||''|'''|''|''|'''||Subject Code: R13102/R13I B. Tech I Semester Supplementary Examinations August - 2015 MATHEMATICS-I (Common to All Branches) Time: 3 hours Max. Marks: 70 Question Paper Consists of Part-A and Part-B Answering the question in Part-A is Compulsory,Three Questions should be answered from Part-B ***** PART-A 1.(a) Find Laplace transform of Dirac Delta function? (b) At the start of an experiment, there are 100 bacteria. If the bacteria follow an exponentialgrowthpatternwithratek=0.02,whatwillbethepopulationafter5hours?Howlong will it take for the population to double? (c) Discuss about Jacobian? (d) Explain about Laplace equation? (e) Find cos (f) Define extreme value? [4+4+4+3+4+3] PART-B 2.(a) Find the maximum and minimum values of x + y + z subject to1z1y1x1= + + (b) Solve 4 3 sin2 [8+8] 3.(a) Solve0dxdye x y 1y 21= + +) ( ) (tan (b) Using Laplace transforms, solve 2 5 3sin3. [8+8] 4.Solve the following heat problem for the given initial conditions. 22xuktu= ) ( ) , ( x f 0 x u = 0 t 0 u = ) , ( 0 t L u = ) , ( (a) ||

\| =Lx6 x f sin ) ( (b) ||

\| ||

\| =Lx 47Lx 912 x f sin sin ) ( [8+8] Page 1 of 2 Set No - 1 ||''|'''|''|''|'''||Subject Code: R13102/R13 5.(a) Find the Laplace transforms of the given functions (i)21 cos att `) (ii) ) sin( ) sinh( t 2 3 t 2 3 + (b) A bottle of soda pop at room temperature (920F) is placed in a refrigerator where thetemperature is 640F. After half an hour the soda pop has cooled to 810F. (i) What is the temperature of the soda pop after another half hour? (ii) How long does it take for the soda pop to cool to 700F? [8+8] 6.(a) Form the differential equation form . (b) Solve . [8+8] 7. A tightly stretched string with fixed end points 0 and is initially at rest in itsequilibrium position. If it is vibrating by giving to each of its points a velocity ,find the displacement of the string at any distance x from one end at any time t. [16] ***** Page 2 of 2 Set No - 1 ||''|'''|''|''|'''||Subject Code: R13102/R13I B. Tech I Semester Supplementary Examinations August - 2015 MATHEMATICS-I (Common to All Branches) Time: 3 hours Max. Marks: 70 Question Paper Consists of Part-A and Part-B Answering the question in Part-A is Compulsory,Three Questions should be answered from Part-B ***** PART-A 1.(a) State convolution theorem? (b) Find the Laplace transform of Heavisides function? (c) Discuss about Bernoullis equation? (d) Find (e) Explain about one dimensional heat equation? (f) Explain chain rule of partial differentiation? [3+4+3+3+5+4] PART-B 2.(a) Solve the following IVP and find the interval of validity for the solution. , ) ( 0dxdy1 x y 2 x 9 xy 22 2= + + + 3 0 y = ) ( (b) Determine the orthogonal trajectories of the family of circles x 2 + (y c) 2 = c 2 tangent to the x- axis at the origin. [8+8] 3.(a) Supposethatthepopulationofacolonyofbacteriaincreasesexponentially.Atthestart ofanexperiment,thereare6,000bacteria,andonehourlater,thepopulationhas increased to 6,400. How long will it take for the population to reach 10,000? Round your answer to the nearest hour. (b) Find and classify all the critical points ofxy 3 y x 4 y x f3 3 + + = ) , ( [8+8] 4.(a) Find the Laplace transforms of the given functions. (i)9 t 5 e e 6 t f3 t 3 t 5 + + =) ((ii)) cos( ) cos( ) ( t 6 e t 6 e t gt 3 t 3 + = (b) Find the Laplace transform of f(t) = |t 1| + |t + 1|, t 0[8+8] 5.(a) Solve cos cos ? (b) Using Laplace transforms, solve222 5 sintd y dyy e tdt dt+ + = . [8+8] Page 1 of 2 Set No - 2 ||''|'''|''|''|'''||Subject Code: R13102/R13 6.(a) Solve 2 3 0. (b) Solve tan tan[8+8] 7. A tightly stretched string with fixed end points x = 0 and x = p is initially in a position given by. sinpxy y30=If it is released from rest from this position, find the displacement y(x, t). [16] ***** Page 2 of 2 Set No - 2 ||''|'''|''|''|'''||Subject Code: R13102/R13I B. Tech I Semester Supplementary Examinations August - 2015 MATHEMATICS-I (Common to All Branches) Time: 3 hours Max. Marks: 70 Question Paper Consists of Part-A and Part-B Answering the question in Part-A is Compulsory,Three Questions should be answered from Part-B ***** PART-A 1.(a) Define saddle point? (b) Explain about law of natural decay? (c) Find (d) Give the statement of Convolution theorem? (e) Explain about wave equation? (f) Define functional dependence? [3+4+3+4+5+3] PART-B 2.(a) Show that the functions , arefunctionally dependent. (b) Form the partial differential equation by eliminating the arbitrary functionfrom: 0 xy 2 z z y x2 2 2 2= + + ) , ([8+8] 3.(a) Solve 1 (b) Solve the differential equation 4 tan2 [8+8] 4.(a)Find the inverse transform of each of the following. (i)21 s 8 ss 3 1s F2+ += ) ( (ii)10 s 3 s7 ss G2 += ) ( (b) A bottle of soda pop at room temperature (720F) is placed in a refrigerator where thetemperature is 440F. After half an hour the soda pop has cooled to 610F. (i) What is the temperature of the soda pop after another half hour? (ii) How long does it take for the soda pop to cool to 500F? [8+8] 5.(a) Using Laplace transforms, solve 2 5 sin2 (b) Find the orthogonal trajectories of the confocal and coaxial parabolas 21 [8+8] Page 1 of 2 Set No - 3 ||''|'''|''|''|'''||Subject Code: R13102/R13 6. A homogeneousrod of conductingmaterial of length 100 cm has its ends kept at zerotemperature and the temperature initially is, 0 ; 0 50100 ; 50 100 Find the temperature u(x, t) at any time. [16] 7.(a)Obtain the Taylors series expansion of sin x in powers of 4x (b)Solve 22yx xx ydxdy cos sintan = [8+8] ***** Page 2 of 2 Set No - 3 ||''|'''|''|''|'''||Subject Code: R13102/R13I B. Tech I Semester Supplementary Examinations August - 2015 MATHEMATICS-I (Common to All Branches) Time: 3 hours Max. Marks: 70 Question Paper Consists of Part-A and Part-B Answering the question in Part-A is Compulsory,Three Questions should be answered from Part-B ***** PART-A 1.(a) Form the differential equation from (b) Explain about Newtons law of cooling? (c) Find the inverse Laplace transform of

(d) Obtain Maclaurins series for (e) Solve (f)Find the particular integral of 2 [3+3+4+4+4+4] PART-B 2.(a) Solve 2. (b) Solve 1 1 [8+8] 3.(a) Use a convolution integral to find the inverse transform of the following transform. 2 2 2a s1s H) () (+= (b) Write the following function (or switch) in terms of Heaviside functions and its Laplacetransform.

[8+8] 4.(a) Solve the following differential equation. 4 sec 2. (b) Investigate for the maxima and minima, if any of 1 . [8+8] Page 1 of 2 Set No - 4 ||''|'''|''|''|'''||Subject Code: R13102/R13

5. A homogenous rod of conducting material of length 100 cm has its ends kept at zerotemperature and the temperature initially is , 0 ; 0 5 100 ; 50 100 Find the temperature u(x, t) at any time.[16] 6.(a) An object cools from 1200 to 950 F in half an hour when surrounded by air whosetemperature is 700F. Find its temperature at the end of another half an hour. (b) Form the differential equation from 2 3 2 3. [8+8] 7.(a) Solve 8 (b) Expand the function , log 1 in terms of x and y up to the terms of 3rd

degree using Taylors thereom? [8+8] ***** Page 2 of 2 Set No - 4 ||''|'''|''|''|'''||CodeNo: R10102/R10 SetNo. 1I B.TechISemester Supplementary Examinations,Aug. 2015MATHEMATICS-I(CommontoCivilEngineering,Electrical&ElectronicsEngineering,MechanicalEngineering,Electronics&CommunicationEngineering,ComputerScience&Engineering,ChemicalEngineering,Electronics&InstrumentationEngineering,Bio-MedicalEngineering,InformationTechnology,Electronics&ComputerEngineering,AeronauticalEngineering,Bio-Technology,AutomobileEngineering,MiningandPetroliemTechnology)Time: 3hours MaxMarks: 75AnsweranyFIVEQuestionsAllQuestionscarryequalmarks

1. (a) Solve(3x2y2+ x2) dx + (2x3y + y2) dy= 0(b) Ifthetemperatureofabodychangesfrom1000Cto700Cin15minutes,ndwhenthetemperatureofthebodywillbe400C,giventhatthetemperatureofthesurroundingsis300C [7+8]2. (a)Solved2ydx2 6dydx+ 9 y= ex+ e3x(b)Solve(D2+ 16)y= 0 [8+7]3. (a)Findthemaximumof15xyz4x+2y+4zgiventhatxyz=8.(b)Findthemaximumofx3y2zgiventhatx+3y+4z=10. [8+7]4. (a)Tracethecurvey= (x 2) (x + 3) (x 4) ..(b)Tracethecurver=12-sin. [8+7]5. (a) Findthesurfaceofthesolidgeneratedbytherevolutionofcardioidr=a(1-cos)abouttheinitialline.(b) Findthesurfaceofthesolidgeneratedbytherevolutionoftheellipsex2+ 4y2= 16aboutitsMajoraxis. [8+7]6. (a) Evaluate rdrdover the region bounded by the cardioid r=a(1+cos) andoutsidethecircler=a.(b) ChangetheorderofIntegration&evaluate

4a0

2axx24adydx [8+7]7. (a) Findtheangleofintersectionofthespheresx2+y2+z2=4andz=x2+y2+3atthepoint(2,-1,1).(b) Provethatdivgradrn=n(n+1)rn2. [8+7]8. (a)Ifsissurfaceofspherewithtwounitsradiusthenshowthat

S r.Nds=32(b) Evaluate

c f.drwhere f = 3x2i + (2xz-y) j + z k along the straight line C from(0,0,0)to(2,1,3). [8+7]

1of1||''|'''|''|''|'''||CodeNo: R10102/R10 SetNo. 2I B.TechISemester Supplementary Examinations,Aug. 2015MATHEMATICS-I(CommontoCivilEngineering,Electrical&ElectronicsEngineering,MechanicalEngineering,Electronics&CommunicationEngineering,ComputerScience&Engineering,ChemicalEngineering,Electronics&InstrumentationEngineering,Bio-MedicalEngineering,InformationTechnology,Electronics&ComputerEngineering,AeronauticalEngineering,Bio-Technology,AutomobileEngineering,MiningandPetroliemTechnology)Time: 3hours MaxMarks: 75AnsweranyFIVEQuestionsAllQuestionscarryequalmarks

1. (a)Solve(x2y3+ xy)dydx= 1(b) Find the orthogonal trajectory of the family of curves r = a(1 +Sin),whereaisaparameter [8+7]2. (a)Solve(D23D + 2)y= ex(b)Solve(D4a4) y= 0 [8+7]3. (a)IfU=f

yxxy,zxxz

,P.T.x2 fx+ y2 fy+ z2 fz= 0.(b)Expandu = xyinpowersof(x-1)and(y-1)uptothirddegreeterms. [8+7]4. (a)Tracethecurver=a(1+cos2).(b)Tracethecurver=3+2cos. [8+7]5. (a) Findthelengthof thearcof thesemi-cubical parabolaay2=x3fromthevertextotheordinatex=5a.(b) Findtheareaof thesurfaceof revolutiongeneratedbyrevolvingonearcofthecurvey=sinxaboutthex-axis. [8+7]6. (a) Evaluate

v dxdydzwhereVistheniteregionof spaceformedbytheplanesx=0,y=0,z= 0and2x + 3y + 4z= 12.(b) Evaluate R ydxdywhereRistheregionboundedbytheParabolasy2= 4x and x2= 4y. [8+7]7. (a) Provethat {f(r)r}=0(b) Findaunitvectorwhichisperpendiculartothesurfaceoftheparaboloidofrevolutionz=x2+y2atthepoint(1,2,5). [8+7]8. (a) If f = y i + z j + x k, nd the circulation of f round the curve C, where c is thecirclex2+y2=0,z=0.(b)If f =(x+y2)i 2xj +2yzk, evaluate

s f.NdswhereSisthesurfaceof theplane2x+y+2z=6intherstoctant. [8+7]

1of1||''|'''|''|''|'''||CodeNo: R10102/R10 SetNo. 3I B.TechISemester Supplementary Examinations,Aug. 2015MATHEMATICS-I(CommontoCivilEngineering,Electrical&ElectronicsEngineering,MechanicalEngineering,Electronics&CommunicationEngineering,ComputerScience&Engineering,ChemicalEngineering,Electronics&InstrumentationEngineering,Bio-MedicalEngineering,InformationTechnology,Electronics&ComputerEngineering,AeronauticalEngineering,Bio-Technology,AutomobileEngineering,MiningandPetroliemTechnology)Time: 3hours MaxMarks: 75AnsweranyFIVEQuestionsAllQuestionscarryequalmarks

1. (a) Solvedydx+ y Sec x = tanx(b) Findtheorthogonaltrajectoryofthefamilyofcurvesx2+ y2= 2 a x,whereaisaparameter [8+7]2. (a)Solvey116y1+ 25y= e2x+ x + Sinx(b)Solvey1113y11+ 4y= 0, y(0) = 1, y1(0) = 8, y11(0) = 4 [8+7]3. (a) Show that U = x2eycosh z, V = x2eysinh z, w = x2+y2+z2xyyzzxarefunctionallydependent.(b) Determine whether the functions U =xyz, V =yzx, W=zxyare dependent. Ifdependentndtherelationshipbetweenthem. [8+7]4. (a)Tracethecurvex2+ y2= xy.(b)Tracethecurvey2(2a x) = x3. [8+7]5. (a) Find the surface area generated by rotating the arc of the catenary y= a coshxafromx=otoaaboutthex-axis.(b) Findthevolumeof thesolidgeneratedbyrevolvingaboutthex-axisof theloopofthecurvey2= x2 (a+x)ax. [8+7]6. (a) Evaluate

10

1z0

1yz0xyz dxdydz.(b) Evaluate

(x+y+z)dxdydz taken over the volume bounded by the planesx = 0, x = 1; y= 0, y= 1; and z = 0, z = 1. [8+7]7. (a) Findtheunitnormalvectortothelevelsurfacesx2y+2xz=4atthepoint(2,-2,3)(b) Findthe directional derivative of the functionxy2+yz2+zx2alongthetangenttothecurvex=t,y=t2,z=tatthepoint(1,1,1) [8+7]8. Verify divergence theorem for 2x2yi-y2j+4xz2k taken over the region of rst octantofthecylindery2+z2=9andx=2. [15]

1of1||''|'''|''|''|'''||CodeNo: R10102/R10 SetNo. 4I B.TechISemester Supplementary Examinations,Aug. 2015MATHEMATICS-I(CommontoCivilEngineering,Electrical&ElectronicsEngineering,MechanicalEngineering,Electronics&CommunicationEngineering,ComputerScience&Engineering,ChemicalEngineering,Electronics&InstrumentationEngineering,Bio-MedicalEngineering,InformationTechnology,Electronics&ComputerEngineering,AeronauticalEngineering,Bio-Technology,AutomobileEngineering,MiningandPetroliemTechnology)Time: 3hours MaxMarks: 75AnsweranyFIVEQuestionsAllQuestionscarryequalmarks

1. (a) Solve(x2+ y3+ 6x) dx + (y2x) dy= 0(b) If thepopulationof acountrydoublesin50years, inhowmanyyearswillittriple, assumingthattherateofincreaseisproportional tothenumberofinhabitants? [8+7]2. (a)Solved2ydx2+y= 0 giventhaty(0) = 2, y

2

= 2(b)Solved2ydx2 6dydx+ 9 y= 5e2x[8+7]3. (a)Ifx=ercos ,y=ertan ,showthatJ

x,yr,

J

r,x,y

= 1.(b)FindTaylorsseriesexpansionofthef(x,y)=sin2xaboutx=4. [8+7]4. (a) Tracethecurver = 4.(b) Tracethecurver=14+ 2 sin . [8+7]5. (a) Findthesurfaceof thesolidgeneratedbyrevolutionof thelemniscater2=a2cos2abouttheinitialline.(b) Showthatthewholelengthofthecurvex2(a2x2) = 8a2y2is a2. [8+7]6. (a)Evaluate (x + y)2dx dy.overtheareaboundedbytheellipsex2a2+y2b2= 1.(b) Transform the following to Cartesian form and hence evaluate

0

a0r3sin drd.[8+7]7. (a) Provethat.

r

1r3

=3r4.(b) Find the directional derivative of (x,y,z) = x2yz+4xz2at the point P = (1,-2,-1) in the directional of the normal to the surface f(x,y,z) = xlogz-y2at (-1,2,1).[8+7]8. (a) Iff=yi+zj+xk,ndthecirculationoffroundthecurveC,wherecisthecirclex2+y2=0,z=0.(b) Iff=(x+y2)i2xj+2yzk,evaluate

s f.NdswhereSisthesurfaceoftheplane2x+y+2z=6intherstoctant. [8+7]

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