USMT-06 Mathematics-II Paper-VI PDF

B.Sc. - II (CBCS Pattern) Semester - III USMT-06 - Mathematics-II Paper-VI : Set Theory and Laplace Transform P. Pages : 2 GUG/S/23/11613 *1744* Time :...

B.Sc. - II (CBCS Pattern) Semester - III USMT-06 - Mathematics-II Paper-VI : Set Theory and Laplace Transform P. Pages : 2 GUG/S/23/11613 *1744* Time : Three Hours Max. Marks : 60 _____________________________________________________________________ Notes : 1. Solve all five questions. 2. All questions carry equal marks. UNIT – I 1. a) Prove that (A B) = A B. 6 b) Show that if a, b  R then there is one & only one number x such that x + a = b. 6 This number x is given by x = b + (−a) OR c) Show that x  R  x 2  0. 6 d)  6 Prove that Let An be a sequence of countable sets. Then S = A n is countable. n =1 UNIT – II 2. a) Prove that A fuzzy set F on the universal set R is convex if and 6 only if Fx1 + (1 − ) x 2   min F(x1), F(x 2 )  x1, x 2  R & all  0,1. b) ~ ~ 6 Find A B , where ~ 0.9 0.7 0.2 0.3 A= + + + , 1 3 4 6 ~ 0.1 0.4 0.5 0.8 B= + + + are defined on the universe U = 1, 2,3, 4,5,6 2 3 4 5 OR c) ~ ~ 6 Let A, B  P (U) Then for all   0,1 show that ~ ~ ~ ~ i) A = B if and only if  A =  B ~ ~ ~ ~ ii) A = B if and only if + A = + B d) ~ + 6 Let A  P (U) then prove that  (A) = [(1−) A],  0,1. UNIT – III 3. a) Show that if f (t) is piecewise continuous in every finite interval 0  t  T for every T  0 6 & of exponential order  for t  T then its Laplace Transform F(s) exists  S  . b) Find the Laplace Transform of function cosh at  sin at. 6 OR GUG/S/23/11613 1 P.T.O  c) Find Laplace transform of 6 t cosh t  e t cosh t dt 0 d) Obtain the Laplace Transform of the 6 x + 2x − 3x, x(0) = 0, x (0) = 1 UNIT – IV 4. a) d2x 6 Find the solution by Laplace Transform of + 9x = cos 2t, x (0) = 1, x (  / 2) = −1. dt 2 b) s2 − 6 6 Find the inverse Laplace Transform of s3 + 4s 2 + 3s OR c) Solve equation 6 2 d x dx + 2x = 4t + e3t −3 2 dt dt when x (0) = 1 and x (0) = −1. d) 1 6 Find inverse Laplace Transform of by convolution theorem. 2 (s − 2) (s + 2) 5. Attempt any six. a) Prove that 2 a + b = a + c  b = c  a, b, c  R b) State law of trichotomy. 2 c) Define intersection of two fuzzy set. 2 d) Define union of crisp set. 2 e) Find : L 3t 2 − 2  et  2   f) If L f (t) = F(s) then prove that L eat f (t)  = F(s − a). 2   s 2 − 3s + 4 2 g) Find Inverse Laplace transform of. s3 h) State the convolution theorem. 2 ************ GUG/S/23/11613 2