PYQ EM-III Engineering Mathematics-III B. Tech - 2022 PDF

# DR. BABASAHEB AMBEDKAR TECHNOLOGICAL UNIVERSITY, LONERE ## Winter Examination - 2022 **Course:** B. Tech **Subject Code & Name:** BTBS301 Engineering Mathematics-III **Branch:** Common for All branches **Semester:** III **Max. Marks:** 60 **Date:** 09/03/2023 **Duration:** 3-Hrs **Instructions to...

# DR. BABASAHEB AMBEDKAR TECHNOLOGICAL UNIVERSITY, LONERE ## Winter Examination - 2022 **Course:** B. Tech **Subject Code & Name:** BTBS301 Engineering Mathematics-III **Branch:** Common for All branches **Semester:** III **Max. Marks:** 60 **Date:** 09/03/2023 **Duration:** 3-Hrs **Instructions to the Students:** 1. All the questions are compulsory. 2. The level of 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. 3. Use of non-programmable scientific calculators is allowed. 4. Assume suitable data wherever necessary and mention it clearly. | (Level/CO) | Marks | |---|---| | | 12 | | L3/CO1 | 4 | | L3/CO1 | 4 | | L3/CO1 | 4 | | L3/CO1 | 4 | ## Q.1 Solve Any Three of the following. **(A)** Find Laplace Transform of $e^{-3t} sin2 t$ **(B)** Find Laplace Transform of $f(t) = \begin{cases} 1 \quad 0<t<1\\ 0 \quad 1<t<2 \end{cases}$ where $f(t)$ is periodic function of period 2. **(C)** Evaluate using Laplace Transform: $\int_{0}^{\infty} \frac{cos4t - cos3t}{t} dt$ **(D)** Find Laplace Transform of $(1 + 2t - 3t^2 + 4t^3)H(t-2)$ | | 12 | |---|:---:| | L3/CO2 | 4 | | L3/CO2 | 4 | | L3/CO2 | 4 | | L3/CO2 | 4 | ## Q2 Solve Any Three of the following. **(A)** Find the inverse Laplace transformation of the function. $log (1 + \frac{1}{s^2})$ **(B)** By using convolution theorem find $L^{-1}\{\frac{s}{(s^2+4)(s²+9)}\}$ **(C)** Find the inverse Laplace transformation of the function. $\frac{5s^2-15s-11}{(s+1)(s-2)^2}$ **(D)** Solve using Laplace transformation $y'' + 3y' + 2y = δ(t − 1)$ for which $y(0) = y'(0) = 0$ ## Q.3 Solve Any Three of the following. **(A)** Using Parseval's identity prove that: $\int_{0}^{\infty} \frac{x^2}{(x^2+1)^2} dx = \frac{\pi}{4}$ **(B)** Find the Fourier transform of $f(x) = \begin{cases} 1-x^2 \quad |x|≤1 \\ 0 \quad |x|>1 \end{cases}$ **(C)** Find the Fourier Sine transform $e^{-ax}$, a > 0 **(D)** Find the Fourier cosine transform of the function $f(y) = \begin{cases} cosy \quad 0 < y < a \\ 0 \quad y > a \end{cases}$ ## Q.4 Solve Any Three of the following. **(A)** Form the partial differential equation by eliminating arbitrary constants from $(x − a)^2 + (y − b)^2 = z^2 cot^2 a$ **(B)** Solve the Partial differential equation $x(y − z)p + y(z − x)q = z(x − y)$ **(C)** Use the method of separation of variables to solve $\frac{∂u}{∂x} = 2 + u$ given that $u(x, 0) = 6e^{-3x}$ **(D)** A bar with insulated at its ends is initially at temperature $0^\circ C$ throughout. The end $x = 0$ is kept at $0^\circ C$ for all times and the heat is suddenly applied so that $\frac{∂u}{∂x} = 10$ at $x = t$ for all time. Find the temperature function $u(x, t)$ ## Q.5 Solve Any Three of the following. **(A)** Determine $k$ such that the function $f(z) = e^x cos y + ie^x sin ky$ is analytic. **(B)** Show that $u = x^2 - y^2 − 2xy − 2x + 3y$ is a harmonic function and hence determine the analytic function $f(z)$ in terms of $z$. **(C)** Determine the pole of the function $f(z) = \frac{2z-1}{z(z+1)(z-3)}$ and also find the residue at each pole and sum of all residues. **(D)** Evaluate $\int_{C} \frac{sin \pi z^2 + cos \pi z^2}{(z-1)^2 (z-2)} dz$, Where C is the circle $|z| = 4$ ## Q.6 Attempt the following. **(A)** Apply Cauchy's integral Formula to evaluate $\int_{C} \frac{e^{-z}}{z+1} dz$, where C is the $circle$ $(a)$ |z| = 2 and (b) $ |z| = 1$ **(B)** State Cauchy's residue theorem and evaluate $\int_{C} tanz dz$, where C is the circle |z| = 2. ***End***