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Bangabasi Evening College MATHEMATICS-HONOURS B. Sc(H)/Part-III Paper- V Module IX Analysis Answer any two Questions 1. (a) Prove or disprove: The range of any convergent sequence in is a compact set. e dt (b) If e denoted by the equation 1 , prove that 2 e 3 . 1 t (c) If S is a closed and bounded set of real numbers, then prove that every cover of S has a finite subcover. (d) Show that log( 1 x) log( 1 x) log( 1 x 2 ) log( 1 x 4 ) converges for x 1 . 2. (a) If a function f : [a, b] has a bounded derivative on [a,b], then prove that f is a bounded variation on [a,b]. (b) Construct a real-valued function on a compact interval which is continuous but is not of bounded variation on that interval. (c) If f is monotonic on [a,b], prove that f is bounded variation on [a,b] and Vab ( f ) f (b) f (a) . 3. (a) Let f : [a, b] be a bounded function such that f is continuous on [a,b] except at finitely many points in the open interval (a,b). Prove that f is Riemann integrable on [a,b]. (b) Let f : [a, b] be a continuous function. Show that the function F : [a, b] x defined by F ( x) f (t )dt , x [a, b] is differential in its domain. a (c) Show that for k 2 1 , 1/ 2 dx 1 2 2 2 0 6 (1 x )(1 k x ) 6 1 k 2 / 4) x2 e (d) Prove that lim x 0 1 t 0 x2 dt e. Module X Group-A Answer one Question from each Section Section-I 1. (a) Prove that a linear transformation T of a vector space V of dimension n into a vector space W of dimension m over a same field F can be represented by m n matrix A over F with respect to a basis in V and a basis W. (b) The matrix representation of a linear transformation T : 3 3 is 1 0 0 0 3 1 0 2 1 3 relative to the standard ordered basis of . Find the explicit representation of T and matrix representation of T relative to the ordered basis {( 0,1,2), (1,0,1), (2,1,1)} . 2. (a) Let V and W be vector spaces of finite dimension over a field F and let T : V W be a linear mapping. Prove that T is nonsingular if and only if the matrix of T relative to some chosen bases is nonsingular. (b) A linear transformation T : 3 3 is defined by T ( x, y , z ) ( x y z , 2 x y 2 z , x 2 y z ) . Find KerT and dimensions of KerT and ImT Section-II 3. (a) Let H be a subgroup of a group G such that [G:H]=2. Prove that H is a normal subgroup of G. Is the converse is true? Justify your answer. (b) Prove that an infinite cyclic group is isomorphic to the group (Z, +), ( Z is the set of all integers). 4. (a) Let f : (G,) (G / ,) be a isomorphism of groups. Prove that G / is cyclic if and if only if G is cyclic. (b) Let H and K be two normal subgroups of a group G such that H K {e} , where e is the identity of G. Prove that hk kh , h H and k K . Group-B Differential Equations Answer any one Question d2y y x near the ordinary point x=0. dx 2 (b) (i) State a set of sufficient conditions for existence of Laplace transform. (t 1) 2 , t 1 (ii) If F (t ) , determine the Laplace transform of F(t). 0 t 1 0, (c) Let F(t) be periodic function with period T (>0). Show that 1. (a) Obtain the series solution of T e st F (t )dt 0 1 e sT L[ F (t )] d2y dy 3 y 0 in series near the ordinary point x=0. dx dx 2 (b) State Convolution theorem on Laplace transform and use it to find 1 L1 . 2 ( s 1)( s 1) 2. (a) Solve the equation 3x (c) Solve, by using Laplace transform y (0) 0, y ' (0) 3 . d2y dx 2 dy 2 y 18e x sin 3x , when dx Group-C Tensor Calculus Full Marks-10 Answer any one Question 1. (a) Define contravariant and covariant vector. Show that for the orthogonal transformation x i a ij x j (where a ij a kj ki , Kronecker delta) the laws of transformation of the covariant and the contravariant vectors are identical. (b) Show that the partial derivative of a covariant vector is not a tensor. (c) If a ik are components of a symmetric contravariant tensor prove that g jk 1 a jk [ij , k ] a ij 2 x i 2. (a) The line element of three dimensional Riemannian space ds 2 (dx1 ) 2 ( x1 ) 2 (dx 2 ) 2 (dx 3 ) 2 . 1 Calculate 2 2 i i (b) Show that j log g , where g det( g ij ) 0 and is a i j x i j Christoffel symbol. (c) Sow that the covariant derivative of a tensor of type (1,0) is a tensor of type (1,1). is