Industrial Mathematics — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

DSE-6(iii): Industrial Mathematics Name 6f the Course B.Sc. (H) Mathematics (DSE) Parts of the questions to be attempted together. Ql. (a) Define the Laplace trap.sform of a continuous time system f(£). Find the Laplace transform of f (t) = f7t, € 2 0, where 7t is a fixed natural number. (b) Find the inverse-Laplace transform of F(s) = 2(s+3) (c) (d) Define transfer function of a system. Find the transfer function of a system with equation 3£ + 4± + 5x = p. Find the state-space model of a system with equation i -2* -x = 1. Q2. (a) Define signal flow graph of a system. Explain signal flow graph of a parallel Find Z transform of f(fe) = cz*, fe € N and ci is a positive constant. Find the root locus ifG(s)H(s) = (d) Determine whether the characteristic equation D(s) = s4 + 4s3. + 23s2 40s + 50 = 0 represents a stable system. Q3. (a) (b) (c) Define signal power. Find the signal power of g(€) = 2e-i, f a 0. Find the energy of the signal g(€) ,= 5cos(6€ + 450). What are analog and digital signals? Give examples. -I, 2 S € S 3 . Sketch time signals g(3€) (5) Q4. (a) (b) (C) .Briefly explain a unit impulse signal and give one property of unit impulse function. Among the following six signals, find a pair of orthogon.al signals. Find the correlation coefficient between pulse g3(i) and g4(f) = €-t/5 given below (d) Define orthonormal signals. Give one example in two-dimensional and three-(5) dimensional spaces. Q5. (a) Obtain the governing equations of the heat absorption model. (C) Show that the substitution t4(x, £) = f ("), ry = xf -i does not reduce the partial differential equation # = # to an ordinary differential equation. (7.5) Q6. (a) Use the transformation tt(x,I) = £af(i), where a is constant and f is a function, to reduce the equation #= b£;¥ and show that a/(i)- (b) Solve the equation g = #,x 2 0,tt(0, t) = 0,tt(co, £) = 1. (c) Consider the Fourier series. /(X) =J4o+Z:g=iAncos(¥)+Zg=iBnsin(¥). Show that Bn =