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Posted Date: 11 Nov 2010 Posted By:: CH.Basaveswara Rao Member Level: Silver Points: 5 (Rs. 1)

2009 Acharya Nagarjuna University B.Sc Electronics MathsII Question paper
ACHARYA NAGARJUNA UNIVERSITY B.A. / B.Sc. DEGREE EXAMINATION, MODEL PAPER (Examination at the end of second year, for 2009  2010 onwards) MATHEMATICS  II PAPER II  ABSTRACT ALGEBRA AND REAL ANALYSIS Time : 3 Hours Max. Marks : 100 SECTION  A (6 X 6 = 36 Marks) Answer any SIX questions. 1. Prove that a subgroup H of a group G is a normal subgroup of G iff every left coset of H in G is a right coset of H in G . 2. Prove that every subgroup of a cyclic group is cyclic. 3. Prove that an ideal M of a commutative ring R with unity is maximal iff R / M is a field. 4. State and prove division algorithm for polynomials. 5. Prove that a monotonic sequence is convergent iff it is bounded. 6. Examine the continuity of the function f defined by f (x) = x+ x 1 at x = 0 and 1. 7. State and prove Cauchy's mean value theorem. 8. If f : [a, b]?R is continuous on [a, b] then prove that f is R integrable on [a, b] . SECTION  B (4 X 16 = 64 Marks) Answer any ALL questions. 9.(a) Let H and K be two subgroups of a group G . Prove that H ? K is a subgroup of G iff either H ? K or K ? H . (b) If G = Q {1} and * is defined on G as a *b = a + b  ab then show that (G, *) is an abelian group. OR 10.(a) State and prove Lagrange's theorem on finite groups. (b) Let G be a group. Show that the mapping f :G ?G defined by f (a) = a1 ?a?G is one one onto. Also show that f is an automorphism iff G is abelian. 11.(a) Define the terms integral domain and field. Prove that every finite field is an integral domain. (b) If Q( 2) = {a + b 2 :a,b ?Q} then show that Q( 2) is a field. OR 12.(a) State and prove fundamental theorem of homomorphism of rings. (b) Find the sum and product of the polynomials f (x) = 2x3 + 4x2 + 3x + 2 and g(x) = 3x4 + 2x + 4 over Z5 . 13(a) State and prove D'Alembert's ratio test. (b) Test for convergence of ( ) .....  ? + + + + ? ? ? ? ? = 8S 1 1 1 2 1 3 1 1 n n n n OR 14(a) If f in continuous on [a,b] then prove that f is bounded on [a,b] . (b) If s n n = 1+ + + + 1 1 1 2 1 ! ! ...... ! then show that {s } n converges. 15(a) State and prove Rolle's theorem. (b) Derive the expansion of sin x by using Maclaurin's theorem. OR 16(a) State and prove fundamental theorem of integral calculus. (b) Prove that p p p 4 2 2 0 4 = ? sec x dx = using the first mean value theorem. 2
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