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DR. BABASAHEB AMBEDKAR TECHNOLOGICAL UNIVERSITY, LONERE
End Semester Examination — Winter 2018
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Course: S.Y.B. Tech (All Branches) Semester: 111
Subject Name: Engineering Mathematics-IIT Subject Code: BTBSC301
Max Marks:60 Date:30/11/2018 Duration: 03 Hrs
Instructions to the Students:
- Attempt Any Five questions of the following .All questions carry equal marks.
- Use of non-programmable scientific calculators is allowed.
- Figures to the right indicate full Marks.
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Q. 1. a) Show that,
( b) Find the Laplace transform of !
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te”3%sin 2u [4]
c) Find the Laplace transform of the function
2 0 () =40 ,T sint t>2m 4] Q2. a) Find the inverse Laplace transform of cot™! sz? : [ b) By convolution theorem, find inverse Laplace transform of --- Content provided by FirstRanker.com --- s (s?2+ 1)(s2+4) ¢) By Laplace transform method, solve the [ollowing simultaneous [4] equations --- Content provided by FirstRanker.com --- dx dy = LA = Sint: gi = = E €55 ; + x =sint; giventhat x(0) =1,y(0) = 0. [4] Q.3. a) Find the Fourier transform of --- Content provided by FirstRanker.com --- _[1—-x?, ra = b) Find the Fourier sine transform of e~ ¥, and hence show that x| <1 - [4] --- Content provided by FirstRanker.com --- FirstRanker.com c) Using Parseval’s Identity , prove that B g2 T fo i %= ; W Q.. a) Solve the partial differential equation --- Content provided by FirstRanker.com --- (x? —y2)p + (y? —zx)q = 2> — xy. 41 b) Use method of separation of variables to solve the equation ou = 2@- +u; given that u(x,0) = 6e 3%, dax ot [4] c) Find the temperature in bar of length 2 units whose ends are kept at --- Content provided by FirstRanker.com --- zero temperature and lateral surface insulated if initial temperature is [4] sin (%) + 3sin 5%) Q.5. a) If f(z) is analytic function with constant modulus, show that f(z) is constant . [4] --- Content provided by FirstRanker.com --- b) If the stream function of an electrostatic field is 1 = 3xy? — x3, find the potential function ¢, where f(z) = ¢ + . “ ¢) 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 n through the origin . --- Content provided by FirstRanker.com --- Q6. a) Evaluate ¢ ; (Z—e_zz—)dz , where c is the circle |z| = 3. 4] b) Evaluate §_ tanz dz, where ¢ is the circle |z| = 2. n c) Evaluate , using Cauchy’s integral formula : [4] ¢ 5 %S—zg%) dz around a rectangle with vertices 2 + i, —2 % i. ey --- Content provided by FirstRanker.com --- 2) éc% dz , where C is the circle |z| = 1. edede End sk FirstRanker.com --- Content provided by FirstRanker.com ---
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