Annamalai University Question Paper december 2014 B.Sc. Mathematics:Vector Calculus And Linear Algebra

Annamalai University question paper december 2014 Total No. of Pages: 2 5277
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B.Sc. DEGREE EXAMINATION December 2014
(MATHEMATICS)
(THIRD YEAR)
(PART - III)
710: VECTOR CALCULUS AND LINEAR ALGEBRA
Time: Three hours Maximum: 100 marks
 Answer any FIVE questions (5× 20=100)
1. a) A particle moves along the curve x=1–t3, y=1+t2, z=2t–5. Determine its velocity
and acceleration.
b) Prove that div curl f =0 and curl grad ϕ=0.
2. a) Prove that f =(x2–yz)i
r
+ (y–zx) j
r
+ (z2–xy) k
r
is irrotational.
b) Find div r and curl r
c) If f ax y z i x y z j x y z k
r r r
= ( + 3 + 4 ) + ( − 3 + 3 ) + 3( + 2 − ) is solenoidal, find the value of a
3. a) Evaluate ∫
c
f . rd where f x y i x y j
r r
( ) ( )
2 2 2 2
= + + − and C is the curve y=x2 joining
(0,0) and (1,1).
b) Verify Gauss divergence theorem for the vector function f x yz i x jy k
r r r
( ) 2 2
2 2
= − − +
over the cube bounded by x=0, y=0, z=0, x=a, y=a, z=a.
4. a) Show that ( )( )( )( )
1
1
1
2 3
2 3
2 3
a b b c c a bc ac ab
c c
b b
a a
= − − − + +
b) Prove that 3
2 2 2
2 2 2
2 2 2
2 ( )
( )
( )
( )
abc a b c
c c a b
b c a b
b c a a
= + +
+
+
+
5. Verify Cayley-Hamilton theorem for the matrix










− −
=
7 2 3
2 1 1
2 2 0
A
6. a) Find the inverse of a matrix










−
−
2 2 1
0 1 4
1 2 3
b) Define an orthogonal matrix and prove that the matrix










0 0 1
1 0 0
0 1 0
is orthogonal.
7. a) Find the rank of the matrix










=
2 1 0 7
6 3 4 7
4 2 1 3
A
b) Show that the equations x+y+z=6; x+2y+3z=14; x+4y+7z=30 are consistent and
solve them.
8. a) Define a vector space. Give an example.
b) Prove that the intersection of two subspaces of a vector space is a subspace.
c) Define a linear transformation. Give an example.
9. a)
Prove that
W
V
is a vector space over F.
b) Show that in v3(R), the vectors (1, 0, 0), (0, 1, 0), (1, 1, 1) are linearly independent.
10. a) Let V be vector space over a field F. Let s={v1, v2.......vn} span V and let S′={w1,
w2,......wm} be a linearly independent set of vectors in V. Prove that m≤n.
b) Let V be a finite dimensional vector space over a field F. Let W be a subspace of V.
Prove that dim
W
V
=dim V – dim W.
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