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V.S.B. ENGINEERING COLLEGE, KARUR
Department of Computer Science and Engineering
Academic Year: 2017-2018 (ODD Semester)
Class/Year/Semester: CSE ‘A’ & ‘B’ /II/III
Question Banks
Analog and Digital Communication
UNIT- I
FUNDAMENTALS OF ANALOG COMMUNICATION
PART-A
1.What is modulation?
2.Define amplitude Modulation.
3.Define Modulation index and percent modulation for an AM wave.
4. Give the bandwidth of AM?
5.Draw the spectrum of AM signal.
6.Give the expression for modulation index in terms of Vmax and Vmin.
7.Give the formula for AM power distribution.
8.Give the expression for total current.
9.Define Single sideband suppressed carrier AM.
10.Define AM Vestigial sideband.
11.What are the advantages of single sideband transmission?
12.What are the disadvantages of single side band transmission?
13.What is the advantage of low-level modulation?
14.Define Low-level Modulation.
15.Define High-level Modulation.
16.What is the advantage of low-level modulation?
17.Define image frequency.
18.Define image frequency rejection ratio.
19.Define Heterodyning.
20.Define direct frequency modulation.
21.Define indirect frequency Modulation.
22.Define instantaneous frequency deviation.
23.Define frequency deviation.
24.State Carson rule.
25.Define Deviation ratio.
26.Write down the comparison of frequency and amplitude modulation.
27.Define Phase modulation.
28.What are the advantages of angle modulation and also list its disadvantages.
29.What is Phase deviation ?
30.Give the expression for bandwidth of angle-modulated wave in terms of Bessel‟s table.
31.Define deviation sensitivity for FM and PM and give its units.
32.If a modulated wave with an average voltage of 20Vp changes in amplitude ±5V,
33.An FM transmitter has a rest frequency fc =96MHz and a deviation sensitivity K1 = 4
34.For an FM receiver with an input frequency deviation ∆f=4 kHz and a transfer ratio K= 0.01 V/k
Hz, determine Vout.
UNIT – II
DIGITAL COMMUNICATION
PART-A
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1.What is digital modulation?
2.What is information capacity?
3.Give the expression for Shannon limit for information capacity.
4.Give the Nyquist formulation for channel capacity.
5.Compare QASK and QPSK.
6.What are Antipodal signals?
7.Define minimum shift keying.
8.Give the difference between standard FSK and MSK.
9.What are the advantages of M-ary signaling scheme?
10.What does correlative coding mean?
11.Differentiate coherent and noncoherent methods.
12.Define peak frequency deviation for FSK.
13.Define bit rate.
14.Define Baud rate.
15.Compare binary PSK with QPSK.
16.Define QAM.
17.What is a constellation diagram?
18.Bring out the difference between DPSK and BPSK.
19.What is bandwidth efficiency?
20.What is an Offset QPSK?
21.Mention any four advantage of digital modulation over analog modulation.
22.Define carrier recovery.
23.What is DPSK?
24.What do you mean by ASK?
UNIT – III DIGITAL TRANSMISSION
PART-A
1.State the sampling theorem for band-limited signals of finite energy.
2.What are the advantages of digital transmission?
3.What are the disadvantages of digital transmission?
4.Define pulse code modulation.
5.What is the purpose of the sample and hold circuit?
6.What is the Nyquist sampling rate?
7.What is the principle of pulse modulation?
8.List the four predominant methods of pulse modulation.
9.What is codec?
10.Define and state the causes of fold over distortion.
11.Define overload distortion.
12.Define quantization.
13.Define dynamic range.
14.What is non uniform or nonlinear encoding?
15.What is the advantage and disadvantage of midtread quantization?
16.What is the necessity of companding?
17.What is idle channel noise?
28.How is percentage error calculated?
29.Compare slope overload and granular noise.
30.Give the concept of delta modulation PCM.
31.What is ISI and give its causes.
32.What is an eye pattern?
33.List the significance of eye pattern.
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34.What are the two fold effects of quantizing process?
35.Define quantization error?
36.What is nyquist rate?
37.What is PAM?
38.What do you mean by slope overload distortion in delta modulation?
UNIT – IV DATA COMMUNICATIONS
PART A
1. Define data communication codes.
2. Define error detection.
3. Define Echoplex.
4. Define serial interface.
5. Define parallel interface.
6. What are the advantages of parallel transmission?
7. What is the purpose of data modem?
8. Classify data modems.
9. Define OSI.
10. What are the advantages of bus topology?
11. What are the disadvantages of star topology?
12. Describe LAN.
13. What are the error detection and error correcting codes?
14. Write the difference between serial and parallel interface?
15. What is meant by RS 232 cables?
16. Define Asynchronus modem.
17. Mention the classifications of modem.
18. Define Synchronous Modem.
19. What is forward Error Correction?
20. What is use of CRC Codes?
UNIT- V SPREAD SPECTRUM AND MULTIPLE ACCESS TECHNIQUES
PART A
1. Define spread spectrum.
2. What do you mean by direct sequence spread technique?
3. What are the advantages of spread spectrum modulation?
4. What is frequency hop spreading?
5. What are the applications of Spread spectrum modulation?
6. List the advantages of direct sequence systems
7. List the disadvantages of direct sequence systems
8. List the advantages of frequency hopping systems
9. List the disadvantages of frequency hopping systems
10. Define slow frequency hopping
11. Define fast frequency hopping
12. What is processing gain?
13. What are the properties of maximum length sequence?
14. Mention the classification of multiple access protocols.
15. Compare slow and fast frequency hopping.
16. What is TDMA?
17. What does CEPT stand for?
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18. Give the expression for probability of gain.
19. What are orthogonal codes?
20. What are the two different techniques used in speech coding for wireless communication?
21. What are the two function of fast frequency hopping?
22. What are the features of code Division multiple Accesses?
PART B
UNIT I
1. For an AM DSBFC wave with unmodulated carrier voltage of 10Vp and load resistance of
10 ohms, and m=1determine (i) Power in carrier,(ii)Power in upper and lower
sidebands(iii)Total transmitted power.
2. Derive the expression for total power in an AM DSBFC and draw the power spectrum.
3. For an AM DSBFC transmitter with an unmodulated carrier power Pc=100w that is
simultaneously modulated by 3 modulating signals with co-efficient of modulation m1=0.2,
m2=0.4 and m3=0.5, determine, Total coefficient of modulation, Upper and lower sideband
power, Total sideband power, Total transmitted power and then, Draw the output spectrum.
4. One input to an AM DSBFC modulator is 500 kHz carrier with peak amplitude of 20Vp. The
second input is a 10 kHz modulating signal whose amplitude is sufficient to produce a ±
7.5Vp change in the amplitude of the envelope. Determine the following (i) Upper and lower
side frequency,(ii)Modulation co-efficient and percent modulation.,(iii)Maximum and
minimum amplitude of the envelope,(iv)Draw the output envelope,(v)Draw the output
frequency spectrum.
5. Derive Eusf and Elsf in terms of Vmax and Vmin .
6. Derive the output expression for an AM DSBFC and also draw the AM spectrum.
7. Explain in detail about the Bandwidth requirements of angle-modulated wave.
8. Compare PM and FM.
9. Determine the side band frequencies of an angle-modulated wave.
10. Derive the expression for average power of an angle-modulated wave
UNIT II
1. Explain FSK bit rate, baud, bandwidth and modulation index.
2. Explain on-off keying (OOK) or ASK.
3. Explain QPSK transmitter and receiver.
4. Explain DPSK with an example.
5. Explain BPSK (transmitter and receiver) and also discuss about the bandwidth.
6. Discuss the operation of 16-QAM transmitters and receivers.
7. Explain in detail about carrier recovery.
UNIT III
1. Explain PCM with a neat block diagram.
2. Explain PCM sampling with necessary diagrams and circuits. Write a note on aliasing.
3. What is companding? Explain in detail Analog and digital companding.
4. With a neat block diagram explain Delta modulation. How slope over and granular noise can
be minimized.
5. What is the advantage of DPCM? With a neat block diagram explain transmitter and
receiver.
6. Write notes on ISI and eye pattern.
UNIT IV
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1. Describe in detail about the standards and organization of data communication protocols.
2. With neat diagram explain about the operation of Serial interface.
3. Write an essay about Serial and Parallel communication.
4. Explain the operation of Asynchronous and synchronous modem.
5. Describe the features and purposes of serial Interfaces.
6. Describe the mechanical, electrical and functional characteristics of
RS 232 interface.
7. Explain how vertical, longitudinal and cyclic redundancy checking is
used for detecting the occurrence of errors in data transmission.
8. Write a note on data communication codes.
9. Explain in detail about error detection and correction.
10. Write a note on medium and high speed modem.
11. Explain in detail about serial and parallel interfaces.
UNIT V
1. Describe what a reference burst is for TDMA and explain the following terms: preamble,
carrier recovery sequence, bit timing recovery, unique word and correlation spike.
2. Describe the operation of CEPT primary multiplex frame.
3. Explain in detail the notion of spread spectrum.
4. Explain CDMA and also give the orthogonal condition of the signals in CDMA.
5. Explain direct sequence-spread spectrum with appropriate waveforms and expressions.
6. Explain the generation of pseudo-noise sequence with an example.
7. List the properties of maximum length sequence with examples.
8. With the help of transmitter and receiver diagram explain frequency hopped spread
spectrum and discuss about slow –frequency hop and Fast-frequency hop.
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Computer Architecture
UNIT -I
1. What are the eight great ideas in computer architecture?
2. What are the five classic components of a computer?
3. Define – ISA.
4. Define – ABI.
5. What are the advantages of network computers?
6. Define – Response Time-
7. Define – Throughput.
8. Write the CPU performance equation.
9. If computer A runs a program in 10 seconds, and computer B runs the same program in 15 seconds, how much
faster is A over B.
10. What are the basic components of performance?
11. Write the formula for CPU execution time for a program.
12. Write the formula for CPU clock cycles required for a program.
13. Define – MIPS
14. What are the fields in an MIPS instruction?
15. Write an example for immediate operand.
16. Define – Stored Program Concepts.
17. Define – Addressing Modes.
18. Give the basic performance equation.
19. States the Amdahl‟s Law.
20. Define – Moore‟s Law
21. List the major components of a computer system. (APRIL/MAY 2017)
UNIT -II ARITHMETIC OPERATIONS
1. Add 610 to 710 in binary and Subtract 610 from 710 in binary.
2. Write the overflow conditions for addition and subtraction.
3. What are the floating-point instructions in MIPS?
4. Define – Guard and Round
5. Define – ULP
6. What is meant by sticky bit?
7. Write the IEEE 754 floating-point format.
8. What is meant by sub-word parallelism?
9. 10. Multiply 100010 × 100110
10. What are the steps in the floating-point addition?
11. What are the main features of Booth‟s algorithm?
12. How can we speed up the multiplication process?
13. What is guard bit?
14. What are the ways to truncate the guard bits?
15. Define carry save addition (CSA) process.
16. In floating point numbers when so you say that an underflow or overflow has occurred?
17. What is an n-bit ripple carry adder?
18. Write the Add/subtract rule for floating point numbers.
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19. When can you say that a number is normalized?
UNIT -III PROCESSOR & CONTROL UNIT
1. What is meant by data path element?
2. What is the use of PC register?
3. What is meant by register file?
4. Draw the diagram of portion of data path used for fetching instruction.
5. Define – Sign Extend
6. What is meant by branch target address?
7. Differentiate branch taken from branch not taken.
8. What is meant by delayed branch?
9. What are the three instruction classes and their instruction formats?
10. Write the instruction format for the jump instruction.
11. What is meant by pipelining?
12. What are the five steps in MIPS instruction execution?
13 . Write the formula for calculating time between instructions in a pipelined processor.
14. What are hazards? Write its types. (APRIL/MAY 2017)
15. What is meant by forwarding?
16. What is pipeline stall?
17. What is meant by branch prediction?
18. What are the 5 pipeline stages?
19. What are exceptions and interrupts?
20. Define – Vectored Interrupts
UNIT -IV PARALLELISM
1. What is meant by ILP? (APRIL/MAY 2017)
2. What is multiple issue? Write any two approaches.
3. What is meant by speculation?
4. Define – Static Multiple Issue
5. Define – Issue Slots and Issue Packet
6. Define – VLIW
7. Define – Superscalar Processor
8. What is meant by loop unrolling?
9. What is meant by anti-dependence? How is it removed?
10. What is the use of reservation station and reorder buffer?
11. Differentiate in-order execution from out-of-order execution.
12. What is meant by hardware multithreading?
13. What are the two main approaches to hardware multithreading?
14. What is SMT?
15. Differentiate SMT from hardware multithreading.
16. What are the three multithreading options?
17. Define – SMP.
18. Differentiate UMA from NUMA.
UNIT -V MEMORY AND I/O SYSTEMS
1. What are the temporal and spatial localities of references?
2. What are the various memory technologies?
3. Differentiate SRAM from DRAM.
4. What is meant by flash memory?
5. Define − Rotational Latency
6. What is direct-mapped cache?
7. What are the writing strategies in cache memory?
8. What are the steps to be taken in an instruction cache miss?
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9. Define – AMAT
10. What are the various block placement schemes in cache memory?
11. Define – MTTF and AFR
12. Define – Availability
13. What are the three ways to improve MTTF?
14. Define – TLB
15. What is meant by virtual memory?
16 .Differentiate physical address from logical address.
17. Define – Page Fault
18. What is meant by address mapping?
13-MARK QUESTIONS
UNIT-1
1. Explain the various addressing modes.
2. Discuss the following:
3. Discuss in detail the various measures of performance of a computer.
4. Discuss the following
5. Explain in detail & also differentiate between uniprocessors and microprocessors
6. Explain the various components of a computer
UNIT-II
1.Give the basic organization arithmetic control unit. Draw a flowchart for ALU operation.
2. Describe the hardware implementation of addition and subtraction unit in detail.
3. Explain the Restoring and non- restoring problems and its hardware implementation. Mention its
advantages and
disadvantages.
4. Describe the floating operations
6. Explain the sub word parallelism in detail.
UNIT III 1. Explain the various types of hazards in pipelining.
2. Write notes on super scalar operation.
3. Give the organization of the internal data path of a processor that supports a 4-stage pipeline for
instructions and uses a 3- bus structure and discuss the same.
4. What is pipelining? What are the various hazards encountered in pipelining? Explain in detail.
UNIT-IV
1. Write short notes on Instructions level parallelism.
2. Explain in detail about parallel processing challenges.
3. What is meant by flynn‟s classification.
4. Write short notes on multi core processors.
5. Write short notes on hardware multithreading.
UNIT-V
1. Write notes on memory hierarchy.
2. Write notes on various types of memory technologies
3. What are the various types of cache mapping mechanisms? Explain in detail.
4. Describe the three mapping techniques used in cache memories with suitable Example.
5. Discuss the virtual memory management technique in detail
6. Explain the various secondary storage devices in detail.
7. Describe the data transfer method using DMA.
8. Explain about the interrupts in detail
9. Explain the various methods available to handle multiple devices using interrupts?
10. Write notes on interrupts in operating system?
11. Explain DMA and the different types of bus arbitration mechanisms.
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DATABASE MANGEMENT SYSTEMS
PART A
UNIT I
1. Who is a DBA? What are the responsibilities of a DBA? April/May-2011
2. What is a data model? List the types of data model used. April/May-2011
3. Define database management system. 4. What is data base management system?
5. List any eight applications of DBMS.
6. What are the disadvantages of file processing system?
7. What are the advantages of using a DBMS?
8. Give the levels of data abstraction.
9. Define instance and schema. 10. Define the terms of Data base schemas.
11. What is conceptual schema?
12. Define data model.
13. What is storage manager?
14. What are the components of storage manager?
15. What is the purpose of storage manager?
16. List the data structures implemented by the storage manager. .
17. What is a data dictionary?
18. What is an entity relationship model?
19. What are attributes? Give examples.
20. What is relationship? Give examples
21. Define the terms i) Entity set ii) Relationship set
22. Define single valued and multi valued attributes.
23. What are stored and derived attributes?
24. What are composite attributes?
25. Define null values.
26. Define the terms i) Entity type ii) Entity set
27. What is meant by the degree of relationship set? 28. Define the terms.
29. What does the cardinality ratio specify?
30. Define weak and strong entity sets.
31. What are the two types of participation constraint?
32. Define the terms i) DDL ii) DML
UNIT II SQL & QUERY OPTIMIZATION
1. What is embedded SQL? What are its advantages? April/May-2011
2. What is the difference between tuple relational calculus and domain relational calculus?
April/May-2011
3. Write short notes on relational model.
4. Define the term relation and Domain.
5. Define tuple variable.
6. What is a candidate key?
7. What is a primary key?
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8. What is a super key?
9. Define- relational algebra.
10. What is a SELECT operation? 11. What is a PROJECT operation?
12. Write short notes on tuple relational calculus.
13. Write short notes on domain relational calculus .
14. Define query language.
15. Write short notes on Schema diagram.
16. What is foreign key?
17. What are the parts of SQL language?
18. What are the categories of SQL command?
19. What are the three classes of SQL expression?
20. Give the general form of SQL query.
21. What is the use of rename operation?
22. Define tuple variable.
23. List the string operations supported by SQL.
24. List the set operations of SQL.
25. What is the use of Union and intersection operation?
26. What are aggregate functions? And list the aggregate functions supported by SQL?
27. Define Normalization. 28. List the properties of decomposition. 29. Define 1NF? 30. Define 2 NF? 31. Define 3NF? 32. What is BCNF? 33. What is query? 34. List out the field level constraints that can be associated with relational table. 35. Justify the need for normalization. 36. Name the different type of joins supported in SQL.
UNIT III TRANSACTION PROCESSING AND CONCURRENCY CONTROL
1. What is meant by lossless-join decomposition? APRIL/MAY-2011
2. List the disadvantages of relational database system.
3. What is first normal form?
4. What is meant by functional dependencies?
5. What are the uses of functional dependencies?
6. What meant by trivial dependency?
7. What are axioms?
8. What is meant by computing the closure of a set of functional dependency?
9. What is meant by normalization of data?
10. Define Boyce codd normal form .
11. List out the desirable properties of decomposition.
12. What is the use of group by clause?
13. What is the use of sub queries?
14. What is view in SQL? How is it defined?
15. What is the use of with clause in SQL? 16. List the table modification commands in SQ.
17. List the SQL domain Types.
18. What is the use of integrity constraints?
19. Mention the 2 forms of integrity constraints in ER model.
20. What is trigger?
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21. What are domain constraints?
22. What are referential integrity constraints?
23. What is assertion? Mention the forms available.
24. Give the syntax of assertion.
25. What is the need for triggers?
26. List the requirements needed to design a trigger.
27. Give the forms of triggers.
28. What does database security refer to? 29. List some security violations (or) name any forms of malicious access.
30. List the types of authorization.
31. What is authorization graph? 32. List out various user authorization to modify the database schema.
33. What are audit trails? 34. Mention the various levels in security measures.
35. Name the various privileges in SQL.
36. Mention the various user privileges.
UNIT IV TRENDS IN DATABASE TECHNOLOGY
1. What are the ACID properties? APRIL/MAY-2011 2. What are two pitfalls (problem) of lock-based protocols? APRIL/MAY-2011
3. What is transaction?
4. What are the two statements regarding transaction?
5. What are the properties of transaction?
6. What is recovery management component?
7. When is a transaction rolled back?
8. What are the states of transaction?
9. List out the statements associated with a database transaction.
10. What is a shadow copy scheme?
11. Give the reasons for allowing concurrency.
12. What is average response time? 13. What are the two types of serializability?
14. Define lock.
15. What are the different modes of lock?
16. Define deadlock.
17. Define the phases of two phase locking protocol.
18. Define upgrade and downgrade.
19. What is a database graph?
20. What are the two methods for dealing deadlock problem?
21. What is a recovery scheme?
22. What are the two types of errors?
23. What are the storage types?
24. Define blocks.
25. What is meant by Physical blocks?
26. What is meant by buffer blocks?
27. What is meant by disk buffer?
28. What is meant by log-based recovery?
29. What are uncommitted modifications?
30. Define shadow paging.
31. Define page.
32. Explain current page table and shadow page table.
33. What are the drawbacks of shadow-paging technique?
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34. Define garbage collection. 35. Differentiate strict two phase locking protocol and rigorous two phase locking protocol.
36. How the time stamps are implemented?
37. What are the time stamps associated with each data item?
UNIT V ADVANCED TOPICS
1. What are the advantages and disadvantages of indexed sequential file? APRIL/MAY-2011
2. What is database tuning? APRIL/MAY-2011 3. Give the measures of quality of a disk. 4. Compare sequential access devices versus random access devices with an example.
5. What are the types of storage devices?
6. Draw the storage device hierarchy according to their speed and their cost.
7. What are called jukebox systems?
8. What is called remapping of bad sectors?
9. Define access time & Seek time.
10. Define average seek time.
11. Define rotational latency time and average latency time.
12. What is meant by data-transfer rate? 13. What is meant by mean time to failure?
14. What are a block and a block number?
15. What are called journaling file systems?
16. What is the use of RAID?
17. How the reliability can be improved through redundancy?
18. What is called mirroring?
19. What is called mean time to repair?
20. What is called bit-level striping?
21. What is called block-level striping?
22. What are the two main goals of parallelism?
23. What are the factors to be taken into account when choosing a RAID level?
24. What is meant by software and hardware RAID systems?
25. Define hot swapping.
26. Which level of RAID is best? Why?
27. Distinguish between fixed length records and variable length records 28. What are the ways in which the variable-length records arise in database systems?
29. Explain the use of variable length records.
30. What is the use of a slotted-page structure and what is the information present in
the header?
31. What are the two types of blocks in the fixed –length representation? Define them.
32. What is known as heap file organization?
33. What is known as sequential file organization?
34. What is hashing file organization?
35. What is known as clustering file organization?
36. What is an index?
37. What are the two types of ordered indices?
38. What are the types of indices?
39. What are the techniques to be evaluated for both ordered indexing and hashing?
40. What is known as a search key?
41. What is a primary index?
42. What are called index-sequential files?
43. What are the two types of indices?
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44. What are called multilevel indices?
45. What are called secondary indices?
46. What are the disadvantages of index sequential files?
47. What is a B+-Tree index?
48. What is B-Tree?
49. What is hashing?
50. How do you create index in SQL? 51. Distinguish between static hashing and dynamic hashing?
52. What is a hash index? 53. What can be done to reduce the occurrences of bucket overflows in a hash file organization?
54. Differentiate open hashing and closed hashing (overflow chaining)
55. What is linear probing?
56. What is called query processing?
57. What are the steps involved in query processing?
58. What is called an evaluation primitive?
59. Define query optimization.
60. What is called a query –execution engine?
61. How do you measure the cost of query evaluation?
62. List out the operations involved in query processing.
63. What are called as index scans?
64. What is called as external sorting? 65. How to form the nested loop join?
66. What is meant by block nested loop join?
67. What is meant by hash join?
68. What is called as recursive partitioning?
69. What is called as an N-way merge?
70. What is known as fudge factor?
PART B
UNIT I INTRODUCTION TO DBMS
1. Explain about the Purpose of Database system &disadvantages of DBMS.
2. Explain about different kinds of data models
3. Briefly explain about Entity-Relationship model.
4. Briefly explain about 1NF, 2NF and 3NF.
5. Briefly explain about Fundamental Relational Algebra operations.
6. Explain about Multi-valued dependencies and Fourth Normal Form.
UNIT II SQL & QUERY OPTIMIZATION
1. Explain about Embedded SQL.
2. Explain about SQL Fundamentals.
3. Explain about data integrity constraints.
4. Explain about Data Definition Language.
5. Explain about query optimization.
6. Discuss about query tuning process.
UNIT III TRANSACTION PROCESSING AND CONCURRENCY CONTROL
1. Explain about Locking Protocols.
2. Explain about Deadlock.
3. Briefly explain about Serializability.
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4. Briefly explain about two phase commit.
5. Briefly explain about Transaction states.
6. What are the issues in concurrency control?
UNIT IV TRENDS IN DATABASE TECHNOLOGY
1. Briefly explain about RAID.
2. Briefly explain about Organization of records in files.
3. Explain about several type of ordered indexes.
4. Briefly explain about B+ tree index file.
5. Explain about Static hashing.
6. Briefly explain about spatial database.
UNIT V ADVANCED TOPICS
1. Explain about Distributed Databases.
2. Discuss briefly about the architecture of Data warehouse.
3. Discuss about access control in DB security.
4. Discuss about object oriented database.
5. Describe about association rules in data mining.
6. Discuss about data mining architecture.
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ENVIRONMENTAL SCIENCE AND ENGINEERING
PART-A
UNIT-I ENVIRONMENT, ECOSYSTEMS AND BIODIVERSITY
ENVIRONMENT
1. Define environment.
2. Write down the components of environment. (May-June 2013) (Nov-Dec 2014)
3. Write the classification of biological environment.
4. What is hazard?
5. Give some important physical hazards and their health effects. Or Give any two examples of
physical hazards. (May - June 2016)
6. Mention some important chemical hazards and their health effects. (Nov-Dec 2016)
7. How are hazards controlled?
ECOSYSTEMS
1. What are the components of Eco system?
2. What are nutrient cycles (or) Bio-geochemical cycle?
3. What is hydrological cycle? (Nov-Dec 2013)
4. What is ecological succession? Mention their types. (Nov-Dec 2008) (Nov-Dec 2013)
5. Why are plants called as producers?
6. What are food chains? (Nov-Dec 2009) (April-May2015) (Nov-Dec 2015)
7. What is food web? (Nov-Dec 2009) (April-May2010)
8. What is ecological pyramid? (April-May2010)
9. What are called Producers?
10. Name the four ecosystems. (April-May 2010)
11. Explain the concept of an ecosystem. (Nov-Dec 2013)
12. Define the terms a) Producers and b) Consumers. (April-May 2008)
13. What is meant by keystone species?
14. What are the characteristics of desert ecosystem? (April-May 2010)
15. What are autotrophic and heterotrophic components of an ecosystem? Give examples. (Nov-Dec
2008)
16. Define primary production and secondary production. (Nov-Dec 2009)
17. How does biome differ from an ecosystem?
18. Define decomposer and give their significance. (Nov-Dec 2012)
19. Define ecology. (May-June 2014)
20. Mention two and secondary consumers in grassland ecosystem. (May-June 2016)
21. How is nitrogen fixed in soil? (April-May 2017)
BIODIVERSITY
1. What is In-situ conservation? (Nov-Dec 2012)
2. What is Ex-situ conservation? (Nov-Dec 2012)
3. Enumerate the human activities which destroy the biodiversity.
16
4. Define the terms a) genetic diversity and b) species diversity (Ap-May2008) (Nov-Dec 2007)
(April-May 2017)
5. What do you understand by the terms flora and fauna?
6. India is a mega diversity nation-Account. (Nov-Dec 2009)
7. What are the two important bio-diversity hot spots in India?
8. Give few examples for endangered and endemic species.
9. Define biodiversity and mention its significance. (April-May2015) (Nov-Dec 2015)
10. Write the classification of biodiversity. (May-June 2014) (Nov-Dec 2016)
11. What are called endangered species? (Nov-Dec 2014)
12. What are the major threats to biodiversity? (Nov-Dec 2014)
13. Explain vulnerable species.
UNIT - II ENVIRONMENTAL POLLUTION
1. Define pollution.
2. What are suspended particulate matters? Give examples.
3. What is photochemical smog? (Nov-Dec 2012)
4. How will you control Air pollution?
5. Define Acid rain. Write its type. (Nov-Dec 2013) (April-May 2015)
6. What is PAN? Give its detrimental effects. (May - June 2016)
7. Define BOD and COD. (Nov-Dec2008) (Nov-Dec 2012) (Nov-Dec 2013)
8. What are point sources and non-point sources of pollution?
9. Give the source of radioactivity.
10. What are the important physical and chemical parameters affecting the quality of water?
(OR) Mention the water quality parameters. (Nov-Dec 2016)
11. What is the role of Citizen in reducing pollution? (Nov-Dec 2009)
12. What is meant by air pollution?
13. List the types of air pollutants. (Nov-Dec2008)
14. Define thermal pollution. (Nov-Dec 2009)
15. Write any four major water pollutants.
16. Name the sources of soil pollution. (May/June - 2013)
17. Write briefly how human activities can introduce thermal pollution in Streams.
18. What are the sources of thermal pollution? (May - June 2016)
19. What is marine pollution? Mention few reasons / sources for marine pollution.(Nov-Dec2014)
(April-May 2017)
20. What is noise pollution? How it is caused? (or) Define the term noise pollution.(Nov/Dec 2013)
(April-May2015) (Nov-Dec 2015) (Nov-Dec 2010)
21. What is disposal?
22. Differentiate between recycling and reuse.
23. What is composting?
24. What are the sources of Urban and Industrial waste?
25. When is a waste said to be hazardous? (Nov-Dec 2009)
26. What are the general methods to removal of heavy metals by adsorption?.
27. What are the causes and effects of ozone layer depletion? (Nov-Dec 2008)
28. What are the major causes of earthquake? (May/June 2014)
29. Mention the effects of ozone on plants. (Nov-Dec 2014)
17
30. Define hazardous waste (Nov-Dec 2014)
31. Define green house effect. (Nov-Dec 2014)
32. Differentiate between sound and noise. (Nov-Dec 2014)
33. How does ozone layer depletion take place? (Nov-Dec 2014)
34. What is the role of individual in preventing pollution? (Nov-Dec 2015)
35. Mention the measures to control thermal pollution caused by industry. (Nov-Dec 2016)
36. Mention the effects of nuclear wastes in humans. (April-May 2017)
UNIT-III NATURAL RESOURCES
1. What are renewable and non-renewable energy resources? Give examples. (Nov-Dec 2008-09).
2. State the environmental effects of extracting and using mineral resources. (May - June 2016)
3. Define sustainable forestry.
4. Define overgrazing.
5. What is desertification? Give two reasons for it. . (Nov-Dec 2009), (Apr-May 2008)
6. What is water logging? Mention about the problems in water logging.
7. What do you mean by environmental impact? (Nov-Dec 2009) (Nov-Dec 2014)
8. Define soil leaching. List the effects of soil leaching.
9. Write any four functions of forests.
10. What are the causes of deforestation? (Nov-Dec 2010)
11. Compare merits and problems of dams
18
. What is meant by soil erosion?
13. Differentiate between deforestation and forest degradation. (Nov-Dec 2007)
14. Enumerate the desired qualities of an ideal pesticide. (Nov-Dec 2007)
15. Write any two adverse effects caused by overgrazing. (Apr-May 2008)
16. Differentiate between renewable and non-renewable energy resources. (Nov-Dec 2012)
17. Mention the various causes of desertification
18. What are the effects of dams on tribal people?
19. What is eutrophication? (Nov-Dec 2012)
20. What is green chemistry?
21. Wood is renewable resources but not coal. Why?
22. Define the term sustainable development. (Nov-Dec 2009)
23. Mention the major environmental impacts of mining. (Nov-2013)
24. What are the changes caused by overgrazing? (Nov-2013)
25. State the use of bio-energy as a non-conventional source of energy.
26. What is environmental biochemistry?
27. What are xenobiotics?
28. What is energy conversion?
29. What is ECO-mark?
30. What is environmental ethics? (Nov/Dec-2013)
31. What is bio-gas? Mention its uses. (Nov-Dec 2016)
32. State the reasons of over exploitations of forest. (May-June 2013)
33. Write the ways of drought management. (May-June 2013)
34. Write the economic importance of forests. (Nov/Dec-2013)
35. Write the problems due to constructions. (May/June-2014)
36. Define non-renewable energy resources. (May/June-2014) (April-May2015) (Nov-Dec 2015)
37. Define the term nuclear energy. (Nov-Dec 2014)
38. Define renewable energy resources. (Nov-Dec 2014) (April-May2015) (Nov-Dec 2015)
39. Write the preventive methods of deforestation. (Nov-Dec 2014)
40. What do you meant by land degradation? What are the reasons for land degradation? (Nov-
Dec 2015) (May - June 2016)
41. What is desertification? (April-May2015)
42. Write any two problems caused by high saline soils? (April-May 2017)
UNIT-IV SOCIAL ISSUES AND THE ENVIRONMENT
1. Define sustainable development. (OR) Explain the term sustainability. (OR) Define sustainable life
style. (Nov-Dec 2007) (Nov-Dec 2012) (Nov-Dec 2013) (May - June 2016) (Nov-Dec 2016)
2. Explain the concept of sustainable development. (Nov-Dec 2009) (Nov-Dec 2008)
3. What is the aim of national committee of environmental planning and co-ordination?
4. Write down the components of Environmental Law.
5. What are the major constitutional provisions in India for environmental protection?
6. State the Article-47 of the Indian Constitution.
7. State the Article 48-A of the Indian Constitution.
8. State the Article 51-A (g) of the Indian Constitution.
9. State the Article 253 of the Indian Constitution.
10. List the major environmental conventions of 20th century.
19
11. Name some of the acts enacted by the Indian Government to protect the environment.
12. What are the common objectives of environmental legislation?
13. List the major environmental protocols of 20th century.
14. Write notes on NGOs.
15. Write notes on „Green Peace‟:
16. List some of the NGOs available in India.
17. How does the community participation involve in protecting the environment?
18. Write short notes on the history of implementation of international treaties and conventions.
19. Write down the purpose / objectives of The Water (Prevention and Control of Pollution)
Act, 1974. (Nov-Dec 2014)
20. Write notes on Manufacture, Storage and Import of Hazardous chemical Rules, 1989.
21. What is meant by rain water harvesting? (Nov-Dec 2012) (Nov-Dec 2014)
22. What is environmental ethics? (Nov/Dec-2013)
23. List the objectives of forest conservation act. (Nov/Dec-2013)
24. What is cyclone? (Nov/Dec 2013) (April-May 2017)
25. What are biomedical wastes?
26. What are landslides? (Nov/Dec 2008) (Nov-Dec 2014)
27. Define floods.
28. What are the advantages of rain water harvesting? (Nov-Dec 2015)
29. Define consumerism. (April-May2015) (Nov-Dec 2015) (Nov-Dec 2016)
30. What do you meant by disaster management? (April-May2015)
31. State any two biomedical waste handling rules. (May - June 2016)
32. Write any four principles of green chemistry. (Nov-Dec 2016)
20
PART B
UNIT I
1. Explain in detail about the Components of environment
2. Discuss the Structure and function of an Ecosystem:
3. Explain the energy flow in ecosystem in food chain and food web.
4. Explain the types, characteristic features, structure and function of Forest ecosystem
5. Explain the types, characteristic features, structure and function of grassland ecosystem
6. What are the classifications (or) levels of biodiversity?
7. Explain In-situ and Ex-situ conservation of biodiversity. (or) What are the measures
recommended for conservation of biodiversity.
8. “India is a mega biodiversity nation” - Explain.
9. . What are the major threats to biodiversity?
UNIT II
ENVIRONMENTAL POLLUTION
1. Mention the sources and effects of various air pollutants
2. Discuss in detail about chemical and photochemical reactions in the atmosphere
3. Explain the mechanism of Ozone layer depletion.
4. What are the sources, effects and control measures of acid rain.
5. Write in detail about global warming.
6. Explain the causes, effects & control measures of Water pollution.
7. What are the sources, effects & control measures of Marine pollution?
8. Briefly describe the sources effects and control of noise pollution.
9.What are the major sources and the measures to be taken to prevent soil pollution?
10. Explain in detail about solid waste management/soil waste management/waste shed
management
11. Case studies related to pollution
(i)The Bhopal gas tragedy:
(ii) Chernobyl nuclear disaster(Nuclear pollution/Nuclear accidents)
UNIT-III
NATURAL RESOURSES
1. What is deforestation? Discuss the ill effects of deforestation
2. Discuss in detail the impacts of modern agriculture. (OR)
3. Explain the role of individual in conservation of natural resources.
4. Discuss briefly on the consequences of overdrawing of ground water.
5. Give an account on conflicts over water.
6. Enumerate the various benefits and drawbacks of constructing dams.
7. What are the environmental impacts of mineral extraction? Explain.
8. What are non-conventional energy/renewable resources?
9. What are conventional/non-renewable energy sources?
10. Explain the role of alternate (renewable) energy sources in environmental impacts.
11. Explain the biochemical degradation of pollutants.
UNIT IV SOCIAL ISSUES AND THE ENVIRONMENT 1. What is rain water harvesting? Name and discuss in brief the types of rain water harvesting.
2. Discuss the water shed management practices.
3. Describe the important waste land reclamation practices.
4. Discuss about the Forest Act and Wildlife act
21
5. State the important provisions in Air Act and in Water Act.
6. State the important provisions in Environment protection Act.
7. Describe the steps involved in management of Bio medical waste.
8. What functions are performed by the central & state pollution control boards?
9. Twelve principles of green chemistry.
10. Write notes on (i) Floods (ii) Cyclones (iii) Earth quake (OR) State the different natural
calamities/disaster and explain any one in detail.
UNIT V HUMAN POPULATION AND THE ENVIRONMENT
1. Explain the population explosion on the environment.
2. Describe in detail about the family planning programme.
3. Write a detailed account on human Rights.
4. Explain the need and importance of Value Education.
5. Describe the causes, preventive measures and effects of HIV/AIDS:
6.What is EIA? Explain the objectives, benefits and process of EIA.
7. Describe in detail about the family welfare and child welfare.
8. Describe the role and applications of IT in Human Health.
22
PROGRAMMING AND DATA STRUCTURE II
PART-A
UNIT I
1. State the characteristics of procedure oriented programming.
2. What are the features of Object Oriented Programming?
3. Distinguish between Procedure Oriented Programming and Object Oriented Programming.
4. Define Object Oriented Programming (OOP).
5. List out the basic concepts of Object Oriented Programming.
6. Define Objects.
7. Define Class.
8. Define Encapsulation and Data Hiding.
9. Define Data Abstraction.
10. Define data members and member functions.
11. State Inheritance.
12. State Polymorphism.
13. List and define the two types of Polymorphism.
14. State Dynamic Binding.
15. Define Message Passing.
16. List out some of the benefits of OOP.
17. Define Object Based Programming language.
18. List out the applications of OOP.
19. Define C++.
20. What are the input and output operators used in C++?
21. What is the return type of main ()?
22. List out the four basic sections in a typical C++ program.
23. Define token. What are the tokens used in C++?
24. Define identifier. What are the rules to be followed for identifiers?
25. State the use of void in C++.
26. Define an enumeration data type.
27. Define constant pointer and pointer to a constant.
28. What are the two ways of creating symbolic constants?
29. Define reference variable. Give its syntax.
30. List out the new operators introduced in C++.
31. What is the use of scope resolution operator?
32. List out the memory differencing operator.
33. Define the 2 memory management operators.
34. List out the advantages of new operator over malloc ().
35. Define manipulators. What are the manipulators used in C++?
23
36. What are the three types of special assignment expressions
37. Define implicit conversion.
38. Define integral widening conversion.
39. What are the control structures used in C++?
40. Define Function Prototyping.
42. What are inline functions?
43. List out the conditions where inline expansion doesn‟t work.
44. Why do we use default arguments?
45. State the advantages of default arguments.
46. Define function overloading.
47. List out the limitations of function overloading.
48. State the difference between structures and class.
49. Define a class.
50. List the access modes used within a class.
51. How can we access the class members?
52. Where can we define member functions?
53. What are the characteristics of member functions?
54. How can an outside function be made inline?
55. What are the properties of a static data member?
56. What are the properties of a static member function?
57. How can objects be used as function arguments?
58. Define friend function?
59. List out the special characteristics of a friend function.
UNIT –II
1. Define Constructor.
2. List some of the special characteristics of constructor.
3. Give the various types of constructors.
4. What are the ways in which a constructor can be called?
5. State dynamic initialization of objects.
6. Define Destructor.
7. Give the general form of an operator function.
8. List some of the rules for operator overloading.
9. What are the types of type conversions?
10. What are the conditions should a casting operator satisfy?
11. What is parameterized constructor?
12. What are the kinds of constructor that we call?
13. What are copy constructors? Explain with examples?
14. Define dynamic constructor?
15. What is the advantage of using dynamic initialization?
16. Define operator overloading?
17. Give the operator in C++ which cannot be overloaded?
18. What are the steps involves in the process of overloading?
19. What are the restriction and limitations of overloading operators?
20. Give a function overload a unary minus operator using friend function?
21. Define unary and binary operator overloading?
22. Explain overloading of new and delete operators?
24
23. Define type conversion?
24. When and how the conversion function exists?
25. Give the syntax for overloading with friend function?
26. Define inheritance?
27. What are the types of inheritance?
28. Give the syntax for inheritance.
29. Define single inheritance.
30. Define multi-level inheritance.
31. Define multiple inheritance.
32. Define Hierarchical inheritance.
33. Define Hybrid inheritance.
34. What is a virtual base class?
35. What is an abstract class?
36. What are the types of polymorphism?
37. Define „this‟ pointer.
38. What is a virtual function?
39. What is a pure virtual function?
40. How can a private member be made inheritable?
41. What are the rules for virtual function?
42. What is RTTI?
43. List out RTTI
UNIT-III
1. Define Templates?
2. What are the advantages of using template?
3. Define Class Template?
4. Function Template?
5. What are the rules for declaring template?
6. What is the syntax used to declare Function Template with multiple parameters?
7. How will you overload Function template?
8. What is Exception Handling?
9. List out the steps of exception handling?
10. Define Errors and its types?
11. Distinguish between syntax and logical errors?
12. What is mean by Synchronous Exceptions?
13. What are the three keywords of exception handling Mechanism?
14. What is the relationship between the try, catch and throw block?
15. Define Unexpected( ) and Terminate( ) Functions?
16. How will you specifying list of Exceptions?
17. What is mean by Catching mechanism?
18. What is meant by Asynchronous Exceptions?
19. Define Member function template?
20. Describe the rules for function template?
21. What is stream and its types?
22. Define Input stream?
23. What are console stream classes in C++?
25
24. What are the file stream classes in C++?
25. What are the file manipulation function in C++?
26. What are the file open modes?
27. What are the error handling function in C++?
28.Define getline( )?
29. What are unformatted output operation in C++?
30. What are the different types of formatted console I/O operations?
31. What do you meant by Manipulators?
32. List out the parameterized manipulators?
33. What are non-parameterized manipulators?
34. Distinguish between width() and precision ()function?
35. Define file?
36. What is the syntax used for closing a file give example?
37. Define Default (or) Random Access?
38. List out the different types of modes used in fine concepts?
39. Define namespace?
40. How will you declare namespace?
41. Define Namespace Alias?
42. Define Namespace Std?
43. What problem does the namespace feature solve?
44. Define Standard Template Library (Stl)?
45. What is a container class? What are the types of container classes?
46. STL containers-What are the types of STL containers?
47. What are the different types of STL Algorithm?
UNIT-IV
1. Define Tree .Give an example.
2. Define root
3. Define length of the path in a tree with an example
4. Write the routine for node declaration in trees.
5. List the applications of trees.
6. Define Binary tree.
7. List the tree traversal applications.
8. Define binary search tree?
9. List the Operations of binary search tree?
10. Define Threaded Binary tree.
11. List the uses of binary tree.
12. State the properties of a binary tree
13. What is meant by binary tree traversal?
14. What are the different binary tree traversal techniques?
15. What are the tasks performed while traversing a binary tree?
16. What are the tasks performed during preorder traversal?
17. What are the tasks performed during inorder traversal?
18. What are the tasks performed during postorder traversal?
19. State the merits of linear representation of binary trees.
20. State the demerit of linear representation of binary trees.
21. State the merit of linked representation of binary trees.
26
22. State the demerits of linked representation of binary trees.
23. What do you mean by general trees?
24.Define ancestor and descendant.
25. Why it is said that searching a node in a binary search tree is efficient than that of a simple
binary tree?
26. What is an expression tree?
27. Define right-in threaded tree
28. Define left-in threaded tree
29. Define AVL Tree. Give Example.
30. Define Balance factor.
31. When AVL tree property is violated and how to solve it?
32. When AVL tree property is violated and how to solve it?
33. Mention the four cases to rebalance the AVL tree.
34. Define Rotation in AVL tree. Mention the two types of rotations.
35. Define Splay Tree.
36. List the types of rotations available in Splay tree.
37. Define B-tree of order M.
38. What do you mean by 2-3 tree.
39. What do you mean by 2-3-4 tree?
40. What are the applications of B-tree?
41. Define binary heaps.
42. What are the applications of priority queues?
43. What do you mean by the term “Percolate up”?
44. What do you mean by the term “Percolate down”?
45. List the Applications of Binaryheap
46. What are red black trees?
47. Explain the properties of Red-black trees.
48. What are binomial heaps.
49. What are the advantages of Fibonacci heaps?
50. What is amortized analysis?
51. Name the techniques used in amortized analysis.
UNIT-V
1. Define Graph.
2. Define adjacent nodes.
3. What is a directed graph?
4. What is an undirected graph?
5. What is a loop?
6. What is a simple graph?
7. What is a weighted graph?
8. Define outdegree of a graph?
9. Define indegree of a graph?
10. Define path in a graph?
11. What is a simple path?
12. What is meant by strongly connected in a graph?
13. When is a graph said to be weakly connected?
14. Name the different ways of representing a graph?
27
15. What is an undirected acyclic graph?
16. What are the two traversal strategies used in traversing a graph?
17. What is a minimum spanning tree?
18. Name two algorithms two find minimum spanning tree
19. Define graph traversals.
20. List the two important key points of depth first search.
21. What do you mean by breadth first search (BFS)?
22. Differentiate BFS and DFS.
23. What do you mean by tree edge?
24. What do you mean by back edge?
25. Define biconnectivity.
26. What do you mean by articulation point?
PROGRAMMING AND DATA STRUCTURES II
13 MARK QUESTIONS WITH HINTS BANK
UNIT - I
1. Write down the characteristics of object oriented programming.
2. Explain the elements of object oriented programming.
3. Describe the applications of OOP technology.
4. What is function overloading? Explain briefly with Program.
5. Write a program to demonstrate how a static data is accessed by a static
member function.
6. What is friend function? What is the use of using friend functions in c++?
Explain with a program.
7. Discuss in detail about default arguments with an example.
UNIT II
28
1. Write a program to implement
2. What is inheritance and explain briefly pointer to derived class.
3. Explain copy constructor and destructor with suitable C++ coding.
4.Explain in detail about constructor with dynamic allocation.
5.With an example explain about operator overloading through friend functions.
6.Explain about Unary Operator and Binary Operator Overloading with program.
7.List out the rules for overloading operators with example.
UNIT III 1. What is the need of Templates?
2. What is uncaught exception function? Give an
3. What are the use of terminate () and Unexpected
functions? Explain with a program
4. How to use multiple catch functions inside a program? Explain with a
program.
5. Write all blocks of exception handling? Explain with a program. 6. Write notes on Formatted and Unformatted Console I/O Operations.
7. Explain about File Pointers and their manipulations with example.
UNIT IV 1. Explain disjoint set in detail.
2. Explain Red Black tree.
3. Explain about splay trees.
4. What are AVL trees? Describe the different rotations defined for the AVL
tree.
5. Explain about the Fibonacci heaps. UINT V
1. Explain Graph traversals with examples.
2. Describe the topological sorting method with suitable examples.
3. What do you know about the breadth first search? Explain
4. Write the Prims algorithm and explain the same.
5. Explain the kruskals algorithm for minimum spanning tree. 6. Discuss any two shortest path algorithms 7. Compare depth first search and depth first search.
29
TRANSFORMS AND PARTIAL DIFFERENTYIAL EQUATIONS
UNIT I PARTIAL DIFFERENTIAL EQUATIONS
PART A
1. Form the p.d.e. by eliminating the arbitrary constants a & B from .z ax by
2. Form the p.d.e. by eliminating the arbitrary constants from z ax by ab . [ Apirl/May
2003]
3. Form the p.d.e. by eliminating the arbitrary constants a and b from
2 2
z x a y b .[April/May2001]
4.Form the partial differential equation by eliminating a and b from
2 2 2 2z x a y b .[Nov/Dec.2004]
5. Eliminate the arbitrary constants a & b from 2 2z x a y b . [April/may2004]
6. Form the p.d.e. by eliminating the constants a and b from n nz ax by . [April/May 2005]
7. Form a partial differential equation by eliminating the arbitrary constants a and b from
the relation 2(2 )(3 )z x a y b .[April/May 2004]
.
8. Obtain partial differential equation by eliminating arbitrary constants a and b from 2 2 2( ) ( ) 1x a y b z .
[Nov/Dec2003]
9. Find the partial differential equation of the family of spheres having their centres on the line x
= y = z.
10. Obtain the p.d.e. by eliminating the arbitrary constants a and b from 2 2(z xy y x a b .
11. Form the p.d.e. by eliminating the arbitrary function from 2 2( )z f x y . [Oct/Nov 1996]
12. Eliminate the arbitrary constants a and b from 2 2z ax by a b . [A/M 2003]
13. Form a partial differential equation by eliminating a and b from 2 2z a x ay b . [ N/D 2008]
14. Form the partial differential equation by e3leminating a and b from ( )z a x y b . [N/D 2008]
15. Form the partial differential equation by eliminating a and b from
2 2 2 2cotx a y b z . [N/D2007]
30
16. Form the p.d.e by eliminating the arbitrary function from 2 2( )z f x y
17. Form the p.d.e. by eliminating the function from the relation xz f
y
. [ A/M, 1996]
18. Form the p.d.e. by eliminating the function f from the relation 2 1
2 logz y f yx
.
19. Form a p.d.e. of eliminating the arbitrary function from f( x-y, x+y+z) = 0. [A/M.2000]
20. Eliminate the arbitrary function f from xy
z fz
and form the partial differential equation.
[ A/M 2004]
Solution:
21. Form the p.d.e. from ( ) ( )z f x t g x t . [March,1996]
22. Eliminate the arbitrary functions f and g from ( ) ( )z f x iy g x iy to obtain a partial
differential
equation involving z,x,y.
23. Solve 1p q . [Oct,1996]
24. Obtain the complete solution of the equation 2z px qy pq . [O/N.,1996]
i.e., Singular integral is xy = 1.
25. Solve 3
2z px qy pq . [ May, 2000]
26. Find the complete integral of 2 2z px qy p q . [ A/M 2001]
27. Find the complete integral of q = 2px. [ A/M 2001]
28.Solve p(1+q) = qz. [N/D1998]
29. Solve p+q = x-y.[Dec.,1996]
30. Solve the equation logyp zyx q . [March,1996& May, 1996]
31. Solve the equation tan tan tanp x q y z . [ May, 1996]
32. Solve the equation 23 2 ' '4 4 0D D D DD z
where ',D Dx y
. [O/N.,1996]
33. Find the solution of 2 2
2 24 0
z z
x y
. [A/M, 2000]
34.Solve 3 3 3 3
3 2 2 32 4 8 0
z z z z
x x y x y y
. [Oct/Nov, 2002]
35. Solve 2 33 ' 13 2 0D DD D z
. [A/M,2003]
36. Find the general solution of 2 2 2
2 24 12 9 0
z z z
x x y y
. { N/D,2003]
31
37. Solve 3 2 2' ' ' 0D DD D D D z . [ A/M,2005]
38. Find the particular integral of 2 2 24 ' 4 ' x yD DD D z e . [ N/D.2003]
39. Find the particular integral of 3 3 32 3
3 2 33 4 x yz z z
ex x y y
. [March,1996]
40. Find the particular integral of 2 ' sin cos2D DD z x y . [A/M1996]
41. Find the particular integral of 2 23 ' 2 ' cos( 2 )D DD D z x y . [A/M.2003]
42. Solve 2 ' ' 1 0D DD D z . [N/D,2004]
43. Find the complete integral of the partial differential equation (1 ) (2 ) 3x p y q z . [N/D
2006]
44. Solve 2
2sin
zy
x
. [M/J,2007]
45. Find the complete integral of z x ypq
pq q p
. [M/J,2007]
46. Find the complete solution of the p.d.e. 2 2 4 0p q pq . [N/D,2007]
47. Find the complete integral of p – q = 0. [N/D,2008]
48. Form the p.d.e. by eliminating the f from the relation 2 2( )z f x y x y
49. Find the p.d.e. of all planes having equal intercepts on the x and y axis. [N/D 2005]
50. Find the particular integral of 2 22 ' ' x yD DD D z e . [N/D,2010]
51. Form a PDE by eliminating the arbitrary constants „a‟ and „b‟ from .
52. Solve
PART B
1. (i) Solve 2 2 2 2 ( )x y yz p x y xz q z x y . [N/D2005]
(ii) Solve 2 2 5' 20 ' sin(4 )x yD DD D z e x y . [N/D2005]
2. (i) Solve 2 2z p q . [N/D2005]
(ii) Solve 2 2 2 3' 6 ' x yD DD D z x y e [N/D2005]
3.(i) Solve 2 2 0y z p xyq xz . [M/J,2006]
(ii) Solve 2 26 ' 5 ' sinhxD DD D z e y xy . [M/J,2006]
4.(i) Solve 2(1 ) (1 )p q q z . [M/J,2006]
(ii) Solve 2 2 24 ' 4 ' x yD DD D z e . [M/J,2006]
5. (i)Find the singular integral of 2 2z px qy p pq q . [N/D,2006], [A/M, 2008]
(ii) Solve 2 25 ' 6 ' sinD DD D z y x . [N/D,2006]
32
6. (i) Solve (3 4 ) (4 2 ) (2 3 )z y p x z q y x . [N/D,2006]
(ii) Solve 2 2 22 ' ' x yD DD D z x y e . [N/D,2006]
7. (i) Form the partial differential equation by eliminating a and b from the expression
2 2 2 2x a y b z c . [M/J,2007]
(ii) Solve 2 22 5 ' 2 ' 5sin(2 )D DD D z x y . [M/J,2007]
8.(i) Solve 2 2 2 2( ) ( ) ( )x y z y x z q z x y . [M/J,2007]
(ii) Solve (1 )p q qz . [M/J,2007]
9.(i) Solve the equation 2 2 2 2 2x y z p xyq zx . [N/D. 2007]
(ii) Solve 2 2 3 4' 2 ' 2 3 x yD DD D z x y e . [N/D. 2007]
10.(i) Solve 2 2 2 2 2z p q x y . [N/D. 2007]
(ii) Solve 2 23 ' 4 ' sinD DD D z x y . [N/D. 2007]
11.(i)Form a partial differential equation by eliminating arbitrary functions from
(2 ) (2 )z xf x y g x y .
[A/M,
2008]
(ii)Solve 2 2 2(1 )p y x qx .[A/M, 2008]
12.(i) Solve 2 2 2 2 2 2x z y p y x z q z y x . [A/M, 2008]
(ii) Solve 3 3
2
3 22 4sin( )x yz z
e x yx x y
. [A/M, 2008]
13.(i) Find the complete integral of 2 2 2 2 2x p y q z . [N/D, 2008]
(ii) Find the singular integral of z px qy pq . [N/D, 2008]
14.(i) Solve 2 2 32 ' ' 2 2 ' 4x yD DD D D D z e . [N/D, 2008]
(ii) Solve 3 2 37 ' 6 ' cos( 2 ) 4D DD D z x y .[N/D, 2008]
15.(i) Find the complete solution of 2pqxy z . [M/J,2009]
(ii) Solvethe equation 2 2' sin( 2 )x yD D z e x y .[M/J,2009]
16. Find the singular solution to the equation 2 21z px qy p q . [M/J,2009]
17.(i) Solve 2 2z px qy p q . [N/D,2009]
(ii) Solve 2 2 22 ' ' sinh( ) x yD DD D z x y e . [N/D,2009]
18.(i) Solve 2 2' 3 3 ' 7D D D D z xy . [N/D,2009]
(ii) Solve 2(1 ) ( )p q q z a .[N/D,2009]
19.(i) Find the partial; differential equation of all planes which are at a constant distance k from
the origin.
[N/D,2009]
(ii) Solve 2 23 ' 2 ' sin(2 )D DD D z x y x y .[N/D,2009]
33
20.(i) Find the general solution of ( ) ( ) ( )x y z p y z x z x y . [N/D,2009]
(ii) Solve 3 2 2 3' ' ' sin 2 cosD D D DD D z x y .[N/D,2009]
21.(i) Solve 2 2 22 ' ' 2 2 ' 3x yD DD D D D z e . [N/D,2009]
(ii) Solve 2 22 ' ' 2 2 ' sin( 2 )D DD D D D z x y . [A/M,2010]
22.(i) Solve 2 2 2x yz p y zx q z xy .[A/M,2010]
(ii) Solve 2 2 2 2p q x y . [A/M,2010]
UNIT-II
PART A
1. Determine nb in the Fourier series expansion of xxf 2
1 in 20 x with period
2 .
2. Define root mean square value of z in bxa .
(OR) Define root mean square value of a function xf over the range (a,b) .
3. If
2
0
,50
,cos
xif
xifxxf
and 2 xfxf for all x,
4. Find the value of na in the cosine series expansion of kxf in the interval (0,10).
5. Find the root mean square value of the function xxf in the interval ( 0, l ).
6. State Dirichlet‟s conditions for a given function to expand in Fourier series.
(OR) State the Dirichlet‟s condition for the convergence of the Fourier series of
f x in 0,2 with period 2 .
7. If the Fourier series of the function 2xxxf in the interval
x is
12
2
sin2
cos4
13 n
nnx
nnx
n
, then find the value of the infinite series
.....3
1
2
1
1
1222
8. Find the Fourier sine series of the function 1xf , x0 .
9. If the Fourier series of the function
2
0
,sin
,0
x
x
xxf is
xxxx
xf sin2
1......
75
6cos
53
4cos
31
2cos21
deduce that
4
2.........
75
1
53
1
31
1
.
10. Does tanf x x possess a Fourier expansion ?
34
11. State Parseval‟s Theorem on Fourier series. (OR) State Parseval‟s identity of Fourier
series.
12. Find nb in the expansion of 2x as a Fourier series in , .
13. If xf is an odd function defined in ( -l , l ) , what are the values of
0a and na ?
14. Find the constant term in the Fourier series corresponding to xxf 2cos expressed in
the interval , .
15. To which value the Half range sine series corresponding to 2xxf expressed in the
interval (0,2) converges to 2x ?
16. If 2xxxf is expressed as a Fourier series in the interval ( - 2, 2 ) to which value
this series converges at 2x ?
17. If the Fourier series corresponding to xxf in the interval 2,0 is
1
0 sincos2
nxbnxaa
nn
,without finding the values of .,,0 nn baa Find the value of
1
2220
2nn ba
a
18. Find the Half range sine series for 2xf in 0 x .
19. If the cosine series for xxxf sin for x0 is given by
2
2
11sin 1 cos 2 cos ,
2 1
n
n
x x x nxn
Prove that 1 1 11 2 ........
1 3 3 5 5 7 2
.
20. What do you mean by Harmonic analysis? (OR) Define Harmonic analysis.
21. In the Fourier expansion
x
x
x
x
xf0
0
,2
1
,2
1in , , find the value of
nb ,the
coefficient of sin nx
22.Find na in expanding xe as Fourier series in , .
23. If
1
03 sincos2
cos ntbntaa
t nn in 20 t , find the sum of the series
2
2 20
12
n na
a b
24. The Fourier series of 2x in (0,2) and that of 22x in (-2,0) are identical or not.
Give reason.
25. Define the value of the Fourier series of f x at a point of discontinuity.
26. If sinhf x x is defined in x , write the values of Fourier coefficients
0a and na .
35
27. If sin sin 2 sin3 sin 42 ...
1 2 3 4
x x x xx
in 0 x , prove that 2
2
1
6n
.
28. The functions tanf x x ,
1sinf x
x
cannot be expanded as a
29. Expand the function 1f x ,0 x as a series of sines.
30. Find the Fourier Cosine series of 2f x cos x , 0 x .
31. 2f x x , 0 2x which one of the following is correct a) an even function
b) an odd function c) neither even nor odd
32. Find the value of f .
33. Determine the value of na in the Fourier series expansion of 3( )f x x in x .
34. Find the Half range Fourier sine series for ( )f x x in (0, )l
35. Find the complex form of the Fourier series of the function ( ) xf x e when x and
( 2 ) ( )f x f x
36. If ( ) 2f x x in (0,4) , then find the value of 2a in the Fourier series expansion.
37. State Parseval‟s identity for the half-range cosine expansion of f(x) in (0,1).
Solution:
38. What do you mean by Harmonic analysis? [A.U April/May ]
39. Find tne root mean square value of the function ( )f x x in the interval (0, )l .
40. Find the Fourier series to represent 2( ) 2f x x in 2 2x
41. Write 0 , na a in the expansion of 3x x as a Fourier series in ( , ) .
42. State Paeseval‟s identity for the half-range cosine expansions of ( )f x in (0,1) June(2006)
43. Find nb in the expansion 2x as a Fourier series in ( , ) . Nov/Dec (2005)
44.If f is an odd function defind in , what are the values of Nov/Dec (2005)
45. Find the value of na in the cosine series expansions of ( )f x k in the interval 0,10
46. If in the interval ,then find the values of in the Fourier series expansion
(A.U May/June 2009)
47. Find the root mean square value of 2f (x) x in the interval (0, ) .(A.U May/June 2008)
48. Expand ( ) 1f x in a sine series in 0 x
49. If 2( )f x x x is expressed as a Fourier series in the interval ( 2,2) to which value this
series
converges at 2x [A.U April/May 2003]
50. Write 0 , na a in the expansions of 3x x as a Fourier series in ( , ) [A.U Nov/ Dec.2003]
51. Find the value of a0 in the Fourier series expansion of f(x)=ex in )2,0( .
52.Find the half range sine series expansion of f(x)=1 in (0,2).
53. Find the root mean square value of in
54. If the function in the interval , then find the constant term of the
Fourier
36
series expansion of the function
55. If the Fourier series expansion of the function is
,
then find the sum of the series .
56. Find the value of the Fourier series of at the point of discontinuity
57. Find the value of in the Fourier series expansion of
UNIT-IV FOURIER TRANSFORM
TWO MARKS
1) Write the Fourier transform pair.
2) If CF S is the Fourier cosine transform of f(x), prove that the Fourier cosine
transform of f(ax) is 1
C
sF
a a
.
3) If F(S) is Fourier transform of f(x), write the Fourier transform of f(x)cos(ax) in terms
of F.
4) State the Convolution theorem for Fourier transforms.
5) If F(S) is Fourier transform of f(x), then find the Fourier transform of f(x-a).
6) If SFs is the Fourier Sine transform of f(x), show that
asFasFaxxfF SSS 2
1cos .
7) Solve the integral equation
exdxxf cos0
.
8) Find the Fourier transform of f(x) if
0
1xf :
:
0
ax
ax .
9) Find the Fourier cosine transform of xe .
10) Find xfxF
n in terms of the Fourier transform of f
11) State Fourier integral theorem.
12) Find the Fourier Sine transform of x
1.
13) Find the Fourier transform of 0,
x
e .
37
14) Find the Fourier cosine integral representation of
0
1xf
1
10
x
x .
15) Find the Fourier Sine transform of f(x)= xe . (May
2006)
Fourier cosine transforms of
2
2 2x ex
17 If sfxfF then give the value of axfF . (May 2006)
18 Find the Fourier transform of
0
1xf
1
1
x
x. (May 2006)
19. Find the Fourier cosine transform of f(x) defined as
P.T asFxfeFiax where sFxfF . (M/J 2007)
19 Write down the Fourier cosine transform pair formulae. (M/J
20 If
sfxfF prove that
sfeaxfFias .
23. Prove that if sFxfF , then sFisa
eaxfF
24. Find the Fourier transform of xf defined by
25. Find the Fourier Cosine transform of
0
xxf
,
,
x
x0
.
State Parseval‟s identity on complex Fourier transforms. (Dec 2008)
26. If sFxfF then prove that asFxfiax
eF . (Dec 2008)
27. State Modulation theorem in Fourier transform
28. Give a function which is self reciprocal under Fourier sine and cosine transforms.
(Dec 2008)
29. If scFxfcF , then prove that scFds
dxxfsF . (Dec 2008)
29)Find the Fourier cosine transform of 0, ae ax .
30. Find the Fourier cosine transform of xe 3
31. 31.Prove that )()(2
1cos)( asFasFaxxfF ccc where CF denotes the Fourier cosines
Transforms )(xf
32. Find the Fourier constants nb for sinx x in ( , )
(June2006)
33. Write change of scale property
34. Write any two property of Fourier sine Transforms and cosine transforms
35. Find the Fourier cosine transform of 0, ae ax .
38
36. Find the fourier transform of 3xe
37. Find
n
n
dx
xfdF
in terms of the Fourier transform of f
38. Find the Fourier sine transform of 5 23 5x xe e
39. Find the Fourier transform sine transforms of 2 2
x
a x
40. Find the Fourier sine transform of , 0axe a and Deduce that 2 2
0
sin2
axssx dx e
s a
.
41. Find then Fourier sine transforms of 21
x
x
42. Define : Finite Fourier Sine transforms
43. Define : Finite Fourier Cosine transforms
44. Define: Inversion formula for Sine Transforms:
45. Define: Inversion formula for Cosine Transforms
46. Find is its finite sine transforms is 1
3
2( 1) p
p
for 1,2,.... 0p and x
47. Find f(x) if its finite Fourier sine transforms is 2 2
1 cos( ) , 1,2,3...0
S
pf p p x
p
UNIT-IV PART-B
1) Find the Fourier transform of f(x) =
0
1
otherwise
xfor 1 .Hence prove that
0 0
2
2
sinsin dx
x
xdx
x
x . (Nov 2002),(Nov/Dec 2003)
2) Find the Fourier transform of
0
sin xxf
x
x
0 . (Nov 2002)
3) Find the Fourier cosine transform of 2,0)2(4)1(3)( nnynyny .Deduce
that
0
8
2 816
2cosedx
x
x ,
0
8
2 816
2sinedx
x
xx .
4) State and prove the Convolution theorem for Fourier transforms.
(Nov 2002)
5) Find the Fourier transform of 0,
aexa
. Deduce that (i)
0
3222 4
1
adx
ax
, (ii)
222
22
sa
asixeF
xa
.
(April 2003)
39
6) Find the Fourier Sine transform of 2
2x
xe
. (April 2003)
7) Find the Fourier cosine transform of 22
xae
.Hence evaluate the Fourier Sine transform
of 22
xaxe
. (Nov/Dec 2006)
8) Find the Fourier transform of 22
xae
. Hence prove that 2
2x
e
is self-reciprocal.
(May 2006), (May 2007)
9) Find the Fourier cosine transform of
0
12
xxf
otherwise
x 10 . Hence prove that
03
.16
3
2cos
cossin dx
x
x
xxx (April 2003)
10) Derive the Parseval‟s identity for Fourier transforms. (April 2003)
11) Find the Fourier Sine and cosine transform of x
e
. Hence find the Fourier Sine
transform of 2
1 x
x
and Fourier cosine transform of 2
1
1
x
.
(Nov/Dec 2003)
12) Show that Fourier transform of
0
22xa
xf :
:
ax
ax
is
3
acosaasin22
. Hence
deduce that
03
.4
cossin dt
t
ttt
(Apr/May 2004)
13) Find the Fourier Sine and cosine transform of
0
2 x
x
xf
:
:
:
2
21
10
x
x
x (Apr/May 2004)
14) If f is the Fourier transform of f(x) , find the Fourier transform of f(x-a) and f(ax).
(Apr/May 2004)
15) Verify Parseval‟s theorem of Fourier transform for the function
x
exf
0:
:
0
0
x
x.
Find the Fourier transform of f(x),
0
12
xxf :
:
1
1
x
x . Hence evaluate
(i)
03 2
coscossin
dxx
x
xxx (ii)
0
2
3 15
cossin ds
s
sss .
(April 2005) (Dec 2008)
16) Find the Fourier Sine transform of 0
ax
eax . (Nov/Dec 2006)
17) Find the Fourier Sine and cosine transform of x
e2
. Hence find the value of the
following integrals (i)
022
4x
dx (ii)
022
2
4dx
x
x . (A.U.M Qu)
40
18) Evaluate (i)
02222
bxax
dx (ii)
022
41 xx
dx using Fourier transform.
(Nov/Dec 2008)
19) Find the Fourier Sine and cosine transform of 1n
x . (May 2006)
20) Using Parseval‟s identity for Fourier cosine transform of ax
e
evaluate
0222
xa
dx .
(Nov/Dec 2007)
21) Find the Fourier Sine transform of 0, ae
ax . Hence find axS xeF
. Hence deduce the
inversion formula. (May/June 2007)
22) Find the Fourier Sine transform of f(x) defined as
0
sin xxf
where
where
ax
ax
0 . (Dec 2008)
23) Find the Fourier transform of
0
x1xf
otherwise
for 1x . Hence find the values of (i)
0
dt
4
t
tsin and (ii)
0
dx
2
x
xsin (Dec 2008)
24) Find the finite sine and cosine transform of 2
x1xf
in the interval ,0 .
(Dec 2008)
25) Find the Fourier transform of
0
xaxf
,
,
ax
ax
. (Dec 2008)
26) Evaluate
0
2x
2b
2x
2a
dx2
x using Parseval‟s identity. (Dec 2008)
27) Find the Fourier transform of xf if
,0
,1xf if
if
0ax
ax
. Hence deduce that
02
dt
2
t
tsin .
28) Find the Fourier Cosine transform of 22
xae for any a>0 and hence prove that
2/xe2 is self-reciprocal under Fourier Cosine transform.
29) Find the Fourier transform of
,0
,2
x2
axf if
if
0ax
ax
. Hence deduce that
04
dt3
t
tcosttsin .
41
30) Find
ax
eCF ,
2x1
1CF and
2x1
xCF . (Hence
CF stands for Fourier Cosine
transform)
UNIT-III APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS
PART-A (TWO MARKS WITH ANSWERS)
1) Classify the following second order partial differential equations:
i)0168644
2
22
2
2
u
y
u
x
u
y
u
yx
u
x
u .
ii) 22
2
2
2
2
y
u
x
u
y
u
x
u .
iii) 0322 22 uuuxxyuuy xyyxyxx.
iv) 07222 yxyyxx uuuuy .
v) 0 yyxx xuu .
2) Classify the partial differential equations t
u
x
u
22
2 1
.
.3)Classify the following PDE
i) xuxuux yyxyxx 4)1( 2 .
ii) 02)1(2 22 xyyxyxx uuyxyuux .
4) What is the constant 2a in the wave equation xxtt uau 2 ?
(or) In the wave equation 2
22
2
2
x
yc
t
y
what does c2 stands for?
5) In the diffusion equation 2
22
x
u
t
u
what does 2 stand for?
6) What is the basic difference between the solutions of one dimensional wave equation and one
dimensional heat equation?
7) What are the possible solutions of one dimensional wave equation?
8) Explain the various variables involved in one dimensional wave equation.
9) A tightly stretched string of length 2L is fastened at both ends. The midpoint of the string is
displaced to a distance „b‟ and released from rest in this position. Write the initial conditions.
10) Write the initial conditions of the wave equation if the string has an initial displacement but
no initial velocity.
42
11) Write the boundary conditions and initial conditions for solving the vibration of string
equation , if the string is subjected to initial displacement f(x) and initial velocity g(x).
12) State one dimensional heat equation with initial and boundary conditions.
13) In steady state conditions derive the solution of one dimensional heat flow equation.
14) An insulated rod of length 60 cm has is ends A and B maintained at C20 and C80
respectively. Find the steady state solution of the rod.
15) A rod 30 cm long has its ends A and B kept at C20 and C80 respectively until steady
state
conditions prevail. Find the steady state temperature in the rod.
16) State Fourier law of heat conduction.
17) What are the possible solutions of one dimensional heat equation?
18) How many boundary conditions are required to solve completely 2
22
x
u
t
u
19) Write the steady state heat flow equation in two dimension in Cartesian & Polar form.
20) Write any two solutions of the Laplace equation obtained by the method of separation of
variables
21) In two dimensional heat flow, the temperature at any point is independent of which
coordinate?
22) Classify the p.d.e 02541 222 yyxyxx uuxuxx
23) State the empirical laws used in deriving one-dimensional heat flow equation.
24) Write the product solutions of 02 uruur rrr.
25) What is the equation governing the two dimensional heat flow steady state and also write its
26) Classify the p.d.e yxey
u
yx
u
x
u 32
2
22
2
2
2
.
27) Write the various possible solutions of the Laplace equation in two dimensions.
28) A infinitely long uniform plate is bounded by the edges lxx ,0 and the ends right angles
to them. The breadth of the edges 0y is l and is maintained at xf . All the other edges are
kept at .0 C Write down the boundary condition in mathematical form.
29) Define steady state. Write the one dimensional heat equation in steady state.
30) Write all the solutions of Laplace equation in Cartesian form, using the method of
separation of variables.
43
31) Verify that atxy coshcosh is a solution of .2
22
2
2
x
ya
t
y
32) A string is stretched and fastened to two point‟s l apart, motion is started by displacing the
string into form 0,0 sinx
y x yl
from which it is released from rest at time t=0 formulate the
B.V.P.
33) A taut string of length L cm fastened at both ends is disturbed from its position of
equilibrium by imparting to each of its points and initial velocity of magnitude ( )xk L x for
0 x L .Formulate the problem mathematically
34) Explain Boundary value problems.
35) Write any two assumptions made in the derivation of one dimensional wave equation.
.36) How many conditions are necessary to solve one dimensional wave equation, one
dimensional heat equation, two dimensional heat equations?
37) Write the initial conditions of the wave equation if the string has an initial displacement.
38) Write the boundary conditions for the following problem.A rectangular plate is bounded by
the lines 0, 0, ,x y x a y b .Its surfaces are insulated .The temperature along 0, 0x y
are kept at 00 C and the others at 0100 C .
39) What is the basic difference between the solutions of one dimensional wave equation and
one dimensional heat equation?
40) State the laws assumed to derive the one dimensional heat equation
41) Write the two dimensional steady state heat conduction equation.
42) Write any two assumptions made while deriving the partial differential equation of
transverse vibrations of a string.
43) A bar of length 50 cm has its ends kept at C20 and C100 until steady state conditions
prevails. Find the temperature t at any point.
44) A rectangular plate is bounded by the lines x=0, y=0, x=a, y=b. Its surfaces are insulated.
The temperature along x=0,y=0 are kept at C0 , other sides are at C100 .Write the boundary
conditions?
45) A rod of 50 cm long with insulated sides has its ends A and B kept at C20 and C70
respectively. Find the steady state temperature distribution of the rod
44
46) Classify uxx=uyy.
47) Classify uxx +4uxy+4uyy – 12ux +uy +7u =x2+y
2
48) Explain the term steady state.
49) A rod 40 cm long with insulated sides has its ends A and B kept at C20 and C60
respectively. Find the steady state temperature at a location 15 cm from A.
50) The ends A and B of a rod 1 cm long have the temperature C40 and C90 respectively
until steady state conditions prevail. Find the steady state temperature in the rod.
51. An insulated rod of length has its ends A and B maintained at respectively.
Find the steady state solution of the rod.
PART-B
1) A tightly stretched string of length ‘ l ’ has its ends fastened at x=0&x=l . The
midpoint of the string is then taken to a height „ h ‟ and then released from rest in
that position. Obtain an expression for the displacement of the string at any
subsequent time.
(Nov 2002)
2) A tightly stretched flexible string has its ends fixed at x=0 and x=l. At time t=0 , string
is
given a shape defined by ),()( 2 xlkxxf where k is a constant , and then released
from
rest. Find the displacement of any point x of the string at any time t > 0.
(April 2003)
3) A tightly stretched string with fixed end points x=0 and x=l is initially in a position
given by
l
xyxy
3
0 sin)0,( . It is released from rest from this position. Find the
displacement at anytime „ t ‟.
(Nov 2004)
4) A tightly stretched string of length „ 2l ‟ has its ends fastened at x=0 , x=2l. The
midpoint of the string is then taken to height „ b ‟ and then released from rest in that
position. Find the lateral displacement of a point of the string at time „ t ‟ from the
instant of release. (May 2005)
45
5) A string of length ‘ l ’ has its ends x=0 , x=l fixed. The point where 3
lx is drawn
aside a small distance „h ‟,the displacement ),( txy satisfies .2
22
2
2
x
ya
t
y
Find
),( txy at any time „t‟.
6) An elastic string of length „ 2l ‟ fixed at both ends is disturbed from its equilibrium
position by imparting to each point an initial velocity of magnitude ).2( 2xlxk Find
the displacement function ),( txy .
(May ‘06)
7) A uniform string is stretched and fastened to two points x=0 and x= l apart. Motion is
started by displacing the string into the form of the curve ),( xlkxy and then
releasing it from this position at time t=0. Find the displacement of the point of the
string at a distance „ x ‟ from one end at time „ t ‟. (A.U. Nov/Dec 2008,
Dec 2008, May/June 2009,Nov/Dec 2010)
8) If a string of length „ l ‟ is initially at rest in its equilibrium position and each of its
points is given a velocity „ v ‟ such that ( )
cxv
c l x
for
for
02
2
lx
lx l
show that the
displacement at any time„ t ‟ is given by
2
3 3
4 1 3 3( , ) sin sin sin sin ...
3
l c x at x aty x t
a l l l l
.
9) A string is stretched between two fixed points at a distance 2l apart and the points of
the string are given initial velocities „ v ‟where
(2 )
cx
lv
cl x
l
in
in
0
2
x l
l x l
„x’ being
the distance from one end point .Find the displacement of the string at any subsequent
time.
46
10) The ends A and B of a rod „ l ‟ cm long have the temperatures 40 C and 90 C until
steady state prevails. The temperature at A is suddenly raised to 90 C and at the same
time that at B is lowered to 40 C . Find the temperature distribution in the rod at time „
t ‟ . Also show that the temperature at the midpoint of the rod remains unaltered for all
time , regardless of the material of the rod.
(April 2003)
11) A metal bar 10 cm long with insulated sides , has its ends A and B kept at 20 C and
40 C until steady state conditions prevail. The temperature at A is then suddenly
raised to 50 C and at the same instant that at B is lowered to 10 C . Find the
subsequent temperature at any point of the bar at any time .
(Nov/Dec 2005)
12) The ends A and B of a rod „ l ‟cm long have their temperatures kept at 30 C and
80 C , until steady state conditions prevail. The temperature at the end B is suddenly
reduced to 60 C and that of A is increased to 40 C . Find the temperature distribution
in the rod after time „ t ‟.
(M/J’ 07)
13) The boundary value problem governing the steady state temperature distribution in a
flat, thin , square plate is given by
2 2
2 20,
u u
x y
0 x a , 0 y a , ( ,0) 0u x ,
3( , ) 4sinx
u x aa
, 0 x a (0, ) 0u y , ( , ) 0u a y , 0 y a .Find the steady-
state temperature distribution in the plate.
(Nov 2002)
14) A rectangular plate with insulated surface is 10 cm wide so long compared to its width
that it may be considered infinite length. If the temperature along short edge y=0 is
given by ( ,0) 8sin10
xu x
when 0 10x , while the two long edges x=0 and
x=10 as well as the other short edge are kept at 0 C , find the steady state temperature
function ( , )u x y .
47
(Nov 2003)
15) An infinitely long rectangular plate with insulated surface is 10 cm wide. The two
long edges and one short edge are kept at zero temperature while the other short edge
x=0 is kept at temperature given by 20
20(10 )
yu
y
for
for
0 5
5 10
y
y
. Find the
steady state temperature in the plate.
(Nov/Dec ‘05,Nov ‘04,Dec ‘08)
16) 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 temperature at short edge y=0 is given by 20
20(10 )
xu
x
for
for
0 5
5 10
x
x
and all
the other three edges are kept at C0 . Find the steady state temperature at any point in
the plate.
(May 2005,Nov/Dec 2010)
17) Find the steady state temperature distribution in a rectangular plate of sides a and b
insulated at the lateral surface and satisfying the boundary conditions
(0, ) ( , ) 0u y u a y for 0 y b , ( , ) 0u x b and ( ,0) ( )u x x a x for 0 x a .
(Nov/Dec 2005)
18) An infinitely long plate in the form of an area is enclosed between the lines
0,y y for positive values of x. The temperature is zero along the edges
0,y y and the edge at infinity. If the edge x=0 is kept at temperature „ Ky(l-y)’ ‟
find the steady state temperature distribution in the plate.
(May 2006)
19) An infinitely long uniform plate is bounded by two parallel edges and an end at right
angle to them. The breadth of this edge x=0 is , this end is maintained at
temperature as 2( )u K y y at all points while the other edges are at zero
temperature . Find the temperature ( , )u x y at any point of the plate in the steady state.
(April/May 2007)
48
20) A rod of length „‘l ’ has its ends „A‟ and „B‟ kept at 0 C and 120 C respectively
until steady state conditions prevail. If the temperature at „B‟ is reduced to 0 C and
kept so while that of „A‟ is maintained, find the temperature distribution in the rod.
(Dec 2008)
21) Find the steady state temperature in a circular plate of radius „a‟ cm, which has one
half of its circumference at 0 C and the other half at 100 C .
(Dec 2008)
22) Find the steady state temperature distribution in a square plate bounded by the lines
0, 0, 20, 20x y x y . Its surfaces are insulated, satisfying the boundary
conditions 0, 20, ,0 0& ,20 20u y u y u x u x x x . (Dec
2008,Nov/Dec 2011)
23) 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.
If the temperature of the short edge y=0 is given by u x for 0 5x and
10 x for 5 10x and the two long edges x=0,x=10 as well as the other short
edges are kept at C0 . Find the temperature u(x,y) at any point (x,y) of the plate in
the steady state. (May/June 2009)
24) A string of length 2l is fastend at both ends. The mid point of the string is taken to a
height
b and then released from rest in that position. Show that the displacement is
1
2 21
8 ( 1) (2 1) (2 1)( , ) sin cos
(2 1) 2 2
n
n
b n x n aty x t
n l l
(Dec 98, Nov/Dec ‘06, Nov/Dec ‘10)
25) 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
temperature at short edge x=0 is given by 10
10(20 )
yu
y
for
for
0 10
10 20
y
y
and the two
long edges as well as and the other three edges are kept at C0 . Find the steady state
temperature distribution in the plate. (May 2011)
49
26) A tightly stretched string with fixed end points x=0 andx=l is initially at rest in its
equilibrium position. If it is set vibrating giving each point a initial velocity , 3x(l-
x)find the displacement. (Nov/ Dec 2011)
UNIT -V
Z-TRANSFORM
PART-A (TWO MARKS WITH ANSWERS)
1. Define two-sided or bilateral Z-transform
2.Define one-sided or unilateral Z-transform .
3.Define Z-transform for discrete values of t.
4. Prove that
2
1
zZ n
z
(or) Find the z transform of {n}.
5. State the final value theorem in Z-transform.
6. Find nz a n
7.Find
!n
aZ
n in Z-transform.
8.Find iatZ e
using Z-transform.
9.State and prove initial value theorem in Z-transform.
10. Find 1
( 1)Z
n n
11.Find the Z-transform of 1 2n n
12.Find the Z-transform of 2n
13.Find
nZ
1 .
14.Evaluate 1
2 7 10
zZ
z z
.
15.Prove that n zZ a
z a
is z a
16.Prove that 1
2( )
n zZ na
z a
.
17.Prove that 11
!zZ e
k
.
50
18.Find 2Z an bn c
.
19.Find the initial and final values of the function 1
2
1( )
1 0.25
zF z
z
.
20.Find the Z-transform of i) ( ) ( )of n f n n ii) ( ) ( )of n u n n iii) 1( ) ( 1)nf n a u n
iv) 1( ) ( )nf n na u n .
21.What is the Z-transform of 1
( )3
n
u n
.
22.State the convolution property of Z-transform.
23.State shifting theorem of Z-transform.
24.Find the Z-transform of 3 cos2
n n .
25.Find the Z-transform of ( , 0).nab a b
26.Prove that ( ) ( ) (0)Z f t T Z f z f
.
27.Find the Z-transform of 1
1n
.
28.Prove that
21
21 nz
Z n az a
29.Find the difference equation from 2ny n A nB
30.Find 1
2 9
zZ
z
.
31.Define unit impulse sequence and find its Z-transform.
32.Define convolution of two sequences.
33.Find the inverse Z-transform of 2
2 4
z
z
34.From the difference equation of 1 0
2 , 1nn ny y y ,find ny in terms of z .
35.Find Z f n ,where f n n for n= 0, 1, 2, ….
36.If F(s) is the fourier transform of f(x), then find the fourier transform of f(x-a) (or) Prove that
F[f(x-a)]= ( )iase F s
51
37.Prove that ( 1) (0)z f n zFz zf (or) Express ( 1)z f n interms of f z (A.U.2007)
38.Form the difference equation by eliminating the constants from 12n
nu a
39.If
2
( )31 1
2 4 4
zF z
z z z
, find f(0).
40.Write the value of the transformation ( )z nf t Answer: ( )dF z
zdz
41. Write the value of the transformation ( 1)nz Answer: 11
zif z
z
42. State the condition for which nz a u n exists. Answer: z a
43. Find the z-transform of cosn .
44.Find the Z-transform of sin2
n
45. Find the inverse Z-transform of 1 2
z
z z .
46. 46. Find Z[-1]-1
47. Show that the Z-transform is linear.
48.Find
21
( )( )
zZ
z a z b
49.Find the Z-Transform of n
kC
50.Prove that
1 1 2
! ! !
n
n n n
51. Find the inverse Z-Transform of .
PART-B
1) Find 3
1
2( 1) ( 2)
zz
z z
using partial fraction. (N/D2005,
Dec2008)
2) Solve the difference equation ( 2) 4 ( 1) 4 ( ) 0y k y k y k where (0) 1, (1) 0.y y
(Nov/Dec 2005)
3) Prove that 1log
1 1
zz z
n z
.
(Nov/Dec 2005)
52
4) State and prove second shifting theorem in Z-transform.
(Nov/Dec 2005)
5) Using convolution theorem evaluate inverse Z-transform of 2
( 1)( 3)
zZ
z z
.
6) Using Z-transform solve ( ) 3 ( 1) 4 ( 2) 0, 2y n y n y n n given that (0) 3, (1) 2.y y
7) Find 2
1
2
( 2)
( 1)( 1)
z z zZ
z z
by using method of partial fraction.
8) Find Z-transform of 1
( 1)( 2)n n .
9) Using convolution theorem evaluate 2
1
( 1)( 2)
zZ
z z
.
10) Find Z-transform of na , cos and e sinn ata n bt .
11) Using the Z-transform method solve 2 2n ny y given that 0 1 0y y .
12) State and prove final value theorem in Z-transform.
(May/Jun 2007)
13) Find the inverse Z-transform of 3
( 1)
( 1)
z z
z
.
14) State and prove first shifting theorem on Z-transform. Also find atz e t .
(Apr/May 2008)
15) Use Z-transform to solve 2 17 12 2n
n n ny y y given 0 1 0y y .
(Dec 2008)
16) Find iatz e and hence deduce the values of cosz at and sinz at .
(Dec 2006)
17) Find 1
2( 1) ( 1)
zZ
z z
.
(May/Jun 2007)
18) Prove that 1p pdZ n z Z n
dz
where p is any positive integer. Deduce that 2( 1)
zZ n
z
and 2
2
3
2
( 1)
zZ n
z
.
19) Find the inverse Z-transform of 22 3
( 2)( 4)
z z
z z
.
(Apr/May 2008)
20) Find the inverse Z-transform of 1 2n n .
(Dec 2007)
21) Using Z-transforms, solve 2 3 1 4 0, 1,y n y n y n n given that 0 3y and
1 2y .
(Dec 2008)
22) Find the Z-transform of the sequence 1
1nf
n
(Dec 2008)
53
23) Find the inverse Z-transform of
2
2
2 4
2
z zF Z
z
using residue theorem.
(Dec 2008)
24) By using convolution theorem, prove the inverse of
2z
z a z b
is
1 11
n
n nb ab a
.
25) By the method of Z – transform solve 2 6 1 9 2ny n y n y n given that 0 0y
and 1 0y .
26) Find the Z – transform of c so n and hence find cosZ n n (May/June 2009).
27) Solve the equation (using Z – transform) 2 5 1 6 36y n y n y n given that
0 1 0y y
28) Find the Z-transform of cos and sinnn . Hence deduce that the transforms of
cos( 1) and a sinnnn (Nov/Dec 2010)
29) Form the difference equation from the relation 3n
ny a b (Nov/Dec 2010)
30) Use Z-transform to solve 2 1 0 14 3 2 with y 0 and y 1n
n n ny y y (Nov/Dec 2010)
31) Solve the difference equation ( 3) 3 ( 1) 2 ( ) 0y n y n y n where
(0) 4, (1) 0 and y(2)=8.y y
32) Derive the difference equation ( 3)n
ny A Bn (April/May 2011)
54
TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS
UNIT WISE IMPORTANT QUESTIONS
1. Form the PDE by eliminating the arbitrary function from the relation
.
2. Form the PDE by eliminating the arbitrary function and g from .
3. Form the P.D.E by eliminating the arbitrary function from
.
4. Form the Partial differential equation by eliminating arbitrary functions and g from
.
5. Find the PDE of all planes which are at a constant distance „K‟ from the origin.
6. Form the PDE by eliminating the arbitrary function and
from . [ A/M „15] R-08
7.
1. Solve: .
2. Find the singular integral of .
3. Find the singular integral of .
4. Solve : .
5. Solve: .
6. Solve: .
7. Solve :
8. Solve: .
Formation of PDE by eliminating funcions
Standard Types
Lagrange’s
Equations
A/M ’15 R-08
A/M ’15 R-08
A/M ’15 R-08
A/M ’15 R-08
55
1. Solve the Lagrange‟s equation .
2. Solve: .
3. Solve : .
4. Solve the partial differential equations .
5. Solve: .
6. Solve: .
7. Solve the Partial differential equation .
1. Solve: .
2. Solve: .
3. Solve: .
4. Solve: .
5. Solve: .
6. Solve : .
7. Solve:
1. Solve: .
2. Solve: .
3. Solve:
4. Solve: .
UNIT II
1. Find the Fourier series expansion of .
2. Find the Fourier series of in of periodicity .
Full Range Fourier series in ,
Linear PDE with second and higher order (Homogeneous)
Linear PDE with second and higher order (Non -Homogeneous)
56
3. Expand as Fourier series in and hence deduce that the sum of
.
4. Find the Fourier series of periodicity 3 for in .
5. Find the Fourier series expansion of also deduce that
.
1. Expand as a full range Fourier series in the interval
.
Hence deduce that .
2. Find the Fourier series expansion of
3. Find the Fourier series of in and hence deduce that .
4. Find the Fourier series expansion of in .
5. Obtain the Fourier series to represent the function and deduce
.
6. Find the Fourier series of in .
7. Find the Fourier series expansion of also deduce that
.
1. Obtain the half range cosine series for in .
2. Find the half-range Fourier cosine series of in the interval .Hence
find the sum of the series .
Odd and Even functions ,
Half-Range Fourier Series in ,
57
3. Obtain the Fourier cosine series expansion of in . And hence find the value
of
4. Find the half range sine series of in .
5. Find the half-range sine series of in the interval . Hence deduce
that the value of the series .
6. Obtain the Fourier cosine series of
1. Expand as a complex form of Fourier series in .
2. Find the complex form of Fourier series of in , where “a” is not an integer
1. Compute the first three harmonics of the Fourier series for from the following data:
x 0
2
f(x) 1.0 1.4 1.9 1.7 1.5 1.2 1.0
2. Find the Fourier series as far as second harmonic represent with period 6 given in
the following data:
x 0 1 2 3 4 5
f(x) 9 18 24 28 26 20
3. The following table gives the variations of a period current over a period. Find the
fundamental and first harmonics of to express as a Fourier series.
x 0
T
f(x) 1.98 1.30 1.05 1.30 -0.88 -0.25 1.98
Full Range Fourier series in Full Range Fourier series in
1 Expand as
Fourier series in and
hence deduce that the sum of
.
1 Expand as Fourier
series in and hence deduce
that the sum of
[A/M‟ 11]
Complex form of Fourier series
Harmonic Analysis
58
[N/D‟14, A/M‟ 08]
2 Find the Fourier series
expansion of
also deduce that
.
[A/M‟08]
2
Find the Fourier series expansion
of
.
[N/D‟13]
Full Range Fourier series in Full Range Fourier series in
1 Find the Fourier series of
periodicity 3 for
in .
[N/D‟11] [N/D‟14]
1
Find the Fourier series expansion
of
also
deduce that
.
[N/D‟12]
2 Find the Fourier series
expansion of
also
deduce that
and .
[N/D‟96] [N/D‟03] [N/D‟07]
Odd and Even functions Odd and Even functions
1 Find the Fourier series of in
and hence deduce that
(i)
(ii)
(iii) .
[M/J‟13] [N/D‟14]
[A/M‟15]
1
Expand
as
a full range Fourier series in the
interval . Hence deduce
that .
[M/J‟14]
2 Obtain the Fourier series to
represent the function
and
deduce .
[M/J‟12]
3 Obtain the Fourier series to
represent the function
2 Obtain the Fourier series
59
of
periodicity .
[A/M‟04][A/M‟15]
of periodicity .
[N/D‟05][M/J‟06]
4 Find the Fourier series
expansion of
[N/D‟13]
3 Find the Fourier series expansion
of
in .
[N/D‟12]
Odd and Even functions Odd and Even functions
1 Obtain the Fourier series to
represent the function
.
[M/J‟07][N/D‟08]
Half-Range Fourier Series in Half-Range Fourier Series in
1 Obtain the half range cosine
series for in
and hence deduce that
.
[N/D‟12] ][N/D‟14]
1
Obtain the Fourier cosine series
expansion of in .
And hence find the value of
[N/D‟11]
2 Find the half-range Fourier
cosine series of
in the interval
.Hence find the sum of
the series
.
[M/J‟12]
3 Find the half-range Fourier
cosine series of
[A/M‟15]
Half-Range Fourier Series in Half-Range Fourier Series in
1 Find the half range sine series
of in .Find the
sum of .
[N/D‟07] [N/D‟10] [N/D‟12]
1
Find the half range sine series of
in .
[A/M‟15]
60
[N/D‟14]
2 Find the half range sine series
of in .
2
Find the half range sine series of
in .
Hence deduce that the value of
the series .
[N/D‟13]
3 Obtain the Fourier sine series of
[N/D‟13]
3
Obtain the Fourier cosine series
of
[M/J „13]
4 Find the half range cosine series
of in
[N/D‟14][M/J‟13]
Complex form of Fourier series Complex form of Fourier series
1 Expand as a
complex form of Fourier series
in .
[A/M‟15][M/J‟07][N/D‟09]
1 Expand as a complex
form of Fourier series in
.
[N/D‟07][N/D‟14]
2 Expand as a
complex form of Fourier series
in .
[M/J‟14]
2 Find the complex form of
Fourier series of in
, where “a” is not an
integer.
[M/J‟13]
Harmonic Analysis Harmonic Analysis
61
1 Compute the first three
harmonics of the Fourier series
for from the following
data:
x 0
2
f(x) 1.0 1.4 1.9 1.7 1.5 1.2 1.0
[A/M‟08][N/D‟13][M/J‟14][N/
D‟13][M/J‟14][N/D‟14][A/M‟1
5]
2 The following table gives the
variations of a period current
over a period. Find the
fundamental and first
harmonics of to express as
a Fourier series.
x 0
T
f(x) 1.98 1.30 1.05 1.30 -0.88 -0.25 1.98
[N/D‟09][N/D‟11][A/M‟15]
3 Find the Fourier series as far as
second harmonic represent
with period 6 given in the
following data:
x 0 1 2 3 4 5
f(x) 9 18 24 28 26 20
[A/M‟06][N/D‟12][M/J‟12][N/
D‟12][N/D‟10][N/D‟06]
62
UNIT III
1. A tightly stretched string has its ends fixed at and initially displaced in a arc
of length and then released from rest. Find the displacement of any point x of the
string at any time t> 0.
2. A string is stretched and fastened to two points x=0& x l apart. Motion is started by
displacing the string into the form 2( )y k lx x from which it is released at time t=0.
Find the displacement of any point on the string at a distance of x from one end time at time
t.
3. If a string of length l is initially at rest in its equilibrium position and each of its points is
given
the velocity 3
0 sinx
vl
, 0<x<l determine the displacement function y(x, t).
4. A string is stretched and fastened to two points x=0& x l apart. Motion is started by
displacing the string by giving each point a velocity 2( )y k lx x from which it is released
at time t=0.Find the displacement of any point on the string at a distance of x from one end
time at time t.
1. The ends A and B of a rod 40 cm long have the temperatures 00 C and 80 C until steady
state conditions prevails. If the temperature at B is suddenly reduced to 400 C and kept so
while that of A is kept to maintained, find the temperature distribution of the rod after time t.
2. Solve the equation subject to the conditions
and .
1. A rectangular plate with insulated surfaces is 20 cm wide and so long compared to its
width
that it may be considered infinite in length without introducing an appreciable error.
If the temperature of the short edge x = 0 is given by u
and the two long edges as well as the other short edge are kept at Find the steady state
temperature distribution in the plate.
2. A square plate bounded by the lines x =0, y =0, x =20 and y =20. Its faces are insulated,
the temperature along the upper horizontal edge is given by u(x, 20) = x(20-x), 0< x < 20
while the other two edges are kept at 0 C . Find the steady state temperature distribution
in the plate.
One dimensional Wave equation (String problem)
One dimensional heat conduction (Rod problem)
Two dimensional heat conduction (Plate Problem)
63
3. A long rectangular plate with insulated surface is l cm wide. If the temperature along one
short edge ( y = 0) is u( x, 0)=k (lx-x2) degrees , for , while the other two long
edges and as well as the other short edge are kept at find the steady state
temperature function.
4. Find the Steady state temperature distribution in a rectangular plate of sides a and b
insulated
at the lateral surface and satisfying the boundary conditions
(0, ) ( , ) 0 0
( , ) 0 ( ,0) ( ) 0
u y u a y for y b
u x b and u x x a x for x a
.
5. An infinitely long rectangular plate is of width 10 cm. The temperature along the short
edge y=0
is given by
105),10(20
50,20
xx
xxu .I f all the other edges are kept at zero temperature,
find the steady state temperature at any point on it.
UNIT IV
1. Solve for from the integral equation
2. Find the Fourier integral representation of defined as
.
1. Find the Fourier Transform of and hence deduce that
(i) (ii) .
2. Find the Fourier transform of .
3. Find the Fourier transform of Hence show that
Fourier Integral theorem
Fourier Transforms and Properties
64
(ii) .
4. Find the Fourier transform of and hence find .
5. Obtain the Fourier transform of .
6. Find the Fourier transform of and hence evaluate
.
7. Find the Fourier transform of hence deduce that
.
1. Using Fourier cosine transform, evaluate .
2. Find the function whose Fourier sine Transform is , .
3. Find the Fourier cosine and sine transform of and hence deduce the
inversion formula.
4. Find the Fourier Cosine transform of .Hence show that the function
self-reciprocal.
5. Find the Fourier sine and cosine transform of and hence prove is self-reciprocal
under Fourier sine and cosine transforms.
6. Find the Fourier sine transform of and hence evaluate Fourier
cosine transform of and .
7. Find the Fourier sine transform of .
1. Use Parseval‟s identity evaluate the following integrals
Fourier sine and cosine transforms & Properties
Convolution theorem and Parseval‟s Identity
65
(i) (ii) where .
2. Evaluate by using Parseval‟s theorem.
3. Derive the Parseval‟s identity for Fourier Transforms.
4. State and prove convolution theorem on Fourier transform.
5. Evaluate using Fourier cosine transforms of and .
UNIT V
1. Find the Z-transforms of and .
2. Find the Z-transform of .
3. Find and hence find .
4. Find the Z – transform of and .
5. Find .
6. If , find and .
7. Find the Z-transforms of and .
8. State and prove the second shifting property of Z-transform.
1. Using convolution theorem, find the Z-1
of .
2. Using convolution theorem find the inverse Z-transform of .
3. Using convolution theorem, find the Z-1
of .
4. Using convolution theorem find .
5. Using convolution theorem find .
6. Evaluate: for .
Z – Transforms
Inverse Z – transforms ( Convolution theorem, Residue theorem and Partial fraction)
66
1. Form the difference equation from .
2. Form the difference equation from
3. Form the difference equation from .
1. Solve: given that , .
2. Solve : , given .
3. Solve : , given
4. Solve the difference equation given that
and .
5. Solve: given .
6. Solve the difference equation given that using
Z – transforms
7. Solve: using Z transforms.
8. Solve: given that , .
Formation of difference equation
Solution of difference equation
