University of Mumbai 2009-3rd Sem B.E Computer Engineering S.EApplied Maths III - Question Paper
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(REVISED COURSE) P4-Exam .-09-21 Con. 3335-09. (3 Hours) |
(2) Attempt any four questions out of remaining six. (3) Non-programmable calculator is allowed. (4) Write the sub questions of the main question collectively, Ttk , k > 0 a 1. (a) Find Z-transform of cos (b) State and prove change of scale property. 5 5 (c) Find A-1 using adjoint method where '1 1 1' 2 3 4 9 (d) Find the complex form of the Fourier series for f(x) = e-x in (-1, 1). 00 (b) Obtain the fourier series for f(x) = x cosx in (-n, -it). 6 8 (c) If A, B, C are non-singular matrices of order n x n then prove that
10 0 0 0 2 0 0 Hence find the inverse of 3 0 4 0 0 4 0 2 3. (a) Using convolution Thm find the Laplace inverse transform of 6 _5_ (s2 + a2) (s2 + b2) (b) Show that every square matrix can be uniquely expressed as the sum of symmetric 6 and skew-symmetric matrices. (c) Find Fourier cosine integral for 8 f(x) = 1 - xs 0 < x < 1 x > 1 0 oo .. , . f f xcosx - sinxj x . Hence evaluate I -- cos dx. [TURN OVER P4-Exam -09-22 Con. 3335-VR-3318-09. 2 4. (a) Using L.T. solve, fLZ + 4. + 8y = 1, y(0) = 0, y'(0) =1- 6 dr dt (b) Find the non-singular matrices P and Q such that PAQ is in normal form. Also find 6 its rank. Where '2 1 4' 3 2 2 A = 7 4 10 1 0 6 (c) Express as a Fourier series, 8 f(x) = ttx , 0 < x < 1 = n (2 - x) , 1 < x < 2. 5. (a) Find L {f (t)} where f (t) = sin 2t, 0 < t < -|, f(t) = 0, < t < n and f(t) = f(t + tc). 6 (b) Obtain half-range cosine series for (c) Find the inverse Z-transform of f(z) = 6 1 (z-3) (z-2)- If region of convergence is (i) I z I < 2 (ii) 2 < I z I < 3 z I > 3. 6. (a) Show that the set of functions
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