MATHEMATICS MODEL PAPER FIFTH SEMESTER PAPER 5 – RING THEORY & VECTOR CALCULUS COMMON FOR B.A & B.Sc (w.e.f. 2015-16 admitted batch)
https://www.computersprofessor.com/2017/10/mathematics-model-paper-fifth-semester.html
SECTION
– A
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I.
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Answer
any five questions. Each question carries five marks.
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5
x 5 = 25 M
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1.
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Prove that every field an integral domain.
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2.
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If R is a Boolean then prove that (i) a + a
= 0, a Î R ii) a + b = 0 Þ a = b.
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3.
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Prove that intersection of two sub rings of
a ring R is also a sub ring of R.
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4
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If f is
ahomomorphism of a ring R into a ring R1 then prove that Ker f is
an ideal of R.
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5.
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Prove that curl (grand f) =
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6.
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If f = xy2
i + 2x2yz j – 3yz2 k the find div f and curl f at the
point (1, –1, 1)
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7.
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If R (u) = (u – u2)
i +2u3j – 3k then find
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8.
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where S is the
surface of the sphere x2 + y2 + z2 = 1
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SECTION – B
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II.
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Answer
the following questions:
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5
x 10 = 50 M
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9.
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a)
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Prove that a finite integral domain is a
field
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(OR)
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b)
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Prove that the characteristic of an
integral domain is either a primer or zero
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10.
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a)
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State and prove fundmental theorem of
homomrphsm of rings.
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(OR)
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b)
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Prove that the ring of integers Z is a
principal ideal ring.
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11.
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a)
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If a = x + y + z, b = x2 + y2
+ z2, c = xy + zx; then prove that [grad a, grad b, grad c] = 0
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(OR)
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b)
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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).
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12.
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a)
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where F = zi + xj +
– 3y2zk and S is the surface x2 +y2 = 16 included in the first octant between z = 0, and z = 5. |
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(OR)
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b)
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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.. |
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13.
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a)
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State and prove Stoke’s theorem
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(OR)
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b)
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around the boundary
C of the region bounded by y2 = 8x and x = 2 by Green’s theorem.
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THE
END