BSc Sem VI 2022 2023 Apr 2023 MATHEMATICS INTEGRAL TRANSFORMS Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate marks

1. Q1 (A) Attempt any One of the following: 8 marks
   • (i) If f(t) is a periodic function of period A, and if exists, then prove that Hence find the Laplace transform of f(t) = and f(t) = f(t+T)
   • (ii) If = Fs) then show that (8) for alln EN
   • (B) Attempt any Two of the following: 12
   • (i) Find L(t sinh
   • (ii) If = then prove that =
   • (iii) Find
2. Q2 (A) Attempt any One of the following: 8 marks
   • i) If F(s) = f(t)) = t) dt then prove the following
   • (ii) If F(s) = f(t) dt is the Fourier transform of f(t) then prove that
   • (B) Attempt any Two of the following: 12
   • (i) Obtain the Fourier integral representation of the function f(t) defined as follows
3. Q0 otherwise
   • (ii) Find the Fourier transform of > 0. Hence show that
   • (iii) Express the function as Fourier sine integral where f(x) = and evaluate ds
4. Q3 (A) Attempt any One of the following: 8 marks
   • (i) Write a short note on one dimensional wave equation (II) Find a bounded solution of = = 0, = 0 =
   • (ii) Let g(t) be a function defined for all t > 0. If f(t) is a function defined by and t)) = — f(t) dt, then
   • (1) state and prove the relation between and (II) verify the above relation for g(t) = 1 and x = s
   • (B) Attempt any Two of the following: 12
   • (ii) Solve the initial value problem y” + + 2y = 2,y(0) = = 1 using Laplace
   • (iii) Solve using Laplace transforms R + a y = V,y(0) = 0 where R, are constants
5. Q4 Attempt any Three of the following: 15 marks
   • (a) If = F(s) = find jim f(t), using Final value theorem
   • (b) Find +
   • (c) Let f(t) = where is defined as = Find F and hence find
   • (d) Let f(t) be a real valued even function and let = F Prove that is a real and even function
   • (e) For what values of the constant c,u = sint sin x is a solution of > 0

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