SET - 1 I B. Tech II Semester Regular Examinations, May 18 Transform Calculus and Fourier Series (Common to all branches) Time: 3 hours Max Marks: 7 PART A Answer ALL questions. All questions carry equal marks. 1 * Marks = Marks 1). a Define the Gamma function. Evaluate 3 b Find c A function f(t) is defined as 1 1 t f ( t). Find its Laplace Transform elsewhere d Expand f(x) = x(-x) as half-range sine series over the interval (,). e Define the Fourier Transform of f ( x) in the range, corresponding inversion formula?. What is the f g Solve the partial differential equation x usual meanings. p, where p and q y q z have their Define the Beta function ( m, n). Express the integral x ( 1 x) 3 dx as a Beta integral. (You need not evaluate) 1 h Find the transfer function U (z) of the recurrence relation u u 8u subject to the conditions u and u 1 n 6 n1 n 1 i Evaluate L 1 s s e stating the property that is required to evaluate it. s 4 j Write the formulae to compute the Fourier coefficients of a periodic function f (x) defined over the range (, l ). Page 1 of 3
SET - 1. a. PART B Answer All questions. All questions carry equal marks. 5 * 1 Marks = 5 Marks 1 3 4 4 Evaluate the improper integral x ( 1 x) 1 dx. Do you need any special property for this evaluation? If so, state that property. Show that dx = 3. a. (i) Using the first shifting property evaluate Lsinh 3t.cos4t (ii Solve y y 1y given y() y() using Laplace Transform [8] (i) Lcos 3 t t ( ii ) Solve y y y t e given y( ) y( ) 1 using Laplace Transform [8] 4. a. (i) Find the Fourier Series for Deduce that ( ii ) Using Z-transforms, solve (i) Develop the half range sine series of the function f ( x) x x valid for x. Determine the value to which the infinite series 1 1 1 1... converges. 3 3 3 3 1 3 5 7 Page of 3
( ii ) Using Z-transforms, solve SET - 1 5. a. (i) Find the Fourier transform of f(x)=. Hence evaluate dx (ii) Using Parseval s identity prove that = (i) Find the Fourier transform of f(x) = prove that =. Using parseval s identity (ii) Using Parseval s identity, show that = 6. a. (i)a rod of length 1 cm has its ends insulated. An initial temperature u( x) u exists in the rod. Write the partial differential equation governing the heat flow in the rod. Clearly state the initial and boundary conditions that apply in this context. (You need not solve) [3] (ii) Identify the partial differential equation u u. Solve this equation subject to the following boundary conditions [7] (i) Solve P.D.E [3] (ii) A tightly stretched string with fixed end points x x y y x and x l is initially in a position given by sin 3 x y y. If it is released from rest l position, find the displacement y ( x, t). [7] Page 3 of 3
SET - I B. Tech II Semester Regular Examinations, May 18 Transform Calculus and Fourier Series (Common to all branches) Time: 3 hours Max Marks: 7 PART A Answer ALL questions. All questions carry equal marks. 1 * Marks = Marks 1). a Find the value of.. b Express the integral in terms of Gamma Function. c Find Laplace Transform of. d Find L 1 3s s e s 16 e Find half range Fourier Sine Series for (x) = π x in [, π]. f Evaluate g Find Fourier Sine Transform of (x) =. h Using the Fourier cosine transform of f(x)= of,find the Fourier sine transform i In heat flow, what is meant by steady state and transient state conditions? j Solve x ( y z) p y ( z x) q z ( x y) PART B Answer All questions. All questions carry equal marks. 5 * 1 Marks = 5 Marks Page 1 of 3
SET -. a. Prove that. 3. a. Evaluate where is the region and the plane. i) Using Laplace Transform, solve given at. ii) Using Convolution Theorem, evaluate. i) Using Laplace Transform, solve y y t cost given y() y() ii) Using Convolution Theorem, evaluate s s a s b 4. a. (i) If (x) = x, <x< = x, <x< Express the function f(x) as a Fourier Series and then show that (x) = ( ii Using Z-transforms Solve given (i) Develop the half range sine series of the function f ( x) x x valid for x. Determine the value to which the infinite series 1 1 1 1... converges. 3 3 3 3 1 3 5 7 ( ii ) Using Z-transforms Solve given Page of 3
SET - 5. a. (i) Find the Fourier transform of f(x)=. Hence evaluate dx (ii) Using Parseval s identity prove that = (i) Find the Fourier sine transform of f(x)=. (ii) Hence show that dx = (m>) Using Parseval s identity, show that = 6. a. (i) Solve [3] (ii) An insulated rod of length l. has its ends A & B maintained at c & respectively until steady state conditions prevails. If B is suddenly reduced to maintainted so, find the temparature distribution [7] & (i) Solve P.D.E [3] (ii) A tightly stretched string with fixed end points x and x l is initially in a position given by sin 3 x y y. If it is released from rest l position, find the displacement y ( x, t). [7] Page 3 of 3