MATHEMATICS MODEL PAPER FIFTH SEMESTER PAPER 5 – RING THEORY & VECTOR CALCULUS COMMON FOR B.A & B.Sc (w.e.f. 201516 admitted batch)
http://www.computersprofessor.com/2017/10/mathematicsmodelpaperfifthsemester.html
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 R^{1} then prove that Ker f is
an ideal of R.


5.

Prove that curl (grand f) =


6.

If f = xy^{2}
i + 2x^{2}yz j – 3yz^{2} k the find div f and curl f at the
point (1, –1, 1)


7.

If R (u) = (u – u^{2})
i +2u^{3}j – 3k then find


8.

where S is the
surface of the sphere x^{2} + y^{2} + z^{2} = 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 = x^{2} + y^{2}
+ z^{2}, c = xy + zx; then prove that [grad a, grad b, grad c] = 0



(OR)



b)

Find the directional derivative of the
function xy^{2} + yz^{3} + zx^{2} along the tangent
to the curve x = t, y = y^{2}, z = t^{3} at the point (1, 1,
1).


12.

a)

where F = zi + xj +
– 3y^{2}zk and S is the surface x^{2} +y^{2} = 16 included in the first octant between z = 0, and z = 5. 


(OR)



b)

If F = (2x^{2} – 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 y^{2} = 8x and x = 2 by Green’s theorem.


THE
END