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CSIR UGC NET 2008 : MATHEMATICAL SCIENCES - PAPER II

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MATHEMATICAL SCIENCES PAPER-II 1. Let {xn} and {yn} be two sequences of real numbers. Prove or disprove each of the statements : 1. 2. If {xnyn} converges, and if {yn} is convergent, then {xn} is convergent. {xn + yn} converges to x+y if {xn} converges to x and {yn} converges to y. If{xn/yn} is convergent, then both {xn} and {yn} are convergent. If {xn} is convergent and {yn} is divergent, then {xnyn} is divergent. 3. 4. 2. Let f : [a,b] be differentiable. (a) Prove that b f 2 (t ) dt = 0 iff f 0 on [a, b]. . a b (b) If f 3 (t ) dt = 0, then for some t0 [a,b], either f ' (t0 ) = 0 or f (t0 ) = 0 . a 3. Let {fn} be a sequence of real-valued Lebesgue integrable functions on such that n =1 f n < . Prove that for any , e n =1 in f n ( x ) converges a.e. to a function g (x) and that g is Lebesgue integrable. 4. Let a = (a1, a2, L , an) n x.a = xi .ai . n and let f(x) = ex.a where x = (x1, x2 L , xn) n , and Compute the directional derivative of f at a point p n in the i =1 direction of h n . 5. (a) Let A = {(x,y) 2 x2 + 2y2 < 37} {(x,y) 2 ex > y} Prove that A is compact. (b) Show that any convex subset C of n is connected. 6. Show that the set {log p p prime number} is linearly independent over . 7. Let V be the vector space of all polynomial functions of degree < n, n > 2, and let D : V V denote the derivative map P a P on V. Show that D is nilpotent and that D is not diagonalizable. 8. Let A and B be two 3 3 complex matrices. Show that A and B are similar if and only if A = B and A = B, where A, B are characteristic polynomials of A, B, respectively and A, B are minimal polynomials of A, B, respectively.

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Additional Info : CSIR UGC (National Eligibility Test) NET New Exam Scheme December 2008 Model Question Paper - Mathematical Sciences Paper II Joint CSIR UGC NET Exam Question Papers
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