 0 − = ′ y . (8)