Download OU B.Sc 2018 May-June 2nd Year 2nd Semester 7114E Mathematics Question Paper

SECTION - A (5 x 4 = 20 Marks)

(Short Answer Type)

Note: Answer any Five of the following questions.

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1. Let G be any group and (ab)² = a² b² for all a, b ? G then show that G is an abelian group.
2. If a, ß ? S5 and a = (1 2 3 4 5), ß = (1 4 5 3 2) then evaluate a?¹ß, aß a?¹ß.
3. If H and K are subgroups of a group G with |H| = 24, |K| = 20 then show that HnK is an abelian group.
4. Determine all group homomorphisms from Z12 to Z20.
5. Define zero divisor in a ring R and find all zero divisors in the ring (Z12, +12, *12).

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6. If I1 and I2 are any two ideals in a ring R, then show that I1 + I2 = {x+y / x ? I1, y ? I2} is always an ideal of R.
7. Show that f(x) = x4 + 3x + 2 has four zeros in Z26.
8. Let R, S be any two rings and f : R ? S is a homomorphism. If R is commutative then show that f(R) is commutative.

SECTION-B (4x15=60 Marks)

(Essay Answer Type)

1. (i) Let G be a group and H is a non-empty subset of G. Then show that H is a subgroup of G if and only if ab?¹ ? H for all a,b ? H. (ii) In the symmetric group S5 find the elements which satisfy x² = identity permutation of S5.

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2. (a) (i) Show that every finite integral domain is a field. (ii) If a is an idempotent element in a ring R then show that 1-a is also idempotent.
3. (b) (i) Define Maximal ideal in a ring R. (ii) Let R be a commutative ring with unity and A be an ideal of R then show that quotient R/A is a field if and only if A is a maximal ideal.
4. (a) Let D be an integral domain. Then show that there exists a field F that contains a subring isomorphic to D.

   OR

   (b) Let F be a field and f(x), g(x) ? F[x] with g(x) ? 0. Then show that there exists unique polynomials q(x) and r(x) ? F[x] such that f(x) = q(x)g(x) + r(x) with either r(x) = 0 or deg r(x) < deg g(x).