Download DBATU B-Tech 3rd Sem and 4th Sem 2018 Dec Engineering Mathematics 1 Question Paper

Download DBATU (Dr. Babasaheb Ambedkar Technological University) B.Tech 3rd Sem and 4th Sem 2018 Dec Engineering Mathematics 1 Question Paper

,
DR. BABASAHEB AMBEDKAR TECHNOLOGICAL UNIVERSITY, LONERE
End Semester Examination ? Winter 2018
Course: S.Y.B. Tech (All Branches) Semester: HI
Subj ect Name: Engineering Mathematics?III Subject Code: BTBSC301
Max Marksz60 Datez30/11/2018 Duration: 03 Hrs
Instructions to the Students:
1. Attempt Any F ive questions of the following .All questions carry equal marks.
2. Use ofnon?programmable scienti?c calculators is allowed.
3. F igures t0 the right indicate full Marks.
Q. 1. a) Show that,
?? sin at 7r
f d=_.
o t Z [4]
b) Find the Laplace transform of
?3u - 4
late sm Zu du. [ l
u
c) Find the Laplace transform of the function
2 ,0 < t < 1:
f(t)=[0 ,7r < t < 27r [4]
sin t , t > 211'
Q'2 ' a) Find the inverse Laplace transform of cot?1 (52E). [4]
b) By convolution theorem, ?nd inverse Laplace transform of
s
(52 + 1)(s2 + 4)' [4]
C) By Laplace transform method, solve the following simultaneous
equations
dx dy [4]
? ? = t ' -?? : I - I = 1 :
dt 3 , dt + x smt, gtven that x(O) ,y(0) O.
Q. 3. a) Find the Fourier transform of
_ 1 ? x2 , lxl S 1
??0?? ,le>1. [4]
b) Find the F ourier sine transform of 6"?, and hence show that
fmm?dx = "9"", m > 0. [41
0 1+x2 2
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c) Using Parseval?s Identity , prove that
f? t2 n
? dt = ? .
0 (t2 + 1)2. 4 4 . [41
Q4. a) Solve the partial differential equation
(x2 ? y2)p + (y2 ? ZXM = z2 ? xy. [4]
b) Use method of separation of variables to solve the equation
Bu au
__ ?3x
a_x ? 2 at + u; given that u(x, 0): 6e . [4]
c) Find the temperature in bar of length 2 units whose ends are kept at
zero temperature and lateral surface insulated if initial temperature is
sin(?7 x) + 3 $1 in (s?Zx). [4]
Q- 5- a) If f (z) is analytic function with constant modulus, show that f (z) is
constant . i [4]
b) If the stream function of an electrostatic ?eld is 1,!) = 3x312 ? x3 , ?nd the
potential function ()5, where f (z) = (p + ill). [4]
c) Prove that the inversion transformation maps a circle in the z-plane into a
circle in w-plane or to a straight line if the circle in the z?plane passes
[4]
through the origin .
Q'6' a) Evaluate s?c?dz, where c is the circle IzI? ? 3. [4]
b) Evaluate 55c tanz dz, where c is the circle lzl = 2. [4]
c) Evaluate ,using Cauchy?s integral formula : [4]
1) ?e L?gszanfj dz around a rectangle with vertices 2 i i , ?2 i i.
2) 56W 5?? :2 3dz where C IS the circle [Zi- ? 1
*ks?r End **>'e
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