set no. 16. a right circular cone of a base diameter 50mm and height 60mm is placed such that one...

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Code No: R10204/R10 Set No. 1

I B.Tech II Semester Supplementary Examinations, August 2014ENGINEERING CHEMISTRY -II

( Common to Civil Engineering, Electrical & Electronics Engineering,Mechanical Engineering, Electronics & Communication Engineering,

Computer Science & Engineering, Chemical Engineering, Electronics &Instrumentation Engineering, Bio-Medical Engineering, Information

Technology, Electronics & Computer Engineering, AeronauticalEngineering, Bio-Technology, Automobile Engineering, Mining and

Petroliem Technology)Time: 3 hours Max Marks: 75

Answer any FIVE QuestionsAll Questions carry equal marks

? ? ? ? ?

1. (a) Differentiated the following with suitable examples.(i) Polymer from monomer(ii) Step polymerization from chain polymerization(iii) Homo polymer from co – polymer(b) Explain the properties which are influenced by structure of the polymer? [6+9]

2. (a) Write a note on the constituents (Compounding) of plastics?(b) Explain the Extrusion moulding of plastics? [10+5]

3. (a) Define vulcanization?(b) How the process of vulatanisation is carried out?(c) What are the advantages of vulcationsation? [2+6+7]

4. (a) Describe the production of carbon nanotubes by laser ablation method?(b) Discuss the applications of fullerenes.(c) Explaine the properties of carbon nanotubes? [8+4+3]

5. (a) What are the different methods of manufacturing cement? Write any one ofthem.

(b) Write the acidic and neutral type of the refractories and their uses. [8+7]

6. (a) Write the informative note on origin of petroleum.

(b) How are lubricants classified? Mention the additives added to the lubricantsand give their functions. [7+8]

7. What are Corrosion inhibitors? Explain with examples how anodic and cathodicinhibitors provide protection against corrosion and explain passivity and its signif-icance. [15]

8. Define green chemistry and outline the twelve basic principles of green chemistry.Explain the Microwave induced method for green systhesis [15]

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1. (a) Give the preparation, properties, and uses of the Flexi glass?

(b) What is the Zeigler – Natta Catalyst? Give its role in the coordination additionmechanism? [5+10]

2. (a) Discuss briefly fabrication of plastic articles?

(b) What is the kelvar? How is it prepared, give its properties and useses? [10+5]

3. (a) Give the method of preparation, properties and application of Buna-S Rubber.

(b) Give the method of preparation properities and application of Buna N Rubber[8+7]

4. (a) Describe the production of carbon nano tubes by arc discharge method

(b) Mention the properties of fullerenes

(c) Write briefly about the physical properties of carbon nanotubes [8+4+3]

5. (a) Explain the manufacture of cement in detail(b) Define and classify refractories with examples [8+7]

6. (a) What is meant by cracking of petroleum? Explain moving bed catalytic crack-ing method of obtaining gasoline.

(b) Indicate the preparation and uses of lithium grease [8+7]

7. (a) State Pilling Bedworth rule. Explain its significance.

(b) Explain the difference in the use of anodic and cathodic coatings for corrosionprevention. [7+8]

8. (a) Discuss the principles (at least seven) of green chemistry.(b) Write briefly with suitable examples the supercritical Fluid extraction methodfor green synthesis. [7+8]

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? ? ? ? ?

1. (a) How are tensile strength and optical properties influenced by the structure ofplastics

(b) What is meant by Coordination polymerization? Explain its mechanism?[9+6]

2. (a) Explain the useses of plastics in the building, construction & medical sector?(b) What is a Thermo plastics resin? Give an example?(c) Define the fibres and matrixes? [6+5+4]

3. What is synthetic Rubber and describe the preparation properties and uses of anytwo synthetic Rubbers. [15]

4. (a) How various types of carbon nano tubes can be formed from grapheme?

(b) Discuss how nano technology useful.

(c) What are the advancements of nano technology in Electronics Field [7+4+4]

5. (a) Give an account of(i)Constituents of Portland cement (ii) Hydration of Portland cement

(b) Write short notes on pyrometric cone test and thermal spalling [8+7]

6. (a) Distinguish between thermal and catalytic cracking.(b) Write short notes on cetane number(c) What is viscosity index of oil? How it is important property? [5+5+5]

7. Define corrosion of metals. What are different types of corrosion? Explain theelectrochemical theory of wet corrosion giving its mechanism. [8+7]

8. (a) What is Green Chemistry? Write briefly about Engineering Applications ofGreen Chemistry?

(b) Discuss any four Principals of the Green Chemistry. [7+8]

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? ? ? ? ?

1. (a) Explain the term polymerization and co-polymerization with suitable exam-ples?

(b) What are the conducting polymers, name four of them & give their engineeringapplications? [9+6]

2. (a) Discuss briefly fabrication of plastic articles?

(b) What is the kelvar? How is it prepared, give its properties and useses? [10+5]

3. (a) Write the structre for natural rubber and Guttapercha and also write thedifference between the two

(b) Why Natural rubber needs vulcanization [8+7]

4. (a) Explain SWNT & MWNT(b) Describe any one method for the production of nanotubes.(c) Descuss the application of fullerenes [5+7+3]

5. (a) Explain the role of gypsum in setting and hardening of cement

(b) Define glazed ceramics

(c) What are refractories? Give an account of any three characteristics of a goodrefractory material [5+5+5]

6. (a) What is octane number? Explain how the structural features of constituenthydrocarbons of gasoline affect the petrol rating.

(b) How are liquid lubricants classified? Discuss the various methods available forrefining mineral oils. [8+7]

7. Write notes on metallic coatings and special paints? [15]

8. What is Green Chemistry and how is it important. Discuss any three Syntheticmethods used in green chemistry [15]

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I B.Tech II Semester Supplementary Examinations, August 2014ENGINEERING DRAWING( Common to All Branches)

Time: 3 hours Max Marks: 75Answer any FIVE Questions

All Questions carry equal marks? ? ? ? ?

1. The distance between two fixed points is 60mm.A point P moves such that the sumof its distance from two fixed points is always a constant and is equal to 80 mm.Draw the locus of P and determine the axes lengths. [15]

2. (a) Draw the projections of a line 70 mm long when it is parallel to both the VPand HP. The line is 40 mm from both HP and VP.

(b) Draw the projections of a line 60 mm long when it is perpendicular to VP andparallel to HP and 20 mm above HP. One end of the line is in VP. [8+7]

3. The top view of a line is 80 mm long and inclined to XY at 450. One end is 20mm above HP and 30 mm in front of VP. The other end is 55 mm above HP and isinfront of VP. What is the true length of the line and its inclination with HP andVP? Also show its traces. [15]

4. A square plane of side 45 mm has its surface parallel to the VP and perpendicularto the HP. Draw its projection when one of the sides is inclined at an angle of 300

to HP and the surface is inclined at 450 to VP. [15]

5. A square prism of side 40mm and 90mm long is resting with one of its generatoron the ground. Two of its rectangular faces are making equal angle with HP andaxis perpendicular to VP. Draw its projections. [15]

6. A right circular cone of a base diameter 50mm and height 60mm is placed suchthat one diameter AB of the base inclined at 450 to HP and another diameter CDof the base parallel to both HP and VP. Draw its projections of cone [15]

7. Draw the front view looking from the direction of X, and top view, side views fromthe following figure. [15]

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8. A hemisphere of radius 30mm is placed centrally on a square slab of side 40mm andthickness 30mm so that the flat circular surface is on the top. Draw the isometricprojection of the solids in the given position. [15]

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1. On a map, 100 cm x 120 cm represents an area of 2400 m2. Draw a diagonal scaleshowing meters, decimeters, and centimeters and to measure up to 3 meters. Showa length of 1.64 meters on it. [15]

2. (a) A line AB 50 mm long is in the VP and parallel to the profile plane. The endA is 20 mm above HP. Draw all the three principal views.

(b) A point 30 mm above XY line is the front view of two points E and F. The topview of E is 40 mm behind the VP, and the top view of F is 50 mm in front ofthe VP. Draw the projections of the two points and state their positions withreference planes and quadrants in which they lie? [8+7]

3. A straight line AB is 80 mm long. Its one end is in VP and the other end is inHP. Its top and front views measure 60 mm and 70 mm respectively. Draw itsprojections and determine its inclinations with the HP and the VP. [15]

4. Draw a rhombus of diagonals 100 mm and 60 mm long with the longer diagonalhorizontal. The figure is the top view of a square of 100 mm long diagonals, witha corner on the ground. Draw its front view and determine the angles made itssurface with the ground. [15]

5. A cube of 50mm side is resting on the HP with its vertical faces equally inclined tothe VP. Draw its projections. [15]

6. Draw the projections of a pentagonal pyramid having side of base 30 mm and lengthof axis 70 mm when it is resting with a triangular face in VP. [15]

7. Draw the orthographic projections to the figure shown below. [15]

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8. Draw the isometric view of the block as shown in figure. [15]

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1. The distance between two fixed points is 65 mm. A point P moves such that thesum of its distance from two fixed points is always a constant and is equal to 100mm. Draw the locus of P and determine the axes lengths. [15]

2. (a) Draw the projections of a line 60 mm long when it is parallel to both the HPand VP. The line is 30 mm from both HP and VP.

(b) Draw the projections of a line 70 mm long when it is perpendicular to VP andparallel to HP and 30 mm above HP. One end of the line is in VP. [8+7]

3. The front view of a line PQ 155 mm measures 135 mm and its top view measures115 mm. The mid-point of the line PQ is 65 mm from both the planes. Draw theprojection of the line PQ. [15]

4. A circular lamina of diameter 70 mm has the end A of the diameter AB in the HPand the end B in VP. Draw its projections when its surface is inclined at 600 to theHP and 300 to the VP. [15]

5. A square prism, side of base 35 mm and height 50 mm rests with its base on HP.Such that one of its rectangular faces is inclined at an angle of 300 to VP. Draw itsprojections. [15]

6. Draw the projections of a pentagonal pyramid side of base 30 mm and altitude 60mm when it is resting on its base on the HP with an edge of base inclined at anangle of 300 to the VP. [15]

7. Draw orthographic projections to the following isometric view. [15]

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8. A sphere of radius 25mm is resting centrally on the top of a square block of side is50mm and thickness is 15mm. Draw the isometric projection of the combination.

[15]

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1. Construct a vernier scale to read distance correct to decameter on a map in whichthe actual distances are reduced in the ratio of 1:20000. The scale should be longenough to measure up to 4 km. Mark on the scale a length of 2.46 km and 3.28km. [15]

2. A line PQ has end P 30 mm above HP and 30 mm infront of VP. End Q is 40 mmabove HP and 50 mm infront of VP. The distance between the end projectors is 50mm. Draw the projections of the line, find true length of the line and its inclinationwith HP and VP. [15]

3. The top view of a 75 mm long line CD measures 50 mm. The mid point of the lineis 50 mm from VP and 75 mm from the HP. The point D is 30 mm from VP. Drawits projections and find its inclinations with the HP and VP. [15]

4. (a) Draw the projections of a regular hexagon as plane of 40 cm side, having oneof its sides on the ground and its surface perpendicular to VP, making anangle of 300? with the ground.

(b) A pentagonal lamina of side 25mm is resting on one of its sides on HP suchthat the surface parrallel is parallel to profile plane. [8+7]

5. A triangular prism, base 40 mm side and height 65 mm is resting on the groundwith one of its rectangular faces such that the axis of the prism is perpendicular tothe profile plane. Draw its projections. [15]

6. A hexagonal pyramid of base side 20 mm and axis 60 mm has a central hole of 30mm dia along the axis to 1/3 rd the height of the pyramid. It is resting on its sideon the H.P. and the axis making an angle 450to the HP. Draw its projections. [15]

7. Draw orthographic projections to the following isometric view. [15]

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8. Draw the isometric view of the bracket as shown in figure. All the dimensions arein mm. [15]

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I B.Tech II Semester Supplementary Examinations, August 2014ENGINEERING PHYSICS -II






? ? ? ? ?

1. (a) What are matter waves. Explain their properties.

(b) Show that the wave length λ associated with an electron of mass, m andKinetic energy, E is given byλ = h√

2mE, where h is planck’s constant.

(c) Calculate the wavelength associated with an electron with energy 2000eV[8+4+3]

2. (a) Explain briefly the classical free electron theory of metals.

(b) Derive an expression for electrical conductivity on the basis of classical freeelectron theory. [8+7]

3. (a) State and explain Bloch theorem in a periodic potential field.

(b) Explain the classification of solids into conductors, semiconductors and insu-lators. [7+8]

4. (a) What is ferromagnetic Hysteresis? Explain retentivity and coercivity. Explainferromagnetic hysteresis on the basis of Domain theory?

(b) Classify & Distinguish ferromagnetic materials on the basis of the hysteresisloop. [8+7]

5. (a) What is the Superconductivity? Explain Meissner effect.

(b) Mention few industrial applications of superconductors.

(c) Calculate the critical current which can flow through a long thin supercon-ducting wire of aluminium of diameter 10−3m. The critical magnetic field foraluminium is 7.9x103amp/m. [6+5+4]

6. (a) What is ionic polarizability? Derive an expression for the ionic polarizability.Explain frequency dependence of ionic polarizability.

(b) If all the molecular dipoles in a 1.0 cm radius water drop are pointed in thesame direction, calculate the intensity of polarization. Dipole moment of thewater molecule is 6 x 10−30 C-m. [9+6]

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7. (a) Derive an expression for carrier concentration in intrinsic semiconductor.

(b) The forbidden gap in pure silicon is 1.1eV. Compare the number of conductionelectrons at temperatures 370C and 270C. [11+4]

8. (a) What are nanomaterials? Why do they exhibit different properties?

(b) Write the applications of Nano Technology. [9+6]

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? ? ? ? ?

1. (a) What is quantum computer? Explain the advantages of quantum computingover classical computation.

(b) Explain briefly about multiple qu bit. [8+7]

2. (a) Mention different mechanisms responsible for electrical resistance in metals.

(b) Find the temperature at which there is 1% probability of a state with anenergy 0.5 eV above Fermi energy level will be occupied [8+7]

3. (a) Discuss the motion of an electron in a periodic potential field and explain theformation of energy bands.

(b) What is the effective mass of an electron? Derive the expression for the effec-tive mass of an electron moving in energy bands of a solid. [7+8]

4. (a) Write the classification of Magnetic Meterial.

(b) Explain hysteresis loop observed in ferromagnetic materials.

(c) The area of the hysteresis loop between B and H is 100 m2. Each unit spacealong the vertical axis represents 0.001 Wb/m2and each unit space along thehorizontal represents 40A/m. Determine the Hysteresis loss per cycle.

[6+5+4]

5. (a) Explain the following:(i)Flux quantization.(ii)Penetration depth.

(b) Explain critical magnetic field of a superconductor as a function of tempera-ture.

(c) For lead, Tc = 7.2K. Calculate the ratio of penetration depth at 3K to thatat 0K. [6+5+4]

6. (a) Define dielectric polarizability and susceptibility.

(b) Derive the expression for electronic polarizability.

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(c) The dielectric constant of helium, measured at 00 C and at one atmosphere is1.0000684; under these conditions the gas contains 2.7x1025atoms/m3. Calcu-late the atomic radius. [4+7+4]

7. (a) Distinguish between conductors, insulators and semiconductors.

(b) The intrinsic carrier density at room temperature in Ge is 2.37X 1019 /m3.If the electron and hole mobilities are 0.38 and 0.18 m2 /v/s respectively,calculate the resistivity. [9+6]

8. (a) What is Quantum Confinement? Explain density of states for various typesof Quantum Confinement.

(b) Explain Ball Milling process and Sol-Gel process in nanomaterials. [8+7]

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? ? ? ? ?

1. (a) Explain the Physical significance of Wave function.

(b) Derive time dependent Schrodinger wave equation. [7+8]



3. (a) State and explain Bloch theorem in a periodic potential field.

(b) Explain the classification of solids into conductors, semiconductors and insu-lators. [7+8]

4. (a) Define the following terms(i)Magnetic permeability(ii) Magnetic susceptibility(iii) Coercivity and(iv) retentivity

(b) Derive the relation between B, H and I

(c) A magnetic material has intensity of magnetization of 1550 A/m and fluxdensity of 0.0022Wb/m2. Calculate the magnetizing force and relativity per-meability of the material. [8+4+3]

5. (a) Write general properties of superconductors.

(b) Draw the magnetization curves for Type-I & Type-II superconductors andmention different regions.

(c) The lead material works as superconductor at a temperature of Tc = 7.26K.If H0= 8 x 105 A/m find critical magnetic field at 5K. [6+5+4]

6. (a)Deduce the relation between polarizability (α) and relative permittivity∈r.(b) Explain briefly the Ferro electricity and Piezoelectricity. [7 + 8]

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7. (a) Explain Hall Effect. Show that for a p-type semiconductor the hall coefficientRH= 1/ne.(b) write any four applications of Hall effect. [11+4]

8. (a) What are the types of Carbon nanotubes? Mention their properties(b) How the physical and chemical properties of nano-particles vary with their size?

[10+5]

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? ? ? ? ?

1. (a) What is a quantum computer? Explain briefly how it works.(b) What is Bloch’s Sphere and describe any two properties of it. [8+7]



3. (a) Explain the formation of allowed and forbidden energy bands on the basis ofthe Kronig- Penny model.

(b) Write short notes on “effective mass” (m∗) of an electron. [8+7]

4. (a) Define the following(i)Magnetic moment(ii) Intensity of magnetization(iii) Magnetizing force(iv) magnetic flux density

(b) Explain the origin of magnetic moment at the atomic level.

(c) Find the relative permeability of a ferromagnetic material if a field of strength110 A/m produces a magnetization of 3300 A/m. [8+5+2]

5. (a) Distinguish the variation of resistivity with temperature in normal and super-conductors.

(b) Mention important property changes that occur in materials when they changefrom normal to superconducting state.

(c) Distinguish energy gap in semiconductors and superconductors. [4+7+4]

6. (a) Describe the various of processes of dielectric polarization.(b) Derive expressions for electronic & ionic polarizabilities. [9+6]

7. (a) Derive an expression for carrier concentration in intrinsic semiconductor.

(b) The forbidden gap in pure silicon is 1.1eV. Compare the number of conductionelectrons at temperatures 370C and 270C. [11+4]

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8. (a) Write a detailed note on nano science and nanotechnology

(b) Describe briefly the fabrication methods of nanomaterials [7+8]

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I B.Tech II Semester Supplementary Examinations, August 2014ENVIRONMENTAL STUDIES

( Common to Civil Engineering, Electrical & Electronics Engineering,Computer Science & Engineering, Electronics & Instrumentation

Engineering, Aeronautical Engineering, Bio-Technology and AutomobileEngineering)



1. Write a detailed note on various Governmental institutions and Organizations inthe field of public awareness and Preservation of natural assets? [15]

2. Discuss the impact of land use on environmental quality? Discuss soil erosionand discuss the factors which influence the extent of soil erosion with preventivemeasures. [15]

3. (a) What is the importance of forests in ecological balance

(b) All matter of production in ecosystem is subject matter of productivity inecosystem? Discuss. [8+7]

4. (a) Enumerate some of the common plants and animal species which are found inIndia

(b) Write briefly about different types of diversity in ecosystems. [8+7]

5. (a) How does marine pollution occur?

(b) What are the International activities to control marine pollution? [8+7]

6. (a) Do you think moral values can prevent pollution in the environment? Explainyour view?

(b) Define climate change? Explain the sources and effects of Global warming?[8+7]

7. (a) What is meant by Environmental Health?(b) Write the causes of population growth?(c) Problems of family welfare programmes [4+6+5]

8. (a) Give a brief account of survey of river ecosystem that you have studied.

(b) What are the types of forests you can observe in the terrestrial Ecosystem[8+7]

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1. (a) Explain why public awareness is needed with regard to Environmental issuesalong with any case study?

(b) Briefly discuss what is the protocol to be followed to make aware the publicabout environmental issues? [8+7]

2. Discuss the reasons for land degradation and suggest steps to control it and explainwhat is Soil management, Shifting cultivation, Desertification and Man inducedlandslides. [15]

3. (a) Explain about different zones of a ocean?(b) Define an Ecosystem, Explain Ecological succession.

[8+7]

4. (a) Write about few birds and their contribution to the environment?

(b) What are the ecological services rendered by forests? Cite the examples ofaesthetic, recreational, economic, historical, cultural and religious values offorest around your place. [5+10]

5. (a) Write short notes on(i)Recycling of waste (ii)Vermicomposting

(b) How solid waste can be managed? [8+7]

6. (a) Write short notes on sources, effects and control measures of(i)Acid rains (ii)Ozone layer depletion?

(b) Give a detailed account of methods used in water conservation? [8+7]

7. (a) Discuss the role of Information Technology in Environment of Human health.

(b) what is the most disturbing factors about population growth in the world?[8+7]

8. (a) What is the use of stratified structure of multilayered vegetation in the forestregion?

(b) Explain the Biotic and Abiotic components present in the different aquaticEcosystems (Salt water, fresh water) [8+7]

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1. (a) What needs to be done to ward off the menace of carbon-emissions?(b) Explain ‘stone cancer’. Discuss the role of ozone layer in the atmosphere. [8+7]

2. Describe the water movement among earth components with a diagrammatic rep-resentation? Explain the present state of water resources and write a detailed noteon over-utilization of surface and ground water. [15]

3. (a) What are decomposers? Explain their role in the ecosystem.(b) Discuss the major feature of a stream (river) that differs from that of a lake.

[7+8]

4. (a) Write about poaching of wildlife in respect of Indian context.(b) Discuss about genetic diversity with examples [8+7]

5. (a) Distinguish between sound, noise and noise pollution?(b) Discuss adverse effects and control of water pollution? [8+7]

6. (a) Define Rain water harvesting? Explain modern and ancient methods of Rainwater harvesting to conserve water?

(b) Explain watershed management reclamation practices? [8+7]

7. (a) What are the major chemical infections to organisms?

(b) Why is life expectancy is dropping in many African countries? [8+7]

8. (a) Give a account of behavior of some common birds.(b) What are the components of river environment? [8+7]

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1. Write a detailed note on various Governmental institutions and Organizations inthe field of public awareness and Preservation of natural assets? [15]

2. (a) Explain what is Cortour farming? Give what are the measures to be takento improve the world’s food production in order to meet the needs of growingpopulation.

(b) Explain the role of forest resources in national economy of India. [8+7]

3. (a) Describe the abiotic and biotic components of a grassland ecosystem(b) With a neat sketch, explain how the element carbon is recycled in nature. [8+7]

4. (a) Explain India As a mega diversity nation.

(b) Explain about indirect values of biodiversity. [8+7]

5. (a) Discuss the ways to minimize radiation exposure in case of nuclear accidents?

(b) Define solid waste? Write the main principles (components) of solid wastemanagement? [8+7]

6. (a) “Population, consumerism & waste production are inter related” – comment?

(b) Discuss the salient features of(i)Air Pollution (Prevention & Control) Act , 1981(ii)Water Pollution (Prevention & Control) Act , 1974 [8+7]

7. Write short notes on

(a) Doubling time

(b) Total fertility rate

(c) Zero population growth

(d) Life-expectancy. [4+4+3+4]

8. (a) What are the types forests in India define them and give proper examples.

(b) What is the role of river water in preparing the EIA report for an Industry?[8+7]

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I B.Tech II Semester Supplementary Examinations, August 2014MATHEMATICAL METHODS

( Common to Mechanical Engineering, Electronics & CommunicationEngineering, Chemical Engineering, Bio-Medical Engineering, InformationTechnology, Electronics & Computer Engineering, Mining and Petroliem

Technology)Time: 3 hours Max Marks: 75


? ? ? ? ?

1. (a) Find rank of A =

2 1 30 1 24 0 2

1−26

by reducing it to Normal Form

(b) Solve by Gauss seidal method, x+4y+15z=24, x+12y+z=26, 10x+y-2z=10[7+8]

2. Find Eigen vectors of A =

2 2 11 3 11 2 2

[15]

3. Reduce the quadratic form 7x2 + 6y2 + 5z2 − 4xy − 4yz to canonical from bydiagonlzation. Also find the nature, index and signature and the linear transfor-mation. [15]

4. (a) Solve the equation x3+2x2+0.4=0 using Newton’s -Raphson’s Method uptothree decimal places.

(b) Show that the iteration scheme φ(x) = −1x2−3

converges and hence find a realroot of f(x)= x3-3x+1=0 near x=3. [8+7]

5. (a) Use gauss forward interpolation formula to estimate f(32), given f(25) =0.2707, f(30) = 0.3027, f(35) = 0.3386, f(40) = 0.3794.

(b) Find the interpolating polynomial f(x) from the table given below.x 0 1 4 5f(x) 4 3 24 39

[8+7]

6. (a) The velocity v of a particle moving in a straight line covers a distance at timet. They are related as shown in the following Table. Find v (x) at x= 10 andx = 15.x 0 10 20 30 40v 45 60 65 54 42

(b) Find the area bounded by the cure y = x3 – x + 1, x – axis between x=0 andx=1.2 by using(i) Trapezoidal Rule (ii) Simpson’ 1/3 rule. [8+7]

7. (a) Solve y1=-xy2, y(o)=2 by modified Euler’s method and hence find y(o.1),y(o.2)

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(b) Solve dydx

= y2−x2

y2+x2 , y(o)=1 by fourth order R-K method and hence find y(o.2),

y(o.4) [8+7]

8. (a) Fit a least square parabola y= a+bx+cx2 to the data (-1,2),(0,1),(1,4)

(b) By the method of least squares fit a straight line to the following datax 5 10 15 15 20y 15 19 23 26 30

[8+7]

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1. (a) Using Echelon form, find rank of A =

1 2 1−2 4 31 0 2

008

(b) Solve system of equations x+y+z=3 , 2x+3y+2z=7 , 4x+2y+3z=9, using

Gauss elimination method. [7+8]

2. Verify Cayley – Hamilton theorem and find A−1if A =

1 0 32 1 −11 −1 1

[15]

3. (a) Define quadratic form, rank and signature. Write the symmetric matrix cor-responding to the quadratic form x1x3 + x2x3 + x1x3 + x1x4 + x2x4 + x3x4.

(b) Discuss the nature of the quadratic form x2−y2 +4z2 +4xy+6xz+2yz [7+8]

4. (a) Using Newton-Raphson’s Method, find a positive root of Cosx-xex=0

(b) Find a real root of f(x) =x+tanx-1=0 in the interval (0, 0.5) by using bisectionmethod. [8+7]

5. The following table gives the population of a town during the last six censuses. Es-timate, using Newton’s interpolation formula, the increase in the population duringthe period 1986 to 1988. [15]year 1911 1921 1931 1941 1951 1961Population (in thousands) 12 15 20 27 39 52

6. (a) Computef′(1)using the given data:

X 1.0 1.5 2.0 2.5 3.0f(x) 27 106.75 324 783.75 1621

(b) Using Simpson’s 3/8th rule evaluate∫ 6

0dx

1+x2 by dividing the range into 6 equalparts [8+7]

7. (a) Solve y1=3x2+1 by Euler’s method and find y at x=2 by taking h=0.5

(b) Solve by fourth order R-K method y1=x-y, y(1)=0.4 and hence find y(1.2)[8+7]

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8. (a) Fit a second degree polynomial to the following data by the method of leastsquaresx 0 1 2 3 4y 1 1.8 1.3 2.5 6.3

(b) Fit a straight line of the form y= a+bx to the following datax 0 5 10 15 20 25y 12 15 17 22 24 30

[8+7]

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? ? ? ? ?

1. (a) Find rank of matrix using Echelon form A =

1 2 −42 −1 38 1 9

567

(b) Solve the equations using Gauss Jordan method

x+5y+z=9 , 2x+y+3z=12, 3x+y+4z=16 [7+8]

2. Verify Cayley – Hamilton theorem and find A−1 if A =

2 2 11 3 11 2 2

[15]

3. (a) Define quadratic form, rank and signature. Write the symmetric matrix cor-responding to the quadratic form x1x3 + x2x3 + x1x3 + x1x4 + x2x4 + x3x4.

(b) Discuss the nature of the quadratic form x2−y2 +4z2 +4xy+6xz+2yz [7+8]

4. (a) Solve the equation x3+2x2+0.4=0 using Newton’s -Raphson’s Method uptothree decimal places.

(b) Find a real root of x = e−x, using Bisection method up to four iterations.[8+7]

5. (a) Find the value of y from the following data at x = 0.47X: 0 1 2 3 4 5Y: 1 2 4 7 11 16

(b) Use Lagrange’s interpolation formula, find f(4) from the following data.x 1 2 5 6 9Y = f(x) 2 8 17 20 35

[8+7]

6. (a) For the function y = f(x) given by the following Table, find y′

at x = 0.04using the Bessel’s formula.x 0.01 0.02 0.03 0.04 0.05 0.06y 0.1023 0.1047 0.1071 0.1096 0.1122 0.1148

(b) Evaluate∫ 4

0e1/xdx by using the Simpson’s 3/8th rule, by dividing the interval

into 3 equal parts. [8+7]

7. (a)Solve y1=y+ex , y(o)=0 by modified Euler’s method and find y(??), y(??)(b) Solve y1=-xy2, y(o)=2 , h=0.2 by R-K method and hence find y(o.2), y(o.4)

[8+7]

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8. (a) Find the best fit of the type y=aebx to the data by the method of least squaresx 1 5 7 9 12y 10 15 12 15 21

(b) Obtain the relation of the form y=abx to the following data by the method ofleast squaresx 2 3 4 5 6y 8.3 15.4 33.1 65.2 127.4

[8+7]

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? ? ? ? ?

1. (a) Find rank of a Matrix using Echelon form where A =

1 −1 20 1 25 3 14

014

(b) Show that equations x+y+z=6 , x+2y+3z=14 x+4y+7z=30 are consistent

and solve them [7+8]

2. Verify Cayley - Hamilton theorem and find A4 if A =

1 2 −12 1 −22 −2 1

[15]

3. Find the transformation which will transform 4x2 + 3y2 + z2−8xy−6yz+ 4zx intoa sum of square and find the reduced from [15]

4. (a) Using Newton-Raphson’s method find the square root of a number and hencefind the square root of 24.

(b) Find a real root of the equation x=e−x, using Bisection method [8+7]

5. (a) Find the value of y from the following data at x = 0.47X: 0 1 2 3 4 5Y: 1 2 4 7 11 16

(b) Use Lagrange’s interpolation formula, find f(4) from the following data.x 1 2 5 6 9Y = f(x) 2 8 17 20 35

[8+7]

6. (a) From the following data find f ’ (0)x 0 1 2 3 4 5f(x) 43 40 38 42 45 50

(b) By considering 4 strips, find the value of∫ 7

3x2 log xdx [8+7]

7. (a) Solve y1=xy1/3,y(1)=1 by Taylor series method and find y(1.1), y(1.2)

(b) Find an approximate value of y for x=0.1, 0.2 if y1=x+y and y(1)=1 byPicard’s method and compare the solution with exact solution. [8+7]

8. (a) Fit a second degree polynomial to the following data by the method of leastsquaresx 0 1 2 3 4y 1 1.8 1.3 2.5 6.3

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(b) Fit a straight line of the form y= a+bx to the following datax 0 5 10 15 20 25y 12 15 17 22 24 30

[8+7]

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I B.Tech II Semester Supplementary Examinations, August 2014MATHEMATICS- II






? ? ? ? ?

1. (a) Find the Laplace transform of cos33t(b) Find the Laplace transform of f(t) = |t− 1|+ |t+ 1|, t≥0 [7+8]

2. (a) Find L−1[s2/(s2 + 4)(s2 + 25)].(b) Find L−1 {tan−1(s+ 1)}. [7+8]

3. (a)Find cosine series and sine series for the function f(x)=x2 in (0,π) find sum ofthe series 1/12-1/22+1/32+. . . .(b) Find cosine series and sine series for the function f(x)=π-x in (0,π) [8+7]

4. Find the finite Fourier sine and cosine transform of f(x) = π3−x+ x2

2πwhere 0<x<π

[15]

5. (a) Solve x2p+y2q=xz(b) Solve (3y+2z)p+(4z-3z)q = -(2x+4y) [8+7]

6. A homogenous rod of conducting material of length 100 cm has its ends kept at zero

temperature and the temperature initially is u(x, o) =

{x; 0 ≤ x ≤ 5100− x; 50 ≤ x ≤ 100

.

Find the temperature u(x, t) at any time. [15]

7. (a) Find the inverse Z-transform of z2

(z− 12)(z− 1

4)

(b) Evaluate Z−1(

zz2+11z+24

). [8+7]

8. (a) Evaluate∫ 1

0x3√

1− x dx,using Beta and Gamma functions.

(b) Prove that∫∞

0x2n−1 e−ax

2dx = Γ(n)

2an, a > 0, n > 0. [8+7]

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? ? ? ? ?

1. (a) Find the Laplace transform of(i) cos (at+b)(ii) sin(at-b)

(b) Find the Laplace transform of(i) e4t cos4t(ii) t4e5t [7+8]

2. (a) Find L−1{

(s2−1)(s2+1)2

}.

(b) Using Laplace transforms, solve(D2 + 1)x = t cos 2t, given thatx = dx

dt= 0 at t = 0. [7+8]

3. Find the half range Fourier cosine series of f(x)= sin(πx/L) in the range o<x<L[15]

4. (a) Find the fourier transform of f(x)defined by f(x)=x, |x|=a =0, |x|>a

(b) Find the fourier transform of f(x)defined by f(x)=cos x, |x|=a, f(x) =0,|x|= a >0 [8+7]

5. (a) Solve p2+q2=x2+y2

(b) Solve yz p – xz q = xy [8+7]

6. A bar 100 cm long with insulated sides has its ends kept at 0oc and 100oC re-spectively until steady state conditions prevail. Then the temperature 0oc is raisedto 30oc and that of 100oc is lowered to 60oc. Find the subsequent temperaturedistribution. [15]


(z− 12)(z− 1

4)

(b) Evaluate Z−1(

zz2+11z+24

). [8+7]

8. (a) Prove that Γ(m)Γ(m+ 1

2

)=

√π

22m−1 Γ(2m)

(b) Prove that∫∞0

xn e−a2x2dx = 1

2an+1 Γ(n+12

),n>-1 [8+7]

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? ? ? ? ?

1. (a) Find the Laplace transform of(i) cosh(at+b)(ii) sint(at-b)

(b) Find the Laplace transform of(i) e4t cos4t(ii) t4e5t [7+8]

2. (a) Find L−1[log(

s2+1s2+9

)].

(b) Find inverse Laplace transform of 1s(s2−a2) [7+8]

3. (a) Express f(x)=x as a half range Fourier cosine series in (– π, π)(b) Express f(x)=x2 as a Fourier series in (– π, π) [8+7]

4. Find the fourier cosine and sine transform of f(x) defined by f(x) = e−ax/x and

hence evaluate that∫∞0

e−ax−e−bx sin sx dxx

[15]

5. (a) Form the partial differential equation by eliminating arbitrary function fromf(x2+y2, x2-z2)=0(b) Solve 2p + 3q = 1 [8+7]

6. The ends A and B of a rod 20cm long have the temperature at 30oc and 80oc untilsteady states prevail. The temperatures of the ends are changed at 40oc and 60ocrespectively. Find the temperature distribution in the rod at time t. [15]


(z− 12)(z− 1

4)

(b) Evaluate Z−1(

zz2+11z+24

). [8+7]

8. (a) Show that Γ(n) = 1n

∫∞0e−x

1n dx, n > 0

(b) Evaluate∫ 1

0x7(1 − x)5dx using Beta and Gamma functions. [8+7]

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? ? ? ? ?

1. (a) Find L(t(3 sin2t – 2 cos2t))(b) Find L(t2 e−2t cost) [7+8]

2. (a) State and prove Convolution theorem for inverse Laplace transform(b) Show that f*g=g*f where f(t)=t3, g(t)=2t [7+8]

3. The intensity of an alternating current after passing through a rectifier is given byf(x) =i0 sin x for 0= x=π, f(x) = 0 for π = x=2π where i0=maximum current andthe period is 2 π. Express f(x) as a Fourier series [15]

4. Find the Fourier cosine and sine transform of f(x) defined by f(x) = e−ax/x and

hence evaluate that∫∞

0e−ax−e−bx sin sx dx

x[15]

5. (a) Solve p2+q2=x2+y2

(b) Solve yz p – xz q = xy [8+7]

6. A tightly stretched string with fixed end points x=0 and x=l is initially at rest inits equilibrium position. If it is vibrating by giving to each of its points a velocityλx(l − x), find the displacement of the string at any distance ‘x’ from one end atany time ‘t’. [15]

7. (a) If Z(un) = u(z), then prove that Z(n un) = −z ddz

[u(z)]

(b) Evaluate Z−1(

z2

z2−5z+6

), using convolution theorem. [8+7]

8. (a) Given∫∞

0xn−1

1+xdx = π

sinnπ, show that Γ(n) Γ(1 − n) = π

sinnπand deduce

Γ(

14

)Γ(

34

)(b) Prove that

∫ π/20

[√tan θ +

√sec θ

]dθ = 1

2Γ(

14

){Γ(

34

)+

√π

Γ( 34)

}[8+7]

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set no. 16. a right circular cone of a base diameter 50mm and height 60mm is placed such that one...

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