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practice PAPER, , ll MATHEMATICS, , , , BAB., ALGEBRA, Maximum Marks 70, Hours, ee ‘A is compulsory: 19 Section B, attempt one question each, Ne ns IV, SECTION -A, Compulsory Question, Deer wat te bt ofall evn ness forms a group under the, i © sary operstion a b= 20+ 2b, ay Show that iG is @ group, then € Gis an idempotent iff a= e, the, identity of G(a Find the right cosets of subsrouP {1,-1) of group {1,-1.4 — 3, tinder multiplication., , ‘Use Fermat's Theorem to determine the remainder when 8!” is, divided by 103, Also state the theorem., (0) How many generators are there of eyclic group of order 10?, (00) Let f'be a homomorphism from group G into group G', then show f, (e)=6" where eis identity of G and e' is the identity of G’, (bil) Let R bea Boolean Ring, then show that xy=yx Vx CR, , (viii) If is left ideal and J be right ideal of the ring R, then show then,, show by an example that + J may not be an ideal of R (2x 8=16), , SECTION -B, Unit-1, [x, 2, (@) Show thatthe s et ofall 2 « 2 matrices of the form lye where, Jian group., , x,y Reals, not both zero simultaneously forms an abe, , ( ct ive, (b) Let G be a group such that (aby" =a" 4 for re come, Pee fx ll cb @. Show i 6 8