Algebra — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

3. Attempt two parts from each question. 1. (a) (i) Solve the equation x3-14x2-84x = 0, giveii that the roots ate in geometrical progression. (i Find an upper limit to the roots of the equation (b) Find all the integral roots of x3 - 5x2 - 2x + 24 = 0. (7.5) (c) Find all the rational roots of 32y' - 6y - 1 = 0. (7:5) 2. (a) Find the value of z, where (b) Find the fourth roots of z = J3 +i and rèpresent them geometrically to show that they lie on a circle of radius 214. (7.5) (c) Solve the equation z'° + (-2 + i)z - 2i = 0. (7.5) using Cardan's method. (7.5) (b) Prove that n? - 2 is never divisible by 4. (7.5) (c) Let a = qb + I where a, b, 9, r e Z. Show that god(a,b) = god(b, I). Find the god of 143 and 481. (7.5) (a) (i) Let a, b, c e Z be such, that a|bc. If a, b are relatively prime, show that alc. (i Find the god of a = 1575, b = 231 and represent it in the form (4+3.5) ma + nb for suitable intégers m and n. (b) Solve the following pair of congruences. If no solution exists, explain why. (C) Let a = x (mod n) and b = y (mod n). Then, show that 5. (a) Define a group. Is Z, under addition module 5, a group? Justify your (2+5.5) (b) Let GL (2, R) = :a,b,c, del, 18,(Al+D, be the set of all 2x2 matrices over R with nonzero determinant. Is GL(2, R) a group with respect to matrix multiplication? Is it Abelian? Justify your answer. (5+1+1.5) Define the Centre Z(G) of a group G. Is Z(G) a sübgroup of G? Is it Abelian? Give reasons to support your answer. (5+2.5) (a) Let G= :a, budE, be a group under aditiom and He G be such that H= Elsatbtotd-0. Prove that Il is a subgroup of - G. What if O is replaced by 1? (b) Define inverse of an element in a group. If G is a group where every element (2+5.5) is its own inverse, show that G is commutative. (c) Define a cyclic group. List down all subgroups of Zo. Are these subgroups cyclic? What are their generators? What is the order of each subgroup?