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   
maths1du
                   
8.
Show that           maths2
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)
Evaluatemaths3
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 
maths4
                 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)
Find maths5
                                                                  
around  the boundary C of the region bounded by y2 = 8x and x = 2 by Green’s theorem.


THE END

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