# MATHEMATICS MODEL PAPER FIFTH SEMESTER PAPER 5 – RING THEORY & VECTOR CALCULUS COMMON FOR B.A & B.Sc (w.e.f. 2015-16 admitted batch)

 SECTION – A I. Answer any five questions. Each question carries five marks. 5 x 5 = 25 M 1. Prove that every field an integral domain. 2. If R is a Boolean then prove that (i) a + a = 0, a Î R                      ii) a + b = 0 Þ a = b. 3. Prove that intersection of two sub rings of a ring R is also a sub ring of R. 4 If f is ahomomorphism of a ring R into a ring R1 then prove that Ker f is an ideal of R. 5. Prove that curl (grand f) = 6. If f = xy2 i + 2x2yz j – 3yz2 k the find div f and curl f at the point (1, –1, 1) 7. If R (u) = (u – u2) i +2u3j – 3k then find 8. where S is the surface of the sphere  x2 + y2 + z2 = 1 SECTION – B II. Answer the following questions: 5 x 10 = 50 M 9. a) Prove that a finite integral domain is a field (OR) b) Prove that the characteristic of an integral domain is either a primer or zero 10. a) State and prove fundmental theorem of homomrphsm of rings. (OR) b) Prove that the ring of integers Z is a principal ideal ring. 11. a) If a = x + y + z, b = x2 + y2 + z2, c = xy + zx; then prove that [grad a, grad b, grad c] = 0 (OR) b) Find the directional derivative of the function xy2 + yz3 + zx2 along the tangent to the curve x = t, y = y2, z = t3 at the point (1, 1, 1). 12. a) where F = zi + xj + – 3y2zk and S is the surface x2 +y2 = 16 included in the first octant between z = 0, and z = 5. (OR) b) If F = (2x2 – 3z) i – 2xy j – 4xk, then evaluate                   where V is the closed region bounded by the planes x = 0, y = 0, z = 0 2x + 2y + z = 4.. 13. a) State and prove Stoke’s theorem (OR) b) around  the boundary C of the region bounded by y2 = 8x and x = 2 by Green’s theorem.

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