OU B.Com Last 10 Years 2010-2020 Question Papers || Osmania University
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DR. BABASAHEB AMBEDKAR TECHNOLOGICAL UNIVERSITY, LONERE
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End Semester Examination — May 2019
Course: B. Tech Sem: III
Subject Name: Engineering Mathematics-III Subject Code: BTBSC301
Max Marks: 60 Date: 28-05-2019 Duration: 3 Hr.
Instructions to the Students:
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- Solve ANY FIVE questions out of the following.
- The level question/expected answer as per OBE or the Course Outcome (CO) on which the question is based is mentioned in () in front of the question.
- Use of non-programmable scientific calculators is allowed.
- Assume suitable data wherever necessary and mention it clearly.
| (Level/CO) | Marks |
|---|---|
| Q.1 Attempt any three. | 12 |
| A) Find L{f(t)}, where f(t) = t2 e3tsinht > Understand | 4 |
| B) Express f(t) in terms of Heaviside's unit step function and hence find its Laplace transform where f(t) = { cost, 0<t<p sin t, t>p Understand | 4 |
| C) Find L{f'(t)}, where f(t) = 2t2 Understand | 4 |
| D) By using Laplace transform evaluate ?08 te-t (1-e-2t) dt Evaluation | 4 |
| Q.2 Attempt the following. | 12 |
| A) Using convolution theorem find L-1 { 1/(s2+4)2} Application | 4 |
| B) Find L-1{f (s)}, where f (s) = cot-1 (s/2) Application | 4 |
| C) Using Laplace transform solve y''' — 3y’ + 2y = 12e-2t;y(0) = 2, y'(0) =6 Application | 4 |
| Q.3 Attempt any three. | 12 |
| A) Express f(t) = {et, 0<t<p; 0, t>p as a Fourier sine integral and hence deduce that ?08 (1-cos(px))/x dx = p/2 Evaluation | 4 |
| B) Using Parseval's identity for cosine transform, prove that ?08 (sinat dt)/(t(a2+t2)) = p/(2a2) (1-e-2?) Application | 4 |
| C) Find the Fourier transform of f(x) = {1—x2, if |x| <1; 0 if |x| > 1}. Hence prove that ?08 ((xcosx—sinx)/x3) cos(x/2) dx = 3p/16 Understand | 4 |
| D) Find Fourier sine transform of 5e-2x + 2x Understand | 4 |
| Q.4 Attempt the following. | 12 |
| A) Form the partial differential equation by eliminating arbitrary function f from f(x +y +z,x2+y2+z2) =0 Synthesis | 4 |
| B) Solve xz(z2 + xy)p — yz(z2 + xy)q = x4 Analysis | 4 |
| C) Find the temperature in a bar of length two units whose ends are kept at zero temperature and lateral surface insulated if the initial temperature is sin(px/2) + 3sin(5px/2) Application | 4 |
| Q.5 Attempt Any three. | 12 |
| A) If the function f(z) = (x2 + axy + by2) + i(cx2 + dxy + y2) is analytic, find the values of the constants a, b, c and d. Understand | 4 |
| B) If f(z) is an analytic function with constant modulus, show that f(z) is constant. Understand | 4 |
| C) Find the bilinear transformation which maps the points z = 0, —i, —1 into the points w = i, 1,0. Understand | 4 |
| D) Prove that the function u = ex(xcosy — ysiny) satisfies the Laplace's equation. Also find the corresponding analytic function. Synthesis | 4 |
| Q.6 Attempt ANY TWO of the following. | 12 |
| A) Evaluate ?C z4z5 dz, where C is the circle |z+1—i| = 2. Evaluation | 6 |
| B) Find the residues of f(z) = 1/(zcosz) at its poles inside the circle |z| = 2. Understand | 6 |
| C) Evaluate ?C (sinmz+cosnz)/(z2(z—2)) dz, where C is the circle |z| = 3. Evaluation | 6 |
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